· lecture notes on universal algebra many-sorted partial algebras preliminary version peter...

Lecture Notes on Universal AlgebraMany-Sorted Partial Algebras

Preliminary version

Peter BurmeisterFachbereich Mathematik

Arbeitsgruppe Allgemeine Algebra und Diskrete Mathematik

Technische Universitat Darmstadt

Schloßgartenstr. 7

D-64289 Darmstadt

Summer 2002

Abstract

Universal Algebra can be considered as one of the three fundamental math-ematical structure theories (besides the theories of ordered sets and topologicalspaces, respectively). Moreover, universal and in particular partial algebras areamong the basic mathematical structures implemented on computers, and thelanguage of universal algebra is fundamental for theoretical computer science.Many-sorted algebras are basically partial algebras, too.

These notes are meant to introduce into a theory of and a language for uni-versal algebras with particular stress on partial algebras including many-sortedpartial algebras. Besides the terminology and constructions from universal alge-bra (homomorphisms, the principle of generation and algebraic (or structural)induction, generalized recursion theorem, epimorphism theorem, free partial al-gebras) also such from logic (existence equations and elementary implications),model theory (preservation and reflection of formulas by mappings) and from(elementary) category theory (factorization systems) prove to be quite usefulfor a good description of the arising concepts, as is shown at the end by theformulation of a “Meta Birkhoff Theorem”.

1

REFERENCES 2

References

[AdHS90] J.Adamek, H.Herrlich, G.E.Strecker. Abstract and Concrete Cate-gories — The Joy of Cats. John Wiley & Sons, Inc., 1990.

[AN79] H. Andreka, I. Nemeti. Injectivity in categories to represent all firstorder formulas. I. Demonstratio Math. 12, 1979, pp. 717–732.

[ABN81] H.Andreka, P.Burmeister, I.Nemeti. Quasivarieties of partial algebras— A unifying approach towards a two-valued model theory for partialalgebras. Studia Sci. Math. Hungar. 16, 1981, pp. 325–372.

[AN82] H. Andreka, I. Nemeti. A general axiomatizability theorem formulated interms of cone-injective subcategories. In: Universal Algebra (Proc. Coll.Esztergom 1977), Colloq. Math. Soc. J. Bolyai, Vol. 29, North-HollandPubl. Co., Amsterdam, 1982, pp. 13–35.

[AN83] H. Andreka, I. Nemeti. Generalization of the concept of variety andquasivariety to partial algebras through category theory. DissertationesMathematicae (Rozprawy Mat.) No. 204, Warszawa, 1983.

[Bi35] G.Birkhoff. On the structure of abstract algebras. Proc. Cambridge Phi-los. Soc. 31, 1935, pp. 433–454.

[Bi67] G.Birkhoff. Lattice Theory. American Mathematical Society, 1967(3rd edition, 1st ed. 1940).

[Bo93] F.Borner. Varieties of Partial Algebras . Beitrage Algebra, Geom., 1996,pp. 259–287.

[B70] P.Burmeister. Free partial algebras. J. reine und angewandte Math. 241,1970, pp. 75–86.

[B82] P.Burmeister. Partial algebras - survey of a unifying approach towardsa two-valued model theory for partial algebras. Algebra Universalis 15,1982, pp. 306–358.

[B86] P.Burmeister. A Model Theoretic Oriented Approach to Par-tial Algebras. Introduction to Theory and Application of Par-tial Algebras – Part I. Mathematical Research Vol. 32, Akademie-Verlag, Berlin, 1986. (Part II : [Re84]).“Not yet debugged LATEX-translation” as pdf-file in the internet athttp://www.mathematik.tu-darmstadt.de/∼burmeister/

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REFERENCES 3

[B93] P.Burmeister. Partial Algebras — An Introductory Survey. In: Alge-bras and Orders (Proceedings of the NATO Advanced Study Instituteand Seminaire de mathematiques superieures on algebras and orders,Montreal, Canada, July 29 – August 9,1991; Eds.: I.G.Rosenberg andG. Sabidussi), NATO ASI Series C, Vol. 389, Kluver Academic Publ.,Dordrecht, London, 1993, pp. 1–70.

[BM94] P.Burmeister, M.Maczynski. Orthomodular (partial) algebras and theirrepresentations. Demonstratio Mathematica 27, 1994, pp. 701–722.

[B95] P.Burmeister. On the equivalence of ECE- and generalized Kleene-equations for many-sorted partial algebras. Contributions to GeneralAlgebra 9 (Proceedings of a conf. on Univ. Alg. and its Applic., Linz1994). Verlag Holder-Pichler-Tempsky, Wien 1995 — Verlag B.G. Teub-ner, Stuttgart, pp. 91–106.

[B02] P.Burmeister. Lecture Notes on Universal Algebra — Many-Sorted Partial Algebras.Fragment of the notes of the lectures held at the Darmstadt Universityof Technology in the years 1996, 1998 and 2000 and by and by updatedin 2002 as ps- and pdf-files — at:http://www.mathematik.tu-darmstadt.de/∼burmeister/

[BuSk81] St.Burris, H.P.Sankappanavar. A Course in Universal Algebra.Springer-Verlag, 1981. In the internet as ps- or pdf-file available at:http://www.thoralf.uwaterloo.ca/htdocs/ualg.html

[C65] P.M.Cohn. Universal Algebra. Harper and Row, 1965 (2nd rev. ed.D.Reidel Publ. Co., Dordrecht, 1981).

[Cr89] W.Craig. Near-equational and equational systems of logic for partialfunctions. I and II. The J. of Symb. Logic 54, 1989, pp. 759–827 andpp. 1181–1215.

[DW02] K.Denecke, S.L.Wismath. Universal Algebra and Applications inTheoretical Computer Science. Chapman and Hall, 2002.

[DvP00] A.Dvurecenskij, S.Pulmannova. New Trends in Quantum Struc-tures. Kluwer, 2000 (Ister Science, Bratislava, 2000).

[EhGoLi89] H.-D.Ehrich, M.Gogolla, U.W.Lipeck. Algebraische Spezifikationabstrakter Datentypen. B.G.Teubner, Stuttgart, 1989.

[EgMa85] H.Ehrig, B.Mahr. Fundamentals of Algebraic Specification 1.Equations and Initial Semantics. Springer-Verlag, 1985.

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REFERENCES 4

[EhMa90] H.Ehrig, B.Mahr. Fundamentals of Algebraic Specification 2.Module Specifications and Constraints. Springer-Verlag, 1990.

[Ev51] T.Evans. Embeddability and the word problem. J. London Math. Soc.26, 1951, pp. 64–71.

[EvsLj97] A.E.Evseev, E.S.Ljapin. The Theory of Partial Algebraic Oper-ations. Kluwer, 1997.

[GoTcWa78] J.A.Goguen, J.W.Thatcher, E.G.Wagner. An initial algebra approachto the specification, correctness, and implementation of abstract datatypes. IBM Research Report RC-6487, 1976; and: Current Trends inProgramming Methodology, IV: Data Structuring (R.Yeh, ed.), PrenticeHall, 1978, pp. 80–144.

[G68] G.Gratzer. Universal Algebra. D.van Nostrand Co., 1968 (2nd ed.:Springer-Verlag, 1979).

[GSchm63] G.Gratzer, E.T.Schmidt. Characterization of congruence lattices of ab-stract algebras. Acta Sci. Math. (Szeged) 24, 1963, 34–59.

[HS73] H.Herrlich, G.E.Strecker. Category Theory - An Introduction.Allyn and Bacon, 1973 (2nd ed.: Heldermann-Verlag).

[Ho70] H.Hoft. Equations in partial algebras. PhD-thesis, Univ. of Houston,Texas, 1970.

[Ho73] H.Hoft. Weak and strong equations in partial algebras. Algebra Univer-salis 3, 1973, pp. 203–215.

[Hl96] R.Holzer. Programmverifikation mit partiellen Algebren als abstrakteDatentypen. Diplomarbeit an den Fachbereichen Informatik und Math-ematik, TH Darmstadt, 1996. Can be downloaded as ps-file from:http://www.mathematik.tu-darmstadt.de/∼holzer

[Ih88] Th.Ihringer. Allgemeine Algebra. Teubner Studienbucher, Stuttgart,1988 (2nd extended ed. 1993).

[J75] R.John. Gultigkeitsbegriffe fur Gleichungen in partiellen Algebren. Dis-sertation, TH Darmstadt, 1975.

[J78] R.John. Gultigkeitsbegriffe fur Gleichungen in partiellen Algebren.Math. Zeitschrift 159, 1978, pp. 25–35.

[Ke70] R.Kerkhoff. Gleichungsdefinierbare Klassen partieller Algebren. Math.Annalen 185, 1970, pp. 112–133.

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REFERENCES 5

[Kla83] H.A.Klaeren. Algebraische Spezifikation. Springer-Verlag, 1983.

[Kl52] S.C.Kleene. Introduction to Metamathematics. North-HollandPubl. Co., 1952.

[KoSp68] S.Kochen, E.P.Specker. The problem of hidden variables in QuantumMechanics. J. Math. Mech. 17, 1968, pp. 59–87.

[Kos94] P.Kosiuczenko. Mal’cev type conditions for partial algebras. AlgebraUniversalis 31, 1994, pp. 467–474.

[Ku64] A.G.Kurosch. Vorlesungen uber allgemeine Algebra. Verlag HarriDeutsch, 1964.

[La69] W.A.Lampe. On related structures of a universal algebra. PhD Thesis,Pennsylvania State University, 1969.

[Lu76] H.Lugowski. Grundzuge der Universellen Algebra. Teubner,Leipzig, 1976.

[Ma73] A.I.Mal’cev. Algebraic Systems. Springer-Verlag, 1973.

[Mar58] E.Marczewski. A general scheme of independence in mathematics. Bull.de l’Acad. Polonaise des Sci., Ser. Math. Astr. et Phys. 6, 1958, pp. 731–736.

[MkMnT87] R.N.McKenzie, G.E.McNulty, W.F.Taylor. Algebras, Lattices, Va-rieties. Volume 1. Wadsworth & Brooks/Cole, 1987.

[Mo69] J.D.Monk. Introduction to Set Theory. McGraw-Hill Book Com-pany, New York, 1969.

[NSa82] I.Nemeti, I.Sain. Cone-implicational subcategories and some Birkhoff-type theorems. Universal Algebra (Proc. Coll. Esztergom 1977), Colloq.Math. Soc. J. Bolyai, Vol. 29, North-Holland Publ. Co., Amsterdam,1982, pp. 535–578.

[Pa79] A.Pasztor. Faktorisierungssysteme in der Kategorie der partiellen Al-gebren — Kennzeichnung von (Homo-)Morphismen. Dissertation, THDarmstadt, 1979; appeared in: Hochschulverlag, Freiburg, 1979.

[Pi68] R.S.Pierce. An Introduction to Abstract Algebras. Holt, Rinehartand Winston, 1968.

[Pu94] S.Pulmannova. A remark on orthomodular partial algebras. Demonstra-tio Math. XXVII, 1994, pp. 687-699.

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REFERENCES 6

[Pu97] S.Pulmannova. Congruences in partial abelian semigroups. Algebraunivers. 37, 1997, pp. 119-140.

[Re84] H.Reichel. Structural Induction on Partial Algebras. Introduc-tion to Theory and Applications of Partial Algebras – PartII. Math. Research, 18, Akademie-V., Berlin, 1984. (Part: I see [B86]).

[Re87] H.Reichel. Initial Computability, Algebraic Specifications, andPartial Algebras. Oxford Science Publications, Oxford: ClarendonPress (Copubl. with Akademie-Verlag, Berlin), 1987. (Revised editionof [Re84]).

[Ru83] L.Rudak. A completeness theorem for weak equational logic. AlgebraUniversalis, 16, 1983, pp. 331–337.

[Sch62] J.Schmidt. Die Charakteristik einer Allgemeinen Algebra. I. Arch.Math. XIII, 1962, pp. 457–470.

[Sch64] J.Schmidt. Die Charakteristik einer Allgemeinen Algebra. II. Arch.Math. XV, 1964, pp. 286-301.

[Sch66] J.Schmidt. Mengenlehre I. Bibliographisches Institut, HTBBand 56/56a, 1966.

[Sch66a] J.Schmidt. A general existence Theorem on partial algebras ans its spe-cial cases. Coll. Math. 14, 1966, pp. 73–87.

[Sch70] J.Schmidt. A homomorphism theorem for partial algebras. Coll.Math. 21, 1970, pp. 5–21.

[Schr1895] E.Schroder. Vorlesungen uber die Algebra der Logik I, II, III.2nd ed. Chelsea 1966 (1st editions 1890, 1891, 1895).

[Sl64] J.S lominski. A theory of extensions of quasi-algebras to algebras.Rozprawy Mat. 40, 1964.

[S72] G.E.Strecker. Epireflection operators vs perfect morphisms and closedclasses of epimorphisms. Bull. Austr. Math. Soc. 7, 1972, pp. 359–366.

[We92] W.Wechler. Universal Algebra for Computer Scientists.Springer-Verlag, 1992.

[Wr78] H.Werner. Einfuhrung in die allgemeine Algebra. Bibliographis-ches Institut, HTB Band 120, 1978.

[Wh98] A.N.Whitehead. A Treatise on Universal Algebra. CambridgeUniversity Press, 1898

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REFERENCES 7

[Wi90] M.Wirsing. Algebraic Specification. In J.van Leeuwen (ed.): Handbookof Theoretical Computer Science, Volume B: Formal Models and Se-mantics, Elsevier 1990 (paperback ed. 1994), pp. 675–788.

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0 Introduction 8

0 Introduction

0.1 General remarks

Universal algebra can be considered as one of the three basic theories of mathematicsin the structure theory of Bourbaki, which comprise:

Ordered Structures, Algebraic Structures, Topological Structures.

Universal algebra provides among others

• a global view onto the common features of the theories of different algebraicstructures;

• a language to unify and simplify the introduction and investigation of newalgebraic structures;

• a good feeling for abstraction;

• a way of thinking in structures.

However, universal algebra does not provide in general a tool to solve special prob-lems for specific algebraic structures, although it allows to distinguish more clearly,which problem is a general one and which problem is specific for the special kind ofstructures under consideration. Among others, the last three features listed abovemake universal algebra so valuable for (theoretical) computer science.

Although there has been already a book “A Treatise on Universal Algebra” byA.N.Whitehead [Wh98] containing the expression “Universal Algebra” in its title, andalthough in the three books “Vorlesungen uber die Algebra der Logik I, II, and III”by Ernst Schroder [Schr1895], already basic ways of universal algebraic thinking weredeveloped, one often rates the paper [Bi35] of Garrett Birkhoff (jr.) as the birth ofUniversal Algebra. This discipline had a large growth during the 1960’s, and startingwith the 1970’s it became one of the fundamental languages of theoretical computersciences. As one of the first books about universal algebra one can consider the book“Lattice Theory” by Garrett Birkhoff (jun.) in 1940. In the second half of the sixtiesseveral books on universal algebra appeared (cf. the bibliography).

0.2 Motivation for the consideration of of partial algebras

If one considers the algebraic structures implemented on computers, one easily realizesthat most of them are partial — even when considered as many-sorted algebras, andthis is one reason that we heavily want to include here the consideration of partialalgebras. Recently also so called test algebras and orthomodular (partial) algebras(which are useful e.g. in quantum logic) — usually considered as ordered sets with aninvolution — have been axiomatized as ECE-varieties of partial algebras and thereforeallowed new insights into their theory (see e.g. [Pu94], [Pu97] and [BM94]).

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0 Introduction 9

With these notes we want — among others — to provide a somewhat easieraccess to a theory of partial algebras than the one of Burmeister [B86] which treatsin a parallel way partialness, many-sortedness (yet in a somewhat unusual way) andpossibly infinitary operations. Since we are well aware of the fact that a great partof the possible readers will consist of computer scientists who rather need a theoryof many-sorted than only of one-sorted partial algebras, we include their treatmentfor finitely many sorts right from the beginning, yet we often consider them justas if they have disjoint carrier sets for different sorts and that they are thereforeonly special cases of homomogeneous partial algebras, what does not cause any lackof generality, since the general treatment of heterogeneous algebras supports such apoint of view. The consideration of many-sortedness often gives better control on thekind of partiality and leaves only few “exceptional cases”.

Let us present in what follows — together with some historical remarks — somemotivations why it is or could be interesting to study a theory of partial algebrasand, should the occasion arise, why to teach their theory to mathematicians and inparticular to computer scientists.

Partial functions have been used in mathematics for a long time, e.g.

– partially defined functions in analysis,

– partial subtraction for natural numbers,

– partial division for integers,

– partial multiplicative inversion in arbitrary fields,

– partial recursive functions in computability theory.

However, this did not lead to an independent general theory for structures withpartial operations. One rather tried in most cases to complete the structures as faras possible, and field theory gave great impulses to the development of model theory.However, in Kleene [Kl52] there already appeared in connection with partial recursivefunctions three kinds of semantics of “equality” for partial (recursive) functions.

Yet finally, in connection with an enforced development of universal algebra andparallel to it of category theory in the 1960’s with growing tendencies to greatest gen-erality, some authors — e.g. J.S lominski (starting about 1964) and J.Schmidt (startingabout 1964, too) — investigated in their papers also the universal algebraic propertiesof partial algebras, what was then continued, not only by their “schools” (B.Wojdy loat Torun, P.Burmeister and H.Hoft at Bonn and Houston, respectively), but also byR.Kerkhoff, V.Poythress and later by H.Andreka, I.Nemeti, I.Sain, A.Pasztor, H.-J.Hoehnke, H.Reichel, H.Kaphengst and some others in the following decade. Someother sources of motivation at that time (may) have been

– the result of T.Evans in 1951 (see [Ev51]) about the equivalence of the solv-ability of the word problem for varieties of total universal algebras and of the

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0 Introduction 10

decidability of the embeddability problem w.r.t.1 the variety under considera-tion for finite partial algebras satisfying — w.r.t. some special semantics (seebelow) — the equations in the equational theory of the variety;

– the use of partial algebras as intermediate structures — and of some resultsabout them — for the proof of the result of G.Gratzer and E.T.Schmidt in[GSchm63] (and later by W.Lampe e.g. in [La69]) that each algebraic lattice isisomorphic to the congruence lattice of some (multi-unary) universal algebra;

– the use of partial algebras as starting structures for the presentation of totalalgebraic structures (by generators and relations) and for the construction of“relatively free algebras over partial relative substructures”;

– the possibility to describe the structure of relatively free algebras by first consid-ering only some part of the structure in order to generate the elements as specialterms in a term algebra — and here one often needs a partial algebraic structure— and then defining the rest of the structure by using the defining equations(identities); this method is now known in computer science as “canonical termrepresentation” or — in another context — as “specification by constructorsand relators” (if one can just partition the set of fundamental operations forthat purpose);

– the fact that (small) categories are also partial algebras and in this connectionwith possibly much “larger” partiality than that observed e.g. for fields (wherethere is only one exceptional case). However, the partiality of the only partialoperation of composition of morphisms is still relatively “tame”, since its domainis the solution set of equations only involving total operations (i.e. the class ofall small categories forms what H.Reichel calls a hep-variety (cf. e.g. [Re84]);— and also the observation that such situations occur quite often in situationswhere partiality arises.

After these first investigations there seemed to be some “motivation crisis”: Sincemost fundamental notions from universal algebra for total algebras splitted into atleast three relevant concepts in the case of partial algebras, “one could not see theforest among the trees”. That is to say, one did not see a starting point for a “nice”and unifying theory (and of important and attractive results). And this lack of a“good” theory also hindered new and interesting applications (there were some e.g.from quantum mechanics, cf. S.Kochen and E.P.Specker, in [KoSp68], but the “weakequality” used there was not transitive).

New motivation for studying partial algebras then came in connection with thesoftware crisis in computer science and the beginning awareness of computer scientists

1“w.r.t.” will be used as abbreviation for “with respect to”.

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0 Introduction 11

for the fact that universal algebra provides a good language and theory for theoret-ical computer science, e.g. for dealing with abstract data types and with program-ming languages and their semantics (see e.g. J.A.Goguen, J.W.Thatcher, E.G.Wagner[GoTcWa78], H.Reichel [Re84] or [Re87], but also H.-D.Ehrich, M.Gogolla, U.W.Lipeck[EhGoLi89], H.Ehrig, B.Mahr [EgMa85] and [EhMa90], H.A.Klaeren [Kla83] and[Hl96]). And, as already mentioned at the beginning of this motivation, in thisconnection one also observed that many — or even most — structures in computerscience are partial, even when considered as many-sorted structures. In particular,since in a computer only finite parts of a — usually infinite — structure can be real-ized and computed, almost every implementation of a computer program representsa partial algebraic structure.

Here we think it to be a task for mathematicians to teach computer scientists andtheir students already universal algebra and in particular a theory of partial algebras.Namely it should not continue that compilers like TURBO PASCAL implement nat-ural numbers in such a way that

32767 + 1 = −32768

instead of an (external) error message informing the user that her/his calculationsgot out of range.

We also do not think it adequate to specify originally partial data types in sucha way that the exceptional cases get meaningful values, e.g. to read “0” (the integerzero) from an empty stack of integers as it is often proposed in books on specifications.

New motivation for a further development of a theory of partial algebras alsocame from a more general category theoretic investigation of first order logic and ameta theorem for Birkhoff type results concerning implicationally definable classes(see the papers of H.Andreka, I.Nemeti and I.Sain, e.g. [AN82], and [NSa82]). Thisresult, of which we shall formulate a more algebraic version at the end of these notes,indicated that the concept of equality which we now call “existence equality” forms agood basis for an equational and implicational theory and in general for an expressivemodel theory for partial algebras, as we shall realize during these notes.

Once a reasonable theory of partial algebras has been developed, new motivationgrows from the general principle that new points of view give new insights, which maybe applied among others in the following cases:

– As the theory of partial algebras with homomorphisms as basic structure pre-serving mappings and existence equations as basic model theoretic concept isdesigned, it lies between the theories of relational structures and the one oftotal algebras, which it comprises in a natural way as a subcategory. It is dis-tinguished from the theory of relational systems in that it allows more easily tospeak about generation — and we think generation to be one of the most im-portant concepts in algebra, since it allows to describe large entities by singlingout a small subset and giving operations and axioms telling how to get (i.e.

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0 Introduction 12

generate) the rest of the structure. In addition our theory of partial algebrasalso allows to speak very easily about definedness and undefinedness of terms.

However, the theory of partial algebras also comprises the one of relationalsystems in a relatively natural way: Given an n-ary relation R on a set A, Rmay be considered as an n-ary partial operation on A, say fR, which has Ras its domain and acts, say, as partial first projection: fR(a1, . . . , an) := a1 forall (a1, . . . , an) ∈ R (and undefined otherwise). The definition of fR can beexpressed by a first order sentence for an n-ary function symbol f as

(∀x1) . . . (∀xn)(fx1 . . . xne≈ fx1 . . . xn ⇒ fx1 . . . xn

e≈ x1).

This shows that the class of all relational systems of similarity type τ can beconsidered as an axiomatic subclass of the class of all partial algebras of type τ ,where even the axioms are elementary implications of a very simple structure.And this is one reason that we do not include the treatment of relations intothese notes, since all the features and concepts observed in connection withpartial algebras also relate to relational systems — this does not mean thatoccasionally a special treatment of relations in their own right might not leadto simpler representations and results than having them always transformedinto partial operations.

The considerations above may also help to understand, why any theory of partialalgebras has to be so rich and full of important fundamental concepts (cf. e.g.our discussion of substructures below, where in addition to the concepts ofrelative and weak relative substructures (also used in the theory of relationalsystems) we have in addition the one of subalgebras (i.e. of relative substructureson closed subsets) which is closely connected with the algebraic concept ofgeneration and the only one which is usually considered for algebras).

Our observation not only gives new insights on the side of partial algebras, butit is also useful e.g. on the side of (total) algebras with relations, the theory ofwhich is now also embeddable into the one of partial algebras. Thus it unifiesand extends the theories of relational systems as well as those of total algebras.

– Another application of the above principle may be the case of many-sorted(partial) algebras. Their theory is usually presented in such a way that thecarrier sets of different sorts may be assumed without loss of generality (oronly by simple modifications) to be disjoint. Then, on the disjoint union of thecarrier sets of different sorts, the original many-sorted structure establishes ina canonical way a partial algebraic structure induced by the specification of themany-sorted similarity type; and on the set, say S, of all sorts, it establisheswhat we shall call the corresponding — partial — sort-algebra. The mapping,which assigns to each element of the carrier set of such a partial algebra its sort,

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0 Introduction 13

then becomes a homomorphism, which is closed iff2 the original many-sortedalgebra is total. In this way the category of many-sorted partial algebras is ina natural way isomorphic to a so-called comma category of the category of allpartial algebras of the corresponding similarity type. W.r.t. this point of viewmany-sorted (partial algebras) have been treated e.g. in P.Burmeister [B86].

– There is a further unifying effect by using partial algebras: By extending thestructure under consideration “in a most general way”, adjoint situations result-ing from forgetting (part of) the structure can be considered within one categoryby using inclusion functors on one side and the principle of universal solutions(relatively free constructions) on the other side, what makes the resulting con-structions in general more easily understandable; and universal solutions haveusually to be studied anyway.

In these notes we want to combine some “different mathematical languages”. Thisshall help us to keep track of the wealth of (possibly) relevant concepts arising e.g.from the concepts in the universal algebra of total algebras, as mentioned above.Therefore we will consider in particular

– the language of universal algebra (generation, freeness, algebraic “construc-tions”, etc.)

– the language of first-order logic based on a concept of existence equations te≈ t′,

which comprises relatively easy ways to speak about

– the scopes and limitations of generation (by term existence statements

te≈ t) including the possibility to express definedness and undefinedness

– the “real generalization” of the concept of equality from total algebras topartial algebras via the concept of ECE-equations (i.e. existentially con-ditioned existence equations) w.r.t. the “usual set of terms”, which is (al-most) equivalent to the one of strong or Kleene equations — the semanticsof which mean the identity of the term operations induced by the termsunder consideration —, if one extends the set of allowed terms by introduc-ing for each ordered pair of sorts a “logical” new binary operation symbol,which is always to be interpreted as total first projection (see e.g. [B95]);

– properties of mappings between partial algebras by

– preservation of first-order formulas (e.g. homomorphisms as mappingspreserving all existence-equations)

– reflection of first order formulas (e.g. closedness as reflection of all

term-existence statements te≈ t, injectivity as reflection of x

e≈ y for

distinct variables x and y, etc.)

2“iff” will always stand for the usual abbreviation of “if and only if”.

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0 Introduction 14

– the elementary language of category theory, e.g. the part on factorization sys-tems as an additional tool to formulate properties of structure preserving map-pings between partial algebras.

We shall present the basic definitions, facts and aspects of the theory of partialalgebras. We restrict considerations here to the finitary many-sorted case with finitelymany sorts, and we shall only occasionally discuss the case of infinitely many sorts.

0.3 Some remarks about set theory

0.3.1 Classes and sets in general

In the second half of the 19th century Georg Cantor gave the following definitionof a set:3

Unter einer”Menge“ verstehen wir jede Zusammenfassung M von

bestimmten wohlunterschiedenen Objekten unserer Anschauung oder un-seres Denkens (welche die

”Elemente“ von M genannt werden) zu einem

Ganzen.

This was later manifested in the axiom of comprehension (e.g. by Frege, 1893):

∃y∀x(x ∈ y ⇔ Φ(x)) ,

where Φ is any formula of set theory.However, some years later several antinomies arising from this definition or axiom,

respectively, were discovered. The most simple and probably most famous one is theone from Bertrand Russell in 1903: Let R := x | x /∈ x. Then, if R is a set,one gets the contradiction4

R ∈ R⇔ R /∈ R .Hence, R must not be considered as a set.

Since in universal algebra one often speaks about the collection of all partialalgebras of a given type or signature (satisfying some axioms), one deals here too withentities which are very large and which cannot be called sets any more. Thereforewe adopt the point of view of Godel, Bernays and von Neumann to have asprimitives class and membership “∈” — and where the above definition applies toclasses, yet where not every class is allowed to be a member of some other class. Andsets are then special classes, inuitively spoken

sets are such classes, which are (or can be) members of other classes.

3Quoted from [Sch66]; try of a translation: By a “set” we understand each combination M ofcertain well distinguished objects of our views or our thinking (which are called the “elements” ofM) to a whole.

4This antinomy is also the essence of many indirect proofs such as the one of the undecidabilityof the so-called “halting problem” in computer science.

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0 Introduction 15

Which classes may be sets is then decided by a system of axioms (see below). Some-times one even wants to extend this hierarchy in such a way to have at least a thirdconcept, often called conglomerates, such that “classes are conglomerates which areallowed to be members of other conglomerates”. This is in particular needed in thesenotes — at least in the background —, when we consider later on operators on classesof partial algebras in connection with axioms or semantical operators like forming theclass of all subalgebras, homomorphic images or direct products of the members of agiven class of partial algebras.

Notation 0.1 We recall here the most elementary notation of set theory:

• Let St be the property of a class to be a set, i.e. to be an element of anotherclass:

St(a) : ∃M (a ∈M),

and let x, y, z, a, b,M,N be variables for classes (lower case letters usually indi-cating that they will actually range over sets because of the context).

• If P is a property making sense for sets or classes, then

a | P(a) := a | P(a) and St(a)

designates the class of all sets having the property P .

• For any classes M and N

M ∩N := a | a ∈M and a ∈ N

designates the intersection of M and N . Moreover, if M is non-empty — insymbols: M 6= Ø —, then⋂

M := a | for all b ∈M one has a ∈ b

designates the intersection over all elements from M ; if one has a family (Ni)i∈Iwith some index set or class I, then⋂

i∈I

Ni := a | for all i ∈ I one has a ∈ Ni

designates the intersection over the family (Ni)i∈I .


M ∪N := a | a ∈M or a ∈ N

designates the union or join of M and N . Moreover,⋃M := a | there exists b ∈M such that a ∈ b

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0 Introduction 16

designates the union over all elements from M ; if one has a family (Ni)i∈I withsome index set or class I, then⋃

i∈I

Ni := a | there exists i ∈ I such that a ∈ Ni

designates the union over the family (Ni)i∈I .


M \N := a | a ∈M and a /∈ N

designates the difference or difference class of M and N .

• For any class M define its power class P(M) as

P(M) := a | a ⊆M = a | a ⊆M and St(a)

to be the class of all subsets of M , and

Pfin(M) := a | a ⊆M and a finite

to be the class of all finite subsets of M .

• For any classes M and N define their ordered pair5 (M, N) to be a class suchthat for any other classes M ′ and N ′ one has

(M, N) = (M ′, N ′) iff M = M ′ and N = N ′ . (1)

This construction can be iterated, such that ((M1, M2), M3) and (M1, (M2), M3))are ordered triples — usually one defines one of the two kinds of bracketingas “normal” and omits all brackets except for the outer one, when this “normalbracketing” is meant, and for any natural number n > 2 one can define in sucha way ordered n-tuples, where one usually also defines some normal form, sowe define here recursively, when not stated otherwise, for n ≥ 2 and classes M1

through Mn+1:

(M1, M2, . . . , Mn, Mn+1) := ((M1, M2, . . . , Mn), Mn+1) (left bracketing) .

• For any classes M and N define their Cartesian product or direct product

M ×N := (a, b) | a ∈M and b ∈ N .5See formula (2) and Definition 0.2 below for the possible definition of ordered pairs, and observe

that only the one in Definition 0.2 has the property of (1) for arbitrary classes while the Kuratowskipair in (2) has it only for proper sets.

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0 Introduction 17

Again, this can be iterated to obtain an n-fold Cartesian product — with respectto some bracketing or in normal form — for n > 2 as the set of all n-tuples ofthe corresponding kind of bracketing, i.e. in particular for classes M1 throughMn

n∏i=1

Mi := M1 × . . .×Mn := (a1, . . . , an) | a1 ∈M1 , and . . . and an ∈Mn .

If M1 = M2 = . . . = Mn = M , then we set

n

×i=1

Mi =: M1 × . . .×Mn =: Mn

and call it the n-th power of the class M . In this connection we define inaddition

M1 := M and M0 := Ø .

• Let M1 through Mn be any classes — for a natural number n ≥ 2 —; then any

subclass R ofn

×i=1

Mi is called an n-ary relation between the sets M1 through

Mn , and if M1 = M2 = . . . = Mn = M and R ⊆ Mn, the R is called an n-aryrelation on M — in particular, for n = 2, R is called a binary relation. Inthis connection any subclass R ⊆M may be called a unary relation on M .

• If R ⊆ Mn is an n-ary relation on some class M , then the support suppR ofR is defined as the set of all elements of M occurring in at least one n-tuple inR:

suppR := a | a ∈M and there are i ∈ 1, . . . , n and

(a1, . . . , an) ∈ R such that ai = a .

• A binary relation R ⊆ M1 ×M2 between two classes M1 and M2 is called thegraph of a partial mapping, say f , out of M1 into M2, if

– for any (m1, m2), (m1, m′2) ∈ R one has m′2 = m2 ,

and we write f : M1 −→M2 for the corresponding partial mapping — what isset theoretically an abbreviation of the triple (M1, R, M2). We write

R := graph f .

Moreover, the set

dom f := m1 | m1 ∈M1 and there is m2 ∈M2 such that (m1, m2) ∈ graph f

is called the domain of the partial mapping f . With this notation we also writef : M1 ⊇ dom f →M2.

If one has, in addition,

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0 Introduction 18

– for each m1 ∈M1 there is m2 ∈M2 such that (m1, m2) ∈ R = graph f ,

then f is called a(n everywhere defined) mapping from M1 into M2. If f is amapping from M1 into M2, then we write f : M1 →M2 .

If f : M1 −→ M2 is any (partial) mapping, m1 ∈ M1 and m2 ∈ M2 such that(m1, m2) ∈ f , then we write m2 =: f(m1) and call it the value of m1 w.r.t. the(partial) mapping f .

• Let f : M1 −→M2 be any partial mapping. f is called

– injective, if, for every a, b ∈M1 , f(a) = f(b) implies a = b;

– surjective or onto, if, for every c ∈ M2 , there exists a ∈ M1 such thatf(a) = c;

– total, if, for every a ∈M1 , there exists c ∈M2 such that f(a) = c;

– bijective, if f is total and injective and surjective.

An injective mappings are also called injection, a surjective mapping a sur-jection, and a bijective mapping a bijection.

• If I is any set, and if (Mi)i∈I any I-indexed family of sets, then define theCartesian product or direct product

×i∈I

Mi := f | f : I → ∪i∈I and for all i ∈ I one has f(i) ∈Mi .

Here f is called a choice function.6 Moreover, one often writes f in form of asequence (ai)i∈I or (ai | i ∈ I), where ai := f(i) for every i ∈ I.

If Mi = M for each i ∈ I, then one writes

×i∈I

Mi =: M I .

Observe that, for finite sets I = i1, . . . , in and classes M1 through Mn thereis always — in a so-called natural way — a bijection between the iterated direct

productn

×i=1

Mi based on the binary direct product — for any fixed kind of

bracketing — and the Cartesian product ×i∈I

Mi as defined in this item. In

what follows we shall usually not distinguish between these different kinds ofdefinitions, although they define different sets.

6Observe that the existence of such choice functions in the case that all classes Mi is guaranteedby the so-called Axiom of Choice — see below —, which is not generally accepted in set theory,since it is not constructive, meaning that it claims the existence of mappings without giving an idea,how one could construct them effectively. However the acceptance of the Axiom of Choice makesmathematics more elegant.

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0 Introduction 19

• Two classes M1 and M2 are called disjoint, if M1 ∩M2 = Ø. The classes in afamily (Mi)i∈I are called pairwise disjoint, if Mi ∩Mj = Ø for any i 6= j in I.

If (Mi)i∈I is a family of pairwise disjoint classes, the one often calls⋃i∈IMi a

disjoint union of this family.

We now can also define the disjoint union of a family (Mi)i∈I of classes whichare not necessarily pairwise disjoint by setting:⋃

i∈I:=⋃i∈I

i ×Mi ,

having artificially made the family pairwise disjoint.

0.3.2 Axioms of set theory

With the above Notation 0.1 we have the following axioms, where in particular thefirst axiom guarantees the existence of all the classes defined in these notations.

Axiom (scheme) of Comprehension: ∃M∀x(x ∈M ⇔ St(x) ∧ Φ(x))

for every first order property Φ in a language for classes (i.e. having as fundamentalnon-logical symbol “∈”), where M is not free in Φ.

Meaning: For every property Φ there is a class M containing exactly the sets withproperty Φ.

Axiom of Extensionality: ∀y(y ∈M ⇔ y ∈ N) =⇒ M = N

Meaning: Two classes are equal iff they contain the same elements.

Axiom of the Null-Set: St(Ø)

Meaning: The empty class is a set (follows from other axioms!).

Axiom of Subsets: St(a) ∧ b ⊆ a =⇒ St(b)

Meaning: Each subclass of a set is again a set.

Axiom of the Binary Union: St(a) ∧ St(b) =⇒ St(a ∪ b)Meaning: The union of two sets is again a set.

Axiom of the One-Element Set: St(a) =⇒ St(a)Meaning: For every set a the class containing exactly a as only element is again aset.

Axiom of Union: St(a) =⇒ St(⋃a)

Meaning: For every set its union is again a set, where⋃a := x | ∃y ∈ a : x ∈ y .

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0 Introduction 20

Inverse of the Axiom of Union: St(⋃a) =⇒ St(a)

Meaning: If the union of a class a is a set, then a is already a set; this axiom impliesin particular the “Power Set Axiom”: St(a) =⇒ St(Pa).

Axiom of Replacement: Function(a) ∧ St(dom a) =⇒ St(rng a)

Meaning: If a is a function (i.e. a set of ordered pairs), and if its domain dom a :=x | ∃y((x, y) ∈ a is a set, then its range rng a := y | ∃x : (x, y) ∈ a is a set, too.

Axiom of Infinity: St(ω0)

Meaning: There exists an infinite set; as a special item for an infinite set the set ω0

of all “natural numbers” is chosen, where a set theoretical definition of ω0 runs asfollows: ω0 :=

⋂x | Ø ∈ x ∧ ∀y(y ∈ x =⇒ y ∪ y ∈ x) .

Axiom of Foundation (Regularity Axiom)∀M(∃x(x ∈M) =⇒ ∃y(y ∈M ∧ y ∩M = Ø))Meaning: Every non-empty class M contains an element which is minimal with re-spect to the ∈-relation; in particular there is no class which possesses an infinitedescending ∈-chain, ∈ is an irreflexive partial order on the class of all sets, and thusno class can be an element of itself.

Axiom of Choice∀M(∀z(z ∈ M =⇒ z 6= Ø) =⇒ ∃a(Function(a) ∧ dom a = M ∧ ∀z(z ∈ M =⇒a(z) ∈ z)))Meaning: Each class of non-empty sets possesses a “choice function”.

Two well known consequences of the Axiom of Choice are the following ones:

(Lemma of Kuratowski-Zorn (or Zorn’s Lemma)) Let (H;⊆) be a non-empty inductive subset of some set M . Then H has a maximal element.7

(Theorem of Well-Ordering) On every set there exists a well-ordering.8

0.3.3 Some useful set theoretical definitions and principles

In what follows we give some set theoretical definitions and results — mainly withoutproofs — which can be found e.g. in [Sch66] and in the lecture notes on “AllgemeineAlgebra” of J.Schmidt at the University of Bonn, 1966.

7A set (H;⊆) is called inductive, if, for every finite subset, say F ⊆ H of H, there exists f ∈ Hsuch that

⋃F ⊆ f .

8A binary relation R on a class M ia called a well-ordering of M , if one has reflexivity (i.e.(a, a) ∈ R for every a ∈M), antisymmetry (i.e. (a, b), (b, a) ∈ R imply a = b), transitivity (i.e.for any a, b, c ∈ M , (a, b), (b, c) ∈ R imply (a, c) ∈ R), totality (i.e. for any a, b ∈ M one has(a, b) ∈ R or (b, c) ∈ R) and the minimal element principle (i.e. every non-empty subclass N ofM has a minimal element, say n0 ∈ N such that (n0, n) ∈ R for every n ∈ N).

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0 Introduction 21

Usually, in mathematics one uses as a set theoretic modelling of an ordered pair(x, y), which has to have the property

(x, y) = (v, w)⇔ x = v ∧ y = w , (2)

the so-called Kuratowski pair

(x, y) := x, x, y .

However, this model of ordered pairs satisfies (1) only in the case when all of x, y, u, vare sets but it does not hold for proper classes z, since then z would be the emptyset. In order to be able to have (1) also for possibly proper classes, one may definein a more complicated way:

Definition 0.2 Let M and N be any classes (i.e. not necessarily sets) and define theordered pair (M,N) as

(M,N) := x | x ∈M ∪ P(y) | y ∈ N =

= x | x ∈M ∪ Ø, y | y ∈ N ,

where the elements of M and N are considered to be sets again, and where P(N) :=Y | Y ⊆ N, Y is a set is the so-called power class (i.e. the class of all subsets)of N and in particular P(y) = Ø, y .

Lemma 0.3 (A set theoretical description of ordered pairs, which also al-lows to form ordered pairs of proper classes): With the above definition of anordered pair one has for any four classes M1, M2, N1, N2 (which may be proper)that

(M1, N1) = (M2, N2) iff M1 = M2 and N1 = N2 .

Let us prove some set theoretical results (following J.Schmidt in his lecture noteson “Allgemeine Algebra” from the University of Bonn), which we shall need below inorder to “double” a set or class:

Lemma 0.4 If M is a set, then P(M) 6⊆M .

This means that outside of any set there are (still enough) other sets — namely(at least some of) the subsets of the given set (cf. the results below).

Proof Let R := x | x is a set, and x 6∈ x . Then we have R ∩ M ⊆ M , andR ∩M is a set (since it is the subclass of the set M (axiom of set theory). ThereforeR ∩M ⊆ P(M). If now P(M) ⊆ M , then we had R ∩M ∈ M . Yet we have for allx ∈M that x ∈ R ∩M iff x ∈M and x 6∈ x, therefore in particular for x := R ∩M :

R ∩M ∈ R ∩M iff r ∩M ∈M and R ∩M 6∈ R ∩M .

Yet this is a contradiction, therefore P(M) 6⊆M .

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0 Introduction 22

Lemma 0.5 P(⋃M) 6∈M for every class M .

Proof If we had P(⋃M) ∈ M , then P(

⋃M) were a set. But then the fact that

P(⋃M) ⊆

⋃x∈M x =

⋃M would yield a contradiction to Lemma 0.4.

Lemma 0.6 For any classes M and N one has N, P(⋃⋃

M) 6∈M .

Proof If we had N, P(⋃⋃

M) ∈ M , then P(⋃⋃

M) were a set, and there-fore P(

⋃⋃M) ∈ N, P(

⋃⋃M) ∈ M . Therefore we had P(

⋃⋃M) ∈

⋃M

contradicting Lemma 0.5.

Lemma 0.7 Let M , N and N ′ be sets. If N, P(⋃⋃

M) = N ′, P(⋃⋃

M) ,then N = N ′ .

Proof Namely, if we had N 6= N ′ , then the above equality would yield N =P(⋃⋃

M) = N ′ (since N as well as N ′ would have to be elements of the other set),but this would contradict the assumption N 6= N ′ .

Lemma 0.8 and Definition (of the double of a set):Let M be any set. Then the assignment x 7→ x, P(

⋃⋃M) with x ∈ V (where V

designates the class of all sets) yields an injection of V into M y | y is a set, and y 6∈M .

Then the restriction of this mapping to M yields a bijection of M onto a set, whichwe shall call the double of M — and which we shall denote by Dbl(M) —, since itis disjoint from M .

Proof Use Lemma 0.7 to realize that the above assignment is injective. FromLemma 0.6 we conclude thatM and Dbl(M) = x, P(

⋃⋃M) | x is a set, and x ∈

M is disjoint from M .

We now use this result to “justify identifications” (i.e. to give them a set theoreticalbackground):

Lemma 0.9 (Pickert – van der Waerden) Let A and M be S-sets, and leti : M → A be an injective S-mapping. Then there exist an S-set, say B, and abijection j : B → A such that (A ∪M) ∩B = Ø.

Observe that, in addition, for the set C := M ∪ (B \ (j−1 i)(M))), the mappingi ∪ (j|(B \ j−1(i(M)))) : C → A is a bijection. This means that there always exists aset C containing M and allowing a bijection onto A extending the mapping i.

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0 Introduction 23

For the following consequence one has to be already familiar with the definitionof a partial Σ-algebra, cf. subsection 1.2

Lemma 0.10 Let A and M be any partial Σ-algebras for some signature Σ, and leti : M → A any injective homomorphism. Then there exist a set C containing M asa subset, a bijection j : C → A extending i (i.e. j|M = i), and an algebraic structure(ϕC)ϕ∈Ω on C such that j is an isomorphism from C := (C, (ϕC)ϕ∈Ω) onto A, and Mis a weak relative subalgebra of C.

Proof The set C and the bijection j with the required properties exist according toLemma 0.9. For each ϕ ∈ Ω we define the partial operation ϕC as follows:

domϕC := (c0, . . . , cτ(ϕ)−1) ∈ Cτ(ϕ) | (j(c0), . . . , j(cτ(ϕ)−1)) ∈ ϕA,

and if (c0, . . . , cτ(ϕ)−1) ∈ domϕC , then

ϕC(c0, . . . , cτ(ϕ)−1) := j−1(ϕA(j(c0), . . . , j(cτ(ϕ)−1)).

Then j is easily seen to be an isomorphism from C onto A. Since i : M → A is ahomomorphism, and since j|M = i, it is easy to realize that M is indeed a relativesubalgebra of C.

This lemma is the basis of all “identifications” used in many parts of mathematics,e.g. when one identifies the real numbers r with the pairs (r, 0) of complex numbers,or with the constant polynomials r within the ring of all real polynomials in onevariable, etc.

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1 Similarity types, partial algebras, first visits to closed subsets, homomorphisms anddirect products 24

1 Similarity types, partial algebras, first visits to

closed subsets, homomorphisms and direct prod-

ucts

Since we often consider heterogeneous — also called: many-sorted — (partial) alge-bras as homogeneous — then also called: one-sorted — partial algebras with addi-tional information, we start with the introduction of the latter concept, yet then weshall mostly restrict considerations to the heterogeneous case, except for the cases ofgreater differences.

1.1 Homogeneous partial algebras

Definition 1.1 (of arities, similarity types, partial operations and partialalgebras):(i) (Arities and similarity types): Let Ω be any set whose elements will be called(fundamental) operation symbols. In addition, let τ : Ω→ N0 be a mapping fromΩ into the set of natural numbers including zero; for ϕ ∈ Ω, τ(ϕ) will be interpretedas the arity of the operation symbol ϕ. If

– τ(ϕ) = 0, then ϕ will be called a nullary operation symbol or a (nullary)constant,

– τ(ϕ) = 1, then ϕ will be called a unary operation symbol,

– τ(ϕ) = 2, then ϕ will be called a binary operation symbol,

– τ(ϕ) = n, then ϕ will be called an n-ary operation symbol.

τ — or more precisely the pair (Ω, τ) — will be called a similarity type or brieflya type.

(ii) ((Fundamental, partial) Operations): Let A be any set, and let n be anynatural number including zero.

An n-ary partial operation ψ on A is a partial function

ψ : An −→ A ,

often represented by a (total) function

ψ : domψ → A

— where “domψ” designates the domain (of definition) of ψ — i.e. such that

domψ ⊆ An := (a1, . . . , an) | a1, . . . , an ∈ A ;

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i.e. ψ is a partial function9 out of An into A.If domψ = An, then ψ is called a total operation on A. If domψ = Ø, then ψ is

called a discrete (partial) operation or an empty (partial) operation on A.10

Observe that, for n = 0, A0 = Ø contains just the empty sequence, andtherefore a partial nullary constant on A is either empty or distinguishes exactly oneelement of A — and it is then usually identified with this element.

By POn(A) we designate the set of all partial n-ary operations on A, i.e.

POn(A) =⋃

D⊆AnAD ,

and we set

PO(A) :=∞⋃n=0

POn(A)

to be the set of all finitary partial operations on A, while

On(A) := A(An)

and

O(A) :=∞⋃n=0

On(A)

designate the set of all total n-ary and the set of all total finitary operations on A,respectively.

(iii) (Partial algebras): Let (Ω, τ) be any similarity type. Then a partialalgebra A of type τ is an ordered pair

(C, (JA(ϕ))ϕ∈Ω) ,

9In some connections, in particular for counting arguments, it is often useful to replace a partialfunction f : A −→ B by a total one: f∞ : A → B ∪ ∞, where it is assumed that ∞ /∈ B andgraph f∞ := graph f ∪ (a,∞) | a ∈ A \ dom f.

10Recall that any (partial) mapping or (partial) function, say f : A −→ B, consists of three parts:its start object A, its target object B and its graph

graph f := (a, f(a)) | a ∈ dom f

(which is sometimes just denoted by f , too). Therefore, if one speaks of a (partial) mapping f , itsstart and target object have to be clear, otherwise one should either use the notation f : A −→ Bindicating all the data, or one should give a triplet

(A, graph f,B)

to describe a mapping f . Such a distinction is particularly necessary, if one has to distinguishdifferent empty partial operations, e.g. the n-ary empty partial operations on a non-empty set A fordifferent natural numbers n.

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where C is any set, called the carrier set of A, and

JA : Ω→ PO(C) , JA(ϕ) ∈ POτ(ϕ)(C) for each ϕ ∈ Ω ,

provides the interpretation of the τ(ϕ)-ary partial operation symbol ϕ as a τ(ϕ)-arypartial operation JA(ϕ) w.r.t. the carrier set C as part of the algebraic structure ofA. JA(ϕ) is called the fundamental (partial) operation of A corresponding tothe operation symbol ϕ.

In general — but not always — we shall use the letter A as the name for thecarrier set of the partial algebra A (i.e. the so-called forgetful functor, which mapseach partial algebra to its carrier set, i.e. which forgets the partial algebraic structure,is indicated by “forgetting” the understroke).

Moreover, JA(ϕ) will be abbreviated by ϕA.This yields a notation for the partial algebra A as

(A, (ϕA)ϕ∈Ω) ,

which seems to be recursive or to contain a self-reference, but which is just meant tobe suggestive. Once this has been understood, the above notation should not causeany confusion. Moreover, for binary operations we shall often use infix notation, i.e.we write

aϕAb instead of ϕA(a, b)

(recall the usual way of writing a + b instead of +(a, b) for the ordinary sum of twonatural, rational, real or complex numbers a and b).

(iv) (The class of all partial algebras of some type): By PAlg(τ) we des-ignate the class of all (homogeneous) partial algebras of type τ , and by TAlg(τ) wedesignate the subclass of all (homogeneous) total algebras of type τ .

Remark 1.2 The language sketched so far is meant to be applicable in most conceiv-able cases and to allow to distinguish structures considered in general. Yet in morespecial situations it will often be more convenient to have some simpler notation.

• If Ω = ϕ1, . . . , ϕn consists only of finitely many operation symbols, we usuallywrite partial algebras A of type τ : Ω→ N0 with carrier set A as

(A ;ϕA1 , . . . , ϕAn ) ,

we write Ω as a sequence (without repetitions):

Ω = (ϕ1, . . . , ϕn )

and we write τ just as the sequence of the corresponding arities:

τ := ( τ(ϕ1), . . . , τ(ϕn) ) .

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• As it is customary in “classical algebra”, we shall often omit the superscriptA, mainly in connection with concrete examples, and in particular, when it isobvious, which particular (partial) algebra is meant. Yet we shall almost alwaystry to distinguish by the notation between an algebraic structure and its carrierset.

Example 1.3 (of the type of fields): The similarity type of fields containing whatcan be specified as (partial) operations can be given as ( (+, n,−, ·, e, −1), (2, 0, 1, 2, 0, 1) ),where n specifies the additive neutral element and e the multiplicative one. Examplesof fields are then the fields — we omit the superscripts on the operations and use asusual “0” instead of “nK” and “1” instead of “eK” —Q = (Q ; +, 0,−, ·, 1, −1) of all ra-tional numbers, R = (R ; +, 0,−, ·, 1, −1) of all real numbers and C = (C ; +, 0,−, ·, 1, −1)of all complex numbers.

Example 1.4 (of the type of vector spaces as homogeneous total algebras):Vector spaces over some field K = (K ; +K , nK ,−K , ·K , eK , (−1)K) can be specifiedas “one-sorted” (“homogeneous”) or as “two-sorted” (“heterogeneous” — see below—) total algebras. The homogeneous similarity type of (left) vector spaces is usuallydescribed as ( (+v, nv,−v, (k·)k∈K), (2, 0, 1, (1)k∈K) ), where k· is the symbol for theunary total operation of multiplication of a vector (from the left, i.e. v 7→ k · v) withthe scalar k ∈ K.

It should be observed that this similarity type does not give “the whole story”of the description of K-vector spaces, since the operations of the field K are notincluded in the specification of the type. We shall see below that the heterogeneousspecification is more adequate in that respect. Yet the treatment of vector spacesin Linear Algebra usually fixes the field and does in this connection not use all thepossibilities of a heterogeneous specification (see the description of homomorphismsbetween heterogeneous (partial) algebras) — yet occasionally changes of the fields areconsidered, too.

Example 1.5 (of homogeneous partial algebras: (small) categories in a ho-mogeneous description): Examples of more or less well known (partial) algebraswill be given in a separate subsection. Yet here we want to present already one ex-ample of — in general — really partial algebras, which will be of particular interestlater on. As a prototype one may think of the set of all m×n-matrices, for all naturalnumbers m,n ≥ 1, over some field or ring R with the usual multiplication, which isallowed, whenever this is possible, and with the square unit matrices as the values ofthe operations Dom∗ and Cod∗ introduced below.

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Namely, we consider small categories — yet here as one-sorted partial algebras— of similarity type

(( Dom∗,Cod∗, ), (1, 1, 2)).

Then a small category M is a partial algebra (M, (Dom∗M ,Cod∗M , M))11, where theelements of M are called morphisms, such that

(C 1) Dom∗M and Cod∗M are total unary operations on M and

M : M ×M −→M

is a binary partial operation on M such that,

for f, g ∈M, the composition gMf is defined in M iff Cod∗M(f) = Dom∗M(g) .

(C 2) For each f ∈ M , the compositions f M Dom∗M(f) and Cod∗M(f) M f alwaysexist and yield f as value.

(C 3) Whenever g M f is defined in M , then Dom∗M(g M f) = Dom∗M(f) andCod∗M(g M f) = Cod∗M(g).

(C 4) If, for any f, g, h ∈ M , the compositions g M f and h M g are defined, then(hM g)M f and hM (gM f) are defined and equal: (hM g)M f = hM (gM f)(and vice versa).

Observe that a category in general is defined in a similar way, yet without requir-ing that M be a set, i.e. M may then be a proper class; however it is then requiredthat, for all f, g ∈M , h ∈M | h M f and g M h exist has to be a set.

As particular examples we may consider the (small) categories of all mappings(respectively partial mappings) between subsets of the sets N of natural numbers, Qof rational numbers, R of real numbers, or of any set M , respectively:For (partial) mappings f = (A, graph f, B) and g : (C, graph g, D) with A,B,C,D ⊆M — usually written as f : A → B and g : C → D, respectively as f : A −→ Band g : C −→ D in the case of partial mappings — we get Dom∗f = (A, idA, A),Cod∗f = (B, idB, B), and the composition of g f is defined as usual for mappingsiff B = C, and if this is the case, then (C, graph g, D) = (B, graph g, D) and

g f = (B, graph g, D) (A, graph f, B) = (A, graph (g f), D) .

11This is not the usual intuitive approach to categories, and a reader, who is not familiar withcategories at all should first read the alternative description below as two-sorted partial structures,since categories mainly model sets or classes of “objects” (first sort) and of all the “morphisms”(usually structure preserving mappings between such “objects”) (second sort), and the followingdescription “forgets the objects”. Namely M stands for the set of morphisms of the category underconsideration and the subset CodM (M) = DomM (M) represents the set of objects via their identitymorphisms.

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Let (P ;≤) be any ordered set (cf. Definition 4.11), then ((a, b) ∈ P | a ≤b ; Dom∗,Cod∗, ) is a (small) category with Dom∗(a, b) := (a, a), Cod∗(a, b) :=(b, b), and (b, c) (a, b) = (a, c) — cf. the transitivity law of ordered sets. (We leavethe verification as an exercise.)

Example 1.6 (of a sketch of a partial algebra): In Figure 1 we give a sketchof an example of a partial algebra A = (A ;ϕA1 , ϕ

A2 , ϕ

A3 , ϕ

A4 ), where we have chosen

τ := (2, 1, 1, 0), with Ω := (ϕ1, ϕ2, ϕ3, ϕ4 ), and:

A := a, b, c, d, e,graphϕA1 := ((a, a), b), ((e, b), d),graphϕA2 := (c, a), (a, e),graphϕA3 := (c, e),

ϕA4 := c.

A :a

ϕA2

ϕA4 = c ϕA3

ϕA1

ϕA1

ϕA2

b

e

d1

2

6

-ZZZZZZZZZZZ~

-

e e

u ee

llll

,,,,

-

Figure 1: Example of a partial algebra

Extremal examples of partial algebras are on the one side the so-called discretepartial algebras, where all fundamental operations are empty:

Adiscrete := (A, (Ø)ϕ∈Ω) .

On each set there is exactly one discrete partial algebra — which can be identifiedwith its carrier set — of a given similarity type.

On the other side all total (universal) algebras are special partial algebras;here all fundamental operations are “everywhere defined”, i.e. (A, (ϕA)ϕ∈Ω) is total,iff domϕA = Aτ(ϕ) for each ϕ ∈ Ω.

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Thus all total algebras like groupoids, semigroups, monoids, groups, rings, semi-lattices, lattices, Boolean lattices and Boolean algebras (Boolean rings) (see section 4of examples, below) are special examples of homogeneous partial algebras.

Fields are proper homogeneous partial algebras where the multiplicative inversion“−1” is the only proper partial operation (all other fundamental operations are total)(similarity type ((+, 0,−, ·, 1,−1 ), (2, 0, 1, 2, 0, 1))).

Observe that if we write occasionally in connection with some partial algebra Athat ϕA(a1, . . . , aτ(ϕ)) = a, then we mean that ϕA(a1, . . . , aτ(ϕ)) exists and has thevalue a.

A very specific and important example of a total algebra of any given similaritytype is the term algebra or word algebra on some given alphabet X.

Definition 1.7 (of homogeneous terms and of the term algebra):(i) (Homogeneous terms): Assume X to be any set, which we call a set of

variables, where we assume the set X of variables to be disjoint from the set Ω ofoperation symbols:

X ∩ Ω = Ø .

Let (X ∪ Ω)∗ be the set of all finite sequences of elements from X ∪ Ω often writtenas “words” l1l2 . . . ln with l1, l2, . . . , ln ∈ X ∪Ω, where n designates the length of sucha word.12 Now we define the set of all terms w.r.t. the type τ and with variables inX as follows:

(T1) x is a term for each variable x ∈ X.

(T2) Let ϕ ∈ Ω be any nullary operation symbol, then ϕ is a term.

(T3) Let ϕ ∈ Ω be any operation symbol with τ(ϕ) 6= 0, and let t1, t2, . . . , tτ(ϕ) beany terms, then ϕt1t2 . . . tτ(ϕ) is a term.13

(T4) Only words formed according to rules (T1), (T2) and (T3) are terms.

The set of all terms of type τ with variables in X is denoted by T (X, τ).(ii) (Homogeneous term algebra): Observe that one easily gets a structure

(ϕT (X,τ))ϕ∈Ω of type τ on T (X, τ) by defining

ϕT (X,τ)(t1, . . . , tτ(ϕ)) := (ϕ, t1, . . . , tτ(ϕ)) (or simply ϕt1 . . . tτ(ϕ) ) ,

12Observe that a word, say l1l2 . . . ln, is just an abbreviation for the n-tuple (l1, l2, . . . , ln) writtenas a word in order to save brackets and commata. It should be noted that this corresponds to theso-called Polish notation for terms below, which can also be given without using brackets.

13If the operation symbols are actually words of the informal language — as it often happens incomputer science —, then we rather have to write this in the longer version as (ϕ, t1, t2, . . . , tτ(ϕ)).If ϕ is a binary operation symbol, then one often also writes (t1ϕt2), i.e. one uses the so-called infixnotation.

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for each operation symbol ϕ ∈ Ω and for any terms t1, . . . , tτ(ϕ) ∈ T (X, τ); in par-ticular, if τ(ϕ) = 0, then ϕT (X,τ) = ϕ (respectively ϕT (X,τ) = (ϕ), if one has to usetuples instead of words — however, then in (T1), for any variable x ∈ X, (x) is aterm (instead of x)).

Example 1.8 (of term algebras): Well-known algebras, which “can be consideredas term algebras”14 are

• the algebra N0 := (N0 ; 0N0 , SUCCN0) is “isomorphic to” the term algebra of type( 0, 1 ) on the empty set of variables (word algebra on the empty alphabet),where N designates the set of all natural numbers including zero (0), and if oneobserves that the natural number n is the n-fold sucessor of the constant 0N0

(with 0N0 := 0 and sets as usual SUCCN0(n) := n+ 1 = n+ SUCCN0(0), for eachnatural number n);

• and similarly N := (N0 ; SUCCN) as “isomorphic to” the term algebra of type(1) on the one-element set X := 0 of variables (with SUCCN := SUCCN0).

1.2 Heterogeneous partial algebras

Since we shall deal from now on mainly with many-sorted (partial) algebras, we listfirst some basic definitions of set-theoretical relations and operations among S-setsfor a given non-empty set S.

Definition 1.9 (of S-sets and of operations with and relations among them):In all the rest of the text let S be any non-empty set (which usually will be assumedto be finite). The elements of S will be called sorts.

(i): Then we define an S-sorted set, briefly called an S-set, as a family (As)s∈S— which we shall usually abbreviate with A, whenever it is obvious that we deal withS-sets rather than with “usual sets”. For s ∈ S the s-th component As is called thephylum of A of sort s or carrier set of sort s of A.15 — If one does not wantto mention explicitely the set S of sorts one may speak in general of a many-sortedset instead of an S-set.

14This means that they are “isomorphic to” a term algebra, what will be defined later in an exactway, but what intuitively means that there exists a bijective mapping between the carrier sets pre-serving the structures in both directions, and therefore isomorphic partial algebras are “algebraicallyindistinguishable”.

15It would be more consequent to write sA for the phylum of sort s of the S-set A instead of As— in analogy to the way how we designate the partial fundamental operations of a partial algebra(cf. subsection 1.1) —, yet the notation As is the more common one in the abstract description ofS-sets (while sA corresponds more to the way of writing the phyla in concrete cases of many-sortedpartial algebras — see Example 1.12 —), and this way fundamental operations and carrier sets canbe better distinguished.

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(ii): Let A = (As)s∈S, B = (Bs)s∈S be any S-sets and let (Ai)i∈I = (((Ais)s∈S)i∈I)be a family of S-sets (I any set). Then the usual set theoretical relations and oper-ations generalize to S-sets in a straight forward way — namely componentwise (orsortwise):

• B is said to be an S-subset of A, iff

Bs ⊆ As for all s ∈ S .

• The set theoretical difference of S-sets A and B is defined as

A \B := (As)s∈S \ (Bs)s∈S := (As \Bs)s∈S .

• If I 6= Ø, then the intersection of S-sets is defined as⋂i∈I

Ai :=⋂i∈I

(Ais)s∈S := (⋂i∈I

Ais)s∈S .

In the case of an empty index set, i.e. of I = Ø, this intersection is only defined,whenever there is a context in which all Ai are supposed to be S-subsets of agiven S-set A, and then this intersection (with empty index set) is defined asA: ⋂

i∈øAi :=

⋂i∈ø

(Ais)s∈S := (Ai)i∈I = A .

• The union of S-sets is defined as⋃i∈I

Ai :=⋃i∈I

(Ais)s∈S := (⋃i∈I

Ais)s∈S .

• The direct product of S-sets is defined as

×i∈I

Ai :=×i∈I

(Ais)s∈S := (×i∈I

Ais)s∈S .

In particular, if Ai = A for each i ∈ I, then one writes AI ( = (AsI)s∈S ) instead

of×i∈I

Ai and calls it the I-th (heterogeneous) power of the S-set A.

• Let A = (As)s∈S and B = (Bs)s∈S be any S-sets. An S-mapping f from A intoB is then a family (fs : As → Bs)s∈S of “usual” mappings.

Example 1.10 (of many-sorted equivalence relations): It is well known that— for one-sorted sets — an equivalence relation R on a set A is a binary relation onA, which is reflexive, symmetric and transitive. In particular it is a subset of A×A.

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Thus an equivalence relation R on an S-set A = (As)s∈S has to be an S-subset ofA× A = (As × As)s∈S, i.e. R has to be an S-set of binary relations:

R = (Rs)s∈S ⊆ (As × As)s∈S

which is reflexive, symmetric and transitive, i.e. each Rs is an equivalence relation onAs for each s ∈ S.

In analogy to the homogeneous case one defines now a heterogeneous or many-sorted similarity type, which is usually — in particular in computer science —called a signature

Definition 1.11 (of heterogeneous similarity types (signatures) and hetero-geneous (partial) operations):

(i) ((Heterogeneous) Signature): By a (heterogeneous) signature Σ we under-stand (here) a quintuple Σ := (S,Ω, τ, η, σ), where

– S is a non-empty set, the elements of which are called sorts,

– Ω is a set, the elements of which are called (fundamental) operation sym-bols;

– τ : Ω→ N0 is a mapping assigning to each operation symbol ϕ ∈ Ω its arity16

– η is a mapping from Ω into the set S∗ (where S∗ designates as usual the setof all finite — empty or non-empty — words over the alphabet S) such thatη(ϕ) = sϕ1 . . . s

ϕτ(ϕ) is the word of input sorts or sorts of entries for the

operation symbol ϕ; thus the length τ(ϕ) of η(ϕ) is just the arity of the operationsymbol ϕ, while for 1 ≤ i ≤ τ(ϕ) the i-th “letter” sϕi indicates that the i-thargument of each realization of ϕ has always to be of sort sϕi — which is thei-th input sort of ϕ; if ϕ is a nullary operation symbol, then η(ϕ) is the emptyword λ;

– σ : Ω → S is a mapping assigning to each operation symbol ϕ ∈ Ω its outputsort or simply sort sϕ.

Thus, for every fundamental operation symbol ϕ ∈ Ω, the pair (η(ϕ), σ(ϕ)) — whichis often given a single name — describes the “input output behaviour” of the corre-sponding heterogeneous fundamental operations.

16τ is usually not mentioned explicitly in the description of a many-sorted signature; yet, becauseof our intended different kinds of presentations of heterogeneous partial algebras it is quite useful tohave it around; moreover, in this way a signature is just an extension of a similarity type. Moreover,σ is usually omitted, too, and then η(ϕ) is defined as the ordered pair consisting of the sequence ofinput sorts of ϕ in the first component and having the output sort of ϕ in the second component.However, we find it more convenient to have denotations for each component available.

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(ii) (Heterogeneous partial algebras): A many-sorted (partial) alge-bra or heterogeneous (partial) algebra A of signature Σ is then defined as((As)s∈S, (ϕ

A)ϕ∈Ω), where (As)s∈S is an S-set17, and for ϕ ∈ Ω with (η, σ)(ϕ) =(sϕ1 . . . s

ϕτ(ϕ), s

ϕ), then ϕA is a (possibly partial) mapping from (or out of) As1ϕ× . . .×Asτ(ϕ)

ϕ — often abbreviated as Aη(ϕ) — into Asϕ , i.e. one has:

ϕA : Aη(ϕ) = Asϕ1 × Asϕ2 × . . .× Asϕτ(ϕ)⊇ dom ϕA → Asϕ .

What has been said about homogeneous partial operations and algebras can alsobe said about heterogeneous ones in an analogous way. In particular, there is againexactly one discrete heterogeneous partial algebra Adiscrete of signature Σ onany given S-set A = (As)s∈S, and A = ((As)s∈S, (ϕ

A)ϕ∈Ω) is a total heterogeneousalgebra of signature Σ, iff, for each ϕ ∈ Ω, one has

domϕA = Asϕ1 × Asϕ2 × . . .× Asϕτ(ϕ)(= Aη(ϕ)) .

Two (homogeneous or heterogeneous) partial algebras of the same signature arecalled similar.

Let w = s1s2 . . . sn ∈ Sn and s ∈ S be given, and let A = (As)s∈S be any S-set,then

PO(w,s)(A) := h | h : As1 × As2 × . . .× Asn −→ As

designates the set of all n-ary partial operations with “input sequence” w = s1s2 . . . snand output sort s (etc.); in particular O(w,s)(A) designates the subset of all totalmany-sorted operations among them.

(iii) (The class of all heterogeneous partial algebras of some signature):By PAlg(Σ) we designate the class of all heterogeneous partial algebras of signatureΣ and by TAlg(Σ) we designate the subclass of all heterogeneous total algebras ofsignature Σ.

If not explicitly stated differently, in what follows we shall always assume that weare given an arbitrary but fixed signature (or similarity type) Σ := (S,Ω, τ, η, σ) (or(Ω, τ)), and that all partial algebras under consideration are of this (same) signature(or type).

Example 1.12 (of heterogeneous partial algebras: (small) categories in theusual two-sorted description): As an example of heterogeneous partial algebras,where the introduction of a second sort does not prevent in general that the structuresare really partial, we give here the “usual” description of small categories as two-sortedalgebras (like for the homogeneous case a formal discription using axioms will be givenin section 4):

17Cf. section 4

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In this case we have S = Ob,Mor , where the elements of carriers of sort Ob arecalled objects, while the elements of carrier sets of sort Mor are called morphisms.

The signature is given as follows (in the notation often used in computer science):Sorts : Ob, MorOperations : : Mor×Mor → Mor

Dom : Mor → ObCod : Mor → Ob

1 : Ob → MorObserve that the arities (i.e. τ) are here specified indirectly.If C is any category, then one traditionally writes Ob(C) instead of COb for the

set of objects, and Mor(C) instead of CMor for the set of morphisms of the categoryC. Moreover the partial operations have to satisfy the following conditions:

(C1) DomC and CodC are total unary mappings from Mor(C) into Ob(C), and C :Mor(C)×Mor(C) −→ Mor(C) is a binary partial operation on Mor(C) such that,for f, g ∈ Mor(C), g C f is defined in C iff CodC(f) = DomC(g).

(C2) For each f ∈ Mor(C), with DomC(f) = A and CodC(f) = B, one has that f C1CA

and 1CB C f always exist and yield f as value.

(C3) Whenever gCf is defined in C, then DomC(gCf) = DomC(f) and CodC(gCf) =CodC(g).

(C4) If, for any f, g, h ∈ Mor(C), g C f and h C g are defined, then (h C g) C f andh C (g C f) are defined and equal: (h C g) C f = h C (g C f).

(C5) For every object A ∈ Ob(C) one has DomC(1CA) = A and CodC(1C

A) = A.

As in the one-sorted description a category in general is defined similarly withoutrequiring that Ob(C) and Mor(C) be sets, i.e. they may then be proper classes; howeverit is then required that for all objects A,B ∈ Ob(C)

Mor(A,B) := h ∈ Mor(C) | DomC(h) = A and CodC(h) = B

has to be a set.A typical example of a category is the category Set, where Ob(Set) is the class of

all sets, Mor(Set) is the class of of all mappings between sets, and if f : A→ B andg : C → D are mappings between sets A,B,C,D ∈ Ob(Set), then DomSet(f) = A,CodSet(f) = B, 1Set

A = idA is the identity mapping from A onto itself, and g Set fexists, iff B = C, and if B = C, then g Set f = g f is the usual composition ofmappings.

A really small category FSN is given by choosing Ob(FSN) to be the set of allfinite subsets of the set N of all natural numbers, and Mor(FSN) is then the set ofall mappings between these objects. The (partial) operations are just the restrictionsto FSN of those from Set.

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Special examples of heterogeneous total algebras of signature Σ = (S,Ω, τ, η, σ)are also given by the heterogeneous term algebras of signature Σ on some S-setX = (Xs)s∈S of variables. We could define them “intuitively” in the same way as wesketched homogeneous term algebras; yet in order to be able to prove their propertiesmore easily, we postpone their definition, see Example 2.5.

1.3 Closed subsets, generation and algebraic (structural) in-duction (1st visit)

The above recursive definition of terms gives rise to a proof technique usually denotedas induction on the structure of terms,18 which we want to replace immediatelyby the more general algebraic induction or, as it is usually called in computerscience, structural induction. For this purpose we need the concept of generation,which needs the one of a closed subset of a partial algebra.

In what follows we always assume a fixed heterogeneous signature Σ = (S,Ω, τ, η, σ)to be given.

Definition 1.13 (of closed subsets and subalgebras of a partial algebra):Let A = ((As)s∈S, (ϕ

A)ϕ∈Ω) be any partial algebra of signature Σ, and let B = (Bs)s∈Sbe an S-subset of the S-set A = (As)s∈S (i.e. one has Bs ⊆ As for each sort s ∈ S).Then B is called a closed subset of the partial algebra A, iff it is closed withrespect to any applications of fundamental operations of A to sequences formed onlyby elements of B, i.e.

(C) for all ϕ ∈ Ω with (η, σ)(ϕ) = (s1s2 . . . sτ(ϕ), s), and for all (a :=) (a1, . . . , aτ(ϕ)) ∈(Bs1×. . .×Bsn)∩domϕA (=Bη(ϕ)∩domϕA) one also has (ϕA(a) =) ϕA(a1, . . . , aτ(ϕ)) ∈Bs.

A partial algebra B = (B, (ϕB)ϕ∈Ω) with carrier set B = (Bs)s∈S is then called a(closed) subalgebra of the partial algebra A = (A, (ϕA)ϕ∈Ω) of the same similaritytype, iff

(c1) B is a closed subset of A;

(c2) for each ϕ ∈ Ω, ϕB is the restriction to Bs1 × . . . × Bsn of the fundamen-tal operation ϕA — with (η, σ)(ϕ) = (s1s2 . . . sn, s) — (in symbols ϕB =ϕA |Bs1×...×Bsn= ϕA |Bη(ϕ)), i.e. one has domϕB = domϕA ∩ (Bs1 × . . . × Bsn)and if (b1, . . . , bn) ∈ domϕB, then ϕB(b1, . . . , bn) = ϕA(b1, . . . , bn).

18Often one also applies here an induction on the length of a term, when one defines as length ofa term the sum of the numbers of all instances of occurrences of variables and operation symbols ina given term (i.e. variables and nullary operation symbols yield terms of length 1, while (ϕ, t1, t2)for a binary operation symbol ϕ and terms t1 of length n1 and t2 of length n2 would be a term oflength 1 +n1 +n2). As an exercise one should formulate the corresponding induction principle (andprove its correctness).

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These two properties are combined in writing

graphϕB = (graphϕA) ∩ (Bs1 × . . .×Bsn × As) .

Remark 1.14 This definition shows that not every subset of a partial algebra Acan be the carrier set of a subalgebra of A. Typical examples are subfields, sub-vectorspaces, subcategories (see above), or e.g. submonoids, subgroups and subrings(etc.) w.r.t. similarity types as given below in the section 4 of examples.

However, in connection with the “classical” definition of groups as algebras withone binary operation satisfying some axioms requiring among others the existence ofa left neutral element and of a left inverse, the closed subsets in the sense of the abovedefinition are only the carrier sets of subsemigroups (i.e. they are only closed w.r.t.this binary operation), but they need not contain the neutral element nor the inverse,since these are not described in that approach by fundamental operations.

Of fundamental importance for the process of “generation” is the following

Lemma 1.15 (The set of closed subsets is closed w.r.t. intersections.) LetA be a partial algebra of signature Σ, and let B be any system of closed (S-) subsetsof A. Then the intersection of all S-sets in B is again a closed subset of A, i.e.

H :=⋂B := (

⋂B∈B

Bs)s∈S

is a closed S-subset of A. — In particular, A =⋂

Ø is a closed subset of A.

Proof With the notation of the lemma let H :=⋂B =

⋂B|B ∈ B, let ϕ ∈ Ω with

(η, σ)(ϕ) = (s1s2 . . . sn, s) and let (b1, . . . , bn) ∈ (Hs1 × . . . × Hsn) ∩ domϕA. SinceH ⊆ B and since B is closed for each B ∈ B, we have ϕA(b1, . . . , bτ(ϕ)) ∈ Bs for eachB ∈ B, thus ϕA(b1, . . . , bτ(ϕ)) ∈ (

⋂B)s = Hs showing that H is a closed subset of A.

This lemma shows that the set of all closed subsets of a partial algebra alwaysforms a so-called closure system. Therefore we recall:

Definition 1.16 (of a closure system and a closure operator):Let us recall that a closure system H on an S-set A is a set of S-subsets of A

which contains A and is closed w.r.t. arbitrary intersections of non-empty subsets ofH,19 i.e.:

If B ⊆ H and B 6= Ø, then⋂B ∈ H .

19If we use the convention⋂

Ø := A, whenever it is known that the intersection only ranges oversystems of subsets of some fixed (S-)set A, then we could just require

⋂B ∈ H for every subsystem

B ⊆ H.

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If H is a closure system on A, then the elements of H are called closed subsets ofA (w.r.t. H).

And a closure operator C on an S-set A is a mapping assigning to each S-subset, say M of A an S-subset CM of A such that it has the following properties forall M,N ⊆ A:

(C1) M ⊆ CM (extensity)

(C2) M ⊆ N ⇒ CM ⊆ CN (monotonicity)

(C3) CCM = CM (idempotency)

If C is a closure operator on A, then the subsets B of A with CB = B are again calledclosed subsets of A (w.r.t. C).

That the two definitions above of closed subsets of some set A are in principle thesame follows from the next result, which shows that the concepts “closure systems”and “closure operators” are in principle “two sides of the same coin” (like the conceptsof “partitions of a set” and of “equivalence relations on a set”, respectively).

Lemma 1.17 (The interconnection between closure systems and closureoperators): Let A = (As)s∈S be an S-set, let H be a closure system and C a closureoperator on A, respectively. Then

(i) The operator CH defined by

CH(B) :=⋂H | B ⊆ H ∈ H = (

⋂Hs | B ⊆ H ∈ H)s∈S ,

for all B ⊆ A, is a closure operator on A.

(ii) The system of S-subsets of A given by

HC := H ⊆ A | C(H) = H

is a closure system on A.

(iii) One always hasC(HC) = C ,

andH(CH) = H .

We leave the straightforward proof as an exercise.

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Corollary 1.18 and Definition (of the closure operator of generation): Foreach partial algebra A the system cs(A) of all closed subsets of A is a closure systemon A, and the corresponding closure operator is denoted by CA — by some authorsalso by < . >A, i.e. for each subset M of A one has:

< M >A:=< M >:= CAM :=⋂H ∈ cs(A) |M ⊆ H = (

⋂Hs|M ⊆ H ∈ cs(A))s∈S .

One says that the partial algebra A is generated by an S-subset M , if CAM = A,and M is then called a generating subset of the partial algebra A.

Moreover, let B be a closed subset of A, and let N be any subset of B, then wesay that N generates B or that N is a generating subset of B, iff CAN = B.

Closely connected with the concept of generation is the proof concept of algebraicor structural induction:

Lemma 1.19 (on algebraic induction or structural induction): Let A be anypartial algebra of signature Σ, and let P be a property making sense for the elementsof A = (As)s∈S. Let M = (Ms)s∈S be any S-subset of A. Then P is true for everyelement of the closed subset CAM , if one can prove:

• Beginning of the induction:P is true for every element m ∈Ms for every sort s ∈ S.

• Induction step:

– P is true for every value ϕA of some nullary operation symbol ϕ ∈ Ω.

– Let ϕ be any operation symbol of arity n = τ(ϕ) 6= 0, and let, for each 1 ≤i ≤ n, ai be an element of sort η(ϕ)i, such that P is true for ai, and suchthat (a1, . . . an) ∈ domϕA, then P is true for the element ϕA(a1, . . . , an).

Proof Since the above items say that the S-subset, say P , of A, which consists ofall elements of A having the property P , is a closed subset containing M , and sinceCAM is the smallest closed subset of A containing M , P must be true for all elementsof CAM .

One might have different opinions, whether to count the statement about thenullary constants to the induction step or to the beginning of the induction. —Observe that the principle of complete induction for natural numbers as well as theinductions on the length of terms respectively on the structure of terms are specialinstances of the principle of structural induction (exercises).

A useful simple result, the straightforward proof of which is left to the reader asan exercise, is the following one:

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Lemma 1.20 Each element outside of a generating set is the value of afundamental operation. Let A = (A, (ϕA)ϕ∈Ω) be any partial algebra of signatureΣ, and let M be a generating subset of A. Then

A = M ∪⋃ϕ∈Ω

ϕA(domϕA) .

Note that the statement of the above lemma can be formulated slightly moregenerally as

CAM = M ∪⋃ϕ∈Ω

ϕA(domϕA ∩ (CAM)η(ϕ))

for any S-subset M of A.20

In connection with partial algebras one has more possibilities to define substruc-tures than when one deals only with total algebras, since besides the “process ofgeneration” one can also consider the “process of restriction” and the “process ofapproximation”, which lead to different kinds of substructures:

Definition 1.21 (of relative and weak subalgebras) and Remarks:

1. As we have already tried to emphasize, the “process of generation of struc-tures” is based on the concept of a (closed) subalgebra, which correspondsto the one of a subalgebra in connection with total algebras. It is based on theconcept of a closed subset of a partial algebra as defined in Definition 1.13.

2. The “process of restriction of structure” to subsets of an algebra corre-sponds to the one of a relative subalgebra:

A partial algebra B = (B, (ϕB)ϕ∈Ω) is called a relative subalgebra of a partialalgebra A = (A, (ϕA)ϕ∈Ω) of the same similarity type, iff

(r1) B is some (arbitrary) S-subset of A,

(r2) for each ϕ ∈ Ω, with (η, σ)(ϕ) = (s1 . . . sτ(ϕ), s) , ϕB is the total restric-tion of ϕA to B (in symbols: ϕB = ϕA||B), what means that domϕB ⊆domϕA, and for any (b1, . . . , bτ(ϕ)) ∈ Bs1 × . . .× Bsτ(ϕ)

, if (b1, . . . , bτ(ϕ)) ∈domϕA and ϕA(b1, . . . , bτ(ϕ)) ∈ Bs, then (b1, . . . , bτ(ϕ)) ∈ domϕB andϕB(b1, . . . , bτ(ϕ)) = ϕA(b1, . . . , bτ(ϕ)). This is briefly indicated by writing:

graphϕB = graphϕA ∩ (Bη(ϕ) ×Bσ(ϕ)).

20Recall that powers of S-sets are formed sortwise.

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This definition shows that every S-subset B of A is the carrier set of exactlyone relative subalgebra B of A, and that the structure of B is uniquely definedby the structure of A and the specification of the subset B (as in the case ofsubalgebras with the only difference that for a subalgebra the carrier set has tobe closed).

3. The “process of approximation” (or “exhaustion”) of algebraic structuresby ”very small” (e.g by finite) pieces (as it is somewhat done e.g. in com-puters) leads to the concept of a weak relative subalgebra: A partial algebraB = (B, (ϕB)ϕ∈Ω) is called a weak relative subalgebra or briefly a weaksubalgebra of some partial algebra A = (A, (ϕA)ϕ∈Ω), iff

(w1) B is some (arbitrary) S-subset of A

(w2) for each ϕ ∈ Ω, the graph of ϕB is contained in the graph of ϕA:

graphϕB ⊆ graphϕA ∩ (Bη(ϕ) ×Bσ(ϕ)) ,

i.e. if ϕB(b1, . . . , bτ(ϕ)) = b in B, then ϕA(b1, . . . , bτ(ϕ)) = b in A. Observethat now each S-subset B of a partial algebra A is the carrier set of atleast one but in general of quite a lot of weak subalgebras of A.

Remark 1.22 In particular one has that every subalgebra of A is a relative subal-gebra of A, and every relative subalgebra of A is a weak subalgebra of A. Moreover,every relative subalgebra on a closed subset is a subalgebra.

1.4 A first visit to homomorphisms and direct products

In what follows it will occasionally be useful for the formulation or proof of some ofthe results to have already the concepts of homomorphisms and of direct productsavailable.

First let us define our tool for the comparison of partial algebras. In section 3.3we shall motivate in more detail, why in particular this concept of homomorphismsas defined below has been chosen

Definition 1.23 (of homomorphisms, closed homomorphisms and isomor-phisms): Let A and B be any partial algebras of the same signature Σ with carriersets A and B, respectively, and let f : A→ B be any mapping.

(a) (Homomorphisms): f is called a homomorphism from A into B — insymbols: f : A → B —, iff for every fundamental operation symbol ϕ ∈Ω with (η, σ)(ϕ) = (s1 . . . sn, s) (n = 0 is allowed) and for every sequence(a1, a2, . . . , an) ∈ As1 × As2 . . . × Asn , the existence of ϕA(a1, a2, . . . , an) in thestart algebra A implies the existence of ϕB(fs1(a1), fs2(a2), . . . , fsn(an)) in the

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target algebra B, and in the case of the existence of ϕA(a1, a2, . . . , an) one hasthat

fs(ϕA(a1, a2, . . . , an)) = ϕB(fs1(a1), fs2(a2), . . . , fsn(an)) .

(b) (Closed homomorphisms): The concept of homomorphisms between partialalgebras is rather “weak” and allows many additional properties, among whichthe following one is of most importance, as we shall see later:

A homomorphism f : A → B is called closed (i.e. a closed homomor-phism)21, iff f satisfies in addition to the properties of a homomorphism that

for every fundamental operation symbol ϕ ∈ Ω with (η, σ)(ϕ) =(s1 . . . sn, s) (n = 0 is allowed) and for every sequence (a1, a2, . . . , an) ∈As1 × As2 . . . × Asn , the existence of ϕB(fs1(a1), fs2(a2), . . . , fsn(an))implies the existence of ϕA(a1, a2, . . . , an) .

(c) (Isomorphisms): A homomorphism f : A → B will be called an isomor-phism between A and B, iff f is bijective (i.e. injective and surjective), andthe inverse mapping f−1 : B → A (with f−1(b) := a iff f(a) = b) is a homo-morphism, too: f−1 : B → A .

Remark 1.24 (i) Observe that the identity mapping on a carrier set A of a partialalgebra A is always a homomorphism and even an isomorphism idA : A→ A.

(ii) Observe that in the case of total algebras of any (but the same) homogeneousor heterogeneous signature Σ every bijective homomorphism is already an iso-morphism (exercise!). However, this is no longer true in the case of partialalgebras:

Let B be a total algebra of signature Σ with a non-empty structure, and letA := Bdiscrete be the discrete partial algebra on the same carrier set B. Thenthe identity mapping on B is a bijective homomorphism from A onto B, but itsinverse is not a homomorphism from B into A.

A very useful observation is the following one, the straightforward proof of whichby algebraic induction is left as an exercise:

Proposition 1.25 (Uniqueness of a homomorphic extension): Let f, g : A→B be homomorphisms between similar partial algebras and let M be any S-subset of

21The expression closed homomorphism is not an optimal choice. In some books. e.g. inGratzer [G68], this kind of homomorphisms to be defined here is called strong homomorphism,yet in the “Torun school of S lominski” strong homomorphisms have a quite different meaning, and inthe papers of J.Schmidt “strong homomorphism” had still another meaning. As we shall see belowin subsection 3.3 the denotation term reflecting homomorphism would be most adequate butwould perhaps be too long for such a fundamental concept. Nevertheless, by and by, this name orits abbreviation TR-homomorphism or something similar will get into use.

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A. If fs(m) = gs(m) for each m ∈ Ms and for each s ∈ S (in symbols f |M = g|M),then fs(m) = gs(m) for each m ∈ (CAM)s and for each s ∈ S (i.e. f |CAM = g|CAM).

If, in particular, M is a generating subset of A, then f |M = g|M implies f = g .

While the formation of substructures and homomorphic images (i.e. targetalgebras of surjective homomorphisms) depends on one given partial algebra and canonly yield a “smaller structure” — as far as the carrier set is concerned — than theone under consideration, the formation of direct products provides a tool to constructlarger algebras from the given ones:

Definition 1.26 (Direct products): Let (Ai)i∈I be any family of partial algebrasof the same signature Σ = (S,Ω, τ, η, σ), where Ai = (Ai, (ϕ

Ai)i∈I) for each i ∈ I.Define

×i∈I

Ai := (×i∈I

(Ai)s)s∈S := ( a : I →⋃i∈I

(Ai)s | a(i) ∈ (Ai)s for each i ∈ I )s∈S

to be the S-set of all so-called choice functions a = (ai)i∈I of the family (Ai)i∈I for the

different sorts. And on A :=×i∈I

Ai we define a partial algebraic structure (ϕA)ϕ∈Ω as

follows:For ϕ ∈ Ω, with (η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s), set

domϕA := (a1, . . . , aτ(ϕ)) ∈ Aη(ϕ) | (a1(i), . . . , aτ(ϕ)(i)) ∈ domϕAi for each i ∈ I

and if (a1, . . . , aτ(ϕ)) ∈ domϕA, then set

ϕA(a1, . . . , aτ(ϕ)) := (ϕAi(a1(i), . . . , aτ(ϕ)(i)) | i ∈ I),

i.e. (ϕA(a1, . . . , aτ(ϕ)))(i) := ϕAi(a1(i), . . . , aτ(ϕ)(i)) for each i ∈ I.

We define∏

i∈I Ai := ( (×i∈I

(Ai)s)s∈S, (ϕA)ϕ∈Ω ) = (×

i∈IAi, (ϕ

A)ϕ∈Ω) to be the di-

rect product of the family (Ai)i∈I (often it is also denoted by×i∈I

Ai).

For each j ∈ I we denote by prj :×i∈I

Ai → Aj, a 7→ a(j) ∈ (Aj)s (a ∈×i∈I

(Ai)s ,

for some s ∈ S) the j-th (canonical) projection.

It is a straightforward exercise to prove

Lemma 1.27 (All projections are homomorphisms.) Let∏

i∈I Ai be the directproduct of the family (Ai)i∈I of partial algebras of the same signature Σ. Then each

projection prj :×i∈I

Ai → Aj is a homomorphism prj :∏

i∈I Ai → Aj .

Moreover, whenever at least one of the domains of the fundamental operationsof∏

i∈I Ai is properly enlarged, then at least one of the projections is no longer ahomomorphism.

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Remark 1.28 Observe that the operations in a direct product are defined “compo-nentwise”, as one defines e.g. addition and scalar multiplication in a vector space Kn

over some field K.

See section 7 for more details on products.

Lemma 1.29 Graphs of homomorphisms are closed subsets of the directproduct of start and target algebra.Let f : A→ B be a homomorphism between partial algebras of the same signature Σ.Then graph f := (a, f(a)) | a ∈ A is always a closed subset of the direct productAπB .

Remark 1.30 One should observe that the other implication is not true: Let A bea total and B a discrete algebra, where As as well as Bs — for some s ∈ S — have atleast two elements, and let their signature Σ be non-trivial. Then AπB is a discretenontrivial partial algebra, in which each subset is closed. Therefore, each graph of anS-mapping from A into B is a closed subset of AπB , yet not all of these graphs willbe the graph of a homomorphism from A into B.

However, in the particular case that B is total the converse is true, too; thestraightforward proof is left as an exercise:

Lemma 1.31 (Closed graphs of mappings belong to homomorphisms, ifthe target is total.) Let A and B be partial algebras, and let f : A → B be anyS-mapping such that graph f is a closed subset of AπB . If B is total, then f is ahomomorphism from A into B: f : A→ B .

Lemma 1.32 (On subgraphs of homomorphisms): Let f : A → B be a homo-morphism, g : dom g → B be any mapping out of A into B such that graph g ⊆graph f , and let dom g be the relative subalgebra of A on the domain of g. Then

(i) g : dom g → B is a homomorphism.

(ii) graph g is a closed subset of AπB, iff dom g is a closed subset of A.

Proof (i) is obvious. In order to show (ii) assume first that graph g is closed inAπB, and let (a1, . . . , aτ(ϕ)) ∈ domϕA ∩ (dom g)η(ϕ). Then (g(a1), . . . , g(aτ(ϕ))) =(f(a1), . . . , f(aτ(ϕ))) ∈ domϕB, since f is a homomorphism. Since graph g is closed,(ϕA(a1, . . . , aτ(ϕ)), ϕ

B(g(a1), . . . , g(aτ(ϕ)))) ∈ graph g, showing that ϕA(a1, . . . , aτ(ϕ)) ∈dom g and that dom g is closed in A. Conversely, if dom g is closed in A, a similarargument — using that f is a homomorphism — shows that graph g is closed inAπB.

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2 Partial Peano algebras, terms and term operations 45

Corollary 1.33 (Connection between the generation of graphs of homo-morphisms and the closure of subsets): For a homomorphism f : A → B anda subset M of A the following statements are equivalent:

(i) M generates A, i.e. CAM = A.

(ii) graph (f |M) generates graph f in AπB:

CAπB graph (f |M) = graph f.

2 Partial Peano algebras, terms and term opera-

tions

2.1 Partial Peano algebras and recursion theorems

Actually all proofs of statements which particularly refer to terms do not refer tothe special way, how the terms are defined, yet to some general properties of thatdefinition. These essential properties are collected in the following definition of so-called partial Peano algebras:

Definition 2.1 (of a partial Peano algebra): Let P be any partial algebra ofsignature Σ, and let X be any S-subset of P . We say that P is a partial Peanoalgebra with Peano basis X (briefly: partial Peano algebra on X), if the followinggeneralized Peano axioms are valid in P :

(P1) For every ϕ ∈ Ω with (η, σ)(ϕ) = (s1s2 . . . sn, s), and for every (a1, . . . , aτ(ϕ)) ∈domϕP one has ϕP (a1, . . . , an) /∈ Xs (i.e. X ∩

⋃ϕ∈Ω ϕ

P (domϕP ) = Ø).

(P2) For any ϕ, ψ ∈ Ω with (η, σ)(ϕ) = (s1s2 . . . sn, s) and (η, σ)(ψ) = (s′1s′2 . . . s

′m, s

′),and for any (a1, . . . , an) ∈ domϕP , (b1, . . . , bm) ∈ domψP one has thatϕP (a1, . . . , an) = ψP (b1, . . . , bm) implies ϕ = ψ — hence one has in particu-lar m = n — and ai = bi for 1 ≤ i ≤ n (i.e. each partial operation of P isinjective and any two distinct partial operations have disjoint ranges).

(P3) (Axiom of Induction) CPX = P (, i.e. X generates P ).

Remarks 2.2 (i) A total Peano algebra (on X) is simply called a Peano algebra— often also term algebra or word algebra (see below).

(ii) A discrete partial algebra D is always a partial Peano algebra on its carrier setD.

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(iii) Let N0 := 0, 1, 2, . . . be the set of natural numbers, and let SUCC : N0 → N0 bethe successor function, i.e. SUCC(n) := n+1 for each n ∈ N0. Then (N0 ; SUCC)is a Peano algebra on 0, and the generalized Peano axioms are just the usualPeano axioms for natural numbers.

Example 2.3 (of a total binary term algebra as the algebra of all iteratedordered pairs over some set): In order to make it easier below to prove the gen-eralized Peano axioms for arbitrary term algebras, we proceed in two steps. Namely,a good background for proving the generalized Peano axioms is the set of all iteratedordered pairs of elements of a given set M . Therefore we define for any homogeneousset M :

• T1(M) := M ;

• Let n ≥ 2 be any natural number, and assume that T1(M) up to Tn−1(M) arealready defined. Then set

Tn(M) :=⋃n−1k=1(Tk(M)× Tn−k(M)) =

= (w1, w2) | w1 ∈ Tk(M), w2 ∈ Tn−k(M), 1 ≤ k ≤ n− 1 .

• Finally, set

T (M) :=∞⋃n=1

Tn(M) ,

and define on T (M) a binary total operation as follows:

w1 w2 := (w1, w2) for all w1, w2 ∈ T (M) .

(T (M),) is then a total algebra of type τ = (2) .In what follows we need the condition that M does not contain any sequence of

its elements. This can be formulated as

M ∩∞⋃n=2

Tn(M) = Ø . (3)

If this condition is not satisfied right from the beginning one can pass to a set, sayM ′, which allows a bijection onto M , and for which condition (3) is satisfied.22

22This can be achieved by choosing a set N , which is larger than any set occurring in M and bysetting M ′ := (m,N) | m ∈ M , where one may define an ordered pair of sets by the so-calledKuratowski-pair:

(a, b) := a, a, b ,

or by using the pair from Definition 0.2.

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Lemma 2.4 The binary algebra T (M) := (T (M),) from Example 2.3 is atotal Peano algebra on M , whenever M satisfies condition (3).

Proof Obviously, (P1) is satisfied because of condition (3).Assume that v1 v2 = w1w2 for any v1, v2, w1, w2 ∈ T (M). Then v1 = w1 and

v2 = w2 by the definition of ordered pairs. This shows that (P2) is satisfied.In order to show CT (M)M = T (M) (⊇ suffices) we use natural induction23 on n:

• T1(M) = M ⊆ CT (M)M by assumption.

• Assume that T1(M), . . . , Tn−1(M) ⊆ CT (M)M , and let t = (w1, w2) ∈ Tn(M).

Then (w1, w2) = w1 w2 ∈ CT (M)M , since w1, w2 ∈⋃n−1k=1 Tk . And therefore

also Tn(M) ⊆ CT (M)M .

This shows that (P3) is satisfied, too. Thus, T (M) is a Peano algebra, and it isobvious that it is total.

Example 2.5 (of the term algebra as a total Peano algebra of signature Σ):Let (S,Ω, τ, η, σ) be any heterogeneous signature, and let X be any S-set of variablessuch that

Xs ∩Xs′ = Xs ∩ Ω = Ø for all s, s′ ∈ S with s 6= s′ .

LetM := (Ω ∪

⋃s∈S

Xs) .

designate the homogeneous disjoint union of all these sets — forget that they mayhave different sorts —, and assume that M satisfies in addition condition (3).

For n ∈ N with n ≥ 3 and t1, . . . , tn ∈ T (M) set

• t1 t2 t3 := t1 (t2 t3) if n = 3, and

• t1 . . . tn := t1 (t2 . . . tn) , if n > 3 (“right bracketing”).

Define a mapping σ : T (M)→ S recursively by setting:

• If t ∈M , then set σ(t) := s , iff t ∈ Xs or t ∈ Ω and σ(t) = s .

• If t = t1 t2 , then define σ(t) := σ(t1) .

Because of our assumptions on M , and since T (M) is a Peano algebra, σ is uniquelydefined this way on all of T (M).24

For s ∈ S defineT (M)s := t ∈ T (M) | σ(t) = s .

23Because of the special recursive definition of T (M) this seems here more adequate than algebraicinduction.

24See Theorem 2.8 with target S, the sort algebra.

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This yields an S-set structure on T (M): T (M)S := (T (M)s)s∈S , and we may define,for ϕ ∈ Ω and for ti ∈ T (M)η(ϕ)i (1 ≤ i ≤ τ(ϕ))

ϕT (M)S(t1, . . . , tτ(ϕ)) := ϕ t1 . . . tτ(ϕ) .

By the definition of σ this yields

ϕT (M)S(t1, . . . , tτ(ϕ)) ∈ T (M)σ(ϕ) .

Finally, setT (X,Σ) := CT (M)S

X .

We call T (X,Σ) a (standard model of a) total heterogeneous term algebra onX of signature Σ.

Let us now prove

Proposition 2.6 (Each (standard model of a) heterogeneous term algebrais a total Peano algebra.) For each signature Σ and for each S-set X = (Xs)s∈Sof variables satisfying

Xs ∩Xs′ = Xs ∩ Ω = Ø for s, s′ ∈ S with s 6= s′ (4)

and condition (3), the term algebra T (X,Σ) = ((T (X,Σ)s)s∈S, (ϕT (X,Σ))ϕ∈Ω) as de-

fined in the above example is a total Peano algebra on X .

Proof Obviously T (X,Σ) is total, and (P3) is satisfied by definition.Ad (P1): Since T (M) satisfies (P1), this also follows immediately for T (X,Σ),

since the homogeneous union⋃s∈S Xs is a subset of M , and ϕT (X,Σ)(t1, . . . , tτ(ϕ))

either belongs to⋃∞n=2 Tn(M) , if τ(ϕ) ≥ 1 , or — for τ(ϕ) = 0 — ϕT (X,Σ) ∈ Ω , which

is disjoint from all phyla of X by assumption (4).Ad (P2): Let ϕ, ψ ∈ Ω with (η, σ)(ϕ) = (s1s2 . . . sn, s) and (η, σ)(ψ) = (s′1s

′2 . . . s

′m, s

′),and let (t1, . . . , tn) ∈ dom ϕT (X,Σ), (u1, . . . , um) ∈ dom ψT (X,Σ) such that one hasϕT (X,Σ)(t1, . . . , tn) = ψT (X,Σ)(u1, . . . , um). Then this means that

ϕ (t1 . . . tn) = ψ (u1 . . . um) .

This implies ϕ = ψ — and therefore n = m —, and t1 . . .tn = u1 . . .um , sinceT (M) is a Peano algebra, and recursively, by the same reason, ti = ui for 1 ≤ i ≤ n .This proves (P2).

This shows that T (X,Σ) is indeed a total Peano algebra of signature Σ on X .

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The proof of the following result is a good example for the principle of structuralinduction.

Lemma 2.7 (Each weak subalgebra of a partial Peano algebra is a partialPeano algebra.) Let P = (P, (ϕP )ϕ∈Ω) be any partial Peano algebra of signatureΣ on some S-set X, and let A be any “weak relative subalgebra” of P , i.e. A is anS-subset of P , and, for each ϕ ∈ Ω, one has graph ϕA ⊆ graph ϕP . Then A is apartial Peano algebra on Y := A \

⋃ϕ∈Ω ϕ

A(dom ϕA).

Proof (P1) is satisfied because of the definition of Y ; and (P2) holds in A, becauseit is true in P . Thus it remains to prove (P3): CAY ⊆ A is obvious; it remains toprove A ⊆ CAY , and this is done by proving by algebraic induction for p ∈ P thestatement

P(p) := “p ∈ As for some s ∈ S implies p ∈ (CAY )s”:

Assume first that p ∈ Xs for some s ∈ S. If p /∈ As, then P(p) is trivially true.Therefore assume p ∈ As; then p ∈ Ys, since (P1) holds for P , and because ofthe definition of Y . Next assume p ∈ Ps \ Xs ; then — cf. Lemma 1.20 — p =ϕP (b1, . . . , bτ(ϕ)) for some ϕ ∈ Ω and (b1, . . . , bτ(ϕ)) ∈ domϕP , and let the statementP(bk) be true for every k with 1 ≤ k ≤ τ(ϕ). Moreover, let us assume that p ∈ As(otherwise P(x) is already trivially true). Then either p ∈ Ys — yielding p ∈ (CAY )s—, or there exist ϕ′ ∈ Ω and (b′1, . . . , b

′τ(ϕ′)) ∈ domϕ′A such that p = ϕ′A(b′1, . . . , b

′τ(ϕ′))

(by the definition of Y ). Since graphϕ′A ⊆ graph ϕ′P , and since P satisfies (P2), weget ϕ′ = ϕ and bk = b′k, for 1 ≤ k ≤ τ(ϕ). Thus bk ∈ Aη(ϕ)k for each k with1 ≤ k ≤ τ(ϕ). Therefore, by the induction hypothesis one has bk ∈ (CAY )η(ϕ)k

(1 ≤ k ≤ τ(ϕ)). But this implies p = ϕA(b1, . . . , bτ(ϕ)) ∈ (CAY )s, since CAY is closedin A. This proves A ⊆ CAY . — The other inclusion is trivial.

Later we shall be able to prove that actually every partial Peano algebra canbe obtained “up to isomorphism”, as a weak subalgebra of a term algebra on somesuitable set of variables.

The following results are basic for what follows:

Theorem 2.8 (First Recursion Theorem: Recursion Theorem for partialPeano algebras w.r.t. total algebras: Each mapping from a Peano basisinto a total algebra extends uniquely to a homomorphism from the corre-sponding partial Peano algebra into that total algebra.): Let P be a partialPeano algebra of signature Σ with Peano basis X, let B be a total algebra of signatureΣ, and let f : X → B be any mapping. Then there exists a unique homomorphismf : P → B extending f (i.e. satisfying f |X = f ).

In particular one hasgraph f = CPπB graph f .

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Proof As already indicated we define graph f := CPπB graph f . When we can provethat this relation is the graph of a mapping, then Lemma 1.31 and Corollary 1.33imply that f is indeed a homomorphism from P into B.

We prove the statement by structural induction w.r.t. the property

P(t) := “t ∈ Ps for some s ∈ S implies that fs(t) is uniquely defined”:

Since P is a partial Peano algebra, because of (P1), f can assign a value to an elementx ∈ Xs — or more precisely to the term xs — only in connection with the conditionfs(x) := fs(x), and therefore fs(t) is uniquely defined, whenever t is a variable ofsome sort s.

Now assume that t is not a variable. Then either t = ϕ, where ϕ ∈ Ω is a nullaryconstant and τ(ϕ) = 0 — and in this case (because of (P2), and since B is total)fσ(ϕ)(t) is uniquely defined as fσ(ϕ)(t) = ϕB — or one has t = ϕP (t1, . . . , tτ(ϕ)) forsome fundamental operation symbol ϕ with τ(ϕ) 6= 0, and one may assume that theproperty P is true for every term tk with 1 ≤ k ≤ τ(ϕ). Recall that because ofthe generalized Peano axiom (P2) there is exactly one possibility to represent t inthis way, and by induction hypothesis there exists for each index k a unique elementbk ∈ Bη(ϕ)k such that fη(ϕ)k(tk) = bk. Since B is total, and by the definition of graph f ,fσ(ϕ)(t) is defined as b := ϕB(b1, . . . , bτ(ϕ)) in B. And since this operation is definedand has value b ∈ Bσ(ϕ), we get that fσ(ϕ)(t) is defined and has b as unique value(always because of axiom (P2)).

Corollary 2.9 Any two total Peano algebras on the same Peano basis areisomorphic and in particular isomorphic to a term algebra.

Proof Let P be any total Peano algebra on some Peano basis X ′. In connectionwith Example 2.3 we have argued in footnote 22 that there always exist an S-set Xand a bijection h : X ′ → X such that X satisfies

Xs ∩Xs′ = Xs ∩ Ω = Ø for all s, s′ ∈ S with s 6= s′ ,

which is in one-to-one correspondence with X ′, and which, for M := (Ω ∪⋃s∈S Xs) ,

satisfies the condition (1), i.e.

M ∩∞⋃n=2

Mn = Ø .

We choose such an S-set X with bijection h : X ′ → X . Then h uniquely extends to anobviously surjective homomorphism h : P → T (X,Σ) because of the First RecursionTheorem. Yet, because of Proposition 2.6 T (X,Σ) is a total Peano algebra, too, andtherefore h′ := h−1 also extends to a surjective homomorphism h′ : T (X,Σ) → P .And since h′ h = idP and h h′ = idT (X,Σ), h′ and h are inverse to each other andtherefore isomorphisms.

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Remark 2.10 This shows that every total Peano algebra with Peano basis X canserve as term algebra T (X,Σ) with set X as variables, and we do not have to thinkof the special one given in Example 2.3. However, up to now this is the only one,which we know explicitly. In set theory (cf. e.g. [Sch66] or Lemma 0.10) one canshow that for any partial algebra A generated by some S-set M and for every S-setX in one-to-one-correspondence to M via a bijection h : M → X there exist a set Bcontaining X and a bijection h : A→ B extending h, as well as an algebraic structure(ϕB)ϕ∈Ω on B such that h : A→ B becomes an isomorphism.

Of even more importance for partial algebras is the Recursion Theorem fortotal Peano algebras w.r.t. partial algebras (Second Recursion Theorem)stated below, the proof of which is quite similar, yet because of its importance we alsogive it here and also present another construction which uses the concept of normalone point completion, which we still have to introduce:

Definition 2.11 (of a normal one point (per sort) completion): Let A be anypartial algebra. For each s ∈ S let os /∈ As be a new element. Define

o

A:= (o

As)s∈S := (As∪os)s∈S

as the carrier set of a total extensiono

A of A, where, for each ϕ ∈ Ω, the extended

operation ϕoA is defined as follows: If τ(ϕ) = 0 , then

ϕoA :=

ϕA , if ϕA is defined,oσ(ϕ) , if ϕA is undefined.

And in the case of τ(ϕ) 6= 0 , η(ϕ) =: (s1, . . . , sn) and ask ∈o

Ask for 1 ≤ k ≤ τ(ϕ) wedefine

ϕoA(as1 , . . . , asn) :=

ϕA(as1 , . . . , asn) , if (as1 , . . . , asn) ∈ domϕA

oσ(ϕ) , otherwise.

o

A = (o

A, (ϕoA)ϕ∈Ω is then called the normal one point (per sort) completion of

A .

Observe thato

A need not be generated by A, since some of the new elements neednot be generated by A; in particular, when A is total, then none of the new elements

is generated by A, since then A is a (closed) subalgebra ofo

A. In computer scienceone would say that the normal one point completion provides for each sort an errorvalue of that sort, wich does not yet belong to the original structure. We now statethe second recursion theorem already mentioned above:

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Theorem 2.12 (Second Recursion Theorem: Recursion Theorem for totalPeano algebras w.r.t. partial algebras:Each mapping from the Peano basis X of a total Peano algebra into anypartial algebra extends (uniquely) to a closed homomorphism from an X-generated relative subalgebra of the Peano algebra into the target algebra):

Let P be a total Peano algebra of signature Σ with Peano basis X, let A be apartial algebra of signature Σ, and let v : X → A be any mapping. Then there existsa unique homomorphism v : P ⊇ dom v → A satisfying

(i) v|X = v , i.e. v extends v.

(ii) dom v is a relative subalgebra of P , which is generated by X: Cdom vX = dom v .

(iii) v : dom v → A is a closed homomorphism.

(iv) v : dom v → A is the largest homomorphic extension of v to an X-generatedrelative subalgebra of P with target algebra A.

In particular one can define v in one of the following ways, when v : P →o

A designatesthe homomorphic extension of v according to Theorem 2.8 from P into the normal

one point (per sort) completiono

A of A:

(v) graph v = CPπA graph v (see Figure 2 for an example).

(vi) dom v = v−1(A) = (v−1s (As))s∈S and v = v|v−1(A) .

Proof Intuitively spoken, (v) means that v is defined recursively by starting on Xwith the values of v and then extending it in the following way: Consider ti ∈ Pη(ϕ)i

such that vη(ϕ)i(ti) =: ai is already defined in Aη(ϕ)i (1 ≤ i ≤ τ(ϕ)) and such that(a1, . . . , aτ(ϕ)) ∈ domϕA. Then one defines

vσ(ϕ)(ϕP (t1, . . . , tτ(ϕ)) := ϕA(a1, . . . , aτ(ϕ)) ,

and one does this “as long as possible”.25 One could also formulate this as: v isdefined recursively along the structures of P and A.

We return to the algebraic formulation of this procedure, what means that weconsider v to be defined according to (v). In order to prove that the resulting relationis indeed a partial mapping out of P into A we can proceed in a similar way as inthe proof of Theorem 2.8 by observing that now in A there may be sequences onwhich the operation is not defined. The induction property P will therefore have tobe reformulated as

P(t) := “t ∈ Ps for some s ∈ S implies that vs(t) is either undefinedor it is uniquely defined”.

25Observe that this remark already applies to the proof of Theorem 2.8.

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cccccc

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17th July 2002


We leave the details of such a proof as an exercise to the reader.Set now

D := dom v := p ∈ P | there is a ∈ A such that (p, a) ∈ graph v ⊆ P ,

and let D designate the relative subalgebra of P on D. We show that v is indeed aclosed homomorphism from D into A extending v:Consider ((t1, a1), . . . , (tτ(ϕ), aτ(ϕ))) ∈ (graph v)η(ϕ)1 × . . .× (graph v)η(ϕ)τ(ϕ)

such that

ϕA(a1, . . . , aτ(ϕ)) exists. Since P is total, we get

ϕP×A((t1, a1), . . . , (tτ(ϕ), aτ(ϕ))) ∈ graph v .

Therefore

vσ(ϕ)(ϕP (t1, . . . , tτ(ϕ))) = ϕA(vη(ϕ)1(t1), . . . , vη(ϕ)τ(ϕ)

(tτ(ϕ))) = ϕA(a1, . . . , aτ(ϕ)) .

Moreover, since P is a Peano algebra, this is the only way that ϕP (t1, . . . , tτ(ϕ)) canbecome a member of D. This shows, too, that v is indeed a closed homomorphismfrom the relative subalgebra D of P into A. And that it extends v is obvious fromthe definition. This proves claims (i) and (iii) of the theorem. And from (ii) thereonly remains to show that X generates dom v: Yet one should observe that

CPπA graph v = CDπA graph v ,

since our argumentation from above shows that only the structure of D is neededfor the generation of graph v . Because of what we have already shown so far we canapply Corollary 1.33, which yields that indeed X generates D; and this finally yieldsstatement (ii).

Before we prove (iv), we want to show that the definitions (v) and (vi) yield thesame homomorphic extensions of v:Define C := v−1(A) = (v−1

s (As))s∈S . As a restriction of v to C, v′ := v|C is obviouslya homomorphism from the relative subalgebra C of P on C into A: v′ : C → A .Since, by definition, P ×A is at least a weak relative subalgebra of the total algebra

P ×o

A, we getgraph v ⊆ graph v′ ,

since both have the same target algebra; and we have to show the other inclusion.We do this by algebraic (i.e. structural) induction with respect to the property

P(t) := “If t ∈ (dom v′)s for some s ∈ S, then t ∈ (dom v)s”.

Namely, if t ∈ Xs for some s ∈ S, then P(t) is obviously true. And if t = ϕP ∈ Ps isa nullary constant, then v′(t) ∈ As means that ϕA is defined. But then (ϕP , ϕA) ∈graph v, showing that ϕP ∈ dom v .

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Finally, assume that t = ϕP (t1, . . . , tτ(ϕ)) and that P(ti) is true for 1 ≤ i ≤ τ(ϕ).If t ∈ dom v′, then this means that v′(t) ∈ A . Yet this can only happen, whenv′(ti) ∈ A for 1 ≤ i ≤ τ(ϕ), and (t1, . . . , tτ(ϕ)) ∈ domϕA . By induction assumptionwe get ti ∈ dom v for 1 ≤ i ≤ τ(ϕ), and finally — by the definition of v — t ∈ v.Thus we have proved (vi).

Now (iv) follows immediately from Corollary 1.33. Namely v is indeed the largesthomomorphic extension of v to an X-generated relative subalgebra of P , since thegraph of each such extension, say g, must be contained in C

PπoA

graph v = graph v ,

and must have its values in A, i.e. graph g must be a subset of graph v′ = graph v .

Remark 2.13 and Notation If we wanted to be more precise, we should write vX,Ainstead of just v for the homomorphic extension of v defined in the above theorem,since this partial interpretation of terms induced by v — as we shall often callit — heavily depends on the target algebra A. Yet this concept is so fundamental inwhat follows that we want to have a very short notation. And in general the set ofvariables and the target algebra will always be mentioned and therefore will be clearfrom the context.

2.2 More on terms, term operations

Since one of the main aims of this section is the description of term operations, wealways consider here some total Peano algebra over some Peano basis X as termalgebra T (X,Σ), the elements of which will be called terms. We start here withthe definition of the algebraic quasi-order on any partial algebra A which providesin many contexts a useful tool of describing special substructures (so-called initialsegments). Moreover it will allow us below an easy way to define the concept of asubterm of a term.

Definition 2.14 of the algebraic quasi-order Let A ∈ PAlg(Σ) be any partial

algebra of signature Σ. Then define on⋃s∈SAs (the disjoint union of the phyla — as

sets, disregarding the sorts) relations /A and A as follows:

(i)/A := (a, a′) ∈ As × As′ | there are ϕ ∈ Ω, i0 ∈ 1, . . . , τ(ϕ),

and (a1, . . . , aτ(ϕ)) ∈ dom ϕA such that a = ai0 and a′ = ϕA(a1, . . . , aτ(ϕ)).

(in particular a ∈ As with s = η(ϕ)i0 and a′ ∈ As′ with s′ = σ(ϕ) ).

If a /A a′, then we say that a is an immediate predecessor of a′.

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(ii) A is then defined as the reflexive and transitive hull26 of /A . Thus, by defini-

tion, A is a quasi-order on⋃s∈SAs which is called the algebraic quasi-order

on A.

(iii) If M ⊆ A is any subset, then we denote by ↓M the initial segment of Agenerated by M (with respect to the algebraic quasi-order just defined), i.e.

↓M := a ∈ A | a A m for some m ∈M,

↓m =:↓m for singletons.

By ↓M we designate the relative subalgebra of A with carrier set ↓M .

(iv) If A = T (X,Σ), and if t ∈ As is any term, then the elements of ↓ t are calledsubterms27 of t.

Proposition 2.15 (On partial Peano algebras the algebraic quasi-order isan order allowing only finite chains between two elements): Let A be anypartial Peano algebra on some set X, then:

(i) A is a partial order on A.

(ii) A only allows finite strictly descending chains.

Proof Because of the Peano axioms the relation /A is strict (i.e. a/Aa′ always implies

a 6= a′) and the reflexive and transitive closure cannot generate cycles includingdifferent elements of A. Since no element of X is the value of a fundamental operation,these elements are minimal in the partial order A and for each element a in As (forsome s ∈ S) one needs only finitely many applications of fundamental operations inorder to generate it from X (one may make this precise by structural induction —or see the observation below). Therefore each (maximal) chain between a minimalelement and an arbitrary element is finite and cannot exceed some fixed finite numbergiven by the number of necessary applications of fundamental operations neededaltogether (multiple uses counted separately) to generate a from X.

If one wants to give a more precise argument, then one should look ahead tosubsection 5.1 below and observe that each element of a partial Peano algebra on Xbelongs to a Baire class BnAX for some natural number n (see Definition 5.7 and the

26Let R ⊆ A×A be any binary relation on an S-set A. By the reflexive hull of R one understandsR ∪ (∆As)s∈S — where ∆B := (b, b) | b ∈ B for every set B —, and by the transitive hull, sayT (R), of R one understands the set T (R) := (

⋃∞n=1(Rs)n)s∈S , where (Rs)0 := ∆As and (Rs)n+1 :=

Rs (Rs)n , for n ≥ 0 — where “” stands for the product of relations.It is then easily realized (exercise) that T (R) is the smallest transitive S-relation containing R.

Observe that the reflexive and transitive hull A of /A is then given by⋃∞n=0(/A)n , which is formed

on the one-sorted set⋃s∈SAs even when A is an S-sorted partial algebra.

27It should be obvious that this definition yields the usual concept of a subterm of t as a term —or word — which has to be constructed, before one can obtain the term or word t.

17th July 2002


observations following it), and one should prove by induction that in a partial Peanoalgebra no element of

⋃nk=0 BkAX can be the value of a fundamental operation having

as arguments elements in some BmAX for some m > n .

A very useful observation in connection with the domains of the closed homo-morphic extensions f of mappings f : X → A from a Peano basis of a total Peanoalgebra is the following one (we shall generalize it later to free completions of partialalgebras):

Lemma 2.16 (On partial Peano algebras on X the set of carriers of X-generated relative subalgebras is the same as the one of initial segmentscontaining X): Let X be any S-set, let B be a relative subalgebra of T (X,Σ)) or anypartial Peano algebra such that X ⊆ B. Then the following statements are equivalent:

(i) CBX = B (i.e. X generates B)

(ii) ↓B = B (w.r.t. T (X,Σ))).

Notation 2.17 If either (i) or (ii) of the previous lemma is satisfied under the condi-tions of the lemma, then we say that B (or B) is an X-initial segment of T (X,Σ).28

Moreover, in connection with any mapping f : X → A from X into some partial al-gebra A a homomorphic extension f ′ : T (X,Σ) ⊇ dom f ′ → A (out of T (X,Σ)) froman X-generated relative subalgebra dom f ′ of T (X,Σ) into A is called an X-initialhomomorphic extension of f .

Proof Assume first that X generates B, i.e. that CBX = B, and consider b ∈ Bs forsome s ∈ S. Let us argue by algebraic induction: Then either b ∈ Xs and b cannotbe the value of a fundamental operation. Or b = ϕB(b1, . . . , bτ(ϕ)) for some ϕ ∈ Ωand some suitable argument sequence of elements of B. Yet this is then the onlypossibility that b can be the value of a fundamental operation within T (X,Σ), andtherefore all immediate predecessors belong to B. Since by induction hypothesis wemay assume that all predecessors of any bi (1 ≤ i ≤ τ(ϕ)) belong to B, this showsthat B is indeed an initial segment of T (X,Σ).

Next assume that B is an initial segment of T (X,Σ) containing X. We thenhave to show that CBX ⊇ B (as the nontrivial part), and do this again by algebraic(structural) induction on t ∈ T (X,Σ) for the property:

P(t) : “If t ∈ B , then t ∈ CBX”.X ⊆ CBX by assumption.Consider t ∈ T (X,Σ) \ X, and let t ∈ B. Then t = ϕT (X,Σ)(t1, . . . , tτ(ϕ)) for someϕ ∈ Ω and some suitable argument sequence of elements of T (X,Σ), and assume thatP(ti) for all i ∈ 1, . . . , τ(ϕ) be true. Since B is an initial segment of T (X,Σ), weobtain ti ∈ B for all i ∈ 1, . . . , τ(ϕ). Hence, by the hypothesis of the induction,we get that ti ∈ CBX for all i ∈ 1, . . . , τ(ϕ). Since B is a relative subalgebra ofT (X,Σ), these facts imply t ∈ CBX, what was to be shown.

28This concept will also be extended later to A-generated relative subalgebras of the free comple-tion A := F (X,TAlg(Σ)) of a partial algebra A (see subsection 8.3).

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Definition 2.18 of partial term operations Let X be any S-set29, A ∈ PAlg(Σ)any partial algebra, and t ∈ T (X,Σ)) any term. Then define (X; t)A as a partialmapping out of AX into A as follows:

dom (X; t)A := w ∈ AX | t ∈ dom w,

and w ∈ dom (X; t)A implies (X; t)A(w) := w(t). This means that we have

graph (X; t)A :=

Ø, if AX = Ø , (w, w(t)) | w : X → A and t ∈ dom w , if AX 6= Ø .

(X; t)A is called the (global partial) term operation on A induced by the termt. X is called the set of variables under consideration, while the mapping w : X → Ais sometimes called a valuation of X in A, or an assignment of values in A to thevariables in X.

t ∈ T (X,Σ) is said to be evaluable in A w.r.t. an assignment w : X → A, ifft ∈ dom w (iff w ∈ dom (X; t)A).

Remarks 2.19 (a) Observe that terms have an instructive representation by rootedtrees (consider cf. e.g. [B86], subsections 5.6 and 5.7 for explicit definitions, yetlook at Figure 3 in Example 2.20 for a drawn example of a rooted tree repre-sentation of a term).

(b) The distinction of two cases in the definition of the graph of a term operationstems from the fact that the set X of variables (arguments), on which a termoperation depends, may have for some sort s ∈ S a non-empty carrier set(phylum) Xs , where As is empty. And in such cases the graph becomes empty,since then there does not exist any S-mapping from X into A.

(c) For x ∈ Xs and the case AX 6= Ø the term operation (X;x)A is a total projec-tion, since then every sequence (v(y) | y ∈ Xs)s∈S — as the mapping v : X → Acan be interpreted — is mapped onto v(x) .

(d) Let us consider the partial algebra A from Figure 2 for τ = (ϕ 7→ 1), andconsider the term operation (x;ϕϕϕx)A. Then

graph (x;ϕϕϕx)A = (a1, a7), (a2, a7), (a3, a8), (a4, a8) .

See also Example 2.20 with Figure 3 below.

Example 2.20 Let a homogeneous similarity type τ , a term t := ϕϕ′ϕ′′x and apartial algebra A be given as sketched in Figure 3.

29We allow that X may possibly be an infinite set, although one will usually consider term oper-ations only for finite argument sets or even only for finite sequences. However, our definition of aterm operation does neither depend on the cardinality of the argument sequence nor on the fact thatit is only a set. In any case a term operation will “depend only on finitely many of its arguments”.

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t := ϕ(ϕ′(ϕ′′), x) :

ϕ′

ϕ′′

ϕ

xbb

bb

@

@@

A :a

ϕ′A

ϕ′′A = c

ϕA

ϕA

ϕ′A

b

e

d

6

-ZZZZZZZZZZZ~

-

e e

e ee

llll

,,,,

τ :

ϕ 7→ 2,ϕ′ 7→ 1,ϕ′′ 7→ 0

X = x

Figure 3: Example for the computation of a term-operation

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Then (x; t)A(a) exists and has value b, while (x; t)A is undefined for any otherassignment.30

Lemma 2.21 (Domains and kernels of the closed extensions of composi-tions:) Let X be any set, f : A → B be any homomorphism, and w : X → A anyvaluation. Then

dom w ⊆ dom (f w)∼

(with equality, iff f is closed), and

ker w ⊆ ker (f w)∼,

where, for any S-mapping g : C → D one defines the kernel of g as the S-setker g := ( (c, c′) ∈ Cs × Cs | gs(c) = gs(c

′) )s∈S.

Proof According to the Second Recursion Theorem 2.12, (f w)∼ is the largest X-initial homomorphic extension, while f w is some X-initial homomorphic extension.Therefore the statements of the lemma immediately follow.

Corollary 2.22 Each homomorphism is compatible with each induced term-operation: Let f : A→ B be a homomorphism, t ∈ T (X,Σ), w : X → A and a ∈ A,then (X; t)A(w) = a implies (X; t)B(f w) = f(a) (in particular, if (X; t)A(w) exists,then so does (X; t)B(f w) and one has f((X; t)A(w)) = (X; t)B(f w)).

This can be proved by algebraic induction, but it will follow from Proposition 3.20in subsection 3.3. Therefore we leave it at this place as an excercise.

The following proposition yields another description of the algebraic closure, andits proof is again a good application of structural induction:

Proposition 2.23 Each element of the closure of a set only needs finiteinformation about the gernerating subset and the structure.Let A be any partial algebra and M any generating subset of A.

(i) For every a ∈ A there are a finite subset Ma ⊆ M and a term t ∈ T (Ma,Σ)such that

a = (Ma; t)A(idMaM).

(ii) For every a ∈ A there is a term t ∈ T (M,Σ) such that a = (M ; t)A(idM).

30Observe that we have used here the image a of the assignment v : x → A with x 7→ a assubstitute for the assignment v. If X = x1, . . . , xn (as an S-set), then one uses analoguously thesequence (a1, . . . , an) as a substitute for the assignment v : X → A with v : xi 7→ ai (1 ≤ i ≤ n).

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3 Some basic logical and model theoretic concepts, homomorphisms revisited 61

Proof By structural induction on the elements of A, as already indicated:Ad (i): For a ∈ M choose Ma := a and t := a. Therefore, assume that

a := ϕA(a1, . . . , aτ(ϕ)), and assume that the statement is true for each ai with Mai

and ti for i ∈ 1, . . . , τ(ϕ). Set Ma :=⋃τ(ϕ)i=1 Mai (this includes the case that τ(ϕ) = 0

with⋃0i=1 Mai := Ø) and t := ϕT (Ma,Σ)(t1, . . . , tτ(ϕ)) , (where we consider T (Mai ,Σ)

in a natural way as a subset of T (Ma,Σ) — w.r.t. the canonical injection induced byidMaiMa). Then it is straightforeward to realize (cf. Proposition 3.10 below) that, for

each i ∈ 1, . . . , τ(ϕ), the values of (Mai ; ti)A(idMaiM

) and of (Ma; ti)A(idMaM) are

identical. And therefore it is easy to realize that a = ((Ma; t)A(idMaM). It is obvious

by the inductive construction that Ma is finite.Statement (ii) is proved analogously, and we leave the details as an exercise.

3 Some basic logical and model theoretic concepts,

homomorphisms revisited

3.1 Existence equations, ECE- and QE-equations

It is well known from e.g. semigroups, groups, rings, lattices and Boolean algebrasthat these structures can be axiomatized by equations. We want to develop here ageneralization of this concept to partial algebras.

In connection with partial algebras it seems to be useful to introduce model the-oretic concepts like equations and implications much earlier than in connection withtotal algebras. One particular reason is a motivation of our concept of homomor-phisms and to present unifying treatment of many of its additional properties. Weshall explain later, why we start with “existence equations” (see below) ratherthan with “strong equations” (also called “Kleene-equations”), which refer rightfrom the beginning to the equality of arbitrary partial term operations. We want tomention here only the fact that our approach via existence equations and elementaryimplications formed with them is strongly connected with the category theoretical ap-proach to a model theory for partial algebras developed by H.Andreka, I.Nemeti

and I.Sain e.g. in the papers [AN79], [AN82], [NSa82] and [AN83].

In what follows we shall always consider a fixed heterogeneous signatureΣ = (S,Ω, τ, η, σ) and some S-set X = (Xs)s∈S of variables such that

Xs ∩Xs′ = Xs ∩ Ω = Ø for s, s′ ∈ S with s 6= s′ ;

the homogeneous case will be a special case, when S is only a one-element set.

Definition 3.1 (of existence equations, partial interpretations, satisfactionand validity):

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(i) (E-equations): A pair (X, (t, t′)), where t and t′ are terms from T (X,Σ)s forsome s ∈ S is called an existence equation (briefly E-equation) and written as

(X; te≈ t′)31. Moreover, we write EeqX for the set of all E-equations with variables

in the set X.(ii) (Valuations or assignments): For any S-sorted partial algebra A of sig-

nature Σ an S-mapping v : X → A — i.e. a family (vs : Xs → As)s∈S will be calleda valuation or an assignment (of X in A).

(iii) (Partial interpretation): In connection with each such assignment v ofan S-set X of variables into a partial algebra A of signature Σ one has a (partial)interpretation of terms, v := (vs : Xs → As)s∈S, according to the Second RecursionTheorem 2.12. Because of its fundamental meaning we give here an explicit recursivedefinition (cf. Figure 2 for a homogenous similarity type with one unary fundamentaloperation symbol):

– For each s ∈ S and for each variable x ∈ Xs one has vs(x) := vs(x).

– If ϕ ∈ Ω is any nullary operation symbol, i.e. with τ(ϕ) = 0, of sort sϕ, and ifϕA is defined, then vsϕ(ϕ) exists and has value ϕA.32

– If ϕ ∈ Ω is any non-nullary operation symbol, and if t1, t2, . . . , tτ(ϕ) are termsof the required sorts, for which vη(ϕ)i(ti) exists, say vη(ϕ)i(ti) = ai ∈ Aη(ϕ)i ,(1 ≤ i ≤ τ(ϕ)), and such that ϕA(a1, a2, . . . , aτ(ϕ)) also exists, then vσ(ϕ)(ϕ t1 t2 . . . tτ(ϕ)) exists, too, and takes the value ϕA(a1, a2, . . . , aτ(ϕ)) in Aσ(ϕ).

– The interpretation vs(t) of some term t ∈ T (X,Σ)s exists, if and only if itis defined by one of the three cases above. Because of the Second RecursionTheorem 2.12 we know that v is well defined this way.

In what follows, let, in addition, A be a many-sorted partial algebra of signature

Σ, v : X → A a valuation, and (X; te≈ t′) an E-equation in EeqX .

(iv) (Semantics of E-equations (satisfaction)): We say that A satisfies the

E-equation (X; te≈ t′) w.r.t. the valuation v (in symbols: A |= (X; t

e≈ t′)[v]), iff

— in case of t, t′ ∈ T (X,Σ)s for some s ∈ S —

– the interpretation vs(t) of t by v exists, and

– the interpretation vs(t′) of t′ by v exists, and

31The reference to the set X of variables is included, since this E-equation has to be distinguishedfrom (X ′; t

e≈ t′), when t, t′ ∈ T (X ′,Σ)) (for some further S-set X ′ = (X ′s)s∈S of variables, and

when assignments — see their definition below — starting from X ′ rather than from X are to beconsidered. The semantics of an E-equation will sometimes depend on this reference set, as will beexplained later.

32Observe that this is actually a special case of the next item, yet it may be more instructive toconsider it separately.

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– these interpretations are equal: vs(t) = vs(t′).

(v) (Semantics of E-equations (validity)): We say that the E-equation

(X; te≈ t′) holds (is valid) in the partial algebra A (in symbols: A |= (X; t

e≈ t′)),

iff A satisfies (X; te≈ t′) for every valuation v : X → A , i.e.:33

A |= (X; te≈ t′) iff A |= (X; t

e≈ t′)[v] for every valuation v : X → A.

(vi) (TE-statements): If t and t′ are identical terms, then (X; te≈ t) will be

called a term existence statement (briefly TE-statement), since one has forv : X → A:

A |= (X; te≈ t)[v] if and only if the interpretation of t w.r.t. v exists.

The last part of the above definition shows that in the semantics of E-equations the“diagonal” of T (X,Σ) × T (X,Σ) (:= (T (X,Σ)s × T (X,Σ))s)s∈S) gains importance,

while for total (many-sorted) algebras A the statement A |= (X; te≈ t) is always triv-

ially true. Thus this gives additional expressive power to E-equations in connectionwith partial algebras, and this will become even more obvious in connection withelementary implications.

Let us mention one specialty of these semantics of existence equations: If one

has A |= (X; te≈ t′), and if t′′ is any “subterm” of either t or t′, then one also has

A |= (X; t′′e≈ t′′), and in particular t′′ induces on A a total “term-operation” t′′A, as

will be shown later.Based on existence equations — between terms with variables in some S-set Y

— as atomic formulas one may now build all the well-known formulas of a first orderlanguage L := L(Y,Σ) with equality. And the semantics of existence equations areextended to arbitrary first order formulas in the usual way, i.e.:

Definition 3.2 (of a first order logical language w.r.t. Σ): Let Y be somefixed S-set of variables, in which each component Ys is countably infinite and satisfiesYs ∩ Ys′ = Ys ∩ Ω = Ø for any s, s′ ∈ S with s 6= s′. We assume at some occasionsthat the elements of each Ys have a fixed indexing by the natural numbers: Ys := ys0, ys1, ys2, . . . (s ∈ S), where we occasionally omit the upper index referring to thesort, if this is clear from the context.

Atomic formulas of the first order language L := L(Y,Σ) then are all E-

equations (X; te≈ t′), where t, t′ ∈ T (X,Σ)s for some s ∈ S and for some finite

S-set X of variables with X ⊆ Y .Formulas of L are defined recursively as follows:

• each atomic formula of L is a formula of L;

33Observe that this is equivalent to saying that (X, t)A and (X, t′)A are total and identical termoperations on A.

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• if (X; Φ) is a formula of L, then (X;¬Φ) is a formula of L (negation);

• if (X; Φ) and (X ′; Φ′) are formulas of L, then(X ∪X ′; (Φ ∧ Φ′)) (conjunction),(X ∪X ′; (Φ ∨ Φ′)) (disjunction),(X ∪X ′; (Φ⇒ Φ′)) (implication), and(X ∪X ′; (Φ⇔ Φ′)) (equivalence) are formulas of L;

• if (X; Φ) is a formula of L, and if x ∈ Xs for some s ∈ S, then34 (X \x; (∀x)Φ)and (X \ x; (∃x)Φ) are formulas of L.

• if (X; Φ) is a formula of L, and if X ′ ⊆ Y is any finite S-set of variables, then(X ∪X ′; Φ) ∈ L.

Remark 3.3 In what follows we shall usually restrict our formulas to so-called openformulas or equivalently to universal formulas, i.e. to such formulas, which eithercontain no quantifiers at all, or where all occurring quantifiers are universal (i.e. ofthe form (∀x) sor some variable x ∈ Ys for some sort s ∈ S). Often we shall omit thereference to the set of variables, when it is exactly the set of variables “occurring” inthe formula Φ, which is defined recursively (in the usual way) as follows.

Definition 3.4 (of the set of free variables occurring in a formula):(i) (Variables occurring in a term):We first define the S-set var(t) of variables really occurring in a term t:

• Let t ≡ xs be any variable of sort s for some s ∈ S. Then, for u ∈ S define

var(t)u :=

xs , if u = s ,Ø , otherwise (i.e. if u 6= s).

• If t ≡ ϕ ∈ Ω with τ(ϕ) = 0, then var (t) := (Ø)s∈S .

• If t ≡ (ϕ, t1, t2, . . . , tτ(ϕ)) , with τ(ϕ) 6= 0, then

var (t) :=

τ(ϕ)⋃k=1

var (tk)

as a union of S-sets.

(ii) (Free variables occurring in a formula): Let (X; Φ) be any formula. Wedefine recursively the S-set fvar (Φ) of free variables occurring in Φ:

34Observe that we then abbreviate by X \x the S-set Z, for which Zs = Xs \x and Zs′ = Xs′

for s′ ∈ S \ s.

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• Let Φ ≡ te≈ t′ be an existence equation, i.e. an atomic formula. Then

fvar (Φ) := var (t) ∪ var (t′) .

• Let Φ ≡ ¬ Ψ, then

fvar (Φ) = fvar (¬ Ψ) := fvar (Ψ) .

• Let Φ ≡ Ψ1 1 Ψ2 , where 1 is any of the sentential connectives ∧, ∨, ⇒ or ⇔,then

fvar (Φ) = fvar (Ψ1 1 Ψ2) := fvar (Ψ1) ∪ fvar (Ψ2) .

• Let Φ ≡ (2x)Ψ, where 2 is any of the two quantifier symbols ∀ or ∃, andwhere x ∈ Ys is any variable of sort s ∈ S, then fvar (Φ) = fvar ((2x)Ψ) :=fvar (Ψ) \ x ; or, more precisely, for every sort u ∈ S:

fvar ((2x)Ψ)u :=

fvar(Ψ)s \ x , if u = s ,fvar(Ψ)u , otherwise (i.e. if u 6= s).

Moreover it should be observed, that in the case of one-sorted (homogeneous)partial algebras we shall only need the two cases35 X = Y (here we do not require Xto be finite) and X = Ø.

Remark 3.5 (on (the semantics of) elementary implications):As mentioned before, we shall restrict in this note our considerations mainly to ele-mentary implications, which are “formulas” of the form

(X;∧i∈I

tie≈ t′i ⇒

∧j∈J

t∗je≈ t∗j

′),

which belong to L, if the two sets I and J are finite.For such elementary implications of arbitrary length the semantics are extended

in a natural way:Let A be any partial algebra of signature Σ, and let v : X → A be any valuation.

We say that A satisfies an elementary implication (X;∧i∈I ti

e≈ t′i ⇒

∧j∈J t

∗j

e≈ t∗j

′)

w.r.t. the valuation v (in symbols: A |= (X;∧i∈I ti

e≈ t′i ⇒

∧j∈J t

∗j

e≈ t∗j

′)[v]) , if andonly if

whenever A |= (X; tie≈ t′i)[v] for all i ∈ I, then A |= (X; t∗j

e≈ t∗j

′)[v] for all j ∈ J.

Moreover, such an elementary implication holds (is valid) in A, if and only if it issatisfied in A w.r.t. every valuation v : X → A.

35In the case of homogeneous total algebras only the countably infinite basic set Y of variables isneeded, while the heterogeneous total case needs the same reference sets as the heterogeneous partialcase.

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Definition 3.6 (of ECE- and QE-equations): Particular elementary implicationsare

– ECE-equations (i.e. existentially conditioned existence equations)

(X;∧i∈I

tie≈ ti ⇒ t

e≈ t′),

– QE-equations (i.e. quasi existence equations)

(X;∧i∈I

tie≈ t′i ⇒ t

e≈ t′),

where X and I are finite, and the conclusion consists of one existence equation only.If X or I are allowed to be infinite, we shall speak of infinitary ECE-equationsand infinitary QE-equations, respectively.

In this connection we want to mention that an S-set X = (Xs)s∈S is finite, iff eachset Xs (s ∈ S) as well as the set s ∈ S | Xs 6= Ø are finite. The last requirementis always satisfied, if S is finite, what we usually shall require.

Remark 3.7 (on the semantics of ECE-equations in total algebras):Observe that in connection with total (many-sorted) algebras A a (possibly infinitary)

ECE-equation (X;∧i∈I ti

e≈ ti ⇒ t

e≈ t′) is satisfied or holds in A, respectively, if and

only if (X; te≈ t′) is satisfied respectively holds in A, i.e. in connection with total

algebras the semantics of (possibly infinitary) ECE-equations reduce to those of E-equations.

Examples 3.8 (of ECE-equations): Special elementary implications occur in con-nection with some further equational concepts, which are also frequently used asaxioms for the description of (classes of) partial algebras.

• Weak equations (X; tw≈ t′) (for t, t′ ∈ T (X,Σ)s) are in our approach special

ECE-equations:

(X; tw≈ t′) := (X; t

e≈ t ∧ t′

e≈ t′ ⇒ t

e≈ t′).

• Strong equations (or Kleene equations) (X; ts≈ t′) (for t, t′ ∈ T (X,Σ)s)

are conjunctions of special ECE-equations:

(X; ts≈ t′) := (X; (t

e≈ t⇒ t

e≈ t′) ∧ (t′

e≈ t′ ⇒ t

e≈ t′)).

A strong equation can therefore always be replaced by exactly two ECE-equations.We shall discuss later that for a suitably extended signature including “logical

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projections” the concepts of Kleene equations and ECE-equations are (almost)equivalent (and indeed eqivalent, when no models with the carrier sets of allsorts empty are allowed).

W. Craig has introduced in [Cr89] — for the homogeneous case — an additional“logical” binary operation symbol ε, which has to be interpreted in each partialΣ-algebra, say A, as a total (!) binary first projection (we would say that

always the E-equation (x, y; ε(x, y)e≈ x) has to hold in Ae — where Ae is the

partial algebra obtained from A by adding εA to the fundamental operations ofA). This then has the effect that the expressive power of the Kleene equationsw.r.t. this extended signature, say Σe, is the same as the expressive power ofECE-equations in the original (or the extended) signature — if one forgets aboutthe empty partial algebra.This can be extended to arbitrary heterogeneous signatures Σ by introducingfor each pair (s, t) ∈ S2 such a logical operation symbol εst such that this isinterpreted again in each partial Σ-algebra, say A, as a total (!) binary first

projection (we would say that always the E-equation (x, y; εst(x, y)e≈ x) has

to hold in Ae). See [B95] or subsection 10.2 at the end of these notes for moredetails.

• P. Kosiuczenko has used a combination of E-equations and weak equationsin order to characterize axiomatic classes of partial algebras, in which eachpartial algebra has a “permutable” respectively “distributive lattice of closedcongruence relations”36

Definition 3.9 (of E-, ECE- and QE-varieties): A class K of partial algebras ofsignature Σ is said to be definable by a set A of axioms, iff K is the class of all partialalgebras of signature Σ, in which all axioms (X; Φ) ∈ A hold. If, in particular, K isdefinable by E-, ECE- or QE-equations, then K is called an E-variety, ECE-varietyor QE-variety, respectively.

The characterization of such kinds of varieties by closure properties w.r.t. algebraicconstructions as well as the characterization of “closed sets” of E-, ECE- and QE-equations are some of the aims of these notes.

In what follows it will often be useful to have the following results, concepts andnotations around:

Proposition 3.10 (The interpretation of a term — if it exists — does onlydepend on the values for the valuation of the variables really occurring inthe term.)

Let X ⊆ Y be any set of variables, let t ∈ T (X,Σ)s be any term, let A be anypartial algebra of signature Σ with carier set A, and let v, w : X → A be any valuations

36See [Kos94]; for the definition of closed congruence see below.

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such that v|var (t) = w|var (t) (i.e. v and w take on all variables in var (t) the samevalues), and such that vs(t) exists. Then ws(t) exists and vs(t) = ws(t).

We leave the straight forward proof by structural induction on T (X,Σ) as anexercise.

Definition 3.11 (of a set of variables corresponding to an operation sym-bol): Let ϕ ∈ Ω be a fundamental operation symbol with (η, σ)(ϕ) = (s1 . . . sn, s).Let Iϕu := i | 1 ≤ i ≤ n and u = si , for each u ∈ S.

(a): We say that an S-set Xϕ ⊆ Y is a set of variables “corresponding to η(ϕ)”,if, for each u ∈ S, Xϕ

u = xi ∈ Yu | i ∈ Iϕu such that xi 6= xk for i, k ∈ Iu withi 6= k.37

(b): Moreover, we say that a set X ′ϕ is a set of variables “corresponding to(η, σ)(ϕ)”, if Xϕ ⊆ X ′ϕ is a set of variables corresponding to η(ϕ), and one choosesy ∈ Ys \Xϕ

s such that X ′ϕ := Xϕ ∪ ysτ(ϕ)+1 , what means that

X ′ϕu :=

Xϕs ∪ y , if u = s ,

Xϕu , otherwise (i.e. if u 6= s).

Remark 3.12 (i) (An additional rule to get first order formulas): We add arule to form first order formulas, which we did not include in Definition 3.2:

If (X; Φ) ∈ L(Y,Σ) is a formula, and if X ′ is a further finite S-set with X ⊆ X ′ ⊆Y , then (X ′; Φ) is also a formula in L(Y,Σ).

(ii) (Another notation for term existence statements): It is often usefulto have a shorter notation for term existence statements — which we shall briefly callTE-statements:

Let X ⊆ Y be any (finite) set of variables, let t ∈ T (X,Σ)s be any term of some

sort s ∈ S. Then we abbreviate the TE-statement (X; te≈ t) by (X;∃ t), or (X;D t),

i.e.38

(X;D t) :≡ (X;∃ t) :≡ (X; te≈ t) .

This is particularly useful, if the term t has a longer description.(iii) (On the semantics of first order formulas): We conclude this subsection

by giving a survey on the semantics of the formulas different from the atomic ones.For this purpose let A be a partial algebra of signature Σ with carrier set A, letX ⊆ Y be any finite S-set of variables, let v : X → A be any valuation, and let(X; Φ), (X; Φ′) ∈ L(Y,Σ) be any formulas and y ∈ Ys any variable of sort s ∈ S

37In particular one can choose Xϕu = yui ∈ Yu | i ∈ Iϕu in order to obtain such a set in a normed

form — cf. Definition 3.2 for the indexing of the variables. One could have also taken the firstvariables of each set, but this would make the indexing here much more complicated. Observe that,by definition, one then has yui 6= yuk for i, k ∈ Iu with i 6= k. In connection with X ′ϕu one would thenchoose in addition ysτ(ϕ)+1 ∈ Ys.

38On one hand (X;∃ t) shall remind the user that the statement (X; te≈ t) says something about

the existence of the (partial) interpretation of the term t, while (X;D t) shall remind her that itmeans the definedness of the (partial) inperpretation or of the corresponding term operation.

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• A |= (X;¬Φ)[v] iff it is not true that A |= (X; Φ)[v].

• A |= (X; (Φ ∧ Φ′))[v] iff A |= (X; Φ)[v] and A |= (X; Φ′)[v] .

• A |= (X; (Φ ∨ Φ′))[v] iff A |= (X; Φ)[v] or A |= (X; Φ′)[v](here one uses the nonexclusive meaning of “or”).

• A |= (X; (Φ⇒ Φ′))[v] iff : if A |= (X; Φ)[v] then A |= (X; Φ′)[v] .

• A |= (X; (Φ⇔ Φ′))[v] iff A |= (X; Φ)[v] if and only if A |= (X; Φ′)[v] .

• A |= (X; (∀y)Φ)[v] iff for all valuations w : X∪y → A which may differ fromv only on the variable y (i.e. for which one has w|X\y = v|X\y) it is true thatA |= (X; Φ)[w] .

• A |= (X; (∃y)Φ)[v] iff there exists a valuation w : X∪y → A which (possibly)differs from v only on the variable y (i.e. for which one has w|X\y = v|X\y),and for which one has A |= (X; Φ)[w] .

3.2 Preservation and reflection of formulas

One further application of the formulas defined above together with their semantics isthe classification of many important properties of mappings — e.g. of homomorphisms— between partial algebras by reflection and preservation of formulas.

Definition 3.13 (of preservation and reflection of formulas by mappings):Let A,B be partial algebras of signature Σ, (X; Φ) ∈ L(Σ) a formula w.r.t. a finite S-set X of variables X ⊆ Y and f : A→ B any S-mapping (i.e. f = (fs : As → Bs)s∈S).We say that

(i) f preserves the formula (X; Φ), iff , for every valuation v : X → A , one hasthat

A |= (X; Φ)[v] implies that B |= (X; Φ)[f v];

(ii) f reflects the formula (X; Φ), iff , for every valuation v : X → A , one has thatB |= (X; Φ)[f v] implies that A |= (X; Φ)[v].

This definition can — and will sometimes — also be applied to infinitary elementaryimplications.

The relationship between preservation and reflection of formulas is very close,since one has

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Lemma 3.14 (A formula is reflected by a mapping, iff its negation is pre-served.) Let A,B be partial algebras of signature Σ, let f : A→ B be any mappingand let (X; Φ) be any formula of L(Σ) or an arbitrary elementary implication. Thenthe following statements are equivalent:

(i) f preserves (X; Φ);

(ii) f reflects (X;¬Φ).

The easy proof is left as an exercise.

The following facts about reflection and preservation of formulas are very useful.

Proposition 3.15 (Some results concerning the preservation and reflectionof formulas by a composition of mappings): Let A,B,C be any partial algebrasof signature Σ, let f : A → B and g : B → C be any S-mappings, and let (X; Φ)and (X; Φi) (i ∈ I) be any formulas from L(Σ) or arbitrary elementary implications.Then one has:

(i) If both f and g reflect (preserve) (X; Φ), then so does g f .

(ii) If g f reflects (X; Φ), and if g preserves (X; Φ), then f reflects (X; Φ).

(iii) If g f preserves (X; Φ), and if g reflects (X; Φ), then f preserves (X; Φ).

(iv) If gf reflects (X; Φ), and if f is surjective and preserves (X; Φ), then g reflects(X; Φ).

(v) If g f preserves (X; Φ), and if f is surjective and reflects (X; Φ), then gpreserves (X; Φ).

(vi) If f reflects (X; Φi) for each i ∈ I, then f reflects (⋃i∈I X;

∧i∈I Φi)

and (⋃i∈I X;

∨i∈I Φi).

Proof We prove the properties (i), (ii) and (iv). Then (iii) and (v) will followbecause of Lemma 3.14. The proof of (vi) is left as an exercise. We always assumethe notation from the proposition.

Ad (i): Assume that both f and g preserve (X; Φ), and let v : X → A be anyvaluation such that A |= (X; Φ)[v]. Then one has successively that B |= (X; Φ)[f v]and therefore C |= (X; Φ)[g (f v)]. Because of g (f v) = (g f) v this showsthat g f preserves (X; Φ). The case of reflection then follows by Lemma 3.14.

Ad (ii): Let v : X → A be any valuation such that B |= (X; Φ)[f v]. Since gpreserves (X; Φ), one gets C |= (X; Φ)[g (f v)]. And since g f reflects (X; Φ),one immediately gets that A |= (X; Φ)[v)], what was to be shown.

Ad (iv): Let w : X → B be any valuation such that C |= (X; Φ)[g w]. Sincef : A→ B is surjective, there exists a valuation v : X → A such that f v = w. Sincetherefore g w = g f v, and since g f reflects (X; Φ), we get that A |= (X; Φ)[v)].Since f preserves (X; Φ), we finally get that B |= (X; Φ)[f v)], i.e. B |= (X; Φ)[w)],what was to be shown.

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Example 3.16 (on a characterization of injective mappings): As an examplefor an application of these concepts and results observe the following

Fact: An S-mapping f : A → B between S-sets A and B is injective, iff f

reflects all formulas of a set (xs, ys ;xse≈ ys) | s ∈ S , where xs 6= ys are

variables of sort s for each s ∈ S.The easy proof is left as an exercise.Thus one can easily derive from Proposition 3.15 the well known facts about

composition and cancellation properties for injective mappings, respectively.

3.3 Homomorphisms, closed homomorphisms and isomorphisms(second visit)

With this well known model theoretic concept we can now easily show that ournotion of homomorphism is basically a model theoretic one, which is closely relatedto existence equations:

Definition 3.17 (of model theoretic homomorphisms, term reflecting ho-momorphisms and isomorphisms): Let A, B be partial algebras of signature Σ,and let f : A→ B be any S-mapping.

(i) (Model theoretic homomorphisms): We say that f is a model theoretichomomorphism from A into B (in symbols: f : A→m B), iff f preserves allexistence equations of the form

( x1 . . . xΣ(ϕ), y ;ϕ x1 . . . xτ(ϕ)

e≈ y ) ,

for all ϕ ∈ Ω and for some collection x1, . . . , xτ(ϕ), y of variables in Y formingan S-set of variables corresponding to (η, σ)(ϕ) (cf. Definition 3.11).

(ii) (Term reflecting homomorphisms): A (model theoretic) homomorphismf : A →m B is said to be term reflecting, iff f reflects (in addition towhat has been required above for (model theoretic) homomorphisms) all termexistence statements

( x1 . . . xτ(ϕ) ;ϕ x1 . . . xτ(ϕ)

e≈ ϕ x1 . . . xτ(ϕ) ),

where we may assume that the sequence (x1 . . . xτ(ϕ)) has no repetitions (i.e.xi 6= xk for 1 ≤ i ≤ k ≤ τ(ϕ)).

(iii) (“Abstract definition” of isomorphisms): A homomorphism f : A→ B issaid to be an isomorphism, iff there exists a homomorphism g : B → A suchthat g f = idA and f g = idB, (i.e. iff f is a bijective homomorphism, and itsinverse mapping f−1 is also a homomorphism from B into A) (idA designatesthe identity mapping on A).

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Lemma 3.18 (Equivalence of model theoretic and “algebraic” definitionsof homomorphisms and closed (= term reflecting) homomorphisms): Let Aand B be any partial algebras of signature Σ with carrier sets A and B, respectively,and let f : A→ B be any mapping.

(a) (Homomorphisms): The following statements are equivalent:

(i) f is a model theoretic homomorphism f : A →m B in the sense of Defini-tion 3.17.

(ii) f is a homomorphism in the sense of Definition 1.23.

(b) (Closedness): The following statements are equivalent:

(iii) f is a term reflecting homomorphism as defined in Definition 3.17.

(iv) f is a closed homomorphism in the sense of Definition 1.23.

Proof In what follows let, for a given operation symbol ϕ ∈ Ω with (η, σ)(ϕ) =(s1 . . . sn, s), X

ϕ ⊆ X ′ϕ be S-sets of variables corresponding to η(ϕ) and (η, σ)(ϕ),respectively, such that X ′ϕs = Xϕ

s ∪ y (cf. Definition 3.11).Ad (a): “(i) implies (ii)”: Let f : A→m B be a model theoretic homomorphism.

For ϕ ∈ Ω let (a1, . . . , an) ∈ domϕA and a := ϕA(a1, . . . , an) . By the assumptions onX ′ϕ we can define a valuation w : X ′ϕ → A such that

wsi(xi) = ai , for 1 ≤ i ≤ n ,

andws(y) = a = ϕA(a1, . . . , an) .

Hence the formula (X ′ϕ;ϕ x1 . . . xne≈ y) is satisfied in A w.r.t. w. Since f

preserves this formula, and by the definition of w, it has also to be satisfied by Bw.r.t. f w. And, again by the definition of interpretations, this means that

(fs1(ws1(x1)), . . . , fsn(wsn(xn))) = (fs1(a1), . . . , fsn(an)) ∈ domϕB

and that one has

fs(ϕA(a1, . . . , an)) = fs(a) = fs(ws(y)) = ϕB(fs1(a1), . . . , fsn(an)) .

This proves the “(i) implies (ii)”.Conversely, assume that f is a homomorphism in the sense of Definition 1.23, and

let v : X ′ϕ → A be any valuation such that

A |= (X ′ϕ;ϕ x1 . . . xn

e≈ y)[v] ,

for some ϕ ∈ Ω . Letai := vsi(xi) for 1 ≤ i ≤ n .

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Then ϕA(a1, . . . , an) exists, and one has

vs(y) =: a = ϕA(a1, . . . , an) .

Since we assume (ii), we get the existence of ϕB(fs1(a1), fs2(a2), . . . , fsn(an)) and theequality

fs(a) = fs(ϕA(a1, a2, . . . , an)) = ϕB(fs1(a1), fs2(a2), . . . , fsn(an)) .

This means that

fs(vs(y)) = ϕB(fs1(vs1(y1)), fs2(vs2(y2)), . . . , fsn(vsn(yn))) .

However, this implies that

B |= (X ′ϕ;ϕ x1 . . . xn

e≈ y)[f v] ,

showing that f preserves (X ′ϕ;ϕ x1 . . . xne≈ y) . Since ϕ and v have been

chosen arbitrarily, this implies that f is a homomorphism from A into B. Hence “(ii)implies (i)”.

The proof of the equivalence “(iii) iff (iv)” runs similarly and is left as an exercise.

Remarks 3.19 (i): In what follows we shall switch freely between the model theo-retic and the algebraic characterization of (closed) homomorphisms without mention-ing it explicitly at the corresponding place.

(ii): Let A be a partial algebra of signature Σ. It is easy to prove that everyhomomorphism starting from A is closed iff A is total (exercise!). This implies thatnot necessarily every homomorphism starting from A is closed, if A is really a partialalgebra: Assume that Σ is a nontrivial signature, and take e.g. any total Σ-algebraB with B := A such that the structure of B extends the structure of A: graphϕA ⊆graphϕB for all ϕ ∈ Ω, and let f = idA be the identity mapping on B. Then, trivially,f is a homomorphism, but f is not closed.

(iii): The example in (ii) also shows that not every bijective homomorphism be-tween partial algebras of the same signature has to be an isomorphism.

Proposition 3.20 (Homomorphisms preserve all E-equations.) Let A,B bepartial algebras of signature Σ, and let f : A → B be any S-mapping. Then thefollowing statements are equivalent:

(i) f is a homomorphism from A into B, i.e. f : A→ B.

(ii) f preserves all E-equations (X : te≈ x) for all terms t ∈ T (X,Σ)s and variables

x ∈ Xs (such that x /∈ var (t)), for any s ∈ S and any S-set X of variables.

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(iii) f preserves all E-equations (X : te≈ t′) for all terms t, t′ ∈ T (X,Σ)s, for any

s ∈ S and any S-set X of variables.

Proof Since the implication from (iii) to (i) is obvious, one only has to prove theimplications from (i) to (ii) and from (ii) to (iii). Moreover, the proof of the im-plication from (ii) to (iii) is quite easy: Consider v : X → A , and assume that

A |= (X; t1e≈ t2)[v] for terms t1, t2 ∈ T (X,Σ)s. Choose variables yi ∈ Ys \ Xs

(1 ≤ i ≤ 2) such that y1 6= y2 and a valuation w : X ∪ y1, y2 → A such thatw|X := v , ws(y1) = vs(t1) and ws(y2) = vs(t2) . Then one also has (cf. Proposi-tion 3.10)

ws(y1) = ws(t1) = vs(t1) = vs(t2) = ws(t2) = ws(y2) ,

and by (ii) one has that fs(ws(t1)) and fs(ws(t2)) exist, and that

fs(vs(t1)) = fs(ws(t1)) = fs(ws(y1)) = fs(ws(y2)) = fs(ws(t2)) = fs(vs(t2)) .

This shows that f preserves (X; t1e≈ t2) .

The implication from (i) to (ii) can be proved by structural induction: That f

preserves (X;xe≈ y) for distinct variables x and y of the same sort, and that f

preserves (X;ϕe≈ y) for τ(ϕ) = 0 and y a variable of sort σ(ϕ) is trivial or follows

from the fact that f is a homomorphism. Therefore, assume ϕ ∈ Ω with τ(ϕ) 6= 0with (η, σ)(ϕ) =: (s1 . . . sn, s) , and let t := tn+1 := ϕ t1 . . . tn ∈ T (X,Σ)s beany term such that, for some arbitrary, but fixed valuation v : X → A , one has thatvs(t) exists in A, say with value an+1 := a ∈ As. By the recursive definition of aninterpretation39 vsi(ti) has to exist in A, too, for 1 ≤ i ≤ n, say with value ai ∈ Asi .Let yi ∈ Ysi \ Xsi and yi 6= yk, for 1 ≤ i 6= k ≤ n + 1 and sn+1 := s. And defineX∗ := X ∪

⋃n+1i=1 yi (union as S-sets, i.e. all phyla not mentioned are empty), and

w : X∗ → A with restriction w|X := v and wsi(yi) := vsi(ti) = ai (1 ≤ i ≤ n + 1).

Then A |= (X∗; tie≈ yi)[w] , for 1 ≤ i ≤ n + 1 . Let us assume in connection with

the induction hypothesis that f preserves (X∗; tie≈ yi) , for 1 ≤ i ≤ n . Then our

assumptions yield that B |= (X∗; tie≈ yi)[f w] , for 1 ≤ i ≤ n . Hence (f w)∼si(ti)

exists in B, and one has (f w)∼si(yi) = (f v)∼si(ti) =: bi . Z′ϕ :=

⋃n+11 yi (union

as S-sets) is a set of variables corresponding to (η, σ)(ϕ). Consider the valuationv∗ : Z ′ϕ → A with v∗(yi) := ai for 1 ≤ i ≤ n+ 1 . By assumption

ϕA(a1, . . . , an) = ϕA(v∗s1(y1), . . . , v∗sn(yn)) = v∗s(ϕ y1 . . . yn)

exists with value a = an+1 . Since f is a homomorphism, all interpretations in thefollowing sequence of equalities exist, and one has the equalities stated there:

fs(ϕA(a1, . . . , an)) = ϕB(fs1(v∗s1(y1)), . . . , fsn(v∗sn(yn))) =

= ϕB(fs1(ws1(y1)), . . . , fsn(wsn(yn))) = fs(v∗s(yn+1) =

= fs(ws(t) = ϕB(fs1(ws1(t1)), . . . , fsn(wsn(tn)))

39Cf. Theorems 2.8 or 2.12 or Definition 3.1.(iii).

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4 Examples 75

Hence B |= (X∗; te≈ yn+1)[f w] . Since (f w)∼s (t) = (f v)∼s (t) by Proposition 3.10,

one also has (f v)∼s (t) = (f w)∼s (yn+1) . This shows that f preserves (X∗; te≈ yn+1)

— and also preserves (X ∪ yn+1; te≈ yn+1) —, what was to be shown.

Analogously one has

Proposition 3.21 (Closed homomorphisms reflect all term existence state-ments.) Let A,B be partial algebras of signature Σ, and let f : A → B be anyhomomorphism. Then the following statements are equivalent:

(i) f is a closed homomorphism from A into B.

(ii) f reflects all term existence statements (X : te≈ t) for all terms t ∈ T (X,Σ)s

for any s ∈ S and any S-set X of variables.

Proof It is obvious from the above statements that one only has to prove the impli-cation from (i) to (ii). Since this proof is quite similar to the one of Proposition 3.20we leave the details as an exercise.

From the proof of Proposition 3.20 one can easily realize

Corollary 3.22 (Preservation of E-equations for appropriate subsets of vari-ables): If a mapping f from a partial algebra A into the a partial algebra B (carrier

sets A and B, respectively) preserves an E-equation (X; te≈ t′) (of sort s ∈ S), if Z

with fvar (te≈ t′) ⊆ Z ⊆ X is another set of variables, and if there exists a valuation,

say v, from X into A such that vs(t) and vs(t′) exist, then f also preserves (Z; t

e≈ t′).

4 Examples

Before we consider what we will call the “classical” examples let us observe (again)that an n-ary relation, say RA on some (S-) set A (i.e. any subset RA ⊆ An) can bedescribed by an n-ary partial operation ϕAR on A with dom ϕAR = RA, which satisfies,say, the axiom

ϕRx1 . . . xne≈ ϕRx1 . . . xn ⇒ ϕRx1 . . . xn

e≈ x1 ,

or, equivalently, with the abbreviation for TE-statements,

∃ ϕRx1 . . . xn ⇒ ϕRx1 . . . xne≈ x1 .

Thus the class of all partial algebras of a given similarity type τ contains the class of all“relational systems” of the same type τ as an ECE-subvariety (as already mentionedin the “Motivation”).

Moreover, we shall mostly omit the reference set of variables, if this is just theset of free variables occurring in the formula under consideration. And in the case ofbinary (partial) operations we shall usually use infix notation also for the terms, andin this connection we shall usually omit the brackets at the outside.

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4 Examples 76

4.1 “Classical” one-sorted algebraic structures and modules

Let us first consider some total and partial algebraic structures used in different areasof algebra and mathematics. They are usually based on at least one binary operation,which we shall usually denote by “”.

We start with a definition of special elements w.r.t. a binary operation:

Definition 4.1 Let A := (A; A) be an algebra with a total binary operation A —( 7→ 2).

(i) An element c ∈ A is called

(ln) left neutral, a left unit (deutsch: linksneutral, eine Linkseins), iffone has c A x = x for all x ∈ A;

(rn) right neutral, a right unit (deutsch: rechtsneutral, eine Recht-seins), iff x A c = x for all x ∈ A;

(n) neutral, a unit (deutsch: neutral, ein Einselement), iff c is both leftand right neutral.

(lz) a left zero (element) (deutsch: eine Linksnull), iff A |= c A x = c forall x ∈ A;

(rz) a right zero (element) (deutsch: eine Rechtsnull), iff A |= x A c = cfor all x ∈ A;

(z) a zero element (deutsch: Nullelement), iff cA is both a left and a rightzero element;

– an idempotent element, iff one has c A c = c.

(ii) (li) Let c be a left unit of A. For a ∈ A an element b ∈ A is called a leftinverse of (deutsch: ein Linksinverses von) a w.r.t. A, iff b A a = c;

(ri) A right inverse or inverse w.r.t. (deutsch: Rechtsinverses oder In-verses bzgl.) A and a right unit (or unit) is defined analogously.

– If c ∈ A is a left unit of A and if f : A −→ A is a unary operationon A such that f(x) A x = c for each x ∈ A, then f is called a leftinversion of (deutsch: linksinverse Operation zu) A; right inversionand inversion of A are defined analogously.

In the examples below let

• N be the set of all natural numbers ≥ 1,

• N0 be the set of all natural numbers ≥ 0,

• Z be the set of all integers,

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4 Examples 77

• Q be the set of all rational numbers,

• R be the set of all real numbers,

• C be the set of all complex numbers,

• H be the set of all quaternions, which can be described in matrix form as

H := (

α β−β α

)| α, β ∈ C.

In these sets addition (a, b) 7→ a + b, additive inverse x 7→ −x, the zero-element 0,multiplication (a, b) 7→ ab := a · b and the multiplicative unit 1 are defined as usualrespectively as known from matrix theory.

Examples 4.2 (of classes of “classical” homogeneous total algebras):An algebra A with carrier set A is called a

(a) groupoid (deutsch: Gruppoid oder Magma), iff it is a (total) algebra (A; A)

of type ( 7→ 2), for which one has A |= xye≈ xy (just meaning the totalness

of A).We shall get to know below many groupoids with additional properties, sinceall the total binary operations occurring below endow the corresponding carrierset with a groupoid structure. However a characteristic groupoid, which hasno other properties, is the term algebra T (X, ( 7→ 2)) w.r.t. any set X ofvariables.

(b) semigroup (deutsch: Halbgruppe), iff it is a groupoid (A; A), for which Aisassociative (deutsch: assoziativ),

i.e. one has that A |= x (y z)e≈ (x y) z

(for different variables x, y, z).

(c) commutative (or abelian) semigroup (deutsch: kommutative (oder abelsche)Halbgruppe), iff it is a semigroup, for which A iscommutative (deutsch: kommutativ),

i.e. one has that A |= x ye≈ y x (for different variables x, y).

(d) regular semigroup (deutsch: regulare Halbgruppe), iff it is a semigroup,for which one has

A |= (∀x)(∃y)(x y xe≈ x).

(e) left cancellative semigroup (deutsch: linkskurzbare Halbgruppe), iff itis a semigroup, for which one has

A |= (x ye≈ x z ⇒ y

e≈ z).

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4 Examples 78

In a similar way right cancellative semigroups and cancellative semi-groups are defined.

(f) monoid (deutsch: Monoid), iff it is a (total) algebra (A; A, eA) of type ( 7→2, e 7→ 0), for which (A, A) is a semigroup with neutral element eA.

(g) commutative monoid (deutsch: kommutatives Monoid), iff it is a monoidsuch that (A; A) is a commutative semigroup.

(h) quasigroup (deutsch: Quasigruppe), iff it is a groupoid (A; A), for whichone has

A |= (∀u)(∀v)(∃x)(∃y)(x ue≈ v ∧ u y

e≈ v∧

∧(∀w)(∀z)((w ue≈ v ⇒ x

e≈ w) ∧ (u z

e≈ v ⇒ y

e≈ z)).

Quasigroups can also be defined as total algebras with three binary operations,one being the multiplication A, while the other two provide the unique elementsfrom the axiom above (corresponding axioms as an exercise).

(i) loop (deutsch: eine Loop), iff it is a total algebra (A; A, eA) of type ( 7→2, e 7→ 0), for which (A; A) is a quasigroup, and in which eA is a unit element.

(j) group (deutsch: Gruppe), iff it is a (total) algebra (A; A, eA, fA) of signature( 7→ 2, e 7→ 0, f 7→ 1) such that (A; A, eA) is a monoid and fA is a (left)inversion of A w.r.t. eA. In group theory one usually defines a group to be asemigroup (G; G), in which the following relatively complicated axiom holds40

(∃e)(∀x)(x ee≈ x ∧ (∃y)(x y

e≈ e)).

An abelian group is just a group (A; A, eA, fA), for which (A; A) is an abeliansemigroup.

Let us pause here and consider special examples and some facts about the algebraicstructures mentioned so far:

Proposition 4.3 Every semigroup S = (S, S) can be embedded in a naturalway into a monoid M .

Proof Let e be an element not occurring in S and let M := S ∪ e be the disjointunion of S with e; let a M b := a S b, if a, b ∈ S, else let a M e := a =: e M a. Itis easy to realize that M is a monoid.

40In many books one finds this to be splitted into two axioms

(∃e)(∀x)(x ee≈ x) and

(∀x)(∃y)(x ye≈ e).

Note that “e” in the second axiom has nothing to do with “e” in the first axiom. Do these axiomshold only in groups? Exercise

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4 Examples 79

Example 4.4 (of monoids: transformation monoids): Typical monoids aregiven for any set M by the so-called transformation monoid

T (M) := (MM , , idM) with MM := f | f : M −→M any mapping,

where designates the composition of mappings, and idM the identity mapping onM (m 7→ m for m ∈M).

Proposition 4.5 (Cayley): Every monoid M is isomorphic to a “submonoid”41

of a transformation monoid.

Proof For m ∈M define

λm : M −→M, (x 7→ m M x for any x ∈M).

It is then easy to realize that the mapping m 7→ λm (m ∈M) is an injective monoidhomomorphism from M into the transformation monoid T (M).

This shows that all monoids can be realized as “submonoids” of transformationmonoids, and also all semigroups as “subsemigroups” of some transformation semi-group (MM , M).

With the same argumentation one may easily realize that

Corollary 4.6 Every group (G; G, eG, fG) is isomorphic to a group of bijec-tive mappings from G onto G.

Proof Namely, all λg (g ∈ G) are then bijective mappings and form a group.

Moreover, every group is a loop, but not vice versa. The groups

S(M) := f | f : M −→M is a bijective mapping

of all bijective mappings of a set ono itself — usually called permutations — arecalled symmetric groups. Thus the above result can be stated in the form

Corollary 4.7 Every group is isomorphic to a subgroup of some symmetricgroup.

Examples 4.8 (of special semigroups, monoids, groups): Well known semi-groups are (N0,+), (N0, ·), (Z,+), (Z, ·), (Q,+), (Q, ·), (R,+), (R, ·), (C,+), (C, ·),(H,+), (H, ·), while adding in the additive case 0 and in the multiplicative case 1 asadditional nullary constant transfers them into monoids. Which of the above algebrasare actually groups?

Proposition 4.9 Each transformation semigroup T (M) is regular, while thesubsemigroup of all surjective (injective) mappings from M onto M is right(left) cancellable.

The proof is left as an exercise.

41This means a subalgebra for the type of monoids.

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4 Examples 80

One gets a quasigroup operation on some (finite) set Q, iff in the “multiplica-tion table” in each column and in each row each element of Q occurs exactly once.Quasigroups and loops often occur in geometry and in combinatorics.

Examples 4.10 (of algebraic structures connected with ordered sets42): Weare now listing some algebraic structures, which have close relationship to specialordered sets. An algebra A with carrier set A is called a

(k) semilattice (or: idempotent commutative semigroup) (deutsch: Halb-verband (oder: idempotente kommutative Halbgruppe)), iff it is a com-mutative semigroup, in which each element is idempotent (deutsch: idempo-tent),

i.e. one has that A |= x xe≈ x.

(l) semilattice with unit (deutsch: Halbverband mit Eins(element)), iff itis a commutative monoid, for which (A; A) is a semilattice.

(m) semilattice with zero (deutsch: Halbverband mit Null), iff it is a (total)algebra (A; A, oA) of type ( 7→ 2, o 7→ 0), for which (A, A) is a semilatticewith zero element oA.

(n) lattice (deutsch: Verband), iff it is a (total) algebra A = (A;tA,uA) of type(t 7→ 2,u 7→ 2) such that both (A;tA) and (A;uA) are semilattices, and Asatisfies the two absorption laws (deutsch: Verschmelzungsgesetze):

A |= x u (x t y)e≈ x and A |= x t (x u y)

e≈ x.

(o) distributive lattice (deutsch: distributiver Verband), iff it is a lattice, say,A = (A;tA,uA) which satisfies in addition the two distributive laws (fordistinct variables x, y, z):

x u (y t z)e≈ (x u y) t (x u z) and x t (y u z)

e≈ (x t y) u (x t z).

(p) bounded lattice or 0-1-lattice (deutsch: beschrankter Verband oder 0-1-Verband), iff it is a (total) algebra A = (A;tA,uA, 0A, 1A) of type (t 7→2,u 7→ 2, 0 7→ 0, 1 7→ 0) such that A = (A;tA,uA) is a lattice and both(A;tA, 0A) and (A;uA, 1A) are semilattices with unit, (and (A;tA, 1A) and(A;uA, 0A) are semilattices with zero). Similarly one defines distributive 0-1-lattices.

42See below for the definition of ordered sets and for the interconnection with these algebras.

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4 Examples 81

(q) Boolean lattice (deutsch: Boolescher Verband), iff it is a (total) algebraA = (A;tA,uA, 0A, 1A, ′A) of type (t 7→ 2,u 7→ 2, 0 7→ 0, 1 7→ 0, ′ 7→ 1) suchthat (A;tA,uA, 0A, 1A) is a distributive 0-1-lattice and ′A is a complementa-tion, i.e. one has

A |= x u x′e≈ 0, and A |= x t x′

e≈ 1.

In a Boolean algebra also the following E-equations hold:

(x′)′e≈ x, (x u y)′

e≈ x′ t y′, and (x t y)′

e≈ x′ u y′;

the last two being the so-called DeMorgan’s laws.

As already mentioned earlier all these concepts have close relationship to orderedsets. Let us first observe that semilattices have an equivalent order theoretic descrip-tion.

Definition 4.11 (of ordered sets as relational systems and as partial alge-bras; order theoretic semilattices):

(i) (Ordered sets): An ordered set is a structure (P ;≤) i.e. a set P with abinary relation ≤, which — as mentioned earlier — can also be described bya partial operation (i.e. one has an equivalent description as a partial algebra(P ; P )), which satisfies the following axioms (we formulate them for the relationas well as for its representation by a binary partial operation ∗):Axioms for the order rel.: Axioms for the partial algebra:x ≤ x i.e. ∃ x ∗ x ,

x ≤ y ∧ y ≤ x⇒ xe≈ y i.e. (∃ x ∗ y ∧ ∃ y ∗ x)⇒ x

e≈ y ,

x ≤ y ∧ y ≤ z ⇒ xe≈ z i.e. (∃ x ∗ y ∧ ∃ y ∗ z)⇒ ∃ x ∗ z ,

— ∃ x ∗ y ⇒ x ∗ ye≈ x .

(ii) (Semilattices as ordered sets): If (A; A) is a semilattice, then one maydefine on A a binary relation ≤A for any a, b ∈ A by

a ≤A b iff a A b = b,

which then is indeed an order relation on A (proof?), for which A representsthe “binary supremum operation”. This ordered set (A;≤A) is then also calledan upper semilattice or sup-semilattice.One may also define a “dual” order relation vA by

a vA b iff a A b = a.

Then A means the “infimum operation” on (A;vA) and (A;vA) is called alower semilattice or an inf-semilattice.

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4 Examples 82

Examples 4.12 and Facts

(r) If one defines for an (algebraically defined) semilattice A := (A; A) the (or-dered) upper semilattice (sup-semilattice) as above, and if A has a unit e, thenthis unit becomes the least element of the upper semilattice — an “order zero”—, i.e. one has e ≤A a for all a ∈ A (or if one transfers it into a lower semilattice(inf-semilattice), then the unit e becomes its largest element — an “order one”—, i.e. one has a ≤A e for all a ∈ A).

(s) Analogously, if e is a zero element of the algebraically defined semilattice (A; A),then e is for the related upper semilattice order relation ≤A the largest element(in the dual case the least element).

(t) For (algebraically defined) lattices A = (A;tA,uA) one has an identity betweenthe two order relations induced by the two semilattice structures in the followingway:

a ≤A b (:iff a tA b = b), iff a vA b (:iff a uA b = a).

Proof as an exercise.

(u) Thus lattices and semilattices can always also be defined as special ordered sets.In what follows let — for an arbitrary set M

P(M) := T | T ⊆M be the power set of M , i.e. the set of all subsets of M .

and let Pfin(M) := F | F ∈ P(M) finite be the set of all finite subsets of aset M .

(v) Special semilattices are then given by (P(M);∪), (P(M);∩), (Pfin(M);∪).Observe that the first two examples are order theoretically already lattices —namely both yielding (P(M);⊆), which corresponds to the algebraically defineddistributive 0-1-lattice (P(M);∪,∩,Ø,M) —, where always ∪ and ∩ denotethe set theoretic union respectively intersection. (Pfin(M);∪) still induces alattice order on (Pfin(M), but this lattice no longer has a largest element.Actually (P(M);∪,∩,Ø,M, ′) — where the prime indicates complementation—, is a Boolean lattice. All finite boolean lattices are obtained this way up toisomorphism, while there are infinite Boolean lattices, which are not isomorphicto power sets.

(w) We shall meet lattices in universal algebra as subalgebra and congruence lat-tices of universal algebras, groups are obtained as “automorphism groups” andmonoids as “endomorphism monoids”.

Actually the lattices connected with a universal (partial) algebra have some ad-ditional properties: they are so-called “complete” and “algebraic” lattices:

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4 Examples 83

Definition 4.13 (of complete (semi-) lattices): Let (P ;≤) be an ordered set,and let N be any subset of P . An element u ∈ P is called an upper bound (lowerbound) of N , if n ≤ u (u ≤ n) for all n ∈ N . s ∈ P is called a supremum (aninfimum) of N — in symbols sup(N) or

∨N (inf(N) or

∧N) —, iff s is a least

upper bound (greatest lower bound), i.e. iff s is an upper (lower) bound of N , and ifs′ is any upper (lower) bound of N , then s ≤ s′ (s′ ≤ s). An ordered set P = (P ;≤)is called a complete upper semilattice (complete lower semilattice), iff in Pevery subset N of P has a supremum (an infimum). If P is both a complete uppersemilattice and a complete lower semilattice, then it is called a complete lattice.

Lemma 4.14 (Some facts about complete (semi-) lattices):

(i) Any complete upper or lower semilattice P is itself already a complete lattice,and it has a greatest element (

∨P or

∧Ø) and a least element (

∨Ø or

∧P ).

This shows that∨

Ø ≤∧

Ø, else one always has:

(ii) If N 6= Ø, N ⊆ P , then∧N ≤

∨N .

Observe that the “operations”∨

and∧

have no fixed arities. Therefore theclass of all complete lattices cannot be defined as a class of universal algebrasof some given type according to our general definition.

“Algebraic lattices” will be defined later. One can now also define ordered semi-groups, ordered groups, lattice ordered groups, etc., but we do not go into detailsabout these structures.

Examples 4.15 (Semi-rings, rings and fields): An algebra A with carrier set Ais called a

(x) semi-ring (deutsch: Halbring oder Semiring) iff it is a (total) algebra(A; +A, 0A, A) of type (+ 7→ 2, 0 7→ 0, 7→ 2), such that (A; +A, 0A) is acommutative monoid43 and (A; A, 0A) is a semigroup44 with zero element 0A,and in which the following two distributive laws hold:

u (x+ y)e≈ (u x) + (u y) and (x+ y) u

e≈ (x u) + (y u).

(y) ring (deutsch: Ring), iff it is a (total) algebra (A; +A, 0A,−A, A) of type(+ 7→ 2, 0 7→ 0,− 7→ 1, 7→ 2), such that (A; +A, 0A,−A) is an abelian groupand (A; +A, 0A, A) is a semiring.Some authors do not require A to be associative.

43Sometimes a zero element is not required, and our definition already yields a semi-ring withzero.

44It may be with some authors that — as with rings below — A need not be associative.

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4 Examples 84

(z) commutative ring (deutsch: kommutativer Ring), iff it is a ring(A; +A, 0A,−A, A), in which A is a commutative operation.

(A) ring with unit (deutsch: unitarer Ring), iff it is a (total) algebra(A; +A, 0A,−A, A, 1A) of similarity type (+ 7→ 2, 0 7→ 0,− 7→ 1, 7→ 2, 1 7→ 0)such that (A; +A, 0A,−A, A) is a ring and (A; A, 1A) is a monoid.

(B) skew field or division ring (deutsch: Schiefkorper oder Divisionsring),iff it is a (partial) algebra (A; +A, 0A,−A, A, 1A, −1A) of similarity type (+ 7→2, 0 7→ 0,− 7→ 1, 7→ 2, 1 7→ 0, −1 7→ 1) such that (A; +A, 0A,−A, A, 1A) isa ring with unit and (A \ 0A; A, 1A, −1A) with the operations restricted toA \ 0A is a group.If A is a commutative operation, then a skew field is called a (commutative)field (deutsch: (kommutativer) Korper).

(C) left module over R (deutsch: Links-R-Modul) for some ringR = (R; +R, 0R,−R, R) in “homogeneous” description, iff it is a (total)algebra

M := (M ;⊕M , θM ,M , (r·M)r∈R)

of similarity type (⊕ 7→ 2, θ 7→ 0, 7→ 1, (r· 7→ 1)r∈R), (often written as RM)such that (M ;⊕M , θM ,M) is an abelian group and in M the following axiomshold for all variables r, s which can only be evaluated in the “ring of scalars”(i.e. which are of sort “scalar”) and for all variables v, w, which can only beevaluated in the module (i.e. which are of the sort “vector”45):

(M1) : (r, s : scalar, v : vector; (r + s) · ve≈ (r · v)⊕ (s · v))

(M2) : (r : scalar, v, w : vector; r · (v ⊕ w)e≈ (r · v)⊕ (r · w))

(M3) : (r, s : scalar, v : vector; (r s) · ve≈ r · (s · v)).

In most cases one requires the ring R to be unitary with additional unit (seebelow).If R is a ring with unit, and if in M there also holds the axiom

(M4) : (v : scalar; 1 · ve≈ v),

then M is called a unitary left module over R (deutsch: unitarer Links-R-Modul).Analogously one may define a (unitary) right module over R (written asMR) with scalar multiplication from the right.

45This shows that we cannot really avoid two-sortedness here.

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There also exists a “really” heterogeneous description of left modules — wherethe ring need not be given in advance — as two-sorted algebras (we considerthe unitary case)

M := (SM , VM ; +M , 0M ,−M , M , 1M ,⊕M , θM ,M , ·M),

where the specification can be described as follows in the way often used incomputer science (the axioms indicated by “*” only apply in the case, when onewants to specify a commutative field, — then one calls it a left vectorspace(deutsch: Links-Vektorraum) over some skew field, if it is also unitary):Specification: ModuleSorts : scalar, vectorOperations : + : scalar × scalar → scalar

0 : → scalar− : scalar → scalar : scalar × scalar → scalar1 : → scalar

* −1 : scalar → scalar⊕ : vector × vector → vectorθ : → vector : vector → vector· : scalar × vector → vector

Axioms : (q, r, s : scalar ; q + (r + s)e≈ (q + r) + s)

(q, r : scalar ; q + re≈ r + q)

(q : scalar ; q + 0e≈ q)

(q : scalar ; q + (−q)e≈ 0)

(q, r, s : scalar ; q (r s)e≈ (q r) s)

(q : scalar ; q 1e≈ q)

(q, r, s : scalar ; q (r + s)e≈ (q r) + (q s))

(q, r, s : scalar ; (r + s) qe≈ (r q) + (s q))

* (q : scalar ; ¬qe≈ 0⇒ q (q−1)

e≈ 1)

* ( ; ¬1e≈ 0)

(x, y, z : vector ;x⊕ (y ⊕ z)e≈ (x⊕ y)⊕ z)

(x, y : vector ; x⊕ ye≈ y ⊕ x)

(x : vector ; x⊕ θe≈ x)

(x : vector ; x⊕ (x)e≈ θ)

(q, r : scalar , x : vector ; (q + r) · xe≈ (q · x)⊕ (r · x))

(q : scalar , x, y : vector ; q · (x⊕ y)e≈ (q · x)⊕ (q · y))

(q, r : scalar , x : vector ; (q r) · xe≈ q · (r · x))

(x : vector ; 1 · xe≈ x).

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(D) R-algebra (in the “classical” sense) (deutsch: Algebra oder im alten Sprachge-brauch: hyperkomplexes System) over a commutative ring (or a field) R is amodule (or vectorspace) M over R together with an additional (bilinear) binaryoperation, i.e. one has the specification46

Specification: Algebra = module +Operations : : vector × vector → vector

Axioms : (q : scalar , x, y : vector ; (q · x) ye≈ q · (x y))

(q : scalar , x, y : vector ; x (q · y)e≈ q · (x y))

(x, y, z : vector ; (x⊕ y) ze≈ (x z)⊕ (y z))

(x, y, z : vector ; x (y z)e≈ (x y) z)

(E) (Special examples for the above kinds of algebraic structures):

– (N; +, ·) is a semi-ring without zero, while (N0; +, 0, ·) is a semi-ring withzero.

– Z := (Z; +, 0,−, ·, 1) is an (associative) commutative unitary ring (as weshall realize later, it is the free unitary ring gerated by the empty set —the initial unitary ring).

– (2 · Z; +, 0,−, ·) is a non-unitary ring.

– The set of all sqare n×n-matrices Mn×n(R) with components in a (unitary)ring R and the usual addition and multiplication etc. of matrices is a (uni-tary) ring, which is non-commutative, if n ≥ 2, even if R is commutative,and it is even an algebra (in the classical sense), if R is commutative.

(F) There is a particular class of non-associative rings, the so-called Lie rings, inwhich all ring axioms except associativity hold and in addition the laws:

x xe≈ 0,

(x y) z + (y z) x+ (z x) ye≈ 0 (the Jacobi identity).

Every associative ring (A; +A, 0A,−A, A) gives rise to a Lie ring (A; +A, 0A,−A,A)by defining

aA b := (a A b)− (b A a) for all a, b ∈ A.If one defines on a three-dimensional vectorspace F 3 over some field F — seebelow — with the usual “componentwise defined” additive group the “outerproduct” or “cross product”

(a, b, c)×F 3

(a′, b′, c′) := (b F c′ −F c F b′, c F a′ −F a F c′, a F b′ −F b F a′),

then one gets a Lie ring.

46The last axiom below is not always required — as it has been observed earlier that associativityis not always required for rings.

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(G) If a ring R does not contain a unit, then one may consider the ring with unit(proof as exercise) — in connection with the ring Z of integers —

R′ := ((z, r) | z ∈ Z, r ∈ R; +′, 0′,−′, ′, 1′),

where the operations are defined as follows:

(z1, r1) +′ (z2, r2) := (z1 +Z z2, r1 +R r2),

0′ := (0Z, 0R),

−′(z, r) := (−Zz,−Rr),(z1, r1) ′ (z2, r2) := (z1 ·Z z2, z1 · r2 +R z2 · r1 +R r1 R r2),

1′ := (1Z, 0R),

where one defines for z ∈ Z with z > 0 and for r ∈ R

0Z · r := 0R, z · r := r +R r +R . . .+R r︸︷︷︸z times

, (−Zz) · r := −R(z · r).

(Observe that the last definition only depends on the fact that (R; +R, 0R,−R)is an abelian group, showing that each abelian group is in a natural way aunitary module over Z.) It is then also easy to realize that any left module overR becomes in a natural way a unitary left module over R′.

(H) Let H = (H; H , 1H) be a monoid and R any unitary ring. Then define themonoid ring (actually a unitary R-algebra) R[H] := (R[H]; +, 0,−, ) with

R[H] := f | f : H −→ R, h ∈ H | f(h) 6= 0R is finite,

where for f, g ∈ R[H] one has the binary operations

(f + g)(h) := f(h) +g (h) for each h ∈ H,

(f g)(h) :=∑

uHv=h

f(u) ·R g(v) for all h, u, v ∈ H.

The other operations are defined as they should be defined in order to get aring.47

(I) A special example of the above construction of monoid rings is the polynomialring R[X1, X2, . . . , Xn] over a set X1, X2, . . . , Xn, where the monoid H is the“commutative word monoid”

(X1, X2, . . . , Xn∗c := Xk11 X

k22 . . . Xkn

n | k1, k2, . . . , kn ∈ N0; ·c, ec),

with Xk11 X

k22 . . . Xkn

n ·c Xl11 X

l22 . . . X ln

n := Xk1+l11 Xk2+l2

2 . . . Xkn+lnn .

47One often represents the elements of R[H] by what one calls “formal sums”

f =:∑h∈H

f(h)h , f + g =∑h∈H

(f(h) + g(h))h , (f g) =∑h∈H

(∑

uHv=h

f(u) ·R g(v))h.

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4.2 Some examples of partial algebras

Examples 4.16 (Axioms for small categories as homogeneous and hetero-geneous partial algebras):

(J) As an example for ECE-varieties of homogeneous partial algebras let us list theaxioms for the class of all small categories considered as homogeneous partialalgebras of type

(Ω, τ) := (D,C, , (D 7→ 1), (C 7→ 1), ( 7→ 2))

satisfying the axioms — for which we use in this case prefix notation —:

1. xDxe≈ x, (this implies Dx

e≈ Dx, i.e. D has to be total),

2. Cxxe≈ x, (this implies Cx

e≈ Cx, i.e. C has to be total),

3. yxe≈ yx⇒ Cx

e≈ Dy ∧ C yx

e≈ Cy ∧D yx

e≈ Dx,

4. yxe≈ yx ∧ zy

e≈ zy ⇒ z yx

e≈ zyx.

One might have expected in addition the QE-equation

5. Dye≈ Cx⇒ yx

e≈ yx,

but this one can be proved from the other axioms (exercise), showing that theclass of all small categories is really an ECE-variety and not only a QE-variety.

(K) Usually, however one defines small categories as two-sorted algebras in the waybelow48, where one sort is “Mor”, the one of “morphisms”, while the second oneis “Ob”, the one of “objects” (which are represented in the homogeneous caseby their identity morphisms, i.e. by the values of the operations “D” and “C”— see above)49.

48Since each axiom there only refers to the variables really occurring in it, we do not list themseparately, however m,m′,m′′ are variables of sort Mor and X is a variable of sort Ob.

49For categories C in general the classes of objects and morphisms need no longer be proper sets,however then one requires that for any two objects A and B of the category C and its operationsthat the class MorC(A,B) := f ∈ CMor | Dom

C(f) = A and CodC(f) = B is a set. See. e.g.[AdHS90] or [HS73] for more details.

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Specification: Category (as heterogeneous algebra)Sorts : Ob, MorOperations : : Mor×Mor −→ Mor

Dom : Mor −→ ObCod : Mor −→ Ob

1 : Ob −→ Mor

Axioms : m′ me≈ m′ m ∧m′′ m′

e≈ m′′ m′ ⇒

⇒ m′′ (m′ m)e≈ (m′′ m′) m

m′ me≈ m′ m⇒ Dom(m′ m)

e≈ Dom(m)

m′ me≈ m′ m⇒ Cod(m′ m)

e≈ Cod(m′)

m′ me≈ m′ m⇒ Cod(m)

e≈ Dom(m′)

m 1Xe≈ m 1X ⇒ m 1X

e≈ m

1X me≈ 1X m⇒ 1X m

e≈ m

Dom(1X)e≈ X

Cod(1X)e≈ X

m 1Dom(m)

e≈ m

1Cod(m)m

e≈ m .

(L) (Examples of (non-small) categories):

– Each class of universal partial algebras (as class of objects) of some signa-ture and satisfying some axioms together with the class of all homomor-phisms (as class of all morphisms) between them forms a general category,where for a partial algebra A one has 1A = idA, the identity mapping,where for f : A −→ B one has Dom(f) = A, Cod(f) = B and is compo-sition of homomorphisms.

– Each ordered set (P ;≤) can be considered as a small category, where theobjects are the elements of P , and between two objects u, v ∈ P thereexists at most one morphism, and this exists, iff u ≤ v.

– Another source for small categories are the “path categories” connectedwith some directed simple graph (V,E), where the objects are the vertices(i.e. the elements of V , and the morphisms are the “paths” in this graphof arbitrary length ≥ 0.

– As a last example of small categories let us consider monoids as smallcategories with exactly one object (the monoid) and the elements of themonoid as its morphisms.

We add some further examples of partial algebras derived from applications incomputer science and quantum logic:

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Examples 4.17 (M) Stack automata: The following partial algebras are well-known in computer science:Specification: Stack automata of integersSorts : stack, intOperations : 0 : −→ int

succ : int −→ intpred : int −→ int

empty : −→ stackpush : stack×int −→ stack

top : stack −→ intpop : stack −→ stack

Axioms : 0e≈ 0

pred(succ(n))e≈ n

succ(pred(n))e≈ n

emptye≈ empty

pop(push(s, n))e≈ s

top(push(s, n))e≈ n

In the “initial algebra”50 of the class of stack automata of integers the operations“top” and “pop” are not defined on the constant “empty” (of sort stack).

(N) (Orthomodular partial algebras): In quantum logics orthomodular posetsare used as generalizations of orthomodular lattices, which are already abstrac-tions of the lattices of closed subspaces of Hilbert spaces. Orthomodular posetscan be axiomatized as homogeneous partial algebras of type (Ω := (⊕,′ , 0), τ :=(2, 1, 0)) satisfying the following axioms :

(A0) ∃ 0.

(A1) x′′e≈ x.

(A2) x⊕ x′e≈ 0′. (Notation: 0′ =: 1).

(A3) x⊕ 0e≈ x.

(A4) ∃ (x⊕ y)⇒ x⊕ ye≈ y ⊕ x.

(A5) ∃ ((x⊕ y)⊕ z)⇒ (x⊕ y)⊕ ze≈ x⊕ (y ⊕ z).

(A6) ∃ (x⊕ y) ∧ ∃ (y′ ⊕ z)⇒ ∃ (x⊕ z).

(A7) ∃ (x⊕ y′) ∧ ∃ (x′ ⊕ y)⇒ xe≈ y.

(A8) ∃ (x⊕ y) ∧ ∃ (y ⊕ z) ∧ ∃ (x⊕ z)⇒ ∃ (x⊕ (y ⊕ z)).

50In a category K an object I is called initial, if it allows exactly one morphism into each otherobject of K.

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(A9) ∃ (x⊕ y′)⇒ x⊕ (x⊕ y′)′e≈ y.

It can be shown that in this axiomatization the axiom (A6) already follows fromthe axioms (A1), (A4), (A5) and (A9) (proof as an exercise).

If one defines a binary relation “≤” on an arbitrary orthomodular partial algebraA by

a ≤ b iff ∃ (a⊕ b′) ,

then the above axioms imply that one gets an ordered set (A :≤), from whichthe original partial algebra can be recovered.51

By changing or replacing some of the axioms above one gets other ECE-varietiesof partial algebras used e.g. in “test theory” or in other connections. We do notgive more details here.

• For further examples of heterogeneous partial algebras from computer sciencesee e.g. [Re84].

4.3 A different view at heterogeneous (partial) algebras

The definitions of many-sorted partial algebras and of homomorphisms between themsuggest that:

– Different phyla52 of a many-sorted partial algebra of signature Σ may, withoutloss of generality, be assumed to be disjoint. (Inclusions should be specified byappropriate unary operations.)

– The specification of the signature Σ can be considered as the description of aparticular homogeneous partial algebraic structure (ϕS)ϕ∈Ω of the homogeneoussimilarity type τ = (τ(ϕ))ϕ∈Ω on the set S of sorts, where for each ϕ ∈ Ω, ϕS

is defined only on the sequence (sϕ1 , . . . , sϕτ(ϕ)), and for this it takes the value

sϕ. We shall call this partial algebra S the sort algebra (or specificationalgebra) for the signature under consideration.

– One can replace now the family (As)s∈S of phyla of a many-sorted partial algebra

by its disjoint union, say A∗ :=⋃As, in which case each many-sorted (partial)

operation ϕA becomes a partial operation on A∗, which we shall designate againby ϕA.

– The original partition of A into phyla can be replaced by a mapping, say vA∗ :A∗ → S such that vA∗(a) = s if and only if a ∈ As (a ∈ A∗, s ∈ S). vA∗ will becalled the canonical sort mapping for A∗.

51For details see P. Burmeister and M. Maczynski: Orthomodular (Partial) Algebras and TheirRepresentations; in: Demonstratio Mathematica 27, 1994, pp. 701–722.

52Phylum is another word for “carrier of some sort”.

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Let us observe that a homomorphism f : A −→ B will be called closed, iff freflects all TE-statements of the form

ϕx0x1 . . . xτ(ϕ)−1

e≈ ϕx0x1 . . . xτ(ϕ)−1

for all ϕ ∈ Ω and for distinct variables x0, x1, . . . , xτ(ϕ)−1 — we shall see later that

this is equivalent to the fact that f reflects all TE-statements te≈ t for arbitrary terms

t (depending on arbitrary sets of variables). Observe that homomorphisms betweentotal algebras always have this property.

One then has the following

Theorem 4.18 With the concepts and notations introduced above for any many-sorted partial algebra A the canonical sort mapping vA∗ is always a homomorphism53

vA∗ : (⋃

s∈SAs, (ϕ

A)ϕ∈Ω)→ S;

and this homomorphism is closed if and only if A is a total many-sorted algebra.Conversely, if A∗ is any partial algebra and vA∗ : A∗ → S is a homomorphism,

then A∗ can be considered as a many-sorted partial algebra ((ϕ−1(s))s∈S, (ϕA)ϕ∈Ω)which is total if and only if vA∗ is closed.

Example 4.19 Let us consider the signature for stack automata of integers as pre-sented above. This means that we have two nullary, four unary and one binary(possibly partial) operations, e.g. “push: stack × integer → stack” means that pushis a binary operation with first argument of sort “stack”, second argument of sort“integer” and value of sort “stack”.

A closed homomorphism vA from a partial algebra A into S means that thereexist fundamental constants 0A ∈ v−1

A (integer) and emptyA ∈ v−1A (stack) and

that e.g. the binary partial operation pushA maps each pair (and only such) (c, d)with c ∈ v−1

A (stack) and d ∈ v−1A (integer) onto some element of v−1

A (stack),etc., but that is just how a total two-sorted algebra

A : (v−1A (stack), v−1

A (integer); 0A, succA, predA, emptyA, pushA, popA, topA)

is described.

Thus, we can consider a many-sorted partial algebra of signature Σ and sortalgebra S as a pair (A∗, vA∗ : A∗ → S), where A∗ is a partial algebra of similaritytype τ = (τ(ϕ))ϕ∈Ω

54, and vA∗ is a homomorphism (which is closed if and only if themany-sorted algebra is total).

53Remember that we assume in particular the phyla to be pairwise disjoint.54Here τ(ϕ) is the length of the word in the first component of τ∗(ϕ).

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A homomorphism h : A → B between many-sorted partial algebras then corre-sponds to a homomorphism h∗ : A∗ → B∗ such that vB∗ h∗ = vA∗ , where the graphof h∗ is the disjoint union of the graphs of the mappings hs : As → Bs (s ∈ S);and vice versa: if h∗ : A∗ → B∗ is a homomorphism between partial algebras oftype τ provided with homomorphisms vA∗ : A∗ → S and vB∗ : B∗ → S such thatvB∗ h∗ = vA∗ , then one has that (h∗|As : As → Bs)s∈S is a homomorphism betweenthe corresponding many-sorted partial algebras A and B. If A is total, then h∗ isalways closed. Therefore, if we only deal with total many-sorted algebras, then thehomomorphisms h∗ : A∗ → B∗ are always closed.

In category theory, for any category C and any object S the category C ↓ S, theobjects of which are pairs (C, v : C → S), where C is an object and v is a morphismof C, and where morphisms h : (C, v)→ (D,w) are morphisms h : C → D of C whichsatisfy w h = v, is called a comma category.

Thus we have established an equivalence between the category Alg(Σ) of all partialmany-sorted algebras of signature Σ = (S,Ω, τ ∗) and the comma category Alg(τ) ↓S of the category Alg(τ) of all partial algebras of the corresponding homogeneoustype τ with respect to the sort algebra S.

In order to form coproducts in Alg(τ) ↓ S for a family ((Ai, vAi))i∈I , one formsin Alg(τ) the coproduct ((ιi : Ai →

∐i∈I Ai)i∈I ,

∐i∈I Ai) =: ((ιi)i∈I , A); and the

sort homomorphism vA : A → S is then the homomorphism in the category Alg(τ)induced by the family (vAi)i∈I of sort homomorphisms. Since

∐i∈I Ai carries the

weakest structure such that all ιi (i ∈ I) are still homomorphisms, it is not difficultto realize that in the case of closed vAi (i ∈ I) also vA and all ιi (i ∈ I) are closed.

If one wants to construct in Alg(τ) ↓ S a product of a family ((Ai, vAi))i∈I ,it is easy to realize that this corresponds to the construction of a pullback (P , (pi :P → Ai)i∈I) with respect to the given sink ((Ai, vAi))i∈I . Since the class of all closedhomomorphisms is equal to Λ(Ext), where Ext is the class of all TAlg(τ)-extendableepimorphisms, it is easy to realize that all induced homomorphisms pi (and hencevp = vAi pi (i ∈ I)) become closed, if all vi (i ∈ I) are closed.

These observations extend to all other category theoretical constructions.There is no problem with subalgebras. With respect to congruence relations one

has to observe that ker vA∗ := (a, a′)|vA∗(a) = vA∗(a′) becomes the largest ad-

missable congruence (and equivalence) relation of the pair (A∗, vA∗), and that thecompatibility condition vA∗ = vB∗ h∗ for homomorphisms h∗ : (A∗, vA∗)→ (B∗, vB∗)

implies that always ker h∗ ⊆ ker vA∗ . Observe that ker vA∗ =⋃s∈SAs × As is just

the union of the largest equivalence relations of the phyla.

One advantage of this approach may be the observation that it easily also allowsto handle overloading: one just has to drop the requirement that in the sort algebraeach partial operation is to be defined on exactly one sequence. Everything which has

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been observed so far does not depend on this requirement. Because of the fact thatone uses variables of different sorts (or the corresponding equivalent representationwith a sort mapping) in order to formulate axioms, different instances of the sameoperation can satisfy different axioms.

Example 4.20 Let us consider integers, sequences of integers of length n (for somefixed natural number n) and n × n square matrices of integers with addition andmultiplication as operations:

sorts: int, seqn, matnoperations: +: int × int → int

seqn × seqn → seqnmatn × matn → matn

· : int × int → intint × seqn → seqnmatn × seqn → seqnmatn × matn → matnint × matn → matn

Among the axioms one may formulate

(x, y : int; ·xye≈ ·yx)

(x, y, z : int; · · xyze≈ ·x · yz)

(x, y : int, z : seqn; · · xyze≈ ·x · yz)

etc., while there is no commutativity law for matrix multiplication.

One thing we have already used here, which is of much more importance in themany-sorted than in the one-sorted case, is the fact that with each axiom one has alsoto specify the variables to which it refers — there may be more variables and moresorts specified than actually needed, and in connection with additional sorts or e.g.for existence-equations without free variables this really will have meaning, if emptyphyla are allowed. Namely, if a partial algebra has an empty phylum of sort s, theneach existence-equation which refers to a variable of this sort in its specification istrivially valid in this partial algebra, e.g. in our specification above the axiom

(x, y : int, z : matn; ·xye≈ x)

is trivially satisfied in every algebra of this signature, in which the phylum of sortmatn is empty.

Concerning its model theoretic meaning an axiom of the form

(x, y : int, z : seqn; · · xyze≈ ·x · yz)

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4 Examples 95

can be replaced by the implication

(xe≈ x ∧ z

e≈ z → · · xyz

e≈ ·x · yz)

(ye≈ y is superfluous in the premise), which means: if the variable x (of sort int)

and the variable z (of sort matn) are interpretable, i.e. if the corresponding phyla arenon-empty, then the conclusion has to hold true.

This observation explains, why many-sorted (existence-) equations (also in thetotal case) behave already much like implications rather than like equations. Namely,it may no longer be true that a variety of many-sorted algebras is always generated(w.r.t. the operators H, S and P) by a single algebra.

(The following example should be skipped on first reading.)In addition, in the case of many-sorted partial algebras which allow empty phyla

one has to distinguish, whether the set S of sorts is finite or infinite55. However, wehave to admit that our example below is rather artificial and pathological, and we donot see right now one which is more likely to be realized in computer science:

sorts: n for each natural number noperations: ϕn, ϕ′n :→ n for each n.

Consider a set An|n ∈ N of partial algebras, where — using the usual many-sorted terminology —

An,m :=

ϕm, ϕ

′m, if m 6= n,

Ø, if m = n,

ϕAmn := ϕn, ϕ′nAm := ϕ′n, if m 6= n,

and if m = n, then ϕAmn and ϕ′n

Am are undefined. Let us consider the class

K := HSPAn|n ∈ N.

Then the only partial algebras in K, which have non-empty phyla for all n ∈ N,are isomorphic to the sort algebra for this signature, i.e. K satisfies the existence-equations

(∗) (xn, yn : n(n ∈ N);xme≈ ym)

for each natural number m, while the category theoretical reduced product of thefamily (An)n∈N with respect to the Frechet filter F of cofinite subsets of N:

F = E|E ⊆ N and N \ E is finite 55If S is infinite, then a corresponding many-sorted partial algebra sometimes already behaves as

if it were of infinitary similarity type, and the following is such an example.

17th July 2002

5 Substructures, generation and homomorphisms revisited 96

(i.e. the direct limit of the directed system

(∏n∈E

An, prE,E′ :∏n∈E

An →∏n∈E′

An|E,E ′ ∈ F , E ⊇ E ′)

where prE,E′(an|n ∈ E) := (an|n ∈ E ′)) is containing the algebra B as subalge-bra, where Bn = ϕn, ϕ′n, and ϕBn = ϕn, ϕ′n

B = ϕ′n for each n ∈ N. This showsthat B /∈ K, i.e. K is not closed w.r.t. direct limits, since these do not preserveexistence-equations of the form (∗) which are actually equivalent to implications withan infinite conjunction of existence-equations in the premise (according to our earliertranslation).

In [B86] the approach to many-sorted (partial) algebras via the comma categoryAlg(τ) ↓ S has been carried through in parallel to the development of the theoryof partial algebras.

The observations above motivate that in what follows we restrict considerationsmainly to homogeneous partial algebras.

5 Substructures, generation and homomorphisms

revisited

5.1 Substructures and generation revisited

Remarks 5.1 Characterization of subalgebras by properties of the inclu-sion homomorphisms. While one uses for total (many-sorted) algebras only oneconcept of subobjects – namely the one of subalgebras –, the use of partial algebrasin different contexts or kinds of applications justifies three different basic concepts ofsubobjects, as has already been indicated in Definitions and Remarks 1.21. In thenext subsection we shall see in more detail, how these concepts are related to differentproperties of the inclusion homomorphism. At the moment we can already say that,for a partial algebra B on an S-subset B of a partial algebra A of signature Σ is

• a weak subalgebra of A, iff the inclusion mapping idBA : B → A is an (injective)homomorphism from B into A: idBA : B → A;

• a relative subalgebra of A, iff the inclusion mapping idBA : B → A is a full (orinitial56) (and injective) homomorphism from B into A: idBA : B → A is initial;

56A homomorphism f : B → A is called initial, iff f reflects all E-equations of the form

( x1 . . . xτ(ϕ), y ;ϕ x1 . . . xτ(ϕ)e≈ y ) ,

for all ϕ ∈ Ω and for some collection x1, . . . , xτ(ϕ), y of variables in Y forming an S-set of variablescorresponding to (η, σ)(ϕ) (cf. Definition 3.11 and also Definition 3.17). Then the initial and injectivehomomorphisms are exactly the full and injective homomorphism, which will be defined in the nextsubsection.

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• a subalgebra of A, iff the inclusion mapping idBA : B → A is a closed (andinjective) homomorphism from B into A: idBA : B → A is closed.

Since we deal with finitary partial operations only, we have in addition the follow-ing property of the closure operator CA defined in Corollary 1.18

Proposition 5.2 (The closure operators CA on partial algebras are “alge-braic”.) The closure operators CA on partial algebras A satisfy

(CA) CAM =⋃CAF | F is a finite S-subset of M ,

for every S-subset M of the carrier set of any given partial algebra A. Such closureoperators are called algebraic.

Proof Clearly⋃ CAF | F ⊆ M finite ⊆ CAM by the monotonicity of CA, i.e. by

(C2) of Definition 1.16. For the converse inclusion we have to show thatB :=⋃CAF |

F ⊆ M is finite is a closed subset of A containing M . This is done by structuralinduction: Clearly M ⊆ B, since m ⊆ CAm ⊆ B for each element m ∈ Ms andsort s ∈ S. Assume ϕ ∈ Ω with (η, σ)(ϕ) = (s1 . . . sτ(ϕ), s), and let (b1, . . . , bτ(ϕ)) ∈Bη(ϕ) ∩ domϕA. Then, by induction hypothesis, for each i ∈ 1, . . . , τ(ϕ), there

are finite S-subsets Fi ⊆ M such that bi ∈ CAFi. Obviously F :=⋃τ(ϕ)i=1 Fi ⊆ M

is finite and contains Fi, and therefore bi ∈ CAF for each i ∈ 1, . . . , τ(ϕ). HenceϕA(b1, . . . , bτ(ϕ)) ∈ CAF ⊆ B, showing that B is closed.

Remark 5.3 In addition one gets that the finitely generated closed subsets are ex-actly the (lattice theoretically) compact elements of the complete lattice57 of allclosed subsets of a given partial algebra A where the supremum of a given set G ofclosed subsets is computed as CA

⋃G as usual (exercise).

Definition 5.4 (of the comparison of substructures): The observation of Propo-sition 1.15 can be generalized, if we introduce for arbitrary weak subalgebras B,C ofa partial algebra A:

• B ⊆ C iff B ⊆ C and, for each ϕ ∈ Ω: graph ϕB ⊆ graph ϕC .

• If G is a system of weak subalgebras of a partial algebra A, then we define itsintersection G :=

⋂G as

G := (⋂H | H ∈ G , (

⋂ϕH | H ∈ G )ϕ∈Ω)

with

graph⋂ϕH | H ∈ G :=

⋂ graph ϕH | H ∈ G for every ϕ ∈ Ω .

57An element c of a complete lattice (L;≤) is called compact, iff one has, for all subsets D ⊆ L,that c ≤ sup(D) implies the existence of a finite subset F ⊆ D such that c ≤ sup(F ).

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This is motivated by the

Proposition 5.5 Weak relative and relative subalgebras also form closuresystems. Let A be any partial algebra, and let G be a system of subalgebras, ofrelative subalgebras or of weak relative subalgebras, respectively, of A. Then

⋂G is a

subalgebra, a relative subalgebra or a weak subalgebra of A, respectively, with carrierset B :=

⋂H | H ∈ G .

Proof Let ϕ ∈ Ω with (η, σ)(ϕ) = (s1 . . . sτ(ϕ), s), and (h1, . . . , hτ(ϕ)) ∈ dom⋂ϕH |

H ∈ G . Then

• for each H ∈ G one has (h1, . . . , hτ(ϕ)) ∈ Hη(ϕ), and therefore (h1, . . . , hτ(ϕ)) ∈Bη(ϕ),

• for each H ∈ G one has ϕH(h1, . . . , hτ(ϕ)) = ϕA(h1, . . . , hτ(ϕ)) ∈ Hs, and there-fore ϕA(h1, . . . , hτ(ϕ)) ∈ Bs, and

ϕB(h1, . . . , hτ(ϕ)) := (⋂ϕH | H ∈ G )(h1, . . . , hτ(ϕ))) = ϕA(h1, . . . , hτ(ϕ))).

This already shows that B := (B, (ϕB)ϕ∈Ω) is in each case at least a weak subal-gebra of A. If G consists of subalgebras of A, then Proposition 1.15 tells us that B isa closed subset of A.

It remains to show that B is a relative subalgebra of A if all H ∈ G are relative sub-algebras of A. Thus let (b1, . . . , bτ(ϕ)) ∈ Bη(ϕ) ∩ domϕA, such that ϕA(b1, . . . , bτ(ϕ)) ∈Bs. Then (b1, . . . , bτ(ϕ)) ∈ domϕH for allH ∈ G, and ϕH(b1, . . . , bτ(ϕ)) = ϕA(b1, . . . , bτ(ϕ)) ∈Hs for each H ∈ G, and therefore ϕB(b1, . . . , bτ(ϕ)) exists.

Remark 5.6 While relative and weak relative substructures can also be defined forrelational systems, the concept of subalgebras is specific for partial algebras and totalalgebras and allows nontrivial generation.

Since CAM as defined in Corollary and Definition 1.18 requires the knowledgeof all closed subsets of A containing M , one introduces more-step closure operatorsexhausting CAM in order to be able to “generate locally”.

Definition 5.7 of more-step operators for the closure: Let A be a partial al-gebra and let M be an arbitrary S-subset of A. Define

DAM := M ∪⋃ ϕA(a1, . . . , aτ(ϕ)) | ϕ ∈ Ω and

(a1, . . . , aτ(ϕ)) ∈Mη(ϕ) ∩ domϕA

(the union being taken as union of S-sets, and (η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s) ), and foreach natural number n define recursively

17th July 2002


D0AM := M ,

Dn+1A M := DA(DnAM),

B0AM := M ,

Bn+1A M := Dn+1

A \ DnAM .

BnAM is called the n-th Baire-class of M .

The process of generation from below is then described by

Proposition 5.8 (The closure of a subset is the union of its Baire-classes.)Let A be any partial algebra, and let M ⊆ A be any subset. Then

(i) M ⊆ DnAM ⊆ DmAM ⊆ CAM for any natural numbers n ≤ m.

(ii) CAM =⋃∞n=0DnAM =

⋃∞n=0 BnAM .

(iii) CAM =⋃kn=0DnAM , iff Dk+1

A M = DkAM , iff Bk+1A M = Ø.

Proof It follows by induction on the natural numbers involved and by the propertiesof closure operators thatM ⊆ DnAM ⊆ DmAM ⊆ CAM for any natural numbers n ≤ m,i.e. the statement of (i). That

⋃∞n=0DnAM =

⋃∞n=0 BnAM follows from the above

definition. In order to prove (ii) it therefore suffices to show that B :=⋃∞n=0DnAM is

a closed S-subset of A: Let ϕ ∈ Ω and (b1, . . . , bτ(ϕ)) ∈ Bη(ϕ) ∩ domϕA. Then thereexist natural numbers n1, . . . , nτ(ϕ) such that bi ∈ (DniAM)si for 1 ≤ i ≤ τ(ϕ). Sincen1, . . . , nτ(ϕ) is a finite set, it has a maximum, say n. Therefore, bi ∈ (DnAM)si ,

for 1 ≤ i ≤ τ(ϕ) , and ϕA(b1, . . . , bτ(ϕ)) ∈ (Dn+1A M)s ⊆ Bs, showing that B is closed.

The proof of (iii) is obvious.

Immediately from the definition of Baire-classes one gets the following

Lemma 5.9 (Some facts about “local generation”): Let A be any partial alge-bra, let M be any subset of A and let a ∈ CAM .

(i) If a ∈ BnAM for some n ≥ 1, i.e. if a ∈ CAM \ M , then there are ϕ ∈ Ω,

a1, . . . , aτ(ϕ) ∈ Dn−1A M such that a = ϕA(a1, . . . , aτ(ϕ)). In particular, if τ(ϕ) ≥

1, then there is at least one r ∈ 1, . . . , τ(ϕ) such that ar ∈ Bn−1A M .

(ii) ϕA ∈ DAM for every non-empty nullary constant ϕA of A.

Figure 4 shows this layer model of generation from below in some details.

17th July 2002


b bb b bb

pppppp

a1

a2

ar

aτ(ϕ)−1

aτ(ϕ)

a = ϕA(a1, . . . , ar, . . . , aτ(ϕ))

@@@@@@@@@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@@@@@@@

@@@@@@ @@@@@@@@@@@@@@@@

(((((

(((((

(((((

hhhhh

&&

''

%%

$$

B1AM M = B0

AM

B2AM

BkAM

Bk+1A M

A

Figure 4: Layer model of generation from below (homogeneous case)

Examples 5.10 (i) In the (total) algebra (N0; (0,′ )) of similarity type(0,′ , (0, 0), (′, 1)) with n′ := n + 1 one has B0

AØ = Ø, and BnAØ = n − 1for n > 0, i.e. the process of generation does not stop for any finite naturalnumber n.

(ii) Consider vector spaces over a field F as one-sorted total algebras V := (V,+V , 0V ,−V , (f ·V )f∈F )) of type (+, 0,−∪ f · | f ∈ F , (+, 2), (0, 0), (−, 1) ∪ (f ·, 1) | f ∈ F ), where f ·V (v) := fv for every scalar f and every vectorv. Let M := v1, . . . , vk be a finite basis (i.e. linearly independent generatingsubset) of V with k ≥ 1. Then DnVM = V for n = dlog2ke + 1, where d.e arethe upper Gaussian brackets (dre is the smallest integral number n ≥ r for anyreal number r).

5.2 On some properties of homomorphisms

Remark 5.11 For the comparison of partial algebras we use the concept of homo-morphisms, as we have defined them in subsection 3.3. In particular they are basedon total mappings. They are the “structure preserving” mappings which we usethroughout these notes. This concept has proven to be most useful for our purposes,e.g. of the one of designing a theory of partial algebras. Even on those occasionswhere partial mappings seem to be necessary — see e.g. the valuation mappings— in connection with our development of the theory of partial algebras, the arisingconcepts can be formulated within the category of partial algebras with homomor-phisms as morphisms, when we shall include also the category theoretical concept of

17th July 2002


“factorization systems” (see Definition 6.22).

We have already seen that in connection with the description of relative sub-algebras by properties of the inclusion mapping we need additional properties forhomomorphisms other than closedness. Although “fullness” — to be defined below— cannot be described by reflection or preservation of formulas, if no further prop-ereties are around, we shall need this concept mainly in connection with two furtherproperties, and it has found its way into the literature (then often called “strong-ness”). In connection with injectivity it becomes definable by reflection of formulas(“full and injective” equals “initial and injective”); and in connection with surjectivityit becomes associated to injectivity via “factorization systems”.

Definition 5.12 of full and initial homomorphisms, respectively: Let A andB be any partial algebras of the same signature Σ, and let f : A → B be anyhomomorphism. f is called

(fh) full, iff , for every ϕ ∈ Ω with (η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s), and for all(a1, . . . , aτ(ϕ), a) ∈ Aη(ϕ) we have:

if ϕB(fs1(a1), . . . , fsτ(ϕ)(aτ(ϕ))) = fs(a) (exists and belongs to the image of f),

then there is a (possibly different) sequence (a′1, . . . , a′τ(ϕ)) ∈ dom ϕA such that

fsi(a′i) = fsi(ai) for all 1 ≤ i ≤ τ(ϕ),

i.e. f is full iff f fully induces the structure on its direct image f(A).

(ih) initial, iff f reflects all E-equations of the form

( x1 . . . xτ(ϕ), y ; (ϕ, x1, . . . , xτ(ϕ))e≈ y ) ,

for all ϕ ∈ Ω and for some collection x1, . . . , xτ(ϕ), y of variables in Y formingan S-set of variables corresponding to (η, σ)(ϕ) (cf. Definition 3.11 and alsoDefinition 3.17).58

Lemma 5.13 Every closed and every initial homomorphism is also full.

Proof as an exercise.

As examples that the converse implications do not hold, consider the homo-morphisms in Figure 5. If one considers there instead of f3 the homomorphismf ′3 : A3 → B′3, where B′3 is the relative subalgebra of B3 on f3(A3), then f ′3 isclosed and initial (and therefore also full).

58 The concept of an initial morphism of a category C, where the objects are sets with somestructure, has been introduced by N. Bourbaki: A morphism i : A → B between objects A,B ∈Ob(C) with carrier sets A and B, respectively, and where the morphisms are based on mappingsbetween these underlying sets (such that different morphisms correspond to different mappings, i.e.in a so-called concrete category) is called initial, iff for all mappings f : C → A (C ∈ Ob(C)) f isthe graph of a morphism in C iff i f is the graph of a morphism in C. It is suggested as an exerciseto prove that initial homomorphisms defined above are exactly the initial morphisms in the sense ofBourbaki in the category PAlg(Σ) (cf. Remark 5.15 below).

17th July 2002


ddddddddddd d

dd

ddddddd

dd

'

&

$

%

f1 f2 f3 f4

neither full

nor closed

nor initial

full, but

neither closed

nor initial

initial (and

therefore full,)

but not closed

closed and

full, but

not initial

- - -

-

- - -

-

-

6 6

6

6 6 6

6

6

6

6

6

*

*

Figure 5: Examples of homomorphisms and their properties

Mainly from the results about reflection and preservation of formulas in Proposi-tion 3.15 we get the following

Proposition 5.14 (on properties of homomorphisms): Let A,B,C be partialalgebras of the same signature Σ. Then:

(i) The identity mapping idA : A→ A is always a homomorphism and even closedand initial and an isomorphism idA : A→ A with idA being also its inverse.

(ii) The composition of homomorphisms as well as of closed or initial homomor-phisms, respectively, is always a homomorphism, which is then also closed orinitial, respectively: If f : A → B and g : B → C are closed or initial ho-momorphisms, then g f : A → C, a 7→ g(f(a)) is also a closed or initialhomomorphism, respectively.

(iii) A homomorphism f : A→ B is full and injective, iff f is initial and injective.

(iv) Let f : A→ B and g : B → C be homomorphisms, then one has:

(a) If both f and g are closed, then g f is closed.

(b) If both f and g are initial, then g f is initial.

(c) If both, f and g are isomorphisms, then g f is an isomorphism.

(d) If both f and g are full, then g f need not be full.

(e) If both f and g are full and injective, then g f is full and injective.

(f) If both f and g are full and surjective, then g f is full and surjective.

(v) Let f : A→ B and g : B → C be homomorphisms. Then one has:

(a) If g f is closed or initial, then f is closed or initial, respectively.

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(b) If g f is closed or initial, and if f is surjective, the g is closed or initial,respectively.

(c) If g f is full and injective, then f is full and injective.

(d) If gf is full and injective, and if f is surjective, the g is full and injective.

(vi) If f : A→ B is any mapping, and if A is a discrete partial algebra (ϕA = Ø foreach ϕ ∈ Ω), then f : A→ B is a homomorphism.


Moreover, one should investigate the question (as an exercise), which further prop-erties one has for “full and surjective”.

Remark 5.15 (The category of partial algebras of a given signature:)Statements (i) and (ii) above show that the class PAlg(Σ) of all partial algebras ofsignature Σ as object class together with all homomorphisms between elements ofPAlg(Σ) forms a category, denoted by PAlg(Σ). The subclass TAlg(Σ) of all totalalgebras of signature Σ together with all homomorphisms between them forms thecategory TAlg(Σ) which is well known from the universal algebra of total algebras.

Observe that one also has the subcategories with PAlg(Σ) as object class and theclasses of all closed, all initial, all injective, all surjective, all full and injective, andall full and surjective homomorphisms, respectively, as class of morphisms.

We can now describe isomorphisms between partial algebras in many ways.

Proposition 5.16 Equivalent characterizations of isomorphisms:Let A,B be partial algebras of the same signature Σ, and let f : A → B be anyhomomorphism. Then the following statements are equivalent:

(i) f is an isomorphism.

(ii) f is bijective, and the inverse map f−1 : B → A — f−1(b) = a iff f(a) = b —is also a homomorphism.

(iii) f is a full and bijective homomorphism.

(iv) f is a closed and bijective homomorphism.

(v) f is an initial and bijective homomorphism.

Proof (i) and (ii) are equivalent by the definition of isomorphisms. (v) or (iv)trivially imply (iii), and closed and injective homomorphisms are initial, therefore(iv) implies (v) (see Lemma 5.13, and take also the last statement as an exercise).Thus it remains to show that (ii) implies (iv) and that (iii) implies (ii).

17th July 2002


“(ii) implies (v)”: Since the inverse of f exists, f has to be a bijective map-ping. And since f−1 : B → A is a homomorphism, too, we have, for each ϕ ∈Ω (assume (η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s)), and for each (a1, . . . , aτ(ϕ)) ∈ A

η(ϕ)s1 with

(fs1(a1), . . . , fsτ(ϕ)(aτ(ϕ))) ∈ domϕB, that ((fs1)−1(fs1(a1)), . . . , (fsτ(ϕ)

)−1(fsτ(ϕ)(aτ(ϕ)))) ∈

domϕA. However, ((fs1)−1(fs1(a1)), . . . , (fsτ(ϕ))−1(fsτ(ϕ)

(aτ(ϕ)))) = (a1, . . . , aτ(ϕ)), show-ing that f is closed.

“(iii)⇒ (ii)”: If (iii) is satisfied, then f is bijective and therefore it has an inversef−1 : B → A satisfying f−1 f = idA and f f−1 = idB. Assume ϕB(b1, . . . , bτ(ϕ)) =b in B for some ϕ ∈ Ω ((η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s)). Since f is bijective, thereare unique ai ∈ Asi (1 ≤ i ≤ τ(ϕ)), and a ∈ As such that fsi(ai) = bi for i =1, . . . , τ(ϕ), and fs(a) = b. Since f is full — and by the uniqueness of the ai and a —we get ϕA(a1, . . . , aτ(ϕ)) = a or, equivalently, ϕA((fs1)−1(b1), . . . , (fsτ(ϕ)

)−1(bτ(ϕ))) =

(fs)−1(b), showing that f−1 is a homomorphism.

5.3 Monomorphisms and epimorphisms

In category theory there are (at least) two further properties of morphisms whichare usually considered, and at least the epimorphisms will play an important role inconnection with the description of closed classes of E-, ECE- or QE- equations.

Definition 5.17 Let C = (Ob(C),Mor(C), (,Dom,Cod, 1)) be an arbitrary category.Let A,B ∈ Ob(C) and f : A→ B be an arbitrary morphism in C.

a) f is called a monomorphism of C, if for any object C ∈ Ob(C) and for anytwo morphisms g, h : C → A, f g = f h always implies g = h.

b) f is called an epimorphism of C, if for any object D ∈ Ob(C), and for any twomorphisms u, v : B → D, u f = v f implies u = v.

Remark 5.18 Observe that an isomorphism is always both, a monomorphism andan epimorphism, and in the categories PAlg(Σ) and TAlg(Σ) each bijective homomor-phism is as well a monomorphism as an epimorphism (see the statements below), yetin PAlg(Σ) it need not be an isomorphism, while in TAlg(Σ) a bijective homomor-phism is always an isomorphism, since in this category a homomorphism is alwaysclosed, too.

Proposition 5.19 (In the categories PAlg(Σ) and TAlg(Σ) the monomorphismsare exactly the injective homomorphisms.)In PAlg(Σ) and TAlg(Σ) a homomorphism f : A→ B is a monomorphism iff f is a

injective.

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csCdiscrete

1

PPPPPPPq

\/

u1

PPPqv

A

ea′ea

f-

1

PPPPPPPq

\

/

B

eb

cT (c,Σ)

1

PPPPPPPq

\/

u1

PPPqv

A

ea′ea

f-

1

PPPPPPPq

\

/

B

eb

Case of partial Σ-algebras Case of total Σ-algebras

Figure 6: A non-injective homomorphism is not a monomorphism

Proof Let f : A → B be an injective homomorphism and let g, h : C → A beany two homomorphisms such that f g = f h; then for all c ∈ Cs (s ∈ S)fs(gs(c)) = fs(hs(c)) and injectivity of fs imply gs(c) = hs(c), i.e. g = h; therefore fis a monomorphism.

Conversely, let us assume that f is not injective, i.e. that there are s ∈ S anda, a′ ∈ As, a 6= a′, such that fs(a) = fs(a

′) (compare Figure 6). Then choose

Cu :=

cs , if u = s ,Ø , otherwise (i.e. if u 6= s).

• In the case of the category PAlg(Σ) of partial Σ-algebras set C := (C, (Ø)ϕ∈Ω) tobe the discrete partial algebra on a one-element S-set c. Choose gs(c) := a,hs(c) := a′, and let gu and hu have empty graphs, for all u ∈ S \ s . Then,clearly g, h : C → A are homomorphisms and f g = f h, but g 6= h, i.e.injectivity is necessary for monomorphisms in PAlg(Σ).

• In the case of the category TAlg(Σ) of total Σ-algebras set C := T (C,Σ), theterm algebra on the one-element S-set C. Choose gs(c) := a, hs(c) := a′, andlet gu and hu have empty graphs, for all u ∈ S \ s . Let g∼, h∼ : T (C,Σ)→ Abe the interpretations induced by the valuations g and h, respectively (observethat g∼ and h∼ are total homomorphisms, since A is a total Σ-algebra). Bydefinition one has fg = fh, and therefore, by Proposition 1.25, fg∼ = fh∼,but g∼ 6= h∼, i.e. injectivity is necessary for monomorphisms in TAlg(Σ).

Proposition 5.20 (In the categories PAlg(Σ) and TAlg(Σ) the epimorphismsare exactly the dense homomorphisms.)In PAlg(Σ) and TAlg(Σ) a homomorphism f : A → B is an epimorphism iff f isdense, i.e. iff the image of f generates B: CBf(A) = B. However, in TAlg(Σ) thismeans that the epimorphisms are exactly the surjective homomorphisms.

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A

A

f-

' $' $

B

f(A)

CBf(A) \ f(A)

E

-idCBf(A)

-idE

d

:

' $' $' $

D

f(A)

CBf(A) \ f(A)

E

E ′

u = idBD : B → Dv : B → D; graph v = graph idCBf(A) ∪ graph du f = v f

Figure 7: A non-dense homomorphism is not an epimorphism

Proof Let f : A→ B be dense and assume for u, v : B → D that uf = v f ; then,for all s ∈ S and for all a ∈ As, us(fs(a)) = vs(fs(a)). Since ( fs(a) | a ∈ A )s∈Sgenerates B, the principle of unique homomorphic extension implies that u = v,showing that f is an epimorphism.

For the nontrivial part of the proof assume that f is not dense (compare Figure 7):

Let E := B \CBf(A), and let E ′ be an S-set disjoint from B such that there existsa bijective mapping d : E → E ′. Choose such a fixed bijection and call it d. DefineD := B ∪ E ′, and v : B → D for s ∈ S and b ∈ Bs by

vs(b) :=

b, if b ∈ (CBf(A))s ,ds(b), if b ∈ Es .

Then v is an injective mapping onto E ′ ∪ CBf(A).Finally we provide D with an algebraic structure as follows: For ϕ ∈ Ω (with

(η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s)) set

domϕD := domϕB ∪ (vs1(b1), . . . , vsτ(ϕ)(bτ(ϕ))) | (b1, . . . , bτ(ϕ)) ∈ domϕB.

And if (d1, . . . , dτ(ϕ)) ∈ domϕD, then set:

ϕD(d1, . . . , dτ(ϕ)) :=

ϕB(d1, . . . , dτ(ϕ)), if di ∈ Bsi for

1 ≤ i ≤ τ(ϕ)vs(ϕ

B((vs1)−1(d1), . . . , (vsτ(ϕ))−1(dτ(ϕ)))), else.

Finally, set u = idBD : B → D, us(b) := b (s ∈ S, b ∈ Bs) and consider v as ahomomorphism from B into D. Then, obviously u 6= v (us(e) 6= vs(e) = ds(e) fore ∈ Es and s ∈ S), while u f = v f , showing that f is not an epimorphism andthat density is necessary for epimorphisms.

17th July 2002


Notation In what follows we will denote by

• Hom the class of all homomorphisms in the category PAlg(Σ),

• Epi the class of all epimorphisms in the category PAlg(Σ),

• Mono the class of all monomorphisms in the category PAlg(Σ),

• Iso the class of all isomorphisms in the category PAlg(Σ).

Remarks 5.21 Category theoretical descriptions of subobjects and equal-izers. (i): In category theory a subobject of an object A is a pair (B,m) consistingof an object B and a monomorphism m : B → A, and two subobjects (B,m) and(B′,m′) of A are said to be equivalent, iff there exists an isomorphism i : B → B′

such that m = m′ i. It is easy to see that each subobject of a partial algebra A inPAlg(Σ) is in this way equivalent to (B, idBA), where B is a weak relative subalgebraof A. However, we recall from Remark 5.1

• B is a weak subalgebra of A iff idBA is a monomorphism;

• B is a relative subalgebra of A iff idBA is a full monomorphism;

• B is a subalgebra of A iff idBA is a closed monomorphism.

(ii): While weak subalgebras can be described in a category theoretical setting as sub-objects, there is a corresponding category theoretical characterization of subalgebras(on closed subsets), namely as “equalizers” in PAlg(Σ):

Let C be any category, A,B ∈ Ob(C) and f, g : A → B morphisms from Mor(C).(Ef,g, ef,g) ∈ Ob(C)×Mor(C) is called an equalizer of f and g in C, iff fef,g = gef,g,and, for every C-morphism r : C → A with f r = gr, there is a unique C-morphismr′ : C → Ef,g such that ef,g r′ = r (draw a diagram).

Assume now that f, g : A → B are two homomorphisms in PAlg(Σ). Because ofProposition 1.25, one has for their equalizer (Ef,g, ef,g) in the category PAlg(Σ), thatEf,g is a subalgebra (on a closed subset) of A (exercise).59

Conversely, if C is a subalgebra of A in PAlg(Σ), then one may construct a partialΣ-algebra B on the disjoint union of A with A \ C in the sense of the proof ofProposition 5.20 (e.g. B := A∪(A×A\C, where we introduce a mapping v : A→ Bby setting

graph v := (c, c) | c ∈ C ∪ (a, (A, a)) | a ∈ A \ C;see also Figure 7) in order to realize that (C, idCA) is the equalizer of idA and v asdefined above.(iii): One could also say that the subalgebras represent the subobjects in the categorywith object class PAlg(Σ) and with the class of all closed homomorphisms as class of

59This observation is more or less repeated in Remark 7.11.

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6 Congruence relations, factor algebras, diagram completion I 108

morphisms, while then the relative subalgebras represent the subobjects in e.g. thecategory with object class PAlg(Σ) and with the class of all initial homomorphismsas class of morphisms.

6 Congruence relations, factor algebras, diagram

completion I

While the previous subsections are formulated for heterogeneous partial algebras,the following two subsections are formulated for homogeneous ones with only someremarks concerning the heterogeneous case.

6.1 Congruence relations and closed congruence relations

The formation of subobjects is one way to get new algebraic structures from a givenone. The dual way is to form factor algebras — as representatives of homomorphicimages. The key for an “internal description” of homomorphic images is the conceptof congruence relation.

Definition 6.1 Let A be a partial algebra of signature Σ. And let binary relationθ ⊆ A× A be a binary S-relation on A, i.e. (θs)s∈S ⊆ (As × As)s∈S.

• θ is called a congruence relation on A iff

– θ is an equivalence relation on A (i.e. each θs is an equivalence relation onAs (s ∈ S)), and

– θ is compatible with the fundamental operations of A, i.e. for all ϕ ∈ Ωand for all (ai, bi) ∈ Aη(ϕ)i × Aη(ϕ)i , 1 ≤ i ≤ τ(ϕ) one has:

if (ai, bi) ∈ θη(ϕ)i for all i ∈ 1, . . . , τ(ϕ),and if both (a1, . . . , aτ(ϕ)), (b1, . . . , bτ(ϕ)) ∈ domϕA ,then (ϕA(a1, . . . , aτ(ϕ)), ϕ

A(b1, . . . , bτ(ϕ))) ∈ θσ(ϕ).

Written in matrix notation (or as a rule)

(a1, . . . , aτ(ϕ)), (b1, . . . , bτ(ϕ)) ∈ dom ϕA

(a1, b1) ∈ θη(ϕ)1

...(aτ(ϕ), bτ(ϕ)) ∈ θη(ϕ)τ(ϕ)

(ϕA(a1, . . . , aτ(ϕ)), ϕA(b1, . . . , bτ(ϕ))) ∈ θσ(ϕ)

• θ is called a closed congruence relation on A, iff

– θ is a congruence relation on A; and

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– if (ai, bi) ∈ θsi (1 ≤ i ≤ τ(ϕ)) and, say, (a1, . . . , aτ(ϕ)) ∈ domϕA thenalways (b1, . . . , bτ(ϕ)) ∈ domϕA, too.

Remark 6.2 Observe that congruence relations on A are just the closed subsets ofAπA which are at the same time equivalence relations.

Examples 6.3 (Smallest and largest congruence relations, connections withnormal subgroups and ideals):

• Observe that always ∆A := ( (a, a) | a ∈ As )s∈S and 5A := A × A =(As × As)s∈S are congruence relations on A. .

• Let G = (G; (G, eG,−1G )) be a group, then θ ⊆ G×G is a congruence relationon G iff Nθ := g ∈ G | (g, eG) ∈ θ is a normal subgroup of G; and one then

has (g, h) ∈ θ iff g G h−1G ∈ Nθ.

• For a ring R = (R, (+R, 0R,−R, ·R)) a relation θ ⊆ R × R is a congruencerelation on R iff Jθ := r ∈ R | (r, 0R) ∈ θ is an ideal of R; and one has(r, s) ∈ θ iff r −R s ∈ Jθ.

As a preparation of the collection below of facts concerning (closed) congruencerelations we recall the following description of equivalence relations:

Definition 6.4 (of the product of binary relations): Let A be any (S-) set, andlet R1, R2 ⊆ A×A be two binary relations on A. Then the relational product R2 R1

is defined as the following binary relation:

R2R1 := (a, c) ∈ A×A | there exists b ∈ A such that (a, b) ∈ R1 and (b, c) ∈ R2 .

Iteratively one defines for a binary relation R ⊆ A× A its n-th powers as:

• R0 := 4A = ( (a, a) | a ∈ As s∈S ; this relation is also called the diagonal ofA× A.

• Rn+1 := R Rn , for each natural number n ≥ 1; and

• R−1 := (b, a) | (a, b) ∈ R is called the inverse of the relation R.

Moreover, set

R(+) :=∞⋃n=1

Rn

called the transitive closure of R, and

R(∗) :=∞⋃n=0

Rn,

called the reflexive and transitive closure of R.

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This terminology is motivated by the following

Lemma 6.5 Let A be any (S-) set, and R ⊆ A× A a binary relation on A. Then

(i) R(+) is the smallest transitive relation on A containing R. In particular onehas:

R(+) = ( (a, b) ∈ As × As | there are n ∈ N1 and ai ∈ As with (ai−1, ai) ∈ Rs

for 1 ≤ i ≤ n, and a0 = a, an = b )s∈S .

(ii) R(∗) is the smallest reflexive and transitive relation on A containing R.

(iii) (R ∪R−1)(∗) is the smallest equivalence relation on A containing R.


Lemma 6.6 and Definition (of kernels of homomorphisms) (The congru-ence relations of A form an algebraic closure system on A × A, and theclosed ones form a “principal ideal” of it.)Let A,B be any partial algebras of the same similarity type (Ω, τ). Then

(i) ∆A and 5A are congruence relations of A, and ∆A is always closed, while 5A

is closed, iff , for each ϕ ∈ Ω, ϕA is either total or has an empty graph (oneoperation may be empty, another one total for the same partial algebra).

(ii) Define on A× A a partial algebraic structure

A× A := ((As × As)s∈S; ((γAs×Asa )a∈As)s∈S, (δAs×As)s∈S, (ϕA×A)ϕ∈Ω)

as follows:

– γAs×Asa := (a, a), is a (total) nullary constant, for each a ∈ A, and for eachs ∈ S;

– δAs×As is a total binary operation on As×As defined for all (a, b), (c, d) ∈As × As as:

δAs×As((a, b), (c, d)) :=

(a, d) , if b = c ,(b, a) , otherwise (i.e. if b 6= c);

– for ϕ ∈ Ω with (η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s), and for (ai, bi) ∈ Asi ×Asi, for1 ≤ i ≤ τ(ϕ), one has60

((a1, b1), . . . , (aτ(ϕ), bτ(ϕ))) ∈ domϕA×A , iff (a1, . . . aτ(ϕ)) ∈ domϕA and(b1, . . . , bτ(ϕ)) ∈ domϕA . And if the sequence belongs to the domain ofϕA×A, then one has

ϕA×A((a1, b1), . . . , (aτ(ϕ), bτ(ϕ))) := (ϕA(a1, . . . aτ(ϕ)), ϕA(b1, . . . , bτ(ϕ))) .

60Observe that we repeat here for the convenience of the reader just the definition of the structureof the direct product of A with itself as given in Definition 1.26.

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Then the congruence relations on A are exactly the closed subsets of this newpartial algebra A× A (in the heterogeneous case the definitions.

(iii) In particular one has:

– If G is a set of congruence relations on A, then its intersection⋂G is also

a congruence relation on A; if G = Ø, then we set “as usual”⋂G := A×A.

– The set ConA of all congruence relations on a partial algebra A of signatureΣ forms an algebraic closure system, on A × A (= (As × As)s∈S); thecorresponding algebraic closure operator is denoted by ConA.

(iv) Let θ and θ′ be congruence relations on A, then one has:

(a) If θ′ ⊆ θ and if θ is closed, then θ′ is closed, too.

(b) If both, θ and θ′ are closed, then their supremum θ′′ := ConA(θ ∪ θ′) in(ConA;⊆) is closed, too.In particular (as for total algebras, but contrary to the general case of arbi-trary congruence relations of partial algebras), the supremum of two closedcongruence relations θ and θ′ is the least equivalence relation Ξ containingR := θ ∪ θ′, i.e.

ConA(θ ∪ θ′) = ConAR = R(∗) =∞⋃n=0

(θ′ θ)n ;

therefore

ConA(θ ∪ θ′) = (a, b) ∈ A× A | there are n ∈ N1 and ai ∈ A with

(a2i, a2i+1) ∈ θ and (a2i+1, a2i+2) ∈ θ′

for 1 ≤ i ≤ n, and a0 = a, a2n = b .

(c) There always exists a largest closed congruence relation θc,A in ConA, andeach congruence relation contained in it is closed.

In connection with the properties (a) and (b) above one says that “the closedcongruence relations form a principal ideal — denoted by ConcA — of thecongruence lattice ConA := (ConA;⊆)”.

(v) If f : A→ B is any homomorphism, then ker f := ( (a, a′) ∈ As×As | fs(a) =fs(a

′) )s∈S is a congruence relation on A, called the kernel of f .If f is closed, then ker f is a closed congruence relation.61

61If f is a full and surjective homomorphism (see Definition 5.12), and if ker f is a closed congruencerelation, then f is a closed homomorphism (see Lemma 6.10.(iii) below).

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Remark 6.7 (Action of the closure operator for congruence relations): Asalready mentioned, the statement of (ii) means that the set of congruence relationsof a partial algebra A is a closure system ConA on A×A. The corresponding closureoperator, denoted by ConA, is algebraic, i.e. satisfies for any θ ⊆ A× A

ConAθ =⋂Ψ | Ψ is a congruence relation on A and θ ⊆ Ψ

=⋃ConAθ′ | θ′ ⊆ θ and θ’ is finite .

Proof Ad (i): In each case the properties of a (closed) congruence relation can bechecked directly. If at least one partial operation is neither total nor discrete, thenthe closedness condition can easily be violated (exercise).

Ad (ii): Consider the partial algebra A× A, and let C be a closed subset ofit. Then C is a binary relation on A × A, which is reflexive, since it containsall nullary constants γA×Aa (a ∈ A); and it is symmetric, since (a, b) ∈ C impliesδA×A((a, b), (a, b)) = (b, a) ∈ C . Moreover, C is transitive, since (a, b), (b, c) ∈ Cimply δA×A((a, b), (b, c)) = (a, c) ∈ C . This shows that C is an equivalence relationon A. — Finally, closedness of C w.r.t. the partial operations ϕA×A for each ϕ ∈ Ωjust means that C satisfies the compatibility condition with fundamental operationsof A. This shows that C is indeed a congruence relation on A. It is similarly directto show that each congruence relation on A is a closed subset of A× A.

The proof of (iii) now follows from (ii), Lemma 1.15 and Proposition 5.2.Ad (iv)(a): If two sequences have components pairwise congruent w.r.t. θ′, then

they are pairwise congruent w.r.t. θ, and since θ is assumed to be closed, existence ofan operation on one sequence immediately implies the existence of that operation onthe other one.

Ad (iv)(b): Observe that⋃∞n=0(θ′ θ)n ⊆ ConA(θ′ θ) is obvious, since all con-

gruence relations are equivalence relations. — Assume ϕ ∈ Ω and ai, bi ∈ A with(ai, bi) ∈

⋃∞n=0(θ′ θ)n, for 1 ≤ i ≤ τ(ϕ) such that (a1, . . . , aτ(ϕ)) ∈ domϕA. Then

there are a natural number n (as maximum of the ni for the different pairs, observingthat θ′ θ contains the diagonal of A × A) and elements zki for 1 ≤ k ≤ τ(ϕ) and0 ≤ i ≤ 2n such that

(zk2i, zk2i+1) ∈ θ

and(zk2i+1, z

k2i+2) ∈ θ′ ,

zk0 = ak and zk2n = bk, for 1 ≤ k ≤ τ(ϕ) and 0 ≤ i ≤ n − 1. Since bothcongruences are closed, the fact that (a1, . . . , aτ(ϕ)) ∈ domϕA and therefore that

(z10 , . . . , z

τ(ϕ)0 ) ∈ domϕA implies (recursively) that (z1

i , . . . , zτ(ϕ)i ) ∈ domϕA , for

0 ≤ i ≤ 2n. This finally implies (b1, . . . , bτ(ϕ)) ∈ domϕA. Moreover, since θ and

θ′ are congruence relations this implies (ϕA(a1, . . . , aτ(ϕ)), ϕA(z1

1 , . . . , zτ(ϕ)1 )) ∈ θ and

(ϕA(z12k, . . . , z

τ(ϕ)2k ), ϕA(z1

2k+1, . . . , zτ(ϕ)2k+1)) ∈ θ, for 1 ≤ k < n and (ϕA(z1

2k−1, . . . , zτ(ϕ)2k−1),

ϕA(z12k, . . . , z

τ(ϕ)2k )) ∈ θ′ for 1 ≤ k ≤ n showing that (ϕA(a1, . . . , aτ(ϕ)), ϕ

A(b1, . . . , bτ(ϕ)))

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∈⋃∞n=0(θ′ θ)n, i.e. that

⋃∞n=0(θ′ θ)n is indeed a closed congruence relation. This

shows the other inclusion.Ad (iv)(c): Let ConcA := θ ∈ ConA | θ is closed be the set of all closed con-

gruence relations on A. Then, by (b), each congruence relation of A generated bythe union of a non-empty finite subset of ConcA is again closed. Let θc :=

⋃ConcA .

Obviously θc is reflexive and symmetric (since ∆A ∈ ConcA and all elements of ConcAare symmetric). Since transitivity can be obtained by the closedness w.r.t. the oper-ation δA×A, an argument similar to the following one yields transitivity of θc, i.e. θcis an equivalence relation. — Let ϕ ∈ Ω and (ai, bi) ∈ θc for 1 ≤ i ≤ τ(ϕ) , and let(a1, . . . , aτ(ϕ)) ∈ domϕA. There are θi ∈ ConcA with (ai, bi) ∈ θi , for 1 ≤ i ≤ τ(ϕ) .If τ(ϕ) = 0, nothing has to be shown. Therefore assume that τ(ϕ) ≥ 1. Then, by(b), the supremum, say θ∗ of θi | 1 ≤ i ≤ τ(ϕ) in the congruence lattice (ConA;⊆)is a closed congruence relation, and therefore (b1, . . . , bτ(ϕ)) ∈ domϕA and

(ϕA(a1, . . . , aτ(ϕ)), ϕA(b1, . . . , bτ(ϕ))) ∈ θ∗ ⊆ θc .

This shows that θc is a closed congruence relation on A.Ad (v): Since f(a) = f(a), since f(a) = f(b) implies f(b) = f(a), and since f(a) =

f(b) and f(b) = f(c) implies f(a) = f(c) (i.e. since equality is an equivalence relationon B), ker f is an equivalence relation on A. Let ϕ ∈ Ω, (a1, a

′1), . . . , (aτ(ϕ), a

′τ(ϕ)) ∈

ker f such that the corresponding applications of ϕA exist, then

f(ϕA(a1, . . . , aτ(ϕ))) = ϕB(f(a1), . . . , f(aτ(ϕ))) =ϕB(f(a′1), . . . , f(a′τ(ϕ))) = f(ϕA(a′1, . . . , a

′τ(ϕ))),

showing that ker f is indeed a congruence relation.If f is closed, then the facts that f preserves the structure and reflects existence

imply that ker f is closed.

Example 6.8 (The supremum of non-closed congruence relations is notnecessarily their equivalence theoretic supremum): In order to show that6.6.(iv)(b) does not hold for arbitrary congruence relations consider the followingcongruence relations θ1 and θ2 on the partial algebra A as depicted in Figure 8:A := a, b, c, d, e, Ω := ϕ, τ(ϕ) := 1, graph ϕA := (a, d), (c, e).

θ1 := ∆A ∪ a, b2, θ2 := ∆A ∪ b, c2,⋃∞n=0(θ1 θ2)n = ∆A ∪ a, b, c2, ConA(θ1 ∪

θ2) = a, b, c2 ∪ d, e2 6=⋃∞n=0(θ1 θ2)n.

The importance of closed homomorphisms and therefore of closed congruencerelations will become evident later in connection with model theoretic concepts andamong others with the description of special implicational classes, e.g. of what weshall call ECE-varieties (i.e. varieties defined by existentially conditional existenceequations (see section 8 and subsection 9.2 as well as the end of subsection 9.4)).

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e

e e e

e

d

a b c

e Θ1 :

Θ2 :

6 6

Figure 8: The equivalence generated by two congruences is in general no congruence

6.2 Factor algebras

The role of kernels of homomorphisms is described by the following definition andresults:

Definition 6.9 Let A be any partial algebra, and let θ be any congruence relationon A. Let A/θ := [a]θ | a ∈ A be the set of all equivalence classes [a]θ := b ∈A | (a, b) ∈ θ of elements of A, the so-called quotient set or factor set of A withrespect to θ. The mapping natθ : A→ A/θ, a 7→ [a]θ is called the natural mappingwith respect to θ. The fact that θ is not only an equivalence but also a congruencerelation allows to define on A/θ a partial algebraic structure (ϕA/θ)ϕ∈Ω, making natθa full and surjective homomorphism:

For ϕ ∈ Ω setdomϕA/θ := ([a1]θ, . . . , [aτ(ϕ)]θ) ∈ (A/θ)τ(ϕ) | there is (a′1, . . . , a

′τ(ϕ)) ∈ domϕA

such that (ai, a′i) ∈ θ for 1 ≤ i ≤ τ(ϕ) ,

and if (a1, . . . , aτ(ϕ)) ∈ domϕA, then set

ϕA/θ([a1]θ, . . . , [aτ(ϕ)]θ) := [ϕA(a1, . . . , aτ(ϕ))]θ .

In the case of heterogeneous partial algebras one obviously has to observe to choosethe correct sorts.

Lemma 6.10 and Notation (Factor algebras): With the notation and definitionsfrom above we have:

(i) The value of ϕA/θ([a1]θ, . . . , [aτ(ϕ)]θ) is independent from the choice of the rep-resenting sequence, i.e. ϕA/θ is indeed a τ(ϕ)-ary partial operation on A/θ.

(ii) natθ : A→ A/θ is a full and surjective homomorphism natθ : A→ A/θ.

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A B--f

C?

g (∃h)h f = g ⇔ ker f ⊆ ker g

Figure 9: First Diagram Completion Lemma

(iii) If θ is a closed congruence relation, then natθ is a closed homomorphism.

A/θ is called the factor algebra or quotient algebra of the partial algebra A withrespect to the congruence relation θ, and natθ is called the natural homomorphismor natural projection from A onto A/θ.

Observe that in a many-sorted case one gets — as it should be — a factor algebraA/θ = ((As/θs)s∈S, (ϕ

A)ϕ∈Ω) on an S-set.

Proof Ad (i): The properties of a congruence relation imply directly that forcongruent sequences (a1, . . . , aτ(ϕ)), (a

′1, . . . , a

′τ(ϕ)) ∈ domϕA (i.e. (ai, a

′i) ∈ θ for

1 ≤ i ≤ τ(ϕ)) one has (ϕA(a1, . . . , aτ(ϕ)), ϕA(a′1, . . . , a

′τ(ϕ))) ∈ θ (i.e. both values

define the same element in A/θ).Ad (ii): The definition of A/θ and ϕA/θ for ϕ ∈ Ω implies immediately that natθ is

a full and surjective homomorphism from A onto A/θ. — The proof of the statementconcerning closedness is left as an exercise.

6.3 Diagram Completion Lemma for mappings and full andsurjective homomorphisms

More about the importance of the factor algebras of A as distinguished representativesof full homomorphic images of A (i.e. of full and surjective homomorphisms startingfrom A) can be derived from the following Diagram Completion Lemma for fulland surjective homomorphisms:

Lemma 6.11 First Diagram Completion Lemma (full and surjective homo-morphisms)

(i) Let f : A → B be a surjective and g : A → C be an arbitrary mapping. Thenthere exists a mapping h : B → C such that g = h f , iff ker f ⊆ ker g (seeFigure 9).

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(ii) If, in addition, f is a full (and surjective !) homomorphism, f : A →full B,and if g : A→ C is a homomorphism, then h is a homomorphism, h : A→ C,whenever it exists.

Let h exist (i.e. let ker f ⊆ ker g be true). Then:

(iii) h is uniquely determined by f and g.

(iv) h is injective, iff ker f = ker g.

(v) h is surjective, iff g is surjective.

Let f : A→ B be a full and surjective homomorphism, let g : A→ C be a homomor-phism, and let h exist. Then:

(vi) h is full, iff g is full.

(vii) If g is closed, then h is closed. If f is closed, then h is closed iff g is closed.

(viii) h is an isomorphism iff ker f = ker g and g is a full and surjective homomor-phism.

Proof In what follows let us assume for each item the notation and assumption fromthe lemma without mentioning it again.

Ad (i): Let us first assume that h : B → C exists and satisfies g = h f . Consider(a, a′) ∈ ker f . Then g(a) = (h f)(a) = h(f(a)) = h(f(a′)) = (h f)(a′) = g(a′), i.e.(a, a′) ∈ ker g, showing that ker f ⊆ ker g.

Conversely, assume ker f ⊆ ker g; and assume b ∈ B. Since f is surjective, wehave an a ∈ A with b = f(a). Since we want g = h f to hold, we have to defineh(b) := g(a) (whenever f(a) = b). The assumption “ker f ⊆ ker g” guarantees that his well defined: If, in addition, b = f(a′), then (a, a′) ∈ ker f ⊆ ker g, and thereforealso g(a′) = g(a) = h(b).

Ad (ii): Let h exist and consider (b1, . . . , bτ(ϕ)) ∈ domϕB for some ϕ ∈ Ω. Since

f is full and surjective, there is (a1, . . . , aτ(ϕ)) ∈ domϕA such that f(ai) = bi for1 ≤ i ≤ τ(ϕ). Since g is a homomorphism, we get (g(a1), . . . , g(aτ(ϕ))) ∈ domϕC andϕC(h(b1), . . . , h(bτ(ϕ))) = ϕC(g(a1), . . . , g(aτ(ϕ))) = g(ϕA(a1, . . . , aτ(ϕ))) = h(f(ϕA(a1,. . . , aτ(ϕ)))) = h(ϕB(b1, . . . , bτ(ϕ))).

Ad (iii): This follows from the fact that f (here considered just as a mapping) is anepimorphism in the category with all sets as objects and all mappings as morphisms.

Ad (iv): Assume first, that h is injective, and let (a, a′) ∈ ker g, i.e. h(f(a)) =g(a) = g(a′) = h(f(a′)). Since h is injective, this implies f(a) = f(a′), and thereforeker g ⊆ ker f , while we already know ker f ⊆ ker g.

Conversely, let ker f = ker g be true, and assume h(b) = h(b′) for any b, b′ ∈ B.Let b = f(a) and b′ = f(a′). Then g(a) = h(b) = h(b′) = g(a′), showing (a, a′) ∈ker g = ker f , implying b = f(a) = f(a′) = b′, i.e. the injectivity of h.

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Ad (v): Since the composition of surjective mappings is surjective, surjectivity ofh implies the one of g. If g is surjective, consider c ∈ C. Then c = g(a) = h(f(a)) forsome a ∈ A, showing that h is surjective.

Ad (vi): Assume first that h is full, and consider, for some ϕ ∈ Ω with (η, σ)(ϕ) =:

(s1 . . . sn, s), (a1, . . . , an) ∈ Aη(ϕ) such that (gs1(a1), . . . , gsn(an)) ∈ domϕC . Sincegsi(ai) = hsi(fsi(ai)) for 1 ≤ i ≤ n, and since h is full, there is (b1, . . . bn) ∈ domϕB

such that hsi(bi) = hsi(fsi(ai)) for 1 ≤ i ≤ n. Since f is full and surjective, there is(a′1, . . . a

′n) ∈ domϕA such that fsi(a

′i) = bi for 1 ≤ i ≤ n, i.e. hsi(fsi(a

′i) = hsi(bi) =

gsi(ai), showing that g is full.If, conversely, g is full and (b1, . . . , bn) ∈ Bη(ϕ) such that (hs1(b1), . . . , hsn(bn)) ∈

domϕC , then surjectivity of f , fullness of g, h f = g and the fact that f is ahomomorphism imply in a similar way that one has a sequence (b′1, . . . , b

′n) ∈ domϕB

with hsi(bi) = hsi(b′i) for 1 ≤ i ≤ n, showing that h is full.

(vii) follows from Proposition 5.14.(ii) and (iv) and the fact that f is surjective.Ad (viii): If ker f = ker g and if g is surjective, then h is bijective according to

(iv) and (v). (vi) implies that h is full, whenever g is full, hence Proposition 6.11 (iii)implies that h is an isomorphism, whenever g is full and surjective and ker f = ker gholds. If h is an isomorphism, then clearly g has the same properties as f .

Remarks 6.12 The assumption of surjectivity in (i) and (ii) of the Diagram Com-pletion Lemma above can be weakened, if one only wants the existence of the inducedmapping or morphism, and not its uniqueness. Yet then other, weaker assumptionshave to be made:

(i) In the case of S-mappings the induced mapping h always exists for ker f ⊆ ker g,if one requires:

For each sort s ∈ S one has: If Bs 6= Ø, then Cs 6= Ø . (5)

If S is infinite, then it may be that, in order to prove the existence of h, oneneeds the Axiom of Choice, which says that for any family (Ai)i∈I of non-empty sets Ai one always has a mapping c : I →

⋃i∈I Ai such that c(i) ∈ Ai ,

for every i ∈ I. I.e. c simultaneously chooses an element c(i) from Ai for eachi ∈ I. Observe that there is no algorithm included, how to construct such amapping.

(ii) In the case of homomorphisms, if the requirement of surjectivity is dropped,one can find a condition on the structure of B such that h is a homomorphism,whenever it exists as a mapping with hf = g. Obviously this is a situation dualto the one of the definition of initial homomorphisms in the sense of Bourbaki(cf. footnote 58). Therefore such a homomorphism f will be called — againfollowing Bourbaki — a final homomorphism:

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A homomorphism f : A→ B is called final, iff for every S-mapping h : B → Cinto a partial algebra C of signature Σ, h is a homomorphism h : B → C, iffh f is a homomorphism h f : A→ C .

Then it is not difficult to realize (exercise) that a homomorphism f : A→ B isfinal, iff

For each ϕ ∈ Ω ((η, σ)(ϕ) =: (s1 . . . sτ(ϕ), s)) one has:

graphϕB = (fs1 , . . . , fsτ(ϕ), fs)(graphϕA) . (6)

This means that f is not only full, but that B carries only the least structurewhich is sufficient to allow f to be a homomorphism. This structure is calledthe final structure on B w.r.t. f and A. For an arbitrary mapping f : A→ Bstarting from a partial algebra A of signature Σ B can always be provided withsuch a final structure w.r.t. f and A, if (and only if) ker f is a congruencerelation on A.

It may be that even without such a requirement on the structure of B somemappings h : B → C with h f = g may be homomorphisms, but it cannot beguaranteed, and it will not be true for all mappings from A into B.

(iii) One always has a diagram completion theorem dual to Lemma 6.11:

(i)d Let f : A → B and g : C → B be any S-mappings. Then there existsan S-mapping h : C → A such that f h = g, iff g(C) ⊆ f(A) (i.e.gs(Cs) ⊆ fs(As) for each s ∈ S). If f is not injective, then this will also ingeneral need the Axiom of Choice.

(ii)d Let f : A→ B and g : C → B be any homomorphisms between partial Σ-algebras. If f is an initial homomorphism, then, if there exists a mappingh : C → A such that f h = g, then h is always a homomorphism. Onethen says that A carries the initial structure w.r.t. f and B.

In this connection we want to mention that a homomorphism f : A→ B isinitial, iff its “codomain restriction” f ′ : A → f(A) with the same graphas f but onto the relative subalgebra of B supported by f(A) is closed andeach value of some partial operation ϕf(A) has exactly one preimage (i.e.one has a restricted injectivity, exercise).

Lemma 6.11 (viii) now shows that the natural homomorphism natθ : A → A/θwith respect to a congruence relation θ on A is determined by θ up to unique iso-morphism as a full and surjective homomorphism f : A → B with ker f = θ (thisfact is often called the Homomorphism Theorem). Of great importance is alsothe following consequence of Lemma 6.11:

Lemma 6.13 Factorization Lemma for full and surjective homomorphismsand monomorphisms

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A B

A/ ker f

D′

f

q

full,surj.

m

mono

q′

full,surj.

m′

mono

∼= i

-

QQQQQs

3A

AAAAAAAAAAAAAAU ?

Figure 10: Factorization Lemma

Let f : A → B be any homomorphism. Then there are up to unique isomorphism afull and surjective homomorphism q : A→ D and a monomorphism m : D → B suchthat f = m q (cf. Figure 10).

Proof Let q := natker f : A→ A/ ker f ; then q is a full and surjective homomorphism.Thus, according to Lemma 6.11 and the fact that ker f = ker q there is an injectivehomomorphism m : A/ ker f → B (i.e. a monomorphism because of Proposition 6.10)such that f = m q. Let f = m′ q′ be another fatorization of f , where m′ : D′ → Bis a monomorphism and q′ : A → D′ is a full and surjective homomorphism (seeFigure10).

Since both, m and m′ are injective, we have ker q = ker q′ = ker f , since both,q and q′ are full and surjective, Lemma 6.11 (viii) implies the existence of a uniqueisomorphism i : A/ ker f → D′ such that iq = q′. This yields f = m′q′ = m′iq =m q. Since q is an epimorphism, we get also m′ i = m.

Definition 6.14 Let A and B be partial algebras of the same signature Σ. Then Bis called a

• (weak) homomorphic image of A, if there exists a surjective homomorphismf : A→ B from A onto B;

• full (or strong) homomorphic image of A, if there exists a full and surjectivehomomorphism f : A→ B from A onto B;

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• closed homomorphic image of A, if there exists a closed and surjectivehomomorphism f : A→ B from A onto B.

Moreover, let K be any class of partial algebras of signature Σ. Then we set

• HK := HwK := B ∈ PAlg(Σ)| there are A ∈ K and a surjective homomorphismf : A→ B = class of all (weak) homomorphic images of K-algebras.

• HfK := B ∈ PAlg(Σ)| there are A ∈ K and a full and surjective homomorphismf : A→ B = class of all full homomorphic images of K-algebras.

• HcK := B ∈ PAlg(Σ)| there are A ∈ K and a closed and surjective homomor-phism f : A→ B = class of all closed homomorphic images of K-algebras.

• IK := B ∈ PAlg(Σ)| there are A ∈ K and an isomorphism f : A → B =class of all isomorphic copies of K-algebras.

• SK := ScK := A ∈ PAlg(Σ)| there are B ∈ K and a closed and injectivehomomorphism f : A→ B = class of all isomorphic copies of (closed) sub-algebras of K-algebras.

• SfK := SrK := A ∈ PAlg(Σ)| there are B ∈ K and a full and injective homo-morphism f : A→ B = class of all isomorphic copies of relative subalgebrasof K-algebras.

• SwK := A ∈ PAlg(Σ)| there are B ∈ K and an injective homomorphismf : A→ B = class of all isomorphic copies of weak relative subalgebras ofK-algebras.

• Epi(Σ) := class of all epimorphisms in the category PAlg(Σ).

• Epi(Σ)surj := class of all surjective epimorphisms in the category PAlg(Σ).

• Epi(Σ)surj,full := class of all full and surjective epimorphisms in the categoryPAlg(Σ).

• Mono(Σ) := class of all monomorphisms in the category PAlg(Σ).

• Mono(Σ)full :=Mono(Σ)initial := class of all full (= initial) monomorphisms inthe category PAlg(Σ).

• Mono(Σ)closed := class of all closed monomorphisms in the category PAlg(Σ).

• Closed(Σ) := class of all closed homomorphisms in the category PAlg(Σ).

In the case of total algebras the three kinds of homomorphic images defined abovecoincide, since then each homomorphism is closed and therefore full.

The First Diagram Completion Lemma now allows us to characterize full homo-morphic images. However, this also includes — in connection with Proposition 6.6— a characterization of closed homomorphic images.

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6.4 Isomorphism theorems

The First Diagram Completion Lemma has many interesting consequences. Amongthem are the so-called “isomorphism theorems”. Yet before we can formulate them,we need some definitions and notation.

Definition 6.15 (of quotients of congruence relations, the saturation of sub-sets w.r.t. congruence relations, and the restriction of congruence rela-tions):Let A be any partial algebra of signature Σ, and with carrier set A = (As)s∈S.

(i) Let Θ,Ψ ∈ ConA be any two congruence relations satisfying Θ ⊆ Ψ. Define asquotient of congruences:

Ψ/Θ := ((Ψ/Θ)s)s∈S := ( ([a]Θs , [b]Θs) | (a, b) ∈ Ψs )s∈S (⊆ (A/Θ)2 . (7)

(ii) Let Θ ∈ ConA be any congruence relation, and let B ⊆ A be any S-subset.Define the saturation of B w.r.t. Θ as the set

Θ[B] := (⋃b∈Bs

[b]Θs)s∈S . (8)

By Θ[B] we designate the relative subalgebra of A with carrier set Θ[B].

(iii) Let Θ ∈ ConA be any congruence relation, and let B ⊆ A be any S-subset.Define the restriction of Θ to the relative subalgebra B of A on B as

Θ|B := Θ|B := Θ ∩ (B ×B) = (Θs ∩ (Bs ×Bs))s∈S . (9)

Corollary 6.16 With the notation introduced in the above Definition one always has:

(i) Let Θ ⊆ Ψ in ConA , then Ψ/Θ ∈ ConA/Θ .

(ii) Let Θ ∈ ConA be any congruence relation, and let B be any relative subalgebraof A. Then Θ|B ∈ ConB .

The straight forward proofs are left as an exercise.

Now one can derive from Lemma 6.11 (as an exercise) the following two results:

Theorem 6.17 (Analogon of the First Isomorphism Theorem of E. Noe-ther.):Let A be any partial algebra of signature Σ, let Θ ∈ ConA be any congruence relation,and let B be any relative subalgebra of A (cf. Figure 11).

(i) Then there exists a bijective homomorphism i : B/Θ|B → Θ[B]/Θ|Θ[B] suchthat

i natΘ|B = natΘ|Θ[B] idB,Θ[B] .

(ii) If B is a subalgebra, and if Θ is even a closed congruence relation on A, then iin (i) is an isomorphism.

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B/Θ|B Θ[B]/Θ|Θ[B]-ex. i bij.

B Θ[B]-idB,Θ[B]

?

natΘ|B

?

natΘ|Θ[B]

Figure 11: 1st Isomorphism Theorem

A/Ψ (A/Θ)/(Ψ/Θ)-ex. i ∼=

A A/Θ-natΘ

?

natΨ

?

natΨ/Θ

Figure 12: 2nd Isomorphism Theorem

Theorem 6.18 (Analogon of the Second Isomorphism Theorem of E. Noe-ther.):Let A be any partial algebra of signature Σ, and let Θ,Ψ ∈ ConA be any two congru-ence relations satisfying Θ ⊆ Ψ. Then there exists an isomorphism (cf. Figure 12)

i : A/Ψ→ (A/Θ)/(Ψ/Θ)

such thati natΨ = natΨ/Θ natΘ .

Another isomorphism theorem following from the First Diagram Completion Lemmaconcerns the lattices of congruence relations and closed congruence relations of quo-tients of partial algebras.

Definition 6.19 (Notation for the principal ideal of closed congruences:)Let A be a partial algebra of signature Σ. We have already denoted the set of allcongruence relations on A by ConA. Together with its order structure, by which itbecomes a complete lattice, we denote it by ConA := (ConA;⊆).

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b

b

b

b

b'

&

$

%

PPPPPPPPPPq

∆A

∇A

ker q

qC

ConA

∆B

∇B

ConB

b

b

b

b

b'

&

$

%

PPPPPPPPPPq

∆A

∇A

ker q

qC

ConcA

ConA

∆B

∇B

ConcB

ConB

q is closed

. . . . . . . . . . . .

. . . . . .

Figure 13: Effect of a (closed) quotient homomorphism on the (closed) congruences

Similarly, recalling62 that we denote the “principal ideal” of all closed congruencerelations of A by ConcA, we want to denote it together with its order structure byConcA := (ConcA;⊆).

Theorem 6.20 (The congruence lattice of a quotient algebra is isomorphicto the lattice of all congruences of the start object containing the kernel.)Let q : A → B be any full and surjective homomorphism between partial algebras ofsignature Σ. Then one has (see Figure 13)

(i) q induces an isomorphism

qC : ([ker q,∇A];⊆) := (Θ ∈ ConA | ker q ⊆ Θ ; ⊆)→ ConB

Θ 7→ Θ/ ker q . (10)

(ii) If q is closed, then it also induces an isomorphism

qC : ([ker q, θc,A];⊆) := (Θ ∈ ConcA | ker q ⊆ Θ ⊆ θc,A ; ⊆)→ ConcB

Θ 7→ Θ/ ker q . (11)

Proof As an example for the argumentation with the First Diagram CompletionLemma 6.11 we sketch the proof of part (i), while the one of part (ii) is again left asan exercise (one should use Lemma 6.11.(vii)):

Assume that Γ ∈ ConB. Then ker q ⊆ ker(natΓ q), and obviously qC(ker(natΓ q)) =Γ by the uniqueness statement in Lemma 6.11. Thus qC is surjective.

Moreover, qC is order preserving, since, for ker q ⊆ Θ ⊆ Ψ we have natΨ =nat(Ψ/ ker q)/(Θ/ ker q) natΘ/ ker q q . Therefore one has by Lemma 6.11 Θ/ ker q ⊆

62See Lemma 6.6.(iv).

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A B

Cod (e)

Cod (e′)

f

e

∈ E

m

∈M

e′

∈ Em′

∈M

∼= i

-

QQQQQs

3A

AAAAAAAAAAAAAAU ?

Figure 14: Uniqueness of a factorization

Ψ/ ker q , i.e. qC(Θ) ⊆ qC(Ψ) . — Conversely, if Γ ⊆ Ξ in ConB, then there is — byLemma 6.11 — a homomorphism, say h′ : B/Γ → B/Ξ with h′ natΓ = natΞ , whatshows that

ker(natΓ q) ⊆ ker(h′ natΓ q) = ker(natΞ q).Therefore one has for Θ, Psi ∈ ConA which contain ker q:

Θ ⊆ Ψ iff qC(Θ) ⊆ qC(Ψ) ,showing — altogether — that qC : Θ 7→ Θ/ ker q is a bijective order isomorphism(and therefore also a lattice isomorphism).

6.5 Factorization systems

Remark 6.21 Lemma 6.13 shows us, as we shall realize below, that the pair con-sisting of the class Epifull,surj(Σ) of all full and surjective homomorphisms (= full andsurjective epimorphisms) and the class Mono(Σ) of all injective homomorphisms (=monomorphisms) in the category PAlg(Σ) is an instance of what we shall define belowas a factorization system. Such factorization systems are useful in many respectsfor our category PAlg(Σ). As we shall see below, they relate important properties ofhomomorphisms like “surjective”, “full and surjective”, “epimorphic” and others notyet defined, to properties of homomorphisms defined by the reflection of certain kindsof existence equations like “initial and injective”, “injective”, “closed and injective”and “closed”. Moreover, this concept is closely related to the one of reflection ofimplications by a homomorphism, as we shall see later.

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A B-m

P C-e

?

p

?

q 7→

A B-m

P C-e

?

p

?

qd

Figure 15: Diagonal-fill-in-property

Definition 6.22 Let C be an arbitrary category.

(i) Let E ,M ⊆ Mor(C) be arbitrary classes of morphisms in C The pair (E ,M) issaid to form a factorization system in C (and E is sometimes called the leftfactor (or “epi”-factor) andM is sometimes called the right factor (or “mono”-factor), if the following conditions are satisfied:

(FS1) M E = Mor(C),

(FS2) MM ⊆M, E E ⊆ E ,

(FS3) Iso(C) ⊆M∩ E ,63

(FS4) The factorization of any morphism into an E-morphism followed by anM-morphism is unique up to unique isomorphism, i.e. if f = m e =m′ e′ with e, e′ ∈ E , m,m′ ∈ M, then there is a unique isomorphismi from Cod (e) onto Cod (e′), such that e′ = i e and m = m′ i (seeFigure 14).

(ii) A pair (e,m) of morphisms e : P → C, m : A→ B has the unique-diagonal-fill-in-property, i.e. it satisfies Difip(e,m), if and only if for all morphismsp : P → A and q : C → B satisfying q e = m p there exists a uniquemorphism d : C → A such that d e = p and m d = q (see Figure 15).

Proposition 6.23 Let (E ,M) be a factorization system in the category C. Then:

(i) E ∩M = IsoC.

(ii) E E = E and M M =M.

(iii) If p, p′ : A → B are C-morphisms such that p e = p′ e and m p = m p′ forsome e ∈ E and m ∈M, then p = p′.

(iv) E ×M ⊆ Difip.

63Here Iso(C) designates the class of all isomorphisms of the category C.

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Proof Ad (i): Let f : A→ B, f ∈ E ∩M. Since IsoC ⊆ E ∩M and 1A,1B ∈ IsoC,(1A, f) and (f,1B) are (E ,M)-factorizations of f ; thus there exists by (FS4) a uniqueisomorphism i : B → A such that i f = 1A and f i = 1B, i.e. f is an isomorphism.In connection with (FS2) we get IsoC = E ∩M.

Ad (ii): Because of (FS1) and (FS2), (FS3) implies E ⊆ IsoC E ⊆ E E ⊆ E , andthe same for M; therefore (ii) follows.

Ad (iii): Let the assumptions of (iii) be satisfied, and let (e∗,m∗) and (e′,m′)be (E ,M)-factorizations of p and p′, respectively. By (FS3) e∗ e, e′ e ∈ E andm m∗,m m′ ∈M, and the assumptions implym∗ (e∗ e) = m′ (e′ e), (m m∗) e∗ =(m m′) e′ and, as a consequence, (m m∗) (e∗ e) = (m m′) (e′ e); hence (FS4)implies the existence of unique isomorphisms d, d′, d′′ : Cod(e∗) → Cod(e′) (draw afigure) such that

(a) d (e∗ e) = e′ e and m′ d = m∗;(b) d′ e∗ = e′ and (m m′) d′ = m m∗;(c) d′′ (e∗ e) = e′ e and (m m′) d′′ = m m∗.

From (a) and (b) we get

(d) d (e∗ e) = e′ e and (m m′) d = m m∗

(e) d′ (e∗ e) = e′ e and (m m′) d′ = m m∗.

The uniqueness of the isomorphisms in each case yields that (d) and (c) imply d′′ = d,while (e) and (c) imply d′ = d′′. Hence (a) and (b) imply m′ d = m∗ and d e∗ = e′,thus p = m∗ e∗ = (m′ d) (d−1 e′) = m′ e′ = p′.

Ad (iv): Assume e ∈ E , m ∈ M, let p, q ∈ MorC such that q e = m p, andlet (e′,m′) and (e′′,m′′) be (E ,M)-factorizations of p and q, respectively. Since nowm′′ (e′′ e) = (m m′) e′, (FS3) and (FS4) imply the existence of a unique isomor-phism i satisfying i (e′′ e) = e′ and (m m′) i = m′′. Define d := m′ i e′′; thend e = m′e i e′′ e = m′ e′ = p and m d = m m′ i e′′ = m′′ e′′ = q, i.e. d is adiagonal-fill-in for e and m. If we have another diagonal-fill-in d′ satisfying d′ e = pand m d′ = q, then (iii) implies d = d′, showing the uniqueness, i.e. Difip(e,m).

Remark 6.24 6.23.(iii) shows that even without requiring E ⊆ EpiC orM⊆ MonoC

– which would immediately imply the uniqueness of the diagonal-fill-in in 6.23.(iv)– this uniqueness can be proved, when (E ,M) is a factorization system. Moreover,the pairs (e,m) ∈ E × M share properties of epimorphisms and monomorphisms.In what follows we shall have in most cases E ⊆ EpiΣ, but the factorization system(final homomorphisms, bijective homomorphisms) (exercise) shows that this is notnecessary, while the factorization system (TAlg(Σ)-extendable epimorphisms, closedhomomorphisms), which we shall introduce later,64 and which is the crucial one in

64See Remark 8.13 for the definition of TAlg(Σ)-extendable epimorphisms, and Lemma 8.19 fortheir characterization.

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connection with the interpretations, shows that the “mono”-factor need not consistof monomorphisms. In [Pa79], Beispiel 1.12 and Beispiel 1.13, A. Pasztor has givenexamples of factorization systems in categories C, where neither the “epi”-factor iscontained in EpiC, nor is the “mono”-factor contained in MonoC.

Lemma 6.25 (E ,M) is a factorization system of a category C, iff (FS1), (FS2),(FS3) and (FS5) are true, where

(FS5) E ×M ⊆ Difip.

Proof In Proposition 6.23.(iv) we have already shown that a factorization systemsatisfies (FS5). It remains to show that (FS1), (FS2), (FS3) and (FS5) imply (FS4):

Assume E ×M ⊆ Difip. Let (e,m), (e′,m′) ∈ E×M be two (E ,M)-factorizationsof f : A → B, i.e. m e = m′ e′ = f . Choose p := e′, q := m . Then Difip(e,m′)implies the existence of d : Cod(e)→ Dom(m′) such that d e = e′ and m′ d = m . Bychoosing p′ := e, q′ := m′ , Difip(e′,m) implies the existence of d′ : Dom(m′)→ Cod(e)such that d′ e′ = e and m d′ = m′ . This gives us

(d′ d) e = d′ e′ = e = 1Cod(e)e

andm (d′ d) = m′ d = m = m 1Cod(e)

.

Then Proposition 6.23.(iv) implies that d′ d = 1Cod(e). In a similar way one gets

d d′ = 1Dom(m′). Therefore d and d are isomorphisms inverse to each other, and

(FS4) has been proved.

Definition 6.26 (The Galois connection induced by Difip):Now, Difip ⊆ Mor(C)×Mor(C) is a binary relation and therefore it defines a Galoisconnection between the subclasses of Mor(C). In particular one has, as usual, twooperators, denoted here by Λ and Λop, assigning to any classes E ,M ⊆ Mor(C) newclasses

Λ(E) := m ∈ Mor(C) | Difip(e,m) for all e ∈ E , (12)

and

Λop(M) := e ∈ Mor(C) | Difip(e,m) for all m ∈M . (13)

Since we shall meet Galois connections more often, and since they play an impor-tant role within mathematics and in applications, we give the basic definitions andproperties in a more general setup:

Let G and M be any sets or classes, and let R ⊆ G ×M be any relation. Thenone defines as above two operators ′ and o assigning to each subclass H of G a class

H ′ := m ∈M | (g,m) ∈ R for all g ∈ H , (14)

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and assigning to each subclass N of M a class

N o := g ∈ G | (g,m) ∈ R for all m ∈ N . (15)

Then the pair (′, o) of operators forms a Galois connection, i.e. it has the properties(GC1) and (GC2) of the following lemma.

Lemma 6.27 Let G and M be two sets, R ⊆ G × M a relation and ′ and o theoperators of the corresponding Galois connection as defined in (14) and (15). Asusual, one then has (proof as an exercise) for any subclasses H,H1, H2 ⊆ G andN,N1, N2 ⊆M :

(GC1) H1 ⊆ H2 implies (H2)′ ⊆ (H1)′, andN1 ⊆ N2 implies (N2)o ⊆ (N1)o (antimonotony).

(GC2) H ⊆ H ′o andN ⊆ N o′ (extensity of the composition).

(GC3) H ′o′ = H ′ , andN o′o = N o .These properties yield in particular H ′o′o = H ′o , andN o′o′ = N o′ .Therefore ′o and o′ are idempotent operators. Since one also has by (GC1) that

(GC4) H1 ⊆ H2 implies (H1)′o ⊆ (H2)′o , andN1 ⊆ N2 implies (N1)o′ ⊆ (N2)o′ , we have monotony of the operators ′o and o′.This shows that

(GC5) ′o and o′ always have the properties of closure operators on the classes underconsideration (if these are sets, then these operators are indeed closure operatorsin the usual sense).65

One then has the corresponding relations for the operators Λ and Λop

Proposition 6.28 (Relations between the classes in a factorization system):If (E ,M) is a factorization system in a category C, thenM = Λ(E) and E = Λop(M).

Proof Assume f ∈ Λ(E)) , and let m e = f = f 1Dom(f)for (e,m) ∈ E ×M .

Since one has Difip(e, f), there exists a d : Cod(e)→ Dom(f) (draw a diagram), suchthat d e = 1Dom(f)

and f d = m . Hence

(e d) e = e (d e) = e 1Dom(f)= e = 1Cod(e)

e ,

65In Formal Concept Analysis one usually calls the triple (G,M,R) a formal context. There therelation is usually denoted by I and both operators are denoted by the same symbol ′. The pairs(H ′o, H ′) for arbitrary H ⊆ G are usually called formal concepts.

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andm (e d) = (m e) d = f d = m = m 1Cod(e)

.

Proposition 6.23.(iii) then implies e d = 1Cod(e), showing together with d e =

1Dom(f)that e is an isomorphism, i.e. f = m e ∈ M IsoC ⊆ M because of (FS2)

and (FS3). Hence Λ(E) ⊆ M. (FS5) implies M ⊆ Λ(E) ; therefore equality holds.Λop(M) = E is proved in a similar (dual) way.

In analogy to the cancellation properties of properties of homomorphisms definedby the reflection of formulas we have

Lemma 6.29 (Cancellation properties of morphisms in the “epi”-factor ofa factorization system):Let (E ,M) be a factorization system in a category C. Then one has:

(i) If e′ e, e ∈ E, then e′ ∈ E;

(ii) if m′ m, m′ ∈M, then m ∈M.

Proof Ad (i): Let e′ e, e ∈ E . Consider m ∈ M and p, q ∈ Mor(C) such thatq e′ = m p . Then one also has q (e′ e) = m (p e) . Since e′ e ∈ E , there existsd : Cod(e′) → Dom(m) such that d (e′ e) = p e and m d = q . Therefore we alsohave

m (d e′) = q e′ = m p

and(d e′) e = p e .

Since e ∈ E and m ∈ M , Proposition 6.23.(iii) implies that d e′ = p . This showsthat d is also a diagonal-fill-in for e′ and m. Again with Proposition 6.23.(iii) one canshow that d is unique (by using this time e′ e ∈ E ).

(ii) is shown in a similar (dual) way.

We have introduced the concept of factorization systems, since in the categoryPAlg(Σ) we have a wealth of factorization systems, and in particular such systems,where the “mono”-factor consists of a class of homomorphisms defined by the reflec-tion of some set of elementary implications (often even existence equations). So farwe can prove this for the following cases:

Proposition 6.30 (Some factorization systems in the category of partialalgebras): In the category PAlg(Σ) we have the following factorization systems (cf.Definition 6.14):

(i) (Epi(Σ)surj,full, Mono(Σ));

(ii) (Epi(Σ)surj, Mono(Σ)full) = (Epi(Σ)surj, Mono(Σ)initial);

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7 Some constructions of partial algebras 130

(iii) (Epi(Σ), Mono(Σ)closed) .

Proof (and Remarks): Ad (i): This is an immediate consequence of the Factoriza-tion Lemma 6.13 and earlier results concerning closure properties w.r.t. composition.We want to observe that there are actually two distinguished (Epi(Σ)surj,full, Mono(Σ))-factorizations of a given homomorphism f : A→ B :

e := natker f : A→ A/ ker f in Epi(Σ)surj,full and the induced monomorphism m ;(16)

and

e′ := f ′ : A→ f(A) in Epi(Σ)surj,full and idf(A),B : f(A)→ B in Mono(Σ) , (17)

where f(A) is the weak subalgebra of B on f(A) carrying the final structure w.r.t. f ′

and A, and graph f ′ := graph f .Ad (ii): Here we have a distinguished (Epi(Σ)surj, Mono(Σ)full)-factorization of a

given homomorphism f : A→ B in the following way:

e′′ := f ′′ : A→ f(A) in Epi(Σ)surj and idf(A),B : f(A)→ B in Mono(Σ)full , (18)

where f(A) is the relative subalgebra of B on f(A), and graph f ′′ := graph f .The details of the proof are left as an exercise.

Ad (iii): Here we have a distinguished (Epi(Σ), Mono(Σ)closed)-factorization of agiven homomorphism f : A→ B in the following way:

e′′′ := f ′′′ : A→ CB f(A) in Epi(Σ) and idf(A),B : CB f(A)→ B in Mono(Σ)closed ,(19)

where CB f(A) is the (closed) subalgebra of B generated by f(A), and graph f ′′′ :=graph f .The details of the proof are again left as an exercise.

It should be observed that in all three cases the factorization in connection withthe substructure of B is the most easily accessible one, and in the last two caseswe cannot yet get a “dual” description. Moreover, let us mention that there is afurther factorization system with the class of all closed homomorphisms as “mono-factor”, while the “epi-factor” cannot yet be described in full detail, but it containsall epimorphisms from discrete partial algebras X to an X-generated initial segmentof the term algebra T (X,Σ), i.e. it comprises all factorizations (id

X,dom v: Xdiscrete →

dom v, v : dom v → A) of valuations v : X → A .

7 Some constructions of partial algebras

There are several ways to construct new partial algebras from given ones. Subalge-bras, relative subalgebras, weak relative subalgebras and (full) homomorphic imagesneed only one given algebra. The constructions like products, coproducts, pullbacks,pushouts, direct limits and reduced products considered in this section usually startfrom several, in some cases infinitely many given partial algebras (and possibly need-ing some homomorphisms between them).

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7.1 Products and reduced products

Definition 7.1 (Category theoretical definition of products): In categorytheory a product of a family (Ci)i∈I of objects of a category C is a C-object Ctogether with a family (pi : C → Ci)i∈I of C-morphisms such that for each C-objectB and each family (fi : B → Ci)i∈I of C-morphisms there is exactly one C-morphismf : B → C such that pi f = fi for every i ∈ I. f is often called the inducedproduct morphism and is sometimes denoted by < fi | i ∈ I > .

Remark 7.2 (Why do we start here with the category theoretical definitionof products?) This definition is an abstraction of the concept of cartesian productsin the category Set of all sets as objects and all mappings as morphisms, and fromthe concept of direct products for (total) algebras and in other categories like the oneof topological spaces and continuous mappings.

To start now from the general concept is useful, since all properties, which one canprove for this category theoretical concept of products then holds in all categories,in which this concept is realized. Many further category theoretical constructions ofother category theoretical concepts like pullbacks, projective limits and the generalconcept of a limit, of which all these are subconcepts, are based on the concept ofa product. One general property immediately following from the uniqueness of theinduced morphism in the above definition is the uniqueness of the product up toisomorphism (see below). One has such a uniqueness result for all those categorytheoretical concepts (constructions), where a unique induced morphism among theconstructed objects is required. In the other cases mentioned above and for the so-called dual concepts we shall not mention this separately. The proof is always similarto the one of the following proposition.

Proposition 7.3 (Products are unique up to isomorphisms, whenever theyexist at all.)Let (Ai)i∈I be any family of objects in a category C indexed by some set I, and let(A, (pi : A → Ai)i∈I and (A′, (p′i : A′ → Ai)i∈I be any products in C of the givenfamily. Then there exists a unique isomorphism p : A → A′ such that p′i p = pi forall i ∈ I.

Proof Let us use the assumptions and notation of the proposition. Assume that(A, (pi : A → Ai)i∈I) and (A′, (p′i : A′ → Ai)i∈I) both are products of the family(Ai)i∈I of C-objects. Then both families can also be considered just as families ofC-morphisms into (Ai)i∈I each family having a common start object (compare Fig-ure 16).

Therefore the product properties imply that there are (unique) C-morphisms p :A → A′ and p′ : A′ → A , respectively, such that p′i p = pi and pi p

′ = p′i, for eachi ∈ I. This implies, for all i ∈ I, that

pi (p′ p) = p′i p = pi = pi 1A ,

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A A′

Ai A

p

pi

pi p′p′i1A

-

? ?

@@@@@@@@@R

Figure 16: Uniqueness of products

andp′i (p p

′) = pi p′ = p′i = p′i 1A′ .

By the uniqueness property for an induced product morphism we have thereforep p′ = 1A and p′ p = 1A′ . This shows that p and p′ are isomorphisms inverse to oneanother.

Remark 7.4 (Terminal objects as products of the empty family of objects):In the definition above of a product in a category C it has not been excluded that Imay be an empty set. Let us therefore discuss, what it means that a product withrespect to an empty family of objects exists. Since there are no morphisms to betaken care of, a product of such an empty family, if it exists, is a C-object T suchthat, for each C-object A there exists exactly one C-morphism, say tA : A→ T . Suchan object in C, if it exists, is usually called a terminal object of the category C.According to Proposition 7.3 it is unique up to (unique) isomorphism, whenever itexists.

In the category PAlg(τ) of all partial algebras of type τ with the usual homo-morphisms as morphism a (“the”) terminal object is given by any total algebra ona one-element set (and this is also a terminal element in the category TAlg(τ) of alltotal algebras of type τ and their homomorphisms).

In the category PAlg(Σ) of all many-sorted partial algebras of an S-sorted signatureΣ with the usual homomorphisms as morphisms a terminal object is given by takingthe one-element set s as carrier of sort s for each s ∈ S, and by taking the totalS-sorted algebra of signature Σ on this S-set, i.e. by defining, for every ϕ ∈ Ω, a totalS-sorted operation having exactly one sequence in its domain:

ϕS : (s1, . . . , sτ(ϕ)) 7→ s , if (η, σ)(ϕ) = (s1 . . . sτ(ϕ), s) .

It should be observed, however, that there can be categories with much morecomplicated terminal objects. If one takes e.g. the category Closed(τ) of all partial

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algebras of type τ with all closed homomorphisms between them as morphisms, thena terminal object does not exist in general, if τ specifies at least one at least binaryoperation symbol or a nullary constant. If τ only specifies unary operation symbols,then the terminal object of Closed((1)ϕ∈Ω) always exists, but it is then at least count-ably infinite if Ω 6= Ø (if Ω contains only one operation symbol, then T consists of the“disjoint union” (cf. the definition of a coproduct in the next subsection) of a totalone-element algebra and a partial algebra (N; π), where

π(n) :=

n− 1 , if n 6= 0 ,undefined , otherwise (i.e. if n = 0);

If τ specifies more than one unary operation, it is almost impossible to describe theterminal element in more detail.

Example 7.5 (Products in a category assigned to an ordered set corre-spond exactly the infima in this ordered set.) Let (P ;≤) be an ordered set.Then one can assign to (P :≤) a category P as follows:

• ObP := P ;

• MorP := (a, b) | a ≤ b in (P ;≤) .

• Dom(a, b) := a and Cod(a, b) := b (for a ≤ b).

• (b, c) (a, b) := (a, c), if a ≤ b and b ≤ c.

• 1a := (a, a) for all a ∈ P = ObP.

It is left as an exercise to show that P is indeed a category, and that for a family(ai)i∈I of P a product, if it exists, is uniquely given by

(a := inf ai | i ∈ I , ((a, ai) : a→ ai)i∈I) .

Therefore, if P has products for each set-indexed family of objects, then (P ;≤) is acomplete inf-semilattice (and therefore a complete lattice). In particular, the terminalobject of P (i.e. the product of the empty family of objects), if it exists, correspondsto the largest element of the ordered set (P ;≤).

In subsection 1.4 we have already defined direct products of partial algebras (seeDefinition 1.26). The following proposition relates them to the category theoreticalconcept of products.

Proposition 7.6 (Direct products are the products in the category of par-tial algebras): With the assumptions and the notation from Definition 1.26 onehas:

(∏i∈I

Ai, (pri)i∈I) is a product of the family (Ai)i∈I in the category PAlg(Σ).

And if all Ai are total, then (∏

i∈I Ai, (pri)i∈I) is a product in TAlg(Σ), too.

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Proof Let (fi : B → Ai)i∈I be a family of homomorphisms. In order to get an

S-mapping f from B into×i∈I

Ai which satisfies prj f = fj for each j ∈ I, one has

to define

(∗) fs(b) := ((fi)s(b)) | i ∈ I)

for each b ∈ Bs (s ∈ S). And if f is defined like that, it is easy to check thatf : B →

∏i∈I Ai is a homomorphism satisfying prj f = fj for each j ∈ I; e.g. f is

unique with this property.

There is a straight foreward connection between the kernels of the homomor-phisms in a family with the same start object and the kernel of the induced producthomomorphism, which is often needed:

Proposition 7.7 (The kernel of an induced product mapping is the inter-section of the kernels of the original homomorphisms.)Let I be a non-empty set, let (fi : B → Ai | i ∈ I) be a family of homomorphisms,and let f : B →

∏i∈I Ai be the induced product homomorphism. Then

ker f =⋂i∈I

ker fi = (⋂i∈I

ker(fi)s)s∈S .

Proof For b, b′ ∈ Bs one has (b, b′) ∈ ker fs iff fs(b) = fs(b′), iff (fi)s(b) = (fi)s(b

′)for all i ∈ I, iff (b, b′) ∈ ker(fi)s for each i ∈ I, and for each s ∈ S.

In connection with the description of classes of partial algebras which are describ-able by universal Horn formulas we shall need the concept of a reduced product. Webriefly present here the model theoretic definition, while we shall also give later acategory theoretical one using directed colimits (see Observation 7.30).

Definition 7.8 (Filters) Let I be a set and F a set of subsets of I. F is called afilter on I, if

(F1) I ∈ F

(F2) F1, F2 ∈ F implies F1 ∩ F2 ∈ F

(F3) F ∈ F and F ⊆ F ′ imply F ′ ∈ F .

If one has in addition

(F4) Ø /∈ F ,

then F is called a proper filter on I, and if F also satisfies

(F5) for all F ⊆ I either F ∈ F or I \ F ∈ F ,

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then F is called an ultrafilter on I. If F = N ⊆ I|M ⊆ N for some fixed subsetM of I, then F is called a principal filter generated by M : F =:↑M .

Further important examples of filters are the so-called generalized Frechet fil-ters on I: Let c be any infinite cardinal number and let

Fc := F ⊆ I|#(I \ F ) < c.

If c ≤ #I (#M designating the cardinal number of the set M), then Fc is anon-principal proper filter on I.

While closedness w.r.t. products is one part of the sufficient condition for classesof partial algebras of a many-sorted signature Σ with a finite set S of sorts in orderto be definable by E-equations, one needs for definability by ECE- or QE-equationsor in the case of E-equations for finite languages but infinitely many sorts anotherconstruction, the one of reduced products, which is a mixture between the limitand the colimit concept, as we shall see in the next subsection. Although we shall givethere the more general construction using directed colimits, we present here the “usualmodel theoretic construction” at least for finitely many sorts — yet starting with thehomogeneous case —, in order to indicate the differences in the heterogeneous caseto the homogeneous one.

Definition 7.9 (Reduced product, homogeneous case) and Lemma: Let I beany set, let (Ai)i∈I be a family of partial algebras of the same similarity type (Ω, τ),let F be some filter on I. Moreover, let

A :=×i∈I

Ai, A :=∏i∈I

Ai ,

and for a, b ∈ A defineIa,b := i ∈ I | a(i) = b(i) .

Then we consider on A the following binary relation

ΘF := (a, b)|a, b ∈ A and Ia,b ∈ F

which is — as can be easily verified — an equivalence relation on A. Let AF :=A/ΘF be the corresponding quotient set, and for a ∈ A denote by aF := [a]ΘF its

equivalence class. Given ϕ ∈ Ω, we say that a sequence (a1F , . . . , aτ(ϕ)F) ∈ Aτ(ϕ)F

has a ϕ-representative, if there is some sequence (a′1, . . . , a′τ(ϕ)) ∈ Aτ(ϕ) — which

will then be called a ϕ-representative for (a1F , . . . , aτ(ϕ)F) — such that i ∈ I |(a′1(i), . . . , a′τ(ϕ)(i)) ∈ domϕAi ∈ F , and for 1 ≤ j ≤ τ(ϕ) one has a′jF = ajF . Thenwe define

domϕAF := (a1F , . . . , aτ(ϕ)F) ∈ Aτ(ϕ)F | (a1F , . . . , aτ(ϕ)F) has a ϕ-representative .

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For each i ∈ I fix some arbitrary element ai,0 ∈ Ai — if some Ai = Ø, then A = Øand therefore A/F = Ø 66 — and if

aF := (a1F , . . . , aτ(ϕ)F) ∈ dom ϕAF ,

and if (a′1, . . . , a′τ(ϕ)) ∈ Aτ(ϕ) is a ϕ-representative for aF , then define

ϕAF (a1F , . . . , aτ(ϕ)F) =: aF

for some sequence a ∈ A such that for i ∈ I

a(i) :=

ϕAi(a′1(i), . . . , a′τ(ϕ)(i)) , if this exists,

ai,0 ∈ Ai , otherwise;

Then (AF , (ϕAF )ϕ∈Ω) =: (

∏i∈I Ai)/F is called the reduced product of the family

(Ai)i∈I of partial algebras with respect to the filter F .

Proof Since it is easy to check that ΘF is an equivalence relation on A, we only haveto prove that ϕAF is well defined for each ϕ ∈ Ω. Thus, assume that (a′1, . . . , a

′τ(ϕ))

and (a′′1, . . . , a′′τ(ϕ)) are two sequences ϕ-representing (a′1F , . . . , a

′τ(ϕ)F) as an element

of domϕAF . Then, since F is a filter,⋂Ia′i,a′′i | 1 ≤ i ≤ τ(ϕ) ∩ i ∈ I |

(a′1(i), . . . , a′τ(ϕ)(i)) ∈ domϕAi ∩ i ∈ I | (a′′1(i), . . . , a′′τ(ϕ)(i)) ∈ domϕAi ∈ F by as-

sumption, showing that the resulting sequences representing the value ϕAF (a1F , . . . , aτ(ϕ)F)are equivalent modulo F .

Remarks 7.10 (i) (For partial algebras reduced products have in generala stronger structure than just the quotient structure.) Observe that areduced product of partial algebras, as defined above, has in general a strongerstructure than just the quotient algebra (

∏i∈I Ai)/ΘF . Namely, assume I := N

to be the set of all natural numbers, and F to be the Frechet-filter of cofinitesets on N. Let all carrier sets Ai of Ai (i ∈ I) be non-empty. If, say, A1 hasempty structure, while all Ai are total algebras, then (

∏i∈I Ai)/ΘF has empty

structure, too, while (∏

i∈I Ai)/F is total.

If we only deal with total algebras, then the reduced product is just the corre-sponding quotient algebra.

(ii) (Reduced products, S-sorted case): With the notation from Definition 7.9for a set I, a family (Ai)i∈I of partial algebras of the same signature Σ, and afilter F on I, we define for each s ∈ S

Is :=

i ∈ I | (Ai)s 6= Ø , if this set belongs to F ,I , otherwise.

66However, observe the end of Remark 7.10.(ii).

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If S is finite, then we set

IA,F :=⋂s∈S

Is (∈ F) .

Moreover, considerF ′ := F ⊆ IA,F | F ∈ F ,

which is easily to be seen to be a filter on IF .

Then construct the reduced product

(∏i∈I

Ai)/F := (∏

i∈IA,F

Ai)/F ′ ,

where (∏

i∈IA,F Ai)/F′ is defined in an analogous way as we have constructed

(∏

i∈I Ai)/F in Definition 7.9. Observe that

×i∈IA,F

(Ai)s = Ø , iff i ∈ I | (Ai)s 6= Ø /∈ F .

This construction shows that emptyness of carrier sets for the different sortshas to be treated in the same way as e.g. emptiness or at least undefinednessof partial operations. — Observe that also in the homogeneous case we couldapply this idea of considering the set IA,F := i ∈ I | Ai 6= Ø and of setting(∏

i∈I Ai)/F := (∏

i∈IA,F Ai)/F′ as above. Then, as in connection with totalness

as discussed above, (∏

i∈I Ai)/F would be empty iff IA,F /∈ F .

If S is an infinite set, then the set theoretical construction of reduced productsbecomes more complicated, since then one has to use sequences of differentlengths (the intersection of the sets Is need no longer belong to F), and werefer to the construction using directed colimits, which is presented in the nextsubsection.

Remarks 7.11 and Definitions (of the category theoretical concepts of pull-backs, equalizers and limits in general):These remarks may be skipped on first reading. We present here the definitions andtheir realizations for partial S-sorted algebras of some category theoretical concepts.For the general definitions below let C be an arbitrary category.

(i) (Pullbacks): Let fi : Ai → B for i = 1, 2 be two C-morphisms with the sametarget object B. A pair (P, (pi : P → Ai)i∈1,2) consisting of a C-object P andC-morphisms pi : P → Ai (i ∈ 1, 2) is called a pullback of the pair (f1, f2), iff1 p1 = f2 p2, and if, for all (H, (hi : H → Ai)i∈1,2) consisting of a C-objectH and C-morphisms hi : H → Ai (i ∈ 1, 2) satisfying f1 h1 = f2 h2 , one

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has exactly one C-morphism h : H → P such that p1 h = h1 and p2 h = h2

(draw a diagram).

In PAlg(Σ) a pullback of ((fi : Ai → B)i∈1,2, B) always exists and is given upto isomorphism as follows:

P := (a1, a2) ∈ A1 × A2 | f1(a1) = f2(a2) ,

is a closed subset of A1πA2 (exercise), P is the subalgebra of A1πA2 on P , andpi is the restriction of the canonical projection pri to P (i ∈ 1, 2).

(ii) (Equalizers):67 Let fi : A → B for i = 1, 2 be two C-morphisms. A pair(E; m : C → A) consisting of a C-object E and a C-morphism m : E → A iscalled an equalizer of the (fi)i∈1,2) , if f1 m = f2 m, and if, for all (H, h :H → A)) consisting of a C-object H and a C-morphisms h : H → A satisfyingf1 h = f2 h , one has exactly one C-morphism h′ : H → E such that m h′ = h(draw a diagram).

In PAlg(Σ) an equalizer of (fi : A → B)i∈1,2 always exists and is given up toisomorphism as follows:

E := a ∈ A | f1(a) = f2(a) ,

and E is the subalgebra of A on E, and m is the restriction of the identitymapping idA to E.

(iii) (Limits): Let (Ai)i∈I be a family of C-objects, let (fj : Aij,1 → Aij,2)j∈J bea family of C-morphisms such that ij,1, ij,2 ∈ I for all j ∈ J . A pair (L, (li :L→ Ai)i∈I) consisting of a C-object L and a family of C-morphisms li : L→ Ai(i ∈ I) is called a limit of the pair ((Ai)i∈I , (fj)j∈J) , if fj lij,1 = lij,2 for allj ∈ J , and if, for all (H, (hi : H → Ai)i∈I) consisting of a C-object H andC-morphisms hi : H → Ai (i ∈ I) satisfying fj hij,1 = hij,2 , one has exactlyone C-morphism h : H → L such that li h = hi for all i ∈ I (draw a diagram).

In PAlg(Σ) a limit of ((Ai)i∈I , (fj : Aj1 → Aj2)j∈J) always exists and is givenup to isomorphism as follows:

– If I = Ø, then J = Ø, and the limit is the terminal object in PAlg(Σ).

– Else let(P 1 :=

∏i∈I

Ai , (pri : P 1 → Ai)i∈I),

be the direct product of the given family (Ai)i∈I of partial algebras, andlet

(P 2 :=∏j∈J

Aij,2 , (pij,2 : P 2 → Aij,2)j∈J),

67Cf. also Remark 5.21.(ii).

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be the direct product of the family (Aij,2)j∈J . Consider the two homomor-phisms g1, g2 : P 1 → P 2, where g1 is induced by the family (prij,2 : P 1 →Aij,2)j∈J) , and g2 is induced by the family (fj prij,1 : P 1 → Aij,2)j∈J) ,and let (P , idP,P1 : P → P 1) be the equalizer of g1 and g2, i.e. P is the

subalgebra of P 1 on P := a ∈×i∈I

Ai | g1(a) = g2(a) .

In principle all limits in categories having products and equalizers can be constructedas above. Observe, that our constructions of pullbacks and equalizers do not useall partial algebras involved in order to form the product. Since the limit object isonly determined up to isomorphism, this is possible, since the special constructionsalready take care of the formation of the equalizer, and the objects omitted will notcontribute new elements or additional structure. It is left as an exercise to go throughthe formation of pullbacks and equalizers according to the general construction in (iii)and to produce the corresponding isomorphisms.

7.2 Coproducts and directed colimits

Definition 7.12 (Category theoretical definition of coproducts): In categorytheory a coproduct is the dual concept of a product (see Figure 17), thus, for a family(Aj)j∈J of similar partial algebras a coproduct is a partial algebra A =:

∐j∈J Aj —

note that as in the case of products it is unique up to isomorphism — and a family(ij : Aj → A)j∈J of homomorphisms, called (canonical) injections (although theyneed not be injective), such that for every family (fj : Aj → B)j∈J of homomorphismsthere is a unique homomorphism f : A→ B such that f ij = fj for all j ∈ J .

Before we give the construction of coproducts in the category PAlg(Σ), we definethe concept of Σ′-reducts, which is useful also in other connections.

Definition 7.13 (Σ′-reducts): Let Σ = (S,Ω, τ, η, σ) be any signature and Σ′ =(S ′,Ω′, τ ′, η′, σ′) be a “subsignature”, i.e. one has: S ′ ⊆ S, Ω′ ⊆ Ω, τ ′ := τ |Ω′ ,η′ := η|Ω′ , and σ′ := σ|Ω′ , where one assumes that S ′ is “closed”, i.e. that itcontains all the sorts needed for (η(ϕ), σ(ϕ)) of the operation symbols ϕ ∈ Ω′.If A = ((As)s∈S, (ϕA)ϕ∈Ω) is any partial algebra of signature Σ, then let A′ =((As)s∈S′ , (ϕA)ϕ∈Ω′) be the induced partial algebra of signature Σ′ on (As)s∈S′ , whichis called the Σ′-reduct of A.

If S = S ′ and only Ω′ 6= Ω , then we shall speak of an Ω′-reduct.

Definition 7.14 (Construction of coproducts in PAlg(Σ)):

1. In the case that the signature Σ does not specify any nullary constants thecoproduct object A :=

∐j∈J Aj in the category PAlg(Σ) of all partial algebras

of signature Σ with all their homomorphisms as morphisms is easily describedas follows:

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Let, for each s ∈ S, As be the disjoint union of the sets (Aj)s, i.e.

As :=⋃j∈J

(Aj)s × j .

For ϕ ∈ Ω one has

domϕA := ((a1, j), . . . , (aτ(ϕ), j)) | j ∈ J, (a1, . . . , aτ(ϕ)) ∈ domϕAj ,

and if ((a1, j), . . . , (aτ(ϕ), j)) ∈ domϕA for some j ∈ J , then

ϕA((a1, j), . . . , (aτ(ϕ), j)) := (ϕAj(a1, . . . , aτ(ϕ)), j),

i.e. graphϕA can be considered as the disjoint union of the graphs of the ϕAj

(j ∈ J).

The canonical injection ij : Aj → A is given by a 7→ (a, j) (a ∈ Aj, j ∈ J).

2. Now assume that Σ specifies nullary constants, let Ω0 := ϕ ∈ Ω | τ(ϕ) = 0 ,and Ω′ := Ω \Ω0, let τ ′ := τ |Ω′ be the restriction of τ to Ω′, and similarly, let η′

and σ′ be the restrictions to Ω′ of η and σ, respectively. Thus, for a given family(Aj)j∈J of partial algebras of signature Σ, let (A∗, (ϕA

∗)ϕ∈Ω′) = A∗′ :=

∐j∈J A

′j

together with (i′j : A′j → A∗′) be the construction from above for the Ω′-reducts.Moreover let, for each s ∈ S, R′s ⊆ A∗s × A∗s be the following relation:

R′s := ((ϕAj , j), (ϕAk , k)) | j, k ∈ J, ϕ ∈ Ω0, σ(ϕ) = s, ϕAj and ϕAk exist ,

let Θ := ConA∗′⋃s∈S R

′s be the congruence relation on A∗′ generated by R′ :=⋃

s∈S R′s, and define finally

A :=∐j∈J

Aj := (A∗/Θ, (ϕA∗′/Θ)ϕ∈Ω′ ∪ (ϕA)ϕ∈Ω0) ,

where, for ϕ ∈ Ω0 ,

ϕA :=

[(ϕAj , j)]Θ if ϕAj exists for some j ∈ Jundefined, else;

and (ij := natΘ i′j : Aj → A)j∈J is the family of corresponding canonicalinjections.

Proposition 7.15 (Verification that the construction above yields a co-product): For any family (Aj)j∈J the partial algebra A =

∐j∈J Aj and the family

(ij : Aj → A)j∈J form a coproduct of the family (Aj)j∈J .

The direct checking of the details of the proof is left to the reader as an exercise.It should be observed that the construction is just made in such a way that it works(use the First Diagram Completion Theorem).

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Aj A Ak

B

ij ik

fj ∃!f fk

-

?

@@@@@@R

Figure 17: Characteristic diagram for coproducts

Remark 7.16 For most categories of (partial, many-sorted) algebras the construc-tion of products is “robust”. In particular, in all categories, of which the class ofobjects forms a QE-variety, and which has all homomorphisms between these par-tial algebras as morphisms, it is given as in the previous subsection. The situation,however, is different in the case of coproducts. The carrier set and the operations ofcoproducts heavily depend on the class of (partial, many-sorted) algebras under con-sideration. Observe that the construction above almost always yields a proper partialalgebra, even when all algebras Ai are total (i ∈ I). We shall, however learn soonthat in classes K closed w.r.t. the formation of direct products, isomorphic copies andsubalgebras coproducts always exist and can be obtained as so-called K-universalsolutions of the partial “global” coproducts defined above.68

For example in the case of (additively written) abelian groups (or modules orvector spaces) a copruduct of a family (Ai)i∈I is the so-called direct sum, i.e. thesubalgebra A of the direct product

∏i∈I with carrier set

A := (ai | i ∈ I) | i ∈ I | ai 6= 0Ai is finite .

However, in the case of non-abelian groups coproducts are the so-called free productshaving a quite different construction, which has nothing to do with direct products.

While coproducts are one interesting instance of colimits, directed colimits (inolder terminology: direct limits or inductive limits) are another one.

Definition 7.17 (of directed systems and directed colimits)

68The K-universal K-solution of a partial algebra A — if it exists — is a pair (rA,K : A →F (A,K) , F (A,K)) such that F (A,K) ∈ K and such that for each homomorphism f : A → K withK ∈ K there exists a unique homomorphism f : F (A,K)→ K satisfying f rA,K = f (see below formore details).

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(1) A directed system of partial algebras is any family

AJ := (Aj, fjl : Aj → Al | j, l ∈ J, j ≤ l ) ,

where J := (J,≤) is a directed ordered set, i.e. (J,≤) is an ordered set (cf.Definition 4.11.(i)), which is directed, what means that for any finite subsetF ⊆ J there exists in J an upper bound, i.e. there is some some j ∈ J such thati ≤ j for all i ∈ F ; moreover, for j ≤ l in J , fjl : Aj → Al is a homomorphismsuch that

(a) fjj = idAj for all j ∈ J ,

(b) for i ≤ j and j ≤ l in J one has fjl fij = fil.69

(2) Let AJ := (Aj, fjl : Aj → Al | j ≤ l in J) be a directed system. A directedcolimit — in older terminology a direct limit or inductive limit — of AJ isa pair ((fj : Aj → A)j∈J , A) consisting of the colimit object A and the so-calledcolimiting cocone (fj)j∈J of homomorphisms fj : Aj → A (j ∈ J), such that

(C0) for all i ≤ j in J one has fj fij = fi, and(CL) for any so-called compatible family ((gj : Aj → B)j∈J , B)

consisting of a partial algebra B and a family (gj : Aj → B)j∈Jof homomorphisms satisfying the compatibility condition(C) for all i ≤ j in J one has gj fij = gi ,there exists a unique homomorphism g : A→ Bwith g fj = gj for all j ∈ J .

We write lim→AJ := ((fj : Aj → A)j∈J , A) for the directed colimit of a directedsystem.70

Proposition 7.18 For any signature Σ and for any directed system AJ = (Aj, fjk :Aj → Ak | j ≤ k in J) in PAlg(Σ) there exists the directed colimit lim→AJ = ((fj :Aj → A)j∈J , A), which can be defined as follows (cf. Proposition 7.15):

Let A∗′ :=∐

j∈J A′j be the coproduct in the category PAlg(Σ′), where again A′j is

the Ω′-reduct of Aj for Ω′ = Ω \ Ω0 ( Ω0 := ϕ ∈ Ω | τ(ϕ) = 0), and let, for eachs ∈ S,

Θs := ((a, j), (b, k)) ∈ (⋃j∈J

(Aj)s × j)2 | there is m ∈ J : j, k ≤ m and (20)

(fjm)s(a) = (fkm)s(b) .69Observe that a directed system AJ in the category PAlg(Σ) is a full homomorphic image in

PAlg(Σ) of the category related to the directed ordered set J . Observe that homomorphisms betweencategories — and here we allow them to be large, i.e. the object classes need not be sets — are calledfunctors.

70This definition and notation also applies to arbitrary categories.

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Then Θ := (Θs)s∈S is a congruence relation on A∗′. Set A := A∗/Θ, ϕA := ϕA∗′/Θ

for ϕ ∈ Ω′, and for ϕ ∈ Ω0 set:

ϕA :=

[(ϕAk , k)]Θ, if ϕAk exists for some k ∈ J,undefined, if ϕAk does not exist for any k ∈ J.

such that71

A := lim→AJ := (A∗/Θ, (ϕA

∗′/Θ)ϕ∈Ω′ ∪ (ϕA)ϕ∈Ω0) ,

Finally set fj := natΘ ij : Aj → A, for each j ∈ J , where ij(a) := (a, j) fora ∈ Aj.

Proof We use the same notation as for coproducts, only Θ is defined differently. Wefirst show that Θ is a congruence relation on A∗

′: Since fjj = idAj , Θ is reflexive, while

symmetry of Θ follows from the definition. Let ((a, j), (b, k)), ((b, k), (c, l)) ∈ Θs ,for some s ∈ S . Then the definition of Θ implies the existence of m1,m2 ∈ J suchthat j, k ≤ m1, k, l ≤ m2 and (fjm1)s(a) = (fkm1)s(b) and (fkm2)s(b) = (flm2)s(c).Since J is directed, there is m ∈ J such that m1,m2 ≤ m. Then (fjm)s(a) =(fm1m)s (fjm1)s(a) = (fm1m)s (fkm1)s(b) = (fkm)s(b) = (fm2m)s (fkm2)s(b) =(fm2m)s (flm2)s(c) = (flm)s(c), showing that Θ is transitive. Thus Θ is an equiv-alence relation on A∗ =

⋃j∈J Aj × j.

Let ϕ ∈ Ω′, (η, σ)(ϕ) := (s1 . . . sn, s), j, k ∈ J , ((ar, j), (br, k)) ∈ Θ for 1 ≤r ≤ τ(ϕ) = n and (a1, . . . , an) ∈ domϕAj , (b1, . . . , bn) ∈ domϕAk . First we havefor each pair ((ar, j), (br, k)) an upper bound mr for the indices j and k, where thepair is equalized; and by directedness of J there is an upper bound m in J forall these finitely many indices such that (by a computation as above) one has, for1 ≤ r ≤ n , (fjm)sr(ar) = (fkm)sr(br), and since fjm as well as fkm are homo-morphisms, one has (fjm)s(ϕ

Aj(a1, . . . , an)) = ϕAm((fjm)s1(a1), . . . , (fjm)sn(an)) =ϕAm((fkm)s1(b1), . . . , (fkm)sn(bn)) = (fkm)s(ϕ

Ak(b1, . . . , bn)).This shows that Θ is a congruence relation on A∗

′. And this shows in addition

that each fj is a homomorphism between the Ω′-reducts A′j and A′. It remains toshow that fj also preserves constants, if they exist: Let ϕ ∈ Ω0, and let, for j, k ∈ J ,ϕAj and ϕAk exist, and assume σ(ϕ) := s. Then there is m ∈ J such that j ≤ mand k ≤ m. Then (fjm)s(ϕ

Aj) = (fkm)s(ϕAk), showing that ((ϕAj , j), (ϕAk , k)) ∈ Θs

and that fj and fk also preserve existing constants. By the construction of Θ one hasfj fij = fi , for all i ≤ j in J , and therefore (C0) has been proved.

Ad (CL): Let ((gj : Aj → B)j∈J , B) be a family of homomorphisms compatiblewith AJ (i.e. gk fjk = gj for j ≤ k in J), and for [(a, j)]Θ of sort s define

(*) gs([(a, j)]Θ) := (gj)s(a).If ((a, j), (b, k)) ∈ Θs , then there is m ∈ J such that j, k ≤ m and (fjm)s(a) =

(fkm)s(b), and then (gj)s(a) = (gm)s (fjm)s(a) = (gm)s (fkm)s(b) = (gk)s(b), showingthat the definition of g does not depend on the representative of the congruence class.

71Observe that we denote here by lim→AJ also the colimit object of the directed colimit underconsideration.

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Because of the requirement that g fj = gj for all j ∈ J the above definition(*) is the only possible one in order to achieve this goal. Finally let us show thatg is a homomorphism: Assume for ϕ ∈ Ω , with (η, σ)(ϕ) := (s1 . . . sn, s) , thatϕA([(a1, j1)]Θ, . . . , [(an, jn)]Θ) exists. Then there exists m ∈ J , about which wemay assume (by directedness of J) that jr ≤ m for 1 ≤ r ≤ τ(ϕ) =: n and that((fj1m)s1(a1), . . . , (fjnm)sn(an)) ∈ domϕAm . Then

gs(ϕA([(a1, j1)]Θ, . . . , [(an, jn)]Θ)) =

= gs([(ϕAm((fj1m)s1(a1), . . . , (fjnm)sn(an)),m)]Θ) =

= (gm)s(ϕAm((fj1m)s1(a1), . . . , (fjnm)sn(an))) =

= ϕB((gm)s1 (fj1m)s1(a1), . . . , (gm)sn (fjnm)sn(an)) == ϕB(gs1([(a1, j1)]Θ), . . . , gsn([(an, jn)]Θ)).

Lemma 7.19 (A directed colimit is unique up to unique isomorphism.) LetAJ = (Ai, fij : Ai → Aj | i ≤ j in J) be a directed system with directed colimitsf := ((fj : Aj → A)j∈J , A) and f ′ := ((f ′j : Aj → A′)j∈J , A

′), then there is a uniqueisomorphism g : A→ A′ such that g fj = f ′j for each j ∈ J .

Proof Since both systems f and f ′ are directed colimits, they are in particularcompatible families, and therefore there exist unique homomorphisms g : A→ A′ suchthat g fj = f ′j for each j ∈ J and g′ : A′ → A such that g′ f ′j = fj for each j ∈ J .Then we get for g′g : A→ A and gg′ : A′ → A′ that g′gfj = g′f ′j = fj = idAfjand g g′ f ′j = g fj = f ′j = idA′ f ′j for each j ∈ J . Thus the uniqueness of theinduced homomorphism yields g′ g = idA and g g′ = idA′, showing that g and g′

are isomorphisms inverse to each other.

Lemma 7.20 (Confinal directed subsystems yield the same directed colim-its.) Let AJ := (Aj, fjk : Aj → Ak | j ≤ k in J) be a directed system, with directedcolimit ((fj : Aj → A)j∈J , A) . Let I ⊆ J be any non-empty subset such that,

(cf) for every j ∈ J there is i ∈ I with j ≤ i,and consider I := (I;≤) with i ≤ i′ in I, iff i ≤ i′ in J (i.e. the order on I is therestriction to I of the one on J (one then says that I is confinal with J).

Then I is also a directed ordered set, AI := (Aj, fjk : Aj → Ak | j ≤ k in I) is alsoa directed system — and one calls it confinal with AJ —, and ((fj : Aj → A)j∈I , A)is its directed colimit — with the same colimiting homomorphisms (yet only for i ∈ I)and object as for AJ .

Proof Let us first show that I is directed: Let i, i′ ∈ I; since i, i′ ∈ J there is j ∈ Jsuch that i, i′ ≤ j. By (cf) there is k ∈ I such that j ≤ k. Hence i, i′ ≤ k in I.Since I is non-empty, this proves that I is directed. Therefore AI is indeed a directedsystem.

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Let ((f ′j : Aj → A′)j∈I , A′) be a directed colimit of AI . We want to show that

there is a canonical isomorphism h′ : A→ A′ satisfying h fi = f ′i , for all i ∈ I:(cf) implies that for each j ∈ J there is ij ∈ I with j ≤ ij. Choose such ij for

each j ∈ J and consider the cocone f′ := (f ′ij fj,ij : Aj → A′)j∈J . We claim that it iscompatible with AJ :

Assume j ≤ k in J , and let r ≥ ij, ik in I; then, since f′ is compatible with AI ,we have

f ′ik fk,ik fjk = f ′r fik,r fk,ik fjk = f ′r fij ,r fj,ij = f ′ij fj,ij

showing the required compatibility. Hence there exists a unique homomorphism h′ :A → A′ satisfying h′ fj = f ′ij fj,ij , for each j ∈ J . On the other hand (fi :Ai → A)i∈I is surely a cocone compatible with AI , and hence there is a uniquehomomorphism h : A′ → A satisfying h fi = f ′i , for each i ∈ I. Because of theuniqueness of the induced homomorphisms one easily concludes that h h′ = idA andh′ h = idA′ . This shows that they are isomorphisms inverse to each other.

Remark 7.21 Observe that, for any element j ∈ J of a directed set J the subsetJj := k ∈ J | j ≤ k is directed and confinal with J . Namely, by directedness of J ,J j is directed, too, and for any k ∈ J there is l ∈ J with j, k ≤ l.

Proposition 7.22 (Preservation of some properties of homomorphisms bydirected colimits): Let AJ := (Aj, fjk : Aj → Ak | j ≤ k in J) and BJ := (Bj, hjk :Bj → Bk | j ≤ k in J) be directed systems with lim→AJ := ((fj : Aj → A)j∈J , A),and lim→ BJ := ((hj : Bj → B)j∈J , B), as directed colimits. Moreover, let ((gj : Aj →Bj)j∈J , B) be a family compatible with AJ and BJ — i.e. gj fij = hij gi , wheneveri ≤ j in J . Then there exists g : A → B as the induced colimit homomorphism,induced by the family (hj gj : Aj → B)j∈J such that g fj = hj gj , for all j ∈ J .Moreover, one has:

(i) If all fjk (j ≤ k in J) are isomorphisms, injective, closed, surjective, dense(epimorphisms) or full, respectively, then all the fj have the same property(j ∈ J).

(ii) If all gj (j ∈ J) are isomorphisms, injective, closed, surjective, dense (epimor-phisms) or full, respectively, then g has the same property.

Proof Let us first observe that the family (hj gj : Aj → B)j∈J is compatible withthe directed system AJ : Assume i ≤ j in J . Then

(hj gj) fij = hj hij gi = hi gi .

Thus g exists as required.All the statements except for the one about fullness in (i) and (ii) could be proved

— with some additional information — by using that the class of homomorphisms

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under consideration is either Λ(S) or Λop(S) for some class S of homomorphisms(cf. [B86]); and this would make the results even more general. Yet we do not havethis additional information right now, and it is not difficult to prove the resultsdirectly. Therefore we want to sketch some of these proofs, and we assume thatalways the correct sorts for the elements and morphisms, respectively, are chosen,and that (η, σ)(ϕ) := (s1 . . . sn, s) for ϕ ∈ Ω:

Ad (i): Injectivity: Let fj(a) = fj(b); then there is k ≥ j such that fjk(a) =fjk(b). Injectivity of fjk implies a = b.

Surjectivity: Consider a := [(a, i)]Θ ∈ A, and let k ∈ J . Then there is m ≥ i, kin J . Since fkm is surjective, there is a′ ∈ Ak such that fkm(a′) = fim(a). Hencea = fk(a

′).Fullness (Closedness can be treated in a similar way):

Assume that ϕA(fi(a1), . . . , fi(an)) = fi(an+1). Then there is k ∈ J , and without lossof generality we may assume that k ≥ i (since J is directed and by the definition ofhomomorphisms) that ϕAk(fik(a1), . . . , fik(an)) = fik(an+1) . Since fik is full, there area′r ∈ Ai with fik(ar) = fik(a

′r) , for 1 ≤ r ≤ n+ 1 , such that ϕAi(a′1), . . . , a′n)) = a′n+1

is true in Ai . Then one also has fi(ar) = fi(a′r) , for 1 ≤ r ≤ n+ 1 .

Density (i.e. the case of epimorphisms): Choose j ∈ J and a := [(a, k)]Θ ∈ A , forsome k ∈ J . We have to show that a ∈ CAfj(Aj) : We know that there is l ∈ J suchthat j, k ≤ l. Hence we have a = fk(a) = fl(fkl(a)). Since fjl is dense (by assumtion),we have fkl(a) ∈ CAlfjl(Aj) = Al . Now

a ∈ fl(Al) = fl(CAlfjl(Aj)) ⊆ CAfl(fjl(Aj)) = CAfj(Aj) .

This shows that fj is dense.Isomorphy: The preservation of isomorphy follows, since injectivity, surjectivity

and fullness (or closedness) are preserved.Ad (ii): Injectivity: Assume g(fi(a)) = g(fj(a

′)), for i, j ∈ J . Then there isk ≥ i, j in J , such that one has

hk(gk(fik(a))) = g(fk(fik(a))) = g(fi(a)) = g(fj(a′)) = g(fk(fjk(a

′))) = hk(gk(fjk(a′))) .

Therefore one has l ≥ k in J such that hkl(gk(fik(a))) == hkl(gk(fjk(a′))), and

therefore

gl(fil(a)) = gl(fkl(fik(a))) = hkl(gk(fik(a))) =

= hkl(gk(fjk(a′))) = gl(fkl(fjk(a

′))) = gl(fjl(a′)) .

Since gl is assumed to be injective, this implies fil(a) = fjl(a′); and therefore we get

fi(a) = fj(a′), showing that g is indeed injective.

Surjectivity: Consider b := hk(b) ∈ B. Since gk is surjective, there is a ∈ Ak suchthat b = gk(a) . This implies b = hk(gk(a)) = g(fk(a)) , showing that g is indeedsurjective.

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Fullness (Closedness can be handled in a similar way):Assume that ϕB(g(fi1(a1)), . . . , g(fin(an))) = g(fin+1(an+1)). Without loss of gener-ality we may assume that i1 = . . . = in = in+1 =: i because of the directedness of J .Moreover, we may assume i to be large enough such that ϕBi(gi(a1), . . . , gi(an)) =gi(an+1) . Since gi is assumed to be full, one has a′r ∈ Ar with gi(a

′r) = gi(ar) , for 1 ≤

r ≤ n + 1 such that ϕAi(a′1, . . . , a′n) = a′n+1 . This implies that g(fi(ar)) = g(fi(a

′r))

and ϕA(fi(a′1), . . . , fi(a

′n)) = fi(a

′n+1) , showing that g is full.

Density (i.e. the case of epimorphisms): Let b ∈ B; we want to show that b =hk(b) ∈ CBg(A) . Since gk is assumed to be dense, we have b ∈ CBkgk(Ak) . Then

hk(b) ∈ hk(CBkgk(Ak)) ⊆ CB(hk gk)(Ak) = CB(g fk)(Ak) ⊆ CBg(A) .

Isomorphy: Use the same argumentation as for (i).

With respect to the representation of a partial algebra as directed colimit of “verysmall pieces” in Proposition 7.27 below we give the following definition. As we shallsee below, it will help to characterize in PAlg(Σ) the partial algebras with finite carrierand finite structure, and in TAlgu(Σ)the finitely generated total algebras which arequotients of term algebras w.r.t. a finitely generated congruence relation.

Definition 7.23 (Finite presentability (= strong smallness)): Let C be anycategory, in which directed colimits of directed systems always exist. A C-objectK is called finitely presentable or strongly small, if, for every directed colimit((fi : Ai → A)i∈I , A) of a directed system AI = (Ai; fik : Ai → Ak | i ≤ k in I ) inC,

(fp1) for every C-morphism h : K → A there exist i ∈ I and a C-morphism hi : K →Ai such that fi hi = h ; and

(fp2) for all hj, h′j : K → Aj for some j ∈ J with fj hj = fj h

′j there exists k ≥ j in

J such that fjk hj = fjk h′j .

Before we characterize the finitely presentable partial and total algebras, we needsome additional informations:

Lemma 7.24 Let A be any partial algebra of signature Σ, and let M be a generatingsubset of A. Then there exists for every a ∈ A (i.e. for a ∈ As for some s ∈ S) afinite weak relative subalgebra Ba of A such that

(i) a ∈ Ba = CBa(Ba ∩M)

(ii) Ba is finite, i.e.⋃s∈Ss × (Ba)s is finite, and

(iii) the structure of Ba is finite, i.e.⋃ϕ∈Ωϕ × graph ϕBa is finite.

Definition 7.25 Finite partial algebras B satisfying (ii) and (iii) above will be calledtotally finite.

The proof of this lemma is a good application of the method of algebraic (=struc-tural) induction, hence we give the details.

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Proof (by structural induction, where reference to the correct sorts has to be addedby the reader):

• If a ∈ M , then set Ba := (a, (Ø)ϕ∈Ω) which obviously satisfies (i), (ii) and(iii).

• If a = ψA is a constant, then set Ba := (a, (ϕBa)ϕ∈Ω) such that for ϕ ∈ Ω

graph ϕBa =

((), ψA) if ϕ = ψØ if ϕ 6= ψ.

• Finally, assume that a = ϕA(a1, . . . , aτ(ϕ)) for some ϕ ∈ Ω and (a1, . . . , aτ(ϕ)) ∈domϕA, and that Ba has not yet been defined. Let the statement be true for

each ai, 1 ≤ i ≤ τ(ϕ). Then define Ba :=⋃τ(ϕ)i=1 Bai ∪ a and, for ψ ∈ Ω,

graph ψBa :=

⋃τ(ϕ)i=1 graph ϕBai ∪ ((a1, . . . , aτ(ϕ)), a), if ψ = ϕ,⋃τ(ψ)i=1 graph ψBai , else.

Then it is obvious that a ∈ Ba, and that Ba satisfies the finiteness conditions in(ii) and (iii). Moreover, for Ma := Ba ∩M one has ai ∈ Bai = CBai (Bai ∩M) ⊆CBaMa and therefore also a = ϕA(a1, . . . , aτ(ϕ)) ∈ CBaMa, i.e. Ba = CBaMa.Thus Ba satisfies (i), (ii) and (iii).

Since M generates A these arguments show the truth of the statement for every a ∈ A.

Related to the three different kinds of subobjects which we have discussed at thebeginning there are several directed systems “exhausting” a given partial algebra.

Definition 7.26 and Notation Let A be any partial algebra of signature Σ, andlet M be any generating subset of A. Then consider the following sets:

I1 := CAN | N ⊆M and N is finite ;

I2 := B | B is a relative subalgebra of A, B is finite and B = CB(B ∩M);

I3 := B | B is a totally finite weak relative subalgebra of A, and B = CB(B ∩M).

In each case define for B, B′ ∈ Ik and 1 ≤ k ≤ 3 B ≤ B′ if and only if B is aweak relative subalgebra of B′; and if B ≤ B′ then define fB,B′ := idBB′ : B → B′ tobe the natural embedding of B into B′. Finally, let for 1 ≤ k ≤ 3 Ik := (Ik,≤) andAIk := (B, fB,B′ : B → B′ | B ≤ B′ in Ik ).

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Proposition 7.27 With the above notation one has that each AIk is a directed systemwith directed colimit ((idBA : B → A)B∈Ik , A) for 1 ≤ k ≤ 3.

Proof It is easy to realize that in all three cases AIk is a directed system, 1 ≤ k ≤ 3.The proofs of the statements about AI1

and AI2will be similar to (and even easier

than) the one about AI3, which we shall only give here.

Thus, consider ((fB : B → A)B∈I3 , A) with fB = idBA. (Observe that we alreadyknow from Proposition 7.22 that each fB has to be injective, if it belongs to the col-imiting cocone.) It is obvious that for each B,B′ ∈ I3 we have fB′ fB,B′ = fB. Let((gB : B → C)B∈I3 , C) be a ln16.texfamily compatible with AI3

. In order to defineg : A → C, set g(a) := gB(a), whenever a ∈ B, where B is the carrier of B. ByLemma 7.24 there is at least one such B ∈ I3. Since all fB and fB,B′ are identityinjections, g is well-defined and the definition is the only possible one in order to haveg fB = gB for each B ∈ I3. It remains to show that g : A→ C is a homomorphism:Let ϕA(a1, . . . , aτ(ϕ)) = a exist in A, and set a := (a1, . . . , aτ(ϕ)). With the notationused in the proof of Lemma 7.24 define Ba,ϕA(a) as Ba has been defined there.72

Then Ba,ϕA(a) ∈ I3, g(a) = g(fBa,ϕA(a)

(a)) = gBa,ϕA(a)

(ϕBa,ϕA(a)(a1, . . . , aτ(ϕ))) =

ϕC(gBa,ϕA(a)

(a1), . . . , gBa,ϕA(a)

(aτ(ϕ))) = ϕC(g(fBa,ϕA(a)

(a1)), . . . , g(fBa,ϕA(a)

(aτ(ϕ)))) =

ϕC(g(a1), . . . , g(aτ(ϕ))). This shows that ((fB : B → A)B∈I3 , A) is indeed a directedcolimit of AI3

.

Proposition 7.27 shows that the three kinds of subobjects of partial algebras allow“three different kinds of finiteness” which can be used in connection with generationand exhaustion of structures. In particular, the totally finite weak relative substruc-tures used in AI3

are those, for which there might be hope that they could be rep-resented on computers, although finiteness is even much more restricted there, withthe consequence that every infinite algebra will have — with respect to a generatingsystem — parts, which will never be representable on a computer, while others willbecome representable when the capacity of computers increases.

Lemma 7.28 (Characterization of finitely presentable partial and total al-gebras):

(i) In the category PAlg(Σ) a partial algebra A = ((As)s∈S, (ϕA)ϕ∈Ω) is finitely

presentable, iff it is totally finite in the sense of Definition 7.25, i.e. iff it satisfies

(tf1)⋃s∈S As × s is finite, and

(tf2)⋃ϕ∈Ω graphϕA × ϕ is finite.

72Here we have to include in addition the reference to the application of the operation, since wealso want to generate by and by all of the structure of A.

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(ii) In the category TAlg(Σ) of all total algebras of signature Σ with all homomor-phisms between them as morphisms a (total) algebra A is finitely presentable,iff there are a finite set M and a finite subset R ⊆ T (M,Σ)2 such that A isisomorphic to the factor algebra T (M,Σ)/ConT (M,Σ))R .

Proof Ad (i): Assume that K ∈ PAlg(Σ) satisfies (tf1) and (tf2) (i.e. that K istotally finite), and let ((fi : Ai → A)i∈I , A) be the colimiting cocone of a directedsystem AI = (Ai; fik : Ai → Ak | i ≤ k in I ) in PAlg(Σ). Moreover, let h : K → Abe any homomorphism. We show (fp1) and (fp2) from Definition 7.23:

Ad (fp1): Since K is finite, h(K) is finite, and by directedness of I there is i ∈ Isuch that, for every k ∈ K there is k′ ∈ Ai with h(k) = fi(k

′). Fix such a k′ for eachk ∈ K and define h′ : K → Ai by h′(k) := k′.Assume ϕK(k1, . . . , kτ(ϕ)) = k . The fact that h is a homomorphism, implies thatϕA(h(k1), . . . , h(kτ(ϕ))) = h(k) . Since AI is a directed system with colimit object A,there is j ∈ I, which may be chosen as j ≥ i, such that

(*) ϕA(fij(h′(k1)), . . . , fij(h

′(kτ(ϕ)))) = fij(h′(k)) .

Since the structure of K is finite, j above may even be chosen in such a way that(*) holds for all sequences in K, for which a fundamental operation is defined. Butthat means that hj := fij h

′ : K → Aj is a homomorphism, which obviously satisfiesfj hj = h. Thus K satisfies (fp1) of Definition 7.23.

Ad (fp2): Consider hj, h′j : K → Aj with fj hj = fj h

′j (= h), then the finiteness

of K and the definition of a directed colimit imply that there exists l ≥ j in I suchthat one already has fjl hj = fjl h

′j . This shows (fp2) for K.

Conversely, assume that K is not totally finite, and let AI3be the directed sys-

tem of totally finite weak relative subalgebras of K. Then idK cannot be “factoredthrough” AI3

, since then something would have to be identified, while idK is an iso-morphism. This shows that the finitely presentable partial algebras are exactly thetotally finite ones.

Ad (ii): Let T := T (M,Σ) be the (total) term algebra with set M of variables73

Let idM,A : T (M,Σ) → A be the surjective homomorphism induced by the inclusionmapping idM,A : M → A , where M is considered here as a subset of the term algebra.

Let ΘM,A := ker idM,A . Define

It := (N,R) | N ⊆M finite, and R ⊆ ΘM,A finite . (21)

For (N,R), (N ′, R′) ∈ It define the order relation as

(N,R) v (N ′, R′) iff N ⊆ N ′ and R ⊆ R′ . (22)

73We assume here that M ∩Ω = Ø. Otherwise one has to choose a set M ′ (of the same cardinalityas M) with M ′ ∩ Ω = Ø and to choose a bijection h : M ′ →M instead of idM,A, below.

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Then it is obvious that I t := (It; v) is a directed set. For (N,R) v (N ′, R′) in I t oneobviously has ConT (N,Σ)R ⊆ ConT (N ′,Σ)R

′ ⊆ ΘM,A, and therefore one has — by theFirst Diagram Completion Lemma — homomorphisms

f(N,R),(N ′,R′) : T (N,Σ)/ConT (N,Σ)R→ T (N ′,Σ)/ConT (N ′,Σ)R′ ,

andf(N,R) : T (N,Σ)/ConT (N,Σ)R→ A

induced by idN,N ′ and idN,M , respectively. Observe that we consider here T (N,Σ) asa subalgebra of T (N ′,Σ) for N ⊆ N ′ ⊆M . Then

AIt := (T (N,Σ)/ConT (N,Σ)R, f(N,R),(N ′,R′) | (N,R) v (N ′, R′) in I t ) (23)

is a directed system and

( ( f(N,R) : T (N,Σ)/ConT (N,Σ)R→ A )(N,R)∈It , A ) (24)

is a directed colimiting cocone of AIt , as can easily be realized.We leave the details of the proof of the last statements as an exercise as well as

the proof of the fact that a total algebra is finitely presentable, iff it is isomorphic tosome T (N,Σ)/ConT (N,Σ)R with N and R finite, which follows the line of the proofabove in the partial case.

We collect the statement in part (ii) of the above proof in:

Corollary 7.29 Each total algebra is the directed colimit of a directed sys-tem of finitely presentable total algebras. Each A ∈ TAlg(Σ) with generatingsubset, say M , is a directed colimit of the directed system

AIt := (T (N,Σ)/ConT (N,Σ)R, fij : T (N,Σ)/ConT (N,Σ)R→ T (N ′,Σ)/ConT (N ′,Σ)R′ |

| i =: (N,R), j =: (N ′, R′) and i v j in I t ) ,

where

It := (N,R) | N ⊆M finite, and R ⊆ ΘM,A finite , and

ΘM,A := ker idM,A with idM,A : T (M,Σ)→ A surjective.

Observation 7.30 Directed colimits provide another possibility to define reducedproducts: If (Ai)i∈I is a family of partial algebras of signature Σ, if F is a filter on I,and if we define AF :=

∏i∈F Ai ; and, for F, F ′ ∈ F with F ⊇ F ′, prFF ′ : AF → AF ′ ,

— (ai | i ∈ F ) 7→ (ai | i ∈ F ′) — to be the corresponding projection, then

AF := (AF , prFF ′ : AF → AF ′ | F, F ′ ∈ F , F ⊇ F ′ )

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is a directed system. And if one is in the one-sorted case and all Ai have non-emptycarrier sets, or if one has the many-sorted case with a finite set S of sorts, then onehas lim→AF ∼=

∏i∈I Ai/F .

If, however, in the one-sorted case, Ai = Ø for at least one i ∈ I, and I \ i ∈ I |Ai = Ø ∈ F , then

∏i∈I Ai/F = Ø,74 while

lim→AF = lim

→AF∈F|F⊆I\i∈I|Ai=Ø

is non-empty.The isomorphism between the two objects in the case of finitely many sorts and

with IA,F :=⋂s∈S Is as in Remark 7.10.(ii) is induced by a mapping which can be

defined as follows:

• First recall that, by Lemma 7.20 we get the same directed colimit, if we restrictconsiderations to a confinal directed subsystem. Therefore, prove first that

AIA,FC := (AF , prFF ′ : AF → AF ′ | F, F ′ ∈ F , F ⊇ F ′ ;F, F ′ ⊆ IA,F )

is a confinal subsystem of AF .

• For s ∈ S let as := (as,i | i ∈ IA,F) be a fixed sequence, if Is := i ∈ I | (Ai)s 6=Ø ∈ F , and there is no such sequence, otherwise.

• For F ∈ F ′ := F ′ ⊆ IA,F | F ′ ∈ F , and for (ai | i ∈ F ) ∈ (AF )s (forsome s ∈ S) set (gF )s : (ai | i ∈ F ) 7→ (ai | i ∈ IA,F), where ai := as,i for alli ∈ IA,F \ F . Then gF : AF →

∏i∈IA,F Ai/F

′ is a homomorphism, the family

(gF )F∈F ′ is compatible, and the induced homomorphism from the colimit objectinto

∏i∈IA,F Ai/F

′ is an isomorphism.

We shall refer to the resulting colimit object of this category theoretic constructionof a reduced product as to the category theoretic reduced product. It also yieldsthe “correct” construction, i.e. the one needed later for the Birkhoff-type theorems forECE- and QE-equations, when the set S of sorts is infinite. Therefore, if not stateddifferently, when we speak from now on of a reduced product of some family (Ai)i∈Iwith respect to some filter F on the index set I, then we mean the partial algebraof signature Σ obtained as colimit object of the category theoretic construction. Theother construction will be referred to as model theoretic reduced product.

7.3 Some operators derived from the constructions

Having at hand several ways to construct new partial algebras from given ones, theseconstructions lead to operators on the class of all partial algebras. These operatorsand their “behaviour with respect to other ones” will be of interest in connection withimplicational model theory of partial algebras. (We repeat some “old” ones, too.)

74This applies, if we have not chosen a construction according to the remark close to the end ofRemark 7.10.(ii).

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Definition 7.31 (Further operators): Let K ⊆ PAlg(Σ) be any class of partialalgebras of the same signature Σ. Then we define in addition to Definition 6.14

HK := HwK := B ∈ PAlg(Σ) | there are A ∈ K and a surjective homomorphismf : A→ B = class of all (weak) homomorphic images of K-algebras.

HfK := B ∈ PAlg(Σ) | there are A ∈ K and a full and surjective homomorphismf : A→ B = class of all full homomorphic images of K-algebras.

HcK := B ∈ PAlg(Σ) | there are A ∈ K and a closed and surjective homomorphismf : A→ B = class of all closed homomorphic images of K-algebras.

IK := B ∈ PAlg(Σ) | there are A ∈ K and an isomorphism f : A → B = class ofall isomorphic copies of K-algebras.

SK := ScK := A ∈ PAlg(Σ) | there are B ∈ K and a closed and injective homomor-phism f : A → B = class of all isomorphic copies of (closed) subalgebrasof K-algebras.

SfK := A ∈ PAlg(Σ) | there are B ∈ K and a full and injective homomorphismf : A → B = class of all isomorphic copies of relative subalgebras of K-algebras.

SwK := A ∈ PAlg(Σ) | there are B ∈ K and an injective homomorphism f : A→ B= class of all isomorphic copies of weak relative subalgebras of K-algebras.item[] PK := B ∈ PAlg(Σ) | there are a set I and a family (Ai)i∈I ∈ KI suchthat B ∼=

∏i∈I Ai = class of all products of families of K-algebras.

P+K := B ∈ PAlg(Σ) | there are a non-empty set I and a family (Ai)i∈I ∈ KI

such that B ∼=∏

i∈I Ai = class of all products of non-empty families ofK-algebras.

PrK := B ∈ PAlg(Σ) | there are a set I, a filter F on I and a family (Ai)i∈I ∈ KI

such that B ∼= (∏

i∈I Ai )/F = class of all isomorphic copies of reducedproducts of families of K-algebras.

Pr+K := B ∈ PAlg(Σ) | there are a non-empty set I, a filter F on I and a family(Ai)i∈I ∈ KI such that B ∼= (

∏i∈I Ai)/F = class of all isomorphic copies of

reduced products of non-empty families of K-algebras.

PuK := B ∈ PAlg(Σ) | there are a set I, an ultrafilter U on I and a family(Ai)i∈I ∈ KI such that B ∼= (

∏i∈I Ai)/U = class of all isomorphic copies of

ultraproducts of families of K-algebras.

oK := K ∪ Ø = K and the empty partial algebra.

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eK := K∪ T | T is a total algebra on an S-set with a one-element set for each sorts ∈ S = union of K with the class of all products with an empty index set.

Let O be the semigroup generated by the operators in Definitions 6.14 and 7.31with respect to composition. Then O consists of finite sequences Y1 . . .Yn — for somenatural number n ≥ 1 —, where each Y is one of the above operators. We define forY ,Y ′ ∈ O:

Y ⊆ Y ′ if and only if for every K ⊆ PAlg(Σ) one has YK ⊆ Y ′K.

And we set Y = Y ′, if Y ⊆ Y ′ and Y ′ ⊆ Y . For convenience we add to the generatingset of operators of O the operator i: iK := K for each K ⊆ PAlg(Σ). Thus O is in facta monoid.

Definition 7.32 A class K ⊆ PAlg(Σ) of partial algebras of signature Σ is called

– primitive, if K = HSPK = HwScPK,

– quasiprimitive, if K = SPK = IScPK.

Lemma 7.33 (Some properties of the monoid O of operators): Let Y ∈ O be anoperator on PAlg(Σ), then one has:

(i) iY = Yi = Y and K ⊆ K′ ⊆ PAlg(Σ) implies YK ⊆ YK′.

(ii) i ⊆ Y, and, for V ,W ∈ O, V ⊆ W implies YV ⊆ YW and VY ⊆ WY.

(iii) Y ⊆ Y, and for V ,W ∈ O one has that Y ⊆ V and V ⊆ W imply Y ⊆ W, i.e.“⊆” is a quasi-order relation on O.

(iv) IY = YI.

(v) If Y ∈ H,Hf ,Hc,S,Sr,Sw, I,P ,P+,Pr,Pr+,Pu or if Y ∈ O is a sequencewhich somewhere contains one of these operators, then IY = Y.

(vi) If Y is one of the operators generating O, then YY = Y.

(vii) P = eP+ = P+e = eP = Pe and Pr = ePr+ = Pr+e = ePr = Pre.

(viii) I ⊆ Hc ⊆ Hf ⊆ H,I ⊆ S ⊆ Sr ⊆ Sw,I ⊆ Pu ⊆ Pr,I ⊆ P,P+ ⊆ P ⊆ Pr,P+ ⊆ Pr+ ⊆ Pr.

(ix) Let Y ∈ I,H,Hf ,Hc, V ∈ S,Sr,Sw, W ∈ P,P+,Pr,Pr+,Pu, then onehas: VY ⊆ YV, WY ⊆ YW, and WV ⊆ VW.

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(x) With the notation from (ix) we have that all of the operators Y, V, W, YV,YW, VW and YVW are closure operators on PAlg(Σ), i.e. they are monotonic,extensive and idempotent.

(xi) Pr ⊆ SPPu,Pr+ ⊆ SP+Pu.If S is finite, then Pr+ ⊆ HP+, and Pr ⊆ HP .

Proof We do not prove all details but only those of greatest interest later. Statements(i), (ii), (iii), (iv), (v), (vii) and (viii) follow directly from the definition of the orderrelation in O and of the operators involved, or their proofs are quite obvious. We shallnot discuss (vii) here. Since the combinations of operators occurring in connectionwith the description of E-, ECE- and QE-varieties are closure operators just becauseof the properties of Galois correspondences induced by relations, we also skip mostof the proofs of (x).

Ad (vi): It is obvious that the (closed) homomorphic image of a (closed) homo-morphic image is a (closed) homomorphic image, i.e. HH = H (where H stands forI, Hw or Hc). Similarly, a subalgebra of a subalgebra is again a subalgebra; henceScSc = Sc .PP = P : P ⊆ PP is obvious. Let, for j ∈ J , (Bj =

∏k∈Ij Ak,j, (prk,j : Bj →

Ak,j | k ∈ Ij )) be products. Set I :=⋃j∈J Ij × j . Then there is a “natural

isomorphism” (i.e. one induced by the product property):∏j∈J

Bj∼=∏

(k,j)∈I

Ak,j .

If (fj : C → Bj)j∈J is a family of homomorphisms from C into the Bj , then (prk,j fj :C → Ak,j)(k,j)∈I is a family of homomorphisms from C into the Ak,j Then one hasa unique induced homomorphism f : C →

∏(k,j)∈I Ak,j (choose in particular C :=∏

j∈J Bj , and fj := prj (j ∈ J)). Conversely, a family (gk,j : C → Ak,j )(k,j)∈Iinduces a family (gj :=< gk,j | k ∈ Ij >: C → Bj)j∈J , which then induces a uniquehomomorphism g : C →

∏j∈J Bj . By choosing here C :=

∏(k,j)∈I Ak,j and gk,j :=

prk,j , ((k, j) ∈ I) one gets isomorphisms inverse to one another.PrPr = Pr : The first claim here is, that — with a notation concerning the partial

algebras and index sets as above for products — for filters Fj on Ij (j ∈ J) and afilter F on J one gets a filter on I by setting

G := ⋃j∈F

Fj | F ∈ F and Fj ∈ Fj for each j ∈ F .

One then has to show that the reduced product (∏

j∈J(∏

k∈Ij Ak,j)/Fj)/F is isomor-

phic to (∏

(k,j)∈I Ak,j)/G . We do not go into the details here.

Ad (ix): We discuss here only some of the cases which are most important inlater applications:

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PSc ⊆ ScP : Let I be any index set, and let, for each i ∈ I, Bi be a subalgebraof Ai . Then it easy to check that

∏i∈I Bi is indeed a subalgebra of

∏i∈I Ai .

ScH ⊆ HSc (H standing here for I, Hw or Hc): Let C be a subalgebra of B, andlet f : A→ B be a (closed and) surjective homomorphism. Then f−1(C) is a closedsubset of A, and C is a (closed) homomorphic image (w.r.t. f |f−1(C) of the subalgebraf−1(C) of A.PH ⊆ HP (H standing here for I, Hw or Hc): Let fi : Ai → Bi be (closed and)

surjective homomorphisms, let (B :=∏

i∈I Bi , (qi : B → Bi)i∈I be the product ofthe homomorphic images, and let (A :=

∏i∈I Ai , (pi : A → Ai)i∈I be the product

of the preimages. Then the family (fi pi : A → Bi)i∈I induces a homomorphismf : A → B . We have to show that f is (closed) and surjective (we need the Axiomof Choice):

Let (bi | i ∈ I) ∈×i∈I

Bi . Then there exists a sequence (ai | i ∈ I) ∈×i∈I

Ai with

fi(ai) = bi for each i ∈ I. Therefore one gets f((ai | i ∈ I)) = (bi | i ∈ I), showingthat p is indeed surjective. In a similar way one sees that f is closed, if all fi areclosed (exercise).

The corresponding statements concerning the relationships PrSc ⊆ ScPr andPrH ⊆ HPr follow from Proposition 7.22, when we use the category theoreticaldefinition of reduced products as given in Observation 7.30. We leave the details asexercises.

Ad (x): As an example we only show thatHSP (= HwScP) is a closure operator,by using (iii), (vi), (ix) and the fact that always Y ⊆ YY for all Y ∈ O:HSP ⊆ HS(PH)SP ⊆(ix) H(SH)PSP ⊆(ix) HHS(PS)P ⊆(ix) (HH)(SS)(PP) =(vi)

HSP . This implies equality everywhere.Ad (xi): From Remark 7.10.(i) we know for the homogeneous case that the

structure of the reduced product is in general stronger than the quotient structure ofthe direct product; and from Remark 7.10.(ii) we may conclude this also for the caseof finitely many sorts. That this is no longer the case in general for infinitely manysorts follows from Example 7.34 below.

Example 7.34 (that for infinitely many sorts Pr ⊆ HwP need not hold, evenin the case of total heterogeneous algebras):

Assume the set of sorts to be S := N, the set of all natural numbers, and let Ωcontain, for each sort s ∈ N, exactly one unary operation symbol,

ϕs : s→ s, for all s ∈ S .

For our purpose let us consider a set

A := An | n ∈ N

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of total S-sorted algebras, where, for each n ∈ N,

An,s :=

an,s, bn,s , with an,s 6= bn,s, if s 6= n,Ø, if s = n,

ϕAns (an,s) := bn,s, ϕ

Ans (bn,s) := bn,s if s 6= n , (25)

and if s = n, then graph ϕAns = Ø (n ∈ N , s ∈ S = N) .

Let us consider the primitive class

K := HwScPAn | n ∈ N .

Then the only algebras in K, which have non-empty carrier sets for all s ∈ S = N,are isomorphic to the “sort algebra”

S = (N; (ϕSs : s 7→ s | s ∈ S))

for this signature, since each direct product with non-empty index set of any familyof the algebras An (n ∈ N) would have for at least one sort s an empty carrier set ofsort s, and this fact would carry over to all subalgebras and homomorphic images ofsubalgebras. This means among others that K satisfies the infinite existence-equations

(( y0s , y

1s | s ∈ N) : y0

u

e≈ y1

u) (26)

for each natural number u ∈ S ,

which can also be interpreted as an infinite ECE-equation∧s∈S

(y0s

e≈ y0

s ∧ y1s

e≈ y1

s)⇒ y0u

e≈ y1

u for each sort u ∈ S , (27)

However, let us consider on the other hand the category theoretical reduced prod-uct (cf. Observation 7.30), say

((prE :∏n∈E

An → P )E∈F , P ) ,

of the family (An | n ∈ N) with respect to the Frechet filter

F = E | E ⊆ N and N \ E is finite

of cofinite subsets of N , i.e. the directed colimit with colimit object P of the directedsystem

(∏n∈E

An, prE,E′ :∏n∈E

An →∏n∈E′

An | E,E ′ ∈ F , E ⊇ E ′ ),

where prE,E′(an | n ∈ E) := (an | n ∈ E ′)) means the restriction to the subsequence.

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Observe that, for E = N \ F with a finite set F , the algebra AE :=∏

n∈E Anhas non-empty carrier sets only for the sorts s ∈ F , and for those s also the corre-sponding operations exist and there are at least two different elements. However, bythe formation of the directed colimit all the carrier sets of any sorts of P becomenon-empty and contain at least two different elements for all sorts s ∈ S = N. Thisshows that P /∈ K, i.e. K is not closed with respect to category theoretical reducedproducts (and hence with respect to directed colimits). Observe that all algebrasunder consideration are total.

In particular, this example shows that Pr 6⊆ HwP , if the signature specifies in-finitely many sorts.

It should, however, be observed that this example is more of theoretical interest,since in computer science one will usually try to have enough constants in the sig-nature (and E-equations in the specification) such that all sorts of the models willbecome non-empty, if all of them are defined. Moreover, there will be much more“interconnections” between the different sorts because of fundamental operations as-signing to a sequence of some “sort pattern” an element of an additional sort. Yet inconnection with some specifications it may nevertheless occur that the “sort algebra”S may be covered by infinitely many pairwise disjoint subalgebras, and that there areΣ-algebras for which the carrier sets of the sorts corresponding to such subalgebrasmay occasionally be empty. And in each such situation the above observation applies.— Up to my knowledge a similar observation does not yet occur in any book treatingmany-sorted (partial) algebras.

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8 Free partial algebras, universal solutions and E-equations 159

8 Free partial algebras, universal solutions and E-

equations

8.1 Some descriptive principles of the universal algebraic lan-guage

Remark 8.1 In subsection 3.1 we have presented a first order language allowingto formulate properties (in the form of axioms), which we can choose to characterizeclasses of algebraic objects under consideration; and we defined appropriate semanticsfor it with respect to partial algebras. This required the specification of a signaturebefore doing anything else in order to describe the correct framework and to specifythe important operations to be considered. In subsections 1.3 and 5.1 we have seenthat some (partial) algebraic structures are in some respect already determined bysome — usually relative small — generating sets, once we know to which axiomaticclass it has to belong (e.g. if one knows that an algebra is to be a vector space overthe field of all real numbers, this is determined up to isomorphism by specifying inaddition, how large a minimal generating set has to be). However, with groups thesituation is quite different. Knowing that the algebra has to satisfy the axioms of agroup and that it has a (minimal) generating set consisting of five elements, one stillhas a large number of models, satisfying this specification without being isomorphic.Thus there lacks a further property. The difference between our two examples is thefact that for vector spaces every minimal generating subset is already a basis, i.e. itis a free generating subset of the vector space. This concept of freeness, which weintroduce and discuss in the following subsubsections, yields the fourth principle usedfor the description of structures (sometimes it is “weakened” a little bit to the moregeneral concept of universal solutions — see the end of subsection 8.3). Observe,however, that “freeness” is a second order concept. Nevertheless it is quite useful,since it allows the specification up to isomorphism.

In computer science one usually restricts the generating set in this connectionto the empty set (“missing” generators are introduced via nullary constants) thusconsidering the so called initial object of the class axiomatized by the given axioms.

Therefore we have the following four concepts needed for a specification of astructure up to isomorphism:

• signature (to specify the important operations (and possibly relations)),

• first order axioms (to specify the behaviour of the structure — including inthe partial case references to what should be defined by all means),

• generation (allowing to construct elements from the given generators alongthe required axioms using the specified operations),

• freeness (singling out from the usually very complex axiomatic class one veryspecific object (up to isomorphism); in addition freeness takes care of most

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requirements concerning undefinedness, since in such a free object no instanceof an operation is defined, which is not required to be defined by the axioms).

Having already discussed the specification of signatures, the formulation of axioms(as far as we shall need them), and the principle of generation, we want to give in thefollowing subsections a definition and characterization of the concept of freeness.

8.2 Free partial algebras

In the universal algebraic theory of total algebras freeness is a most fundamentalconcept, since every total algebra of an equational variety is the (naturally closed)homomorphic image of a free algebra in that variety, and if — at least in the homoge-neous case — it has an infinite free generating set then the set of all equations validin the whole variety is identical with the set of all equations valid in that free algebra— for heterogeneous signatures we usually need a family related to the power set of Sof such free algebras. In the theory of partial algebras this is almost true except thatone can only state that every partial algebra in that E-variety is a weak homomorphicimage of a free algebra, yet not necessaryly a full or closed homomorphic image. Andin ECE- or QE-varieties the more general concept of universal solution (see below)becomes of more importance than the one of freeness.

Definition 8.2 of independence and free algebras: Let A be a partial algebraof signature Σ, let K ⊆ PAlg(Σ) be any class of partial algebras, and let M ⊆ A beany S-subset. Then we define:

(i) M is called a K-independent subset or a K-free subset of a partial algebraA (M is K-free in A), if for every partial algebra B of K and for every mappingf0 : M → B there exists a homomorphic extension f : CAM → B with f |M =f0.

(ii) The subclass of PAlg(Σ), defined as

indAM := B ∈ PAlg(Σ)|M is B-free in A

is called the independence class of M with respect to the partial algebra A.

(iii) Let M be a generating subset of A, let K = A, and let M be K-free in A,then we say that A is a (relatively) free partial algebra, freely generatedby M , or — more briefly — a free partial algebra with basis M .

(iv) Let M generate A, let A ∈ K, and let M be K-free in A, then we say that A isa K-free K-algebra, K-freely generated by M (a K-free K-algebra withK-basis M). Since we will show below that a K-free K-algebra with K-basis M

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is determined by M and K up to isomorphism, we shall denote it by F (M,K),if it exists at all.75

Remarks 8.3 and examples of free partial and total algebras

(i) A K-free K-algebra on some set M need not exist in general (i.e. within K). E.g.in the class F of all fields with arbitrary characteristics an F-free field does notexist on any set. However, let p be any prime number or 0, let Fp designatethe class of all fields of characteristic p, and let Fp denote the prime field ofcharacteristic p (with a similarity type containing two nullary constants for 0and 1), then Fp ∼= F (Ø,Fp) ∈ Fp, while for sets M 6= Ø a free field F (M,Fp)does not exist within Fp. (Observe that homomorphisms between fields havealways to be injective.)

(ii) We shall see later that for any primitive or quasi-primitive class K the K-freealgebra F (M,K) always belongs to K for every set M . However, F (M,K) willalways be defined, even if it does not belong to K. Namely, it is more generallydefined as the object of the K-universal solution of Mdiscrete (see subsection 8.3).

(iii) Let A be a nonempty partial algebra generated by the empty set, let K be anyclass of partial algebras of the same signature, and assume that K contains theempty partial algebra; then Ø cannot be K-free in A.

(iv) For every partial algebra A and for every set M the independence class indAMalways contains all total algebras with exactly one element in each phylum.Therefore it can never be empty.

(v) All total free algebras like e.g. free monoids (=word monoids), free groups,free rings (=polynomial rings over the ring Z of integers) are also examplesof (relatively) free partial algebras. In particular, let VK be the class of allvector spaces (homogeneous) over some field K. Then one knows — using theAxiom of Choice — that every vector space V ∈ VK has some basis, say B,and therefore V ∼= F (B,VK), i.e. every K-vector space is VK-freely generated.

(vi) Let C be the class of all (many-sorted) small categories, let M be any setconsidered as a set of morphisms. Then define

Mi := (m, i) | m ∈M for i ∈ 0, 1, 2, 3 ,

OF := M0 ∪M1

andMF := M2 ∪M ∪M3 ,

75See item (ii) of the remarks below. By F (M,K) we shall also denote the free algebraF (M, IScP(K)), which always exists in this quasi-primitive class generated by K, as we shall seelater.

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and assume that M ∩ Mi = Mj ∩ Mi = Ø for i 6= j in 0, 1, 2, 3. ThenF := (OF ,MF ) is the carrier set of the free category F which is C-freelygenerated by M as a set of morphisms, when we define for each m ∈M :

DomF (m) := DomF (m, 2) := CodF (m, 2) := (m, 0),

CodF (m) := CodF (m, 3) := DomF (m, 3) := (m, 1),

dom F :=⋃ (m, (m, 2)), ((m, 3),m), ((m, 2), (m, 2)), ((m, 3), (m, 3)) |

m ∈M,and F is then defined according to the rule (C 2) for small categories given inExample 1.12 of subsection 1.2.

(vii) For every S-set X the Peano-algebra (or term algebra) T (X,TAlg(Σ)) is a freeTAlg(Σ)-algebra TAlg(Σ)-freely generated by X. And Xdiscrete, the discretepartial Σ-algebra on X is a free PAlg(Σ)-algebra PAlg(Σ)-freely generated byX.

Proposition 8.4 Independence classes are always primitive.For any partial algebra A and for every subset M of A the independence class indAMis always a primitive class: HwScP indAM = indAM .76

And if Ms 6= Ø for some s ∈ S, then trivially N ∈ indAM for every N ∈ PAlg(Σ)with Ns = Ø.

Proof The last statement is obvious, since then there is no mapping from M intoN at all.

Assume Ai ∈ indAM for all indices i in some set I, let (P , (pri : P → Ai)i∈I) bea product of the family (Ai)i∈I , and let f : M → P be any mapping. Then eachfi := pri f : M → Ai has, by assumption, a homomorphic extension fi : CAM → Aisuch that fi|M = fi = pri f , for each i ∈ I. The product property then implies thatthere exists a (unique) homomorphism f : CAM → P such that pri f = fi for each

i ∈ I, what immediately implies f |M = f .Consider a subalgebra B of some C ∈ indAM , and let f : M → B be any

mapping. Then f can also be considered as a mapping from M into C and thereforehas a homomorphic extension f ′ : CAM → C. However, its image then belongs to thesubalgebra B of C, and therefore we can also consider it as a homomorphic extensionf : CAM → B of f .

Finally, assume C ∈ indAM , let g : C → B be a surjective homomorphism, andlet f : M → B be any mapping. Then there is (Axiom of Choice, if M is infinite)a mapping h : M → C such that g h = f . By assumption, h has a homomorphicextension h : CAM → C, and therefore f := g h : CAM → B is the requiredhomomorphic extension of f .

76For the closure w.r.t. homomorphic images the Axiom of Choice is needed.

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Corollary 8.5 Some equivalent characterizations of free partial algebras:Let A ∈ PAlg(Σ) with some generating subset M . Then the following statements areequivalent:

(i) A is a free partial algebra with basis M .

(ii) There is some primitive class K containing A such that A is K-freely generatedby M .

(iii) A ∈ indAM .

(iv) HwScPA ⊆ indAM .

Proof (i) ⇒ (ii): If A is a free partial algebra with basis M , then A ∈ indAM , andaccording to Proposition 8.4 K := indAM is a primitive class, and by definition andassumption A is K-freely generated by M .

(ii) ⇒ (iii): If K is a primitive class with A ∈ K such that M K-freely generatesA, then, obviously, M freely generates A, and therefore A ∈ indAM .

(iii) ⇒ (iv): This is an immediate consequence of Proposition 8.4 and items (iii)and (x) of Lemma 7.33.

(iv) ⇒ (i): If HwScPA ⊆ indAM , then in particular A ∈ indAM , and thereforeevery mapping from M into A can be extended to a homomorphism f : A → Ashowing that M freely generates A.

8.3 Free completions

Now we are generalizing the concept of total Peano algebras as follows:

Definition 8.6 of the (absolutely) free completion of a partial algebra: LetA ∈ PAlg(Σ) be any partial algebra. B ∈ TAlg(Σ) is called an (absolutely) freecompletion of A — and then it is often denoted by A — if the following axiomshold:

(FC0) A ⊆ B and for each ϕ ∈ Ω one has graph ϕA ⊆ graph ϕB.

(FC1) For each ϕ ∈ Ω and for each sequence b = (b1, . . . , bτ(ϕ)) ∈ Bη(ϕ)77 one hasϕB(b) ∈ A implies ϕB(b) = ϕA(b) (thus in particular b ∈ dom ϕA ⊆ Aη(ϕ)).

(FC2) For all ϕ, ψ ∈ Ω, for all b = (b1, . . . , bτ(ϕ)) ∈ Bη(ϕ), and for all b′ = (b′1, . . . , b′τ(ψ)) ∈

Bη(ψ) one has: ϕB(b) = ψB(b′) /∈ A implies ϕ = ψ and b = b′.

77Recall thatBη(ϕ) := Bs1 × . . .×Bsτ(ϕ) ,

whenever η(ϕ) = s1 . . . sτ(ϕ) .

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(FC3) (Axiom of Induction) CBA = B (i.e. the completion B of A is minimal).

Remarks 8.7 on the connections between the axioms for absolutely freecompletions and the generalized Peano axioms and on error handling:

(i) The axioms (FC0) through (FC3) above are closely related to the generalizedPeano axioms: (FC0) tells us that A is at least a weak relative subalgebra of B,while (FC1) says that A is even a normal relative subalgebra of B (normalitymeans that no application of a fundamental operation to a sequence with at leastone argument outside of A can have its value in A). This fact corresponds to(P1). (FC2) says that “outside” of A (P2) is satisfied, while (FC3) correspondsto (P3). This observation shows that if we forget for a free completion B of Athe structure of A in B, then we end up with a partial Peano algebra on A (i.e.with Peano basis A and carrier B).

(ii) Absolutely free completions of A are important in connection with a descriptionof all minimal completions of A. Therefore they are a useful tool in connectionwith error handling for the specification of abstract data types.

As an immediate consequence we get:

Corollary 8.8 A total algebra B is a Peano algebra over some subset X iffthe relative subalgebra X of B is discrete and B is the free completion ofX.

The existence of a free completion of any given algebra A can be based upon theexistence of total Peano algebras:

Let P be a total Peano algebra on A, and define for each ϕ ∈ Ω and for eachb = (b1, . . . , bτ(ϕ)) in P η(ϕ) on P :

ϕ#(b) :=

ϕA(b) , if b ∈ dom ϕA

ϕP (b) , else.

Let P# := (P, (ϕ#)ϕ∈Ω) and B := CP#A.

Theorem 8.9 Construction of the absolutely free completion by modifyinga total Peano algebra:B := CP#A is a free completion of A.

Proof (FC0) and (FC1) are given by the definition of the structure of P# and thefact that P satisfies (P1); (FC2) follows from (P2), and (FC3) is a consequence ofthe definition of B as CP#A.

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Remark 8.10 Construction of free completions as quotients of total Peanoalgebras:Consider again the Peano algebra P on A as above. Consider on P the congruencerelation Θ, generated by the relation

R := (ϕA(a1, . . . , aτ(ϕ)), ϕP (a1, . . . , aτ(ϕ)))|ϕ ∈ Ω, (a1, . . . , aτ(ϕ)) ∈ dom ϕA.

Then there is a full and injective homomorphism i : A → P/Θ such that P/Θ isa free completion of the relative subalgebra i(A) of P/Θ ( and i(A) is isomorphicto A by i). Once we have the characterization of free completions by the RecursionTheorem 8.11 for Free Completions below, the proof of this statement is a not sodifficult excercise.

Observe that the construction of free completions in this remark is the morecanonical one, while the first one, used in Theorem 8.9, gives more directly insightinto the structure of the result and can be compared with the modelling of algebrasin computer science (see [GoTcWa78]).

In analogy to the Recursion Theorem for Peano algebras we have the

Theorem 8.11 Recursion Theorem for Free Completions Let A be any partialalgebra, B a free completion of A, and C be any total algebra similar to A. Then, forevery homomorphism f : A → C there exists a unique homomorphism f ′ : B → Cextending f , i.e. f ′|A = f .

Proof Since B is a free completion of A, axioms (FC2) and (FC3) tell us that B∗ :=(B; (ϕ∗)ϕ∈Ω) is a partial Peano algebra on A, when we define ϕ∗ to be the restrictionof ϕB to Bη(ϕ) \ domϕA for each ϕ ∈ Ω. Then the First Recursion Theorem 2.8guarantees the existence of a homomorphic extension f ′ : B∗ → C with f ′|A = f .Since, by assumption, f ′|A = f is already a homomorphism from A into C, f ′ iscompatible with all of the structure of B and therefore a homomorphism f ′ : B → Cextending f .

As in the case of total Peano algebras we get as immediate consequence:

Corollary 8.12 Let A ∈ PAlg(Σ) be any partial algebra. Then there is up to isomor-phism (over idA as restriction to A) exactly one free completion, say A := B, of A.

Definition 8.13 and Remarks (K-extendable and K-universal epimorphismsand K-universal solutions)

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C D-f : K-ext.

E ∈ K

g

@@@@@R

7→

C D-f : K-ext.

E ∈ K

g

@@@@@R

h : h f = g

Figure 18: f is K-extendable

C D′-f ′ : K-ext. epi.

D

f : K-univ.

@@@@@R

7→

C D′-f ′ : K-ext. epi.

D

f : K-univ.

@@@@@R

l : l f ′ = f

Figure 19: f is a K-universal epimorphism.

(i) Because of its “similarity” with a TAlg(Σ)-free partial algebra with TAlg(Σ)-basis A we also denote the absolutely free completion of A by F (A,TAlg(Σ)).Indeed the free completion F (A,TAlg(Σ)) of A together with the canonicalembedding

i = idA,F (A,TAlg(Σ)) = idA,A : A→ F (A,TAlg(Σ)) = A

is a special instance of some more general concept (observe that i is an epimor-phism because of (FC3)):

Let K ⊆ PAlg(Σ) be any class of partial algebras, and let f : C → D be anyhomomorphism in the category PAlg(Σ).

(a) f is called K-extendable, and K is said to be injective w.r.t. f , iff forevery E ∈ K and for every homomorphism g : C → E there exists ahomomorphism h : D → E such that g = h f (cf. Figure 18).

Observe that we can show later78 that the K-extendable epimorphismsrepresent implications, which are valid in K.

(b) The homomorphism f : C → D is called K-universal (a K-universalepimorphism), if it is a K-extendable epimorphism, and if, for everyK-extendable epimorphism f ′ : C → D′, there exists a homomorphisml : D′ → D such that l f ′ = f (cf. Figure 19).

Observe, that then also f ′ is required to be an epimorphism.

(c) If f : C → D is a K-universal epimorphism, then the pair (f,D) is called aK-universal solution of C. And if, in addition, D ∈ K, then D is calleda K-universal K-solution of C.

78See Lemma 9.5.

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(ii) Observe, that for every partial algebra A which is K-freely generated by a subsetM the homomorphism idM,A : Mdiscrete → A is a K-extendable epimorphism;and if A ∈ K, then (idM,A, A) is a K-universal K-solution of Mdiscrete.

Similarly, it is easy to realize, that

(idA,F (A,TAlg(Σ)), F (A,TAlg(Σ))

is a TAlg(Σ)-universal TAlg(Σ)-solution of A.

(iii) From the definition it easily follows that, if f : C → D and f ′ : C → D′ areK-universal, then there is a unique isomorphism l : D′ → D such that lf ′ = f .Thus K-universal solutions are determined up to unique isomorphism, and wewrite D =: F (C,K) and f =: rC,F (C,K).

(iv) For those who know a bit more about category theory, let us observe that thefact that every partial algebra A has a K-universal K-solution is equivalent tothe fact that the embedding functor I : K→ PAlg(Σ) of the full subcategory K

of PAlg(Σ) has a left adjoint.

(v) Our remarks show that a total Peano algebra (term algebra) on X is exactlythe TAlg(Σ)-algebra TAlg(Σ)-freely generated by X, i.e. the TAlg(Σ)-universalTAlg(Σ)-solution of Xdiscrete.

Because of the close relationship between total Peano algebras and free comple-tions (cf. Corollary 8.8) one of the most useful tools for a model theory of partialalgebras can be formulated and proved for free completions in general:

Theorem 8.14 Generalized Recursion Theorem Let f : A → B be any homo-morphism in PAlg(Σ). Then there exists a closed homomorphic extension

(f∼)B =: f∼ : dom f∼ → B

of f such that dom f∼ is an A-generated relative subalgebra of the (absolutely) freecompletion F (A,TAlg(Σ)) = A of A. Moreover, we have:

(i) graph f∼ = CAπB graph f .

(ii) f∼ is the largest homomorphic extension of f to an A-generated relative subal-gebra of F (A,TAlg(Σ)), and it is the only closed one of this kind.

(iii) Let f : A → B be the homomorphic extension of f , which exists according tothe Recursion Theorem 8.11 for Free Completions. Then f∼ = f |dom f∼ anddom f∼ = f−1(B).

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Proof According to the Recursion Theorem 8.11 there exists a homomorphic exten-sion f : A→ B. Moreover, A and B are relative subalgebras of A and B, respectively.Thus — as in the proof of Theorem 2.12 — graph f∼ := CA×B graph f is the graph of

a homomorphic extension of f to an A-generated relative subalgebra dom f∼ of A;and since A is total, f∼ : dom f∼ → B is closed. And, again similar to the proof ofTheorem 2.12, one can show that f∼ is the largest homomorphic extension of f toan A-generated relative subalgebra of A with respect to B, that it is closed in theonly closed one — since all strictly smaller A-initial extension cannot be closed, sincethe domain can only be enlarged as long as the extension is not yet closed. Thus itremains only to show (iii):

Obviously dom f∼ ⊆ f−1(B). Therefore — because of (ii) — we only have to showthat A generates f−1(B). This is done by algebraic induction on q ∈ A with respectto the property

H(q) := “q ∈ f−1(B) implies q ∈ Cf−1

(B)A”:

H(q) is true for all q ∈ A. Let q ∈ f−1(B), q /∈ A. Then q = ϕA( c1, . . . , cτ(ϕ) ).

Let H(ck) be true for all 1 ≤ k ≤ τ(ϕ). Since f(q) ∈ B, we have

f(q) = ϕB( f(c1), . . . , f(cτ(ϕ)) ) = ϕB( f(c1) . . . , f(cτ(ϕ)) ) ,

since B satisfies (FC1). Therefore f(ck) ∈ B, i.e. ck ∈ f−1(B), for each 1 ≤ k ≤ τ(ϕ).

Because of H(ck) we have ck ∈ Cf−1(B)A, and therefore also q = ϕA( c1, . . . , cτ(ϕ)) ) ∈

Cf−1

(B)A.

Definition 8.15 In honour of Jurgen Schmidt, who introduced these concepts in[Sch70], we call, for f : A → B, f∼B := f∼ : dom f∼ → B the closed A-initialextension of f (in F (A,TAlg(Σ))); and e.g. kerf∼ is called the S-kernel (short for“J.Schmidt-kernel”) of f (in symbols: S-ker f (:= kerf∼)).

8.4 Diagram completion II, the Epimorphism Theorem

In the theory of partial algebras and e.g. in connection with their model theory theEpimorphism Theorem below is of much more importance than the homomorphismtheorem, which, however, is an important tool for most of the proofs of the followingresults. As a preparation observe

Lemma 8.16 (Characterization of epimorphisms by their (surjective) ini-tial extensions:) The homomorphism f : A → B is an epimorphism, if and onlyif its greatest A-initial extension f∼ out of A = F (A,TAlg(Σ)) into B is surjective.

Proof Assume that f is an epimorphism. Then one has CBf(A) = B, and sincef∼ : dom f∼ → B maps dom f∼ onto the subalgebra generated by f(A), i.e. onto B,f∼ is surjective.

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D := dom f∼ B--f∼

A

idA,D

@@

@@

@@@I

I

idA,E

E := dom g∼?

?

(∃ idD,E) . . .

(∀) g

@@@@@@@R

Epi 3 f

C?

(∃h)(h f = g) iff ker f∼ ⊆ ker g∼

--g∼

F := F (A,TAlg(Σ))?

?

idE,F

Figure 20: Diagram completion for the epimorphism f

Conversely, assume that f∼ is surjective. Since it maps onto CBf(A), one hasCBf(A) = B, i.e. f is an epimorphism.

Lemma 8.17 (Diagram Completion Lemma for Epimorphisms:) Let f : A→B be an epimorphism and let g : A→ C be any homomorphism (see Figure 20).

a) Then the following statements are equivalent:

(i) There exists a unique homomorphism h : B → C such that h f = g.

(ii) ker f∼ ⊆ ker g∼ (as binary relations on A = F (A,TAlg(Σ))).

b) Let h : B → C exist satisfying h f = g, then one has:

(iii) h is an epimorphism if and only if g is an epimorphism.

(iv) h is surjective, if and only if g∼|dom f∼ is surjective (e.g. if g itself issurjective).

(v) h is injective, if and only if ker f∼ = ker g∼ ∩ (dom f∼)2.

(vi) h is closed, if and only if dom f∼ = dom g∼.

(vii) h is closed and injective, if and only if ker f∼ = ker g∼.

(viii) h is an isomorphism, if and only if g is an epimorphism and ker f∼ =ker g∼.

(ix) h is full, if and only if g∼|dom f∼ is full.

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Proof Ad (a): Consider Figure 20. f∼ is closed by definition. Since f is anepimorphism, f∼ is surjective by Lemma 8.16.

If h exists with h f = g, then h f∼ is an A-initial homomorphic extension ofh f = g, and therefore ker f∼ ⊆ ker(h f∼) ⊆ ker g∼ (compare also the First DiagramCompletion Lemma 6.11).

Conversely, assume that ker f∼ ⊆ ker g∼. In particular, one then has D :=dom f∼ ⊆ dom g∼ =: E, and therefore idD,E exists such that idD,E idA,D = idA,E.Since f∼ is closed and surjective, and since, by assumption, also ker f∼ ⊆ ker(g∼ idD,E) =ker g∼ ∩ D2, the First Diagram Completion Lemma 6.11 implies the existence of aunique homomorphism h : B → C satisfying h f∼ = g∼ idD,E. And therefore, onehas for the restrictions f and g that h f = g.

Ad (b): Assume that h exists as required. Then most of the properties (iii) upto (ix) are consequences of properties (iii) to (viii) of Lemma 6.11 applied to f∼ andg∼ idD,E:

Ad (iii): Since f is an epimorphism, the fact that h is an epimorphism impliesthat g = h f is an epimorphism. Assume that g is an epimorphism and that u h =v h for some u, v : C → F . Then u g = u h f = v h f = v g and the fact that gis an epimorphism implies u = v, showing that h is an epimorphism.

Ad (iv): h is surjective, iff (by Lemma 6.11.(v)) g∼ idD,E = g∼|dom f∼ is sur-jective.

Ad (v): h is injective, iff (by Lemma 6.11.(iv)) ker f∼ = ker(g∼ idD,E) = ker g∼∩E2.

Ad (vi): If h is closed, then h f∼ is closed and therefore equals g∼. Therefore, inparticular, dom f∼ = dom g∼. Conversely, assume dom f∼ = dom g∼ = domh f∼ =dom (h f)∼ = dom (h∼ f∼). This can only happen, if h is already closed and h∼ = h.

(vii) is a combination of (vi) and (v).(viii) is a combination of (vii) and the fact that g is an epimorphism, iff g∼ is

surjective (cf. Lemma 8.16 and Lemma 6.11.(v)).(ix) follows from Lemma 6.11.(vi).

As a corollary in particular of Lemma 8.17.(viii) we get, what J.Schmidt hascalled “General Homomorphism Theorem” (see [Sch70]) and what we call here“Epimorphism Theorem”:

Theorem 8.18 Epimorphism Theorem Let f : A → B and g : A → C be anytwo epimorphisms. Then the following statements are equivalent:

(i) There exists an isomorphism h : B → C such that h f = g.

(ii) ker f∼ = ker g∼.

In what follows we want to characterize TAlg(Σ)-extendable epimorphisms, sincethey form together with the closed homomorphisms a factorization systems which

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subsumes the factorization of valuations, thus motivating that the partial interpre-tations are actually a concept of the category PAlg(Σ) with the usual everywheredefined homomorphisms as morphisms.

Lemma 8.19 Characterization of TAlg(Σ)-extendable epimorphisms: For anepimorphism e : A→ E the following statements are equivalent:

(i) e is TAlg(Σ)-extendable;

(ii) there are an A-initial segment P of A = F (A,TAlg(Σ)) and an isomorphismj : E → P such that j e = idA,P .

In particular, a TAlg(Σ)-extendable epimorphism is full and injective onto a normal79

relative subalgebra of its target algebra.

Proof (ii) ⇒ (i): Assume that (ii) is satisfied. Let g : A → C ∈ TAlg(Σ) be anyhomomorphism. Then g has a homomorphic extension g : A → C, i.e. g idA,A =

g. Set h := (g|P ) j = g idP,A j : E → C (observe that P ⊆ A). Then, since

e = j−1 idA,P , h e = (g|P ) j e = (g|P ) idA,P = g. This shows that e is TAlg(Σ)-extendable.

(ii) ⇒ (i): Assume that e is TAlg(Σ)-extendable. Set P := dom e∼ as a relativesubalgebra of A. Observe that e∼ is surjective, since e is an epimorphism. ThenidA,P : A → P is a TAlg(Σ)-extendable epimorphism because of what we have just

shown. Let j : E → A be the homomorphism such that j e = idA,A, which exists,

since e is TAlg(Σ)-extendable. Then idA : A = P → A is a homomorphic extension

of j e∼ : P → A, and therefore j e∼ = idP . This shows that e∼ is injective. Sinceit is closed and surjective, we can conclude that e∼ : P → E is an isomorphism. Letj : E → P be its inverse. Then j e = idA,P , as was to be shown. Since idA,P is fulland injective, this also applies to e.

Theorem 8.20 Factorization Theorem for Epimorphisms and Closed Mono-morphisms: Let f : A→ B be any homomorphism, E := dom f∼ in F (A,TAlg(Σ)),and e := idA,E : A→ E the natural injection.

(i) Then f is the composition of the TAlg(Σ)-extendable epimorphism e followed bythe closed homomorphism f∼.

(ii) Assume that f = g e′, where e′ : A → C is any TAlg(Σ)-extendable epimor-phism and g : C → B is a closed homomorphism. Then there exists an isomor-phism (a unique one) j : E → C such that j e = e′ and f∼ = g j.

Proof Ad (i): From Lemma 8.19 we infer that e is a TAlg(Σ)-extendable epimor-phism, and, by definition, f∼ is closed.

79See Remark 8.7.(i).

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A B

E = dom f∼

C

f

e

epiTAlg(Σ)-ext.

f∼

closed

e′

epiTAlg(Σ)-ext.

g

closed

∼= j

-

QQQQQs

3A

AAAAAAAAAAAAAAU ?

Figure 21: Unique factorization into TAlg(Σ)-ext. epi and closed homo.

Ad (ii): Consider Figure 21, and let the assumptions be as in the theorem. Sinceg is closed, the Diagram Completion Lemma 8.17 for Epimorphisms, statement (vi),implies that dom dom e∼ = f∼ = dom (e′)∼; moreover, ker e∼ = ∆dom e∼

, and there-fore ker e∼ ⊆ ker(e′)∼. Hence Lemma 8.17 implies the existence of j : E → C suchthat j e = e′. Because of dom e∼ = dom (e′)∼ j is closed. From Lemma 8.19 weknow that there is an isomorphism k : C → P , where P is an A-initial segment of Aand k e′ = idA,P . Then k−1 idA,P = e′, and therefore k−1 = k−1 idP : P → Cis an A-initial homomorphic extension of e′. Hence E = dom e∼ ⊆ P ⊆ (e′)∼

and j e∼ = k|E, and e∼ = idE . Therefore, j = k|E (e∼)−1 = k|E is injective.Moreover, h := g k−1 : P → B is a closed homomorphism such that h idA,P isa closed homomorphism (since g and k−1 are closed) from an A-initial segment ofA into C with h idA,P = f . By the Generalized Recursion Theorem 8.14, state-ment (ii), f∼ is the largest such extension, and it is the only closed one. ThereforeP = E, and k j = idE, and we know that k is an isomorphism. Thus j is anisomorphism inverse to k. Finally, observe that g j = (g k−1) (k j) = f∼, sincef∼ e = f = g e′ = g (j e) = (g j) e, and since e is an epimorphism. This endsthe proof.

We will return later to this corollary (see Proposition 9.13) in order to realizethat the TAlg(Σ)-extendable homomorphisms and the closed homomorphisms aretwo “partners” of a factorization system.

In the theorems 8.17 and 8.18 we have seen, that an epimorphism f : A → Binduces a surjective homomorphism f∼ : dom f∼ → B out of F (A,TAlg(Σ)) onto B.The converse is also true, and we leave the proof as an exercise:

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Proposition 8.21 Epimorphisms stem from closed homomorphisms start-ing from “initial extensions”.Let A be any partial algebra, C an A-generated relative subalgebra of F (A,TAlg(Σ)),θ a closed congruence relation on C, and natθ : C → C/θ the natural projection.Then its restriction to A, i.e. natθ|A : A→ C/θ is an epimorphism.

Definition 8.22 of initial congruences: Recall that the A-generated relative sub-algebras of F (A,TAlg(Σ)) are called A-initial segments of F (A,TAlg(Σ)) = A.Hence the closed congruence relations on A-initial segments of A will be called A-initial congruences of A.

The proof of the following proposition is straightforward and left as an exercise.

Proposition 8.23 Let A ∈ PAlg(Σ) be any partial algebra. Then:

(i) The set of all A-initial segments of the absolutely free completion A = F (A,TAlg(Σ))of A is a closure system on A with smallest element A (and largest element A).

(ii) The set of all A-initial congruences of A is a closure system on its square A2

with smallest element ∆A (= (a, a) | a ∈ A) and greatest element 5A (= A2).

(iii) For any set C of A-initial congruences, we have

dom⋂

C =⋂ dom θ | θ ∈ C ,

where for any relation R ⊆ B ×B we havedom R := a, b ∈ B | (a, b) ∈ R

= a ∈ B | there is b ∈ B such that (a, b) ∈ R or (b, a) ∈ R

(iv) The set of all A-initial congruences is inductive.

Basic for the investigations below is the following

Corollary 8.24 Let (fi : A → Bi)i∈I be a family of homomorphisms, let (f∼i :dom f∼i → Bi)i∈I be the family of their closed A-initial extensions within F (A,TAlg(Σ)),and let f : A→

∏i∈I Bi be the induced homomorphism.

Then, for f∼ : dom f∼ →∏

i∈I Bi we have

dom f∼ =⋂i∈I

dom f∼i ,

ker f∼ =⋂i∈I

ker f∼i .

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For some proofs in what follows one will need estimates about the size of partialalgebras generated by a given set. These can be derived from the following lemma:

Lemma 8.25 A bound for the cardinalities of M-generated partial algebras:Let A be any partial algebra and let M be any generating subset of A; by |B| denotethe cardinality of B for any set B. Then, with ℵ0 being the smallest infinite cardinal,one has for each s ∈ S:

|As| ≤ |F (M,TAlg(Σ))s| = |T (M,Σ)s| ≤ max|Mu|, |Ω|,ℵ0 | u ∈ S =: m.

Proof Since A is a closed homomorphic image of an M -generated relative sub-algebra — namely of dom (idM)∼ — of T (M,Σ), for each s ∈ S, one has |As| ≤|T (M,Σ)s| . Now, assume w.o.l.g. that Ω ∩Ms = Ø = Ms ∩Ms′ , for all s, s′ ∈ S.Set Ω(0) := ϕ ∈ Ω | τ(ϕ) = 0 and M0 := Ω(0) ∪

⋃s∈S(Ms × s). Then it

is easy to realize that |M0| ≤ m (with m as defined in the lemma). For eachϕ ∈ Ω \ Ω(0) and any set B with |B| ≤ m one has |Bτ(ϕ)| ≤ m, and therefore, bysetting D :=

⋃s∈S(Bs∪s, one gets |(DT (M,Σ)B)s| = |(B∪

⋃ϕ∈Ω ϕ

T (M,Σ)(Bη(ϕ)))s| ≤|D| +

∑ϕ∈Ω |Dτ(ϕ)| ≤ m +

∑ϕ∈Ω mτ(ϕ) ≤ m because of the definition of m. Since

T (M,Σ)s = (M ∪⋃∞n=0DnT (M,Σ))s one may conclude that |T (M,Σ)s| ≤ m, for each

s ∈ S, as was to be shown.

8.5 On the existence of universal solutions

Definition 8.26 of the characteristic of a class of partial algebras: Let A ∈PAlg(Σ) be any partial algebra, and let K ⊆ PAlg(Σ) be any class of partial algebras.Then we define in analogy to the case of rings the A-characteristic of K, charAK as

charAK :=⋂kerf∼ | f : A→ B for some B ∈ K,

i.e. as the intersection of all closed A-initial congruences of F (A,TAlg(Σ)), whichare the S-kernels of homomorphisms into K-algebras starting from A (see J.Schmidt[Sch62] and [Sch64] for his corresponding concept for total algebras). If A = Mdiscreteis a discrete partial algebra, then we simply write charMK instead of charMdiscreteK.

The usual characteristic of a ring is then the (Z; +, 0,−)-characteristic of its ad-ditive group.

Then we have the following

Theorem 8.27 Characterization theorem of K-universal solutions Let A beany partial algebra of signature Σ and let K be any class of partial algebras of signatureΣ.

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(i) Then(natcharAK

|A, (dom charAK)/charAK)

is a K-universal solution (rA,F (A,K), F (A,K)) of A.

(ii) F (A,K) is isomorphic to a subalgebra of a direct product of K-algebras, i.e. onealways has F (A,K) ∈ ISP K.

(iii) Let A = A be discrete, and let K contain for every sort s ∈ S, for which As hasat least two elements an algebra, say K(s), for which the carrier (K(s))s of sorts has at least two elements, and for which there exists an S-mapping from Ainto K(s). Then dom charAK is freely and K-freely generated by A.

Proof Ad (i): Let e : A → B be a K-extendable epimorphism, let K ∈ K, and letf : A → K be any homomorphism. Since e is a K-extendable epimorphism, thereis a homomorphism h : B → K such that h e = f . By the Diagram CompletionLemma 8.17 for Epimorphisms this means that ker e∼ ⊆ ker f∼. The definition ofcharAK then implies — since K ∈ K was arbitrarily chosen — that ker e∼ ⊆ charAK.Hence e is ((dom charAK)/charAK)-extendable, since again Lemma 8.17 implies theexistence of a homomorphism, say l : B → (dom charAK)/charAK such that l e =natcharAK

|A .

On the other hand the definition of charAK and Lemma 8.17 imply that natcharAK|A

is a K-extendable epimorphism. This shows that natcharAK|A is K-universal.

Ad (ii): Define ΘK := ker f∼ | f : A → K for some K ∈ K , and observethat this is a set. For each θ ∈ ΘK choose some Kθ ∈ K, for which there is ahomomorphism, say gθ : A → K, such that S-ker gθ = ker(gθ)

∼ = θ, (Axiom ofChoice). Let K0 := Kθ | θ ∈ ΘK Then

charAK =⋂ S-ker gθ | Kθ ∈ K0 =

⋂ S-ker g | g : A→ Kθ for some Kθ ∈ K0 .

Then it is easy to realize, using Corollary 8.24 and Lemma 8.17, that F (A,K) =(dom charAK)/charAK is isomorphic to the subalgebra of

∏(Kθ | Kθ ∈ K0, f : A →

Kθ) =: B0 generated by f(A), where f : A → B0 is the unique homomorphisminduced by the set (family)

f | f : A→ Kθ andKθ ∈ K0 =⋃

Kθ∈K0

Hom(A,Kθ)

of homomorphisms.Ad (iii): The assumptions guarantee that for each s ∈ S and for all m,n ∈ Ms

with m 6= n there exist a K ∈ KG and a mapping f : M → K such that f(m) 6= f(n).Hence (charAK) ∩ M2 = ∆M . This implies that the the restriction to M of thehomomorphism f as constructed in part (ii) of this proof is injective. Hence thereexists an isomorphic copy of (dom charMdiscrete

K)/charMdiscreteK, which contains M as

a K-free generating set.

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Corollary 8.28 Quasiprimitive classes contain all universal solutions.For every quasiprimitive (i.e. ISP- closed) class K of partial algebras of signatureΣ, K-universal K-solutions exist for all partial algebras of signature Σ and can beconstructed as in the above theorem.

If one has only K = ISP+(K), then the K-universal K-solution of A exists, if Aallows at least one homomorphism into some K-algebra.

Remark 8.29 A “classical” proof of the existence of universal solutions inquasi-primitive classes.80 Because of its importance for the general theory we giveanother proof for the existence of K-universal K-solutions in a quasiprimitive class K

(i.e. K = ISP(K)), which is the basis of corresponding proofs in category theory:Let A ∈ PAlg(Σ) be any partial algebra, and let K ⊆ PAlg(Σ) be any quasiprimitive

class of partial algebras. Let m be the bound on the cadinalities of A-generated partialΣ-algebras. Let M be any set of cardinality m, and set

KM := K | K ∈ K, and K ⊆M .

And for any K ∈ K set

HK := f | f : A→ K = HomPAlg(Σ)(A,K) , and I :=

⋃K∈K

HK .

Obviously, I is a set, since for each subset K ⊆M of M there is only a set of differentpartial algebras of signature Σ on K. Moreover, let

(∏

K∈KM

∏f∈HK

K =∏

K∈KM

KHK =: KA,K,M , (prf )f∈I)

be the direct product of all these K-algebras w.r.t. all the homomorphisms from Ainto any of these algebras; and fK : A → KA,K,M the corresponding homomorphisminduced by I. By assumption one has KA,K,M ∈ K.

Let FA,K,M be the subalgebra ofKA,K,M generated by fK : FA,K,M := CKA,K,MfK(A) ;

and in what follows we denote the restriction of fK to FA,K,M by fA,K . We claim that

(fA,K : A→ FK, FK)

is a K-universal K-solution of A:81

Obviously, by construction, fA,K is now an epimorphism, and FK ∈ K, since K isquasiprimitive. Let B ∈ K be any partial algebra, let g : A ∈ B be any homo-morphism, and let C be the subalgebra of B generated by g(A). Then |C| ≤ m, and

80Cf. e.g. G.Birkhoff [Bi35] in connection with the proof of the existence of the free algebra ina variety,, J.Schmidt [Sch66a] in connection with proof of the existence of universal solutions (likehere) or H.Herrlich and G.Strecker [HS73], where such a kind of proof is used in order to constructthe left adjoint of some functor.

81See Figure 22.

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K 3 B FA,K,M := CKA,K,MfK(A)

_

idFK,KA,K,M

A

g

bbFFFFFFFFFF fA,K

66 66mmmmmmmmmmmmmmm

g0

xx

xx

xfK

((QQQQQQQQQQQQQQQ

KM 3 COO

OO ∏

K∈KMK =: KA,K,Mprg0

oo

Figure 22: Existence of the K-universal solution

therefore, there exists a subset K ⊆M with the same cardinality as that of C; and bythe Lemma 0.9 of Pickert and van der Waerden there exist an algebraic struc-ture K on K and an isomorphism i : C → K. This means that i g ∈ I. Let pig bethe restriction of prig to FA,K,M . Then it is easy to realize that idCB pig fA,K = g ,showing that fA,K is a K-extendable epimorphism. Since FA,K,M ∈ K it has even tobe K-universal.

Remarks 8.30 (among others concerning the initial algebra approach in[GoTcWa78] for abstract data types):

(i) In particular the above results apply to all E-, ECE- and QE-varieties, which arealways quasiprimitive (cf. section 9).

(ii) The above results imply among others, that quasiprimitive classes K ⊆ PAlg(Σ)with all homomorphisms are epireflective subcategories of the categoryPAlg(Σ) of all partial algebras of signature Σ and with all homomorphisms be-tween them as morphisms, i.e. the objects of the universal solutions exist withinthe class, and the K-universal homomorphisms are always epimorphisms.

(iii) In particular, in quasiprimitive classes K initial algebras (which allow exactlyone homomorphism into any other K-algebra) always exist within this class andare isomorphic to F (Ø,K). This is a very important result for computer sciences,since there often the so called initial algebra semantics is used, which definesan abstract data type defined by a given “specification” (through a signa-ture Σ and a set A of axioms) to be the initial (partial) algebra F (Ø,Mod(A))of the class Mod(A) of all (partial) Σ-algebras satisfying all axioms from A(whenever this exists).

(iv) Observe that the classes F of all fields and Fp of all fields of characteristic p (pa prime or 0) are not closed w.r.t. (direct) products, since every direct productwith an at least two element index set will contain zero divisors. Thus, in

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general, universal solutions will not exist within these classes. In particular,F has no initial object. However, for the class Fp of all fields of some fixedcharacteristic p the so-called prime field Fp (generated by Ø) is an initialobject of that class (while F (M,Fp) /∈ Fp, whenever M 6= Ø. We conjecturethat F (M,Fp) is then isomorphic to the commutative ring (with partial inverse)Fp[X] of all polynomials with the set X as set of variables, the elements of Fpas coefficients, and where only the non-zero elements from Fp have an inverse(we leave a proof (or a proof that the conjecture is wrong) as an exercise)).

Lemma 8.31 For epimorphisms K-extendability is equivalent to IScP(K)-extendability, and the same for universality.Let K ⊆ PAlg(Σ), and let e : P → C be any epimorphism.

(i) e is K-extendable iff e is IScP(K)-extendable.

(ii) e is K-universal iff e is IScP(K)-universal.

Proof It is obvious that every IScP(K)-extendable or IScP(K)-universal epimor-phism is also K-extendable or K-universal, respectively, since K ⊆ IScP(K). In bothcases it remains to show the other direction. And here, obviously, nothing has to beproved for isomorphic copies.

Ad (i): Subalgebras: Assume that A ∈ K, that B is a subalgebra of A, and thatf : P → B is any homomorphism. Let f ′ := idB,A f : P → A be the homomorphismwith the same graph as f . By assumption there exists a homomorphism, say g′ : C →A, such that g′ e = f ′. We claim that g′(C) ⊆ B.

Proof of the claim: We have (g′ e)(P ) = (idB,A f)(P ). Since e is an epimorphism,we have C = CCe(P ). And therefore

g′(C) = g′(CCe(P )) ⊆ CA(g′ f)(P ) = CA(idB,A f)(P ) = CAf(P ) ⊆ CAB = B.Therefore there exists a homomorphism g : C → B with graph g = graphg′, whichsatisfies g e = f , what was to be shown.

Products: Consider a family (Ai)i∈I ∈ KI for any indexset I, and some ho-momorphism f : P →

∏i∈I Ai =: A. Then (pri f : P → Ai)i∈I is a family of

homomorphisms from P into K-algebras. By assumption there exists, for each j ∈ I,a homomorphism, say gj : C → Aj such that gj e = prj f . Since (A =

∏i∈I Ai, (prj :

A → Aj)j∈I) is a product in PAlg(Σ), there exists exactly one homomorphism, sayg : C → A such that, for all j ∈ I, one has pri g = gj. Therefore, for all j ∈ I,pri g e = gj e = prj g. This implies — by the uniqueness of the induced productmorphism — that g e = g, i.e. g is the desired morphism. Ad (ii): Since universalepimorphisms are defined via the extendable epimorphisms, and these are in bothcases the same, we immediately get (ii) as a consequence of (i). Observe that theobjects of K-universal solutions always belong to IScP(K) by Theorem 8.27.(ii) or byRemark 8.29.

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8.6 Fully invariant congruence relations

For a set M , F (M,K) is the K-free partial algebra on rM,K(M), which is “closest” toK, and it has a further description, using the concept of a fully invariant congruencerelation:

Definition 8.32 of fully invariant congruence relations:Let A be a partial algebra, θ any congruence relation on A. Then θ is called a fullyinvariant congruence relation, if and only if for any endomorphism f of A (i.e.for any homomorphism f : A → A) one has (fs(a), fs(b)) ∈ θs for every (a, b) ∈ θsand for every s ∈ S (i.e. (f × f)(θ) ⊆ θ).

Remark 8.33 Observe that in the case that A is a partial Peano algebra on somesubset X — i.e. (up to isomorphism) a relative subalgebra of the term algebra T (X,Σ)—, then “full invariance” of a congruence relation θ on A just means (is equivalentto) the substitution property:

(SP) Assume that p(x1, . . . , xn), q(x1, . . . , xn)) ∈ θ for any terms p, q ∈ T (X,Σ),which depend at most on the variables x1, . . . , xn ∈ X (we suppress here any ref-erence to sorts). Let, moreover, t1, . . . , tn ∈ T (X,Σ). Then (p(t1, . . . , tn), q(t1, . . . , tn)) ∈θ.

Namely this is just how an endomorphism, which maps xi to ti for 1 ≤ i ≤ n acts on(p, q) in T (X,Σ). (Exercise)

Theorem 8.34 Some results related to freeness and fully invariant congru-ence relations:

(i) Let A be a free partial algebra freely generated by M , and let θ be a fully invariantcongruence relation on A.

(a) Then A/θ is freely generated by M/θ := [m]θ | m ∈M.(b) And if (A/θ)s has at least two elements (for some s ∈ S), then (natθ)s|Ms :

Ms → (M/θ)s is bijective.

(c) Moreover, M is A/θ-free in A.

(ii) Let f : A → B be a homomorphism which maps some generating subset M ofA into some B-free subset N of B such that f |Ms is injective, whenever Bs

has at least two elements. Then ker f is a fully invariant congruence relationon A.

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(iii) 82 For every class K of partial algebras of signature Σ and for every S-set X,the X-initial congruence relation charXK of dom charXK is closed and fully in-variant, and dom charXK is freely and K-freely generated by X.

Proof We use the notation and assumptions of the theorem.Ad (i).(a): Let f : M/θ → A/θ be any mapping. We have to show that it

extends to an endomorphism of A/θ. In order to use our assumptions, we choose— in the case of an infinite S-set M this indeed needs the Axiom of Choice — amapping83, say g from M into A such that, for every s ∈ S and for every m ∈ Ms ,we have

A

natΘ

∃2g //__________ A

natΘ

MR2

dHHHHHHHHHH

∃1g::v

vv

vv

natΘ|M

M/Θ

f ##GG

GGM m

wwwwwwww

A/Θ∃3f

//_________ A/Θ

Figure 23: The quotient of a free algebra w.r.t. a fully invariant congruence is free

natθsgs(m)) = [gs(m)]θs = fs([m]θs) = (fs natθs)(m). (28)

Since M freely generates A, there is an endomorphism g : A→ AU extending g, i.e.such that g|M = g. As natθ is full and surjective, θ is fully invariant, and g is anendomorphism of A, we have for all (a, b) ∈ A2

s (s ∈ S):

If (a, b) ∈ θs , ( i.e. [a]θs = [b]θs) , then [g(a)]θs = [g(b)]θs , ( i.e. (a.b) ∈ (ker(natθ g))s)(29)

i.e. we have θ = ker natθ ⊆ ker(natθ g). Hence, by the First Diagram CompletionLemma 6.11, there exists an endomorphism f : A/θ → A/θ such that f natθ =

82In about 1968 the following facts — published in [B70] — have been the starting point otthinking of charXK as some kind of equational theory for the class K. It still needed several yearsuntil the name “existence equation” was introduced. In In particular the paper [AN83] of Andrekaand Nemeti, of which a preprint already existed about 1975, was a great help to introduce theconcept.

83See Figure 23.

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natθ g . We still have to show that it extends f :For each m ∈Ms (s ∈ S) one has by (28)

fs([m]θs) = (fs natθs)(m)) = (natθs gs)(m) = (natθs gs)(m) =

= (fs natθs)(m) = fs([m]θs). (30)

This shows, indeed, that f is indeed a homomorphic extension of f .Ad (i).(b): If Ms has only one element, then nothing is to be proved. Therefore

assume that there are, say m,m′ ∈ Ms, with m 6= m′. (m,m′) ∈ θs would immedi-ately imply (a1, a2) ∈ θs, for every a1, a2 ∈ As by the freeness of M in A and thefull invariance of θ, since we could map m to a1, m′ to a2 and extend this to anendomorphism of A. Hence As/θs can have at least two different elements only if notwo elements from Ms are congruent modulo θ, i.e. only when natθs is injective.

Ad (i).(c): Let g : M → A/θ be any mapping. Since gs|Ms is injective for everys ∈ S, for which As/θs has an at least two-element phylum (see (i).(b) above), thereis a mapping h : M/θ → A/θ such that g = h natθ|M . By the freeness of M/θ inA/θ one has an endomorphism h : A/θ → A/θ extending h. But then h natθ extendsg, showing that M is A/θ-free in A.

Ad (ii): Let the assumptions in (ii) be satisfied, and let h be any endomorphismof A.84 Assume (a, a′) ∈ ker f . We have to show that (f h)(a) = (f h)(b): Since

A

f

h // A

f

MQ1

bEEEEEEEEE

h|M<<yyyyyyyyy

f |M

f(M)

∃1g

33

33

33

33_

N

∃2g′ ""EE

EE

EM m

∃3|yyyyyyyyy

Bg

//_________ B

Figure 24: If the quotient is free, then the kernel is fully invariant.

f |Ms is injective, whenever Bs, has at least two elements, we can find a mappingg : f(M) → B such that g f |M = f h|M (g : f(m) 7→ f(h(m)), m ∈M). Next we

84See Figure 24.

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extend g to a mapping g′ : N → B, which then can be extended to a homomorphismg : B → B, i.e. g|f [M ] = g. Hence g f |M = g f |M = f h|M , implying g f = f h,and (f h)(a) = (g f)(a) = (g f)(b) = (f h)(b), what we wanted to show.

Ad (iii): That charXK is a closed congruence relation on its domain follows fromProposition 8.23.(ii), and that F := dom charXK is generated by X follows fromProposition 8.23.(i). From Theorem 8.27 we know that (natcharXK

|X, (F/charXK) is

a K-universal solution of Xdiscrete, and by Lemma 8.31.(ii) it is also an IScP(K)-universal solution of Xdiscrete. Since therefore F (X,K) := F/charXK) ∈ IScP(K) byRemark 8.29, this implies that X/charXK is also K-free in F/charXK, and thereforealso IScP(K)-free. By Theorem 8.34.(ii) we can conclude — because of F/charXK) ∈IScP(K) — that charXK is a fully invariant congruence relation of F = dom charXK.We still have to show that X freely generates F :

Proof that X freely generates F : Let f : X → F be any mapping. Observethat F is a relative subalgebra of T (X,Σ); and recall that r := natcharXK

: F →F (X,K) is a closed and surjective homomorphism. Consider the closed X-initialextension f∼ : dom f∼ → F . Then r f∼ is a closed and X-initial extension ofr f . And by the definition of F as the intersection of all such domains we obtainthat F ⊆ dom (r f∼) = dom f∼. This shows that f∼|F is an endomorphism of Fextending f , and that therefore, indeed, X freely generates F .

Corollary 8.35 Let A be a free partial algebra of signature Σ and M a free generatingsubset of A. Then the mapping85

m 7→ [m]charMA

induces an isomorphism between A and (dom charMA)/charMA. Thus, up toisomorphism the free partial algebras A (A-)freely generated by M are in one-to-one correspondence to the closed and fully invariant congruence relations in relativesubalgebras of F (M,TAlg(Σ)) = T (M,Σ), which are freely generated by M .

8.7 E-varieties and the characterization of primitive classesand of free partial algebras

In subsection 3.1 we have introduced existence equations (briefly called E-equations)and their semantics.86 In this subsection we mainly want to characterize their modelclasses.

One tool for this project is the following Tacking Lemma, which could be derivedfrom results in section 7,87 but we prove it here directly.

85Observe that we write below only M , when we mean the discrete partial algebra Mdiscrete onM . Recall that F (Mdiscrete,TAlg(Σ)) ∼= T (M,Σ).

86See Definition 3.1.(i), (iv) and (v).87Cf. Proposition 7.27.

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Lemma 8.36 Tacking Lemma:Let A and B be partial Σ-algebras, M ⊆ A a generating subset, and f : M → B suchthat there is a system, say M ⊆ P(M), of subsets of M , which is closed w.r.t. finiteunions and covers M (i.e.

⋃M = M), and such that for every N ∈ M there exists

a homomorphic extension fN : CAN → B of f (i.e. fN |N = f |N). Then f has a

homomorphic extension f : A→ B.

Proof Let g :=⋃ graph fN | N ∈M . From the fact that CA is an algebraic closure

operator, one can easily derive that A = a ∈ A | there is b ∈ B such that (a, b) ∈g =

⋃ dom fN | N ∈ M , since by the assumptions that M is closed w.r.t. finite

unions88 every finite subset of M is contained in some N ∈ M. It remains to showthat g is the graph of a homomorphism, i.e. that, for (a, b), (a, b′) ∈ g one always hasb = b′, and that it is a subalgebra of AπB.

Therefore, assume first that (a, b), (a, b′) ∈ g. Then there are sets N, N ′ ∈ M

such that fN(a) = b and fN ′(a) = b′. But — because of Proposition 1.25 — thehomomorphism fN∪N ′ coincides with fN on CAN and with fN ′ on CAN ′. Therefore

b = fN(a) = fN∪N ′(a) = fN ′(a) = b .

This shows that g is the graph of a mapping, say f .That f is a homomorphism is shown in a similar way: Assume (a1, . . . , aτ(ϕ)) ∈

domϕA, for some ϕ ∈ Ω. Then there are sets Ni ∈ M such that ai ∈ CANi (i =

1, . . . , τ(ϕ)). Set N :=⋃τ(ϕ)

1 Ni . Then (a1, . . . , aτ(ϕ)) ∈ CAN , and therefore also

ϕA(a1, . . . , aτ(ϕ)) ∈ CAN . Since fN is a homomorphism, we get f(ϕA(a1, . . . , aτ(ϕ))) =

fN(ϕA(a1, . . . , aτ(ϕ))) = ϕB(fN(a1), . . . , fN(aτ(ϕ))) = ϕB(f(a1), . . . , f(aτ(ϕ))). Thisshows that f is a homomorphism. It is obvious that f extends f because of itsdefinition.

Notation 8.37 The system of S-sets derived from an S-set w.r.t. the powerset of S and the S-support of an S-set and the operators connected withE-equations:

(i) Recall that we have chosen in Definition 3.2 a fixed set Y of variables such thatYs is countably infinite for each s ∈ S. This we shall also use in what followsin this sense.

(ii) Let M be an arbitrary S-set. Then we shall denote, for S ′ ⊆ S, by M|S′ theS-set ((M|S′)s)s∈S , where

(M|S′)s :=

Ms if s ∈ S ,Ø else

88Actually it is sufficient that M is an upwards directed set of subsets of M , i.e. (see Defini-tion 7.17.(i)) there exists for each finite subset F ⊆M an F ∈M such that

⋃F ⊆ F .

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Moreover, we defineM||P(S) := M|S′ | S ′ ⊆ S

to be the set of all such S-sets derived from M , and

M||Pfin(S) := M|S′ | S ′ ⊆ S , S ′ finite

to be the set of all such S-sets derived from M w.r.t. finite subsets of S.

(iii) Let M be again any S-set. Then we define its S-support as

supp SM := s ∈ S |Ms 6= Ø .

Very often we shall abbreviate supp SM just by SM .

(iv) Finally, let X ⊆ Y be any S-set of variables, G ⊆ L(Y,Σ) any set and (Gi | i ∈I) ∈ (P(L(Y,Σ))I (I an arbitrary set) any family of sets of first order formulas— below in particular of E-equations —, and let K ⊆ PAlg(Σ) be any class ofpartial algebras. We introduce the operators

Eeq :=⋃T (X,Σ)2 | X ⊆ Y is finite ,

EeqX K := (t, t′) ∈ T (X,Σ)2 | K |= te≈ t′,

Eeq K :=⋃EeqXK | X ⊆ Y is finite ,

ModG := A ∈ PAlg(Σ) | A |= G,Mod (Gi | i ∈ I) := A ∈ PAlg(Σ) | A |= Gi for all i ∈ I.

(v) The model class of any set of E-equations is called an E-variety or an existencevariety.

From Theorem 8.27 and the Definition 3.1 one easily obtains the

Corollary 8.38 The E-equational theory (for a set of variables) of a classof algebras equals its characteristic for this set of variables.For every subset X of Y and for any class K ⊆ PAlg(Σ) of partial algebras we havethat

EeqX K = charXK .

As a preparation of Lemma 8.40 we consider the following examples:

Examples 8.39 The K-free algebra on a subset M of N need not be iso-morphic to a subalgebra of the K-free algebra on N, when M and N havedifferent S-supports.

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1. Let S := s, u , Ω = Ø, and let K be the class of all S-sets (Ms,Mu) such thatMu has at most one element, whenever Ms is nonempty, i.e.

K = Mod ( (x s , y, y′ u ) ; ye≈ y′ ) .

Let N := ( (x1 , x2 s , y1 , y2 u ) and M := ( ( Ø , y1 , y2 u ) , i.e. supp SM =u 6= s, u = supp SN . Then one has

F (M,K) = ( ( Ø , y1 , y2 u ) = M and F (N,K) = ( (x1 , x2 s , y u ) 6= N ,

and F (M,K) is not isomorphic to a “subalgebra” of F (N,K), since the mappingfrom F (M,K) to F (N,K) induced by the inclusion from M int N is not injective.

2. Let S := s, u , Ω = ϕ with τ(ϕ) = 0 and σ(ϕ) = u and let

K := Mod ( (x s , Ø ) ; Dϕ ) .

Let N := ( (x s , Ø ) and M := ( ( Ø , Ø ) . Then again the S-supports aredifferent, and we have

F (M,K) = ( ( Ø, Ø ); Ø) and F (N,K) = ( ( x s , ϕF (M,K) u ); ϕF (M,K) ) .

This shows that the homomorphism from F (M,K) to F (N,K) induced by theinclusion from M int N need not be closed.

3. The reader is asked to find as an exercise an example, where the induced homo-morphism is neither closed nor injective.

4. Let K := TAlg(Σ) for any signature Σ, and let M ⊆ N be any two S-sets.Then always F (M,K) ∼= CF (N,K)M , i.e. F (M,K) is then always isomorphic toa subalgebra of F (N,K).

Lemma 8.40 The K-free algebra on a subset M of N is only then alwaysisomorphic to a subalgebra of the K-free algebra on N, when M and Nhave the same support.Consider any class K ⊆ PAlg(Σ) and any two S-sets M and N and an injective S-mapping i : M → N . Moreover, let M ′ and N ′ be the images of M and N in F (M,K)and F (N,K), respectively. Then there exists a homomorphism ι : F (M,K) →F (N,K) such that rN,K i = ι rN,K.89 But only in the case of supp SM = supp SNthe homomorphism ι is always closed and injective.

Proof The existence of the homomorphism ι follows from Lemma 8.31. If supp SM =supp SN , then there is a mapping, say g : N → M such that g|M = idM . Thisinduces a homomorphism, say g′ : F (N,K) → F (M,K) such that g′ ι = idF (M,K).This implies, by Proposition 3.15, that ι is closed and injective. On the other handwe have seen in Examples 8.39 that in the case of different supports F (M,K) neednot be isomorphic to a subalgebra of F (N,K).

89Recall that rN,K : N → F (N,K) designates the K-universal epimorphism starting from thediscrete partial algebra on N , and rM,K is defined analogously.

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Theorem 8.41 E-varieties are always primitive classes.For every class K of partial Σ-algebras one has that K = ModEeqK always impliesK = HwScPK .

Proof Consider ι :≡ (X; te≈ t′) ∈ EeqK.

HwK ⊆ K : Consider A ∈ K, and let f : A → B be any surjective homomor-phism. Moreover, let v : X → B be any assignment. Since f is surjective, thereis an assignment w : X → A such that v = f w. Since A |= ι, we have t, t′ ∈dom v∼ ⊆ dom (f v)∼ = domw∼, and v∼(t) = v∼(t′). But this immediately impliesthat w∼(t) = w∼(t′), i.e. B |= ι[v]. Since v was arbitrarily chosen, this proves B ∈ K.We shall not repeat this last argument in the next cases.ScK ⊆ K : Consider A ∈ K, let B be any subalgebra of A, and let v : x→ B be any

assignment and v∼ : dom v∼ → B its induced partial interpretation. Let v′ : X → Abe the assignment with the same graph as v. Since v∼ maps dom v∼ onto a subalgebraof B, which is also a subalgebra of A, also v′∼ has the same domain and graph asv∼. Since A |= ι, we have that v∼(t) and v∼(t′) are defined and v∼(t) = v∼(t′), i.e.B |= ι[v].PK ⊆ K : Consider Ai ∈ K (i ∈ I), for an arbitrary set I, let B :=

∏i∈I Ai, and

let v : X → B be any assignment. Then vi := pri v : X → Ai are assignments,too, and by assumption, (t, t′) ∈ ker vi

∼ for each i ∈ I. By Corollary 8.24 we haveker v∼ =

⋂i∈I ker vi

∼, and therefore (t, t′) ∈ ker v∼, hence B |= ι. This finishes theproof.

Remark 8.42 It has been shown in [B95] that the converse of the statement inTheorem 8.41 is only then always true, when S is finite. In the case of an inifiniteset S of sorts the operator P of the formation of products has to be replaced by theoperator Pr of the formation of reduced products.90 We shall discuss this later inmore detail.

One argument in the proof leeds us to the following statement:

Lemma 8.43 Independence is hereditary only in a restricted sense.Let K ⊆ PAlg(Σ) be any class of partial algebras, and let A be a Σ-algebra with aK-free subset M .

(i) Then every subset N of M with supp SN = supp SM is also K-free in A.

(ii) If Σ is homogeneous, then every subset of M is always K-free in A iff K containsonly total algebras or Σ contains no constants.

90Cf. Definition 7.9 and Remarks 7.10.

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Proof Ad (i): Let K ∈ K, N ⊆M with supp SN = supp SM , and let f : N → K beany mapping. Because of supp SN = supp SM one can define a mapping f ′ : M → Ksuch that f ′|N = f . By assumption f ′ has a homomorphic extension f ′ : CAM → K,

an obviously f ′|CAN : CAN → K is then a homomorphic extension of f .Ad (ii): Let S have only one element. Then we can only distinguish between

empty and non-empty sets. Let K contain only total algebras; then it can contain theempty total algebra iff Σ has no constants. Therefore, if M 6= Ø is K-independent,then Ø is K-independent, too.

If Σ is homogeneous and contains a constant, then consider K := TAlg(Σ) ∪ Ø and A := T (X,Σ) for some non-empty set X. Then X is K-independent T (X,Σ),but this is not true for its subset Ø.

These results enables us to formulate and prove the following

Proposition 8.44 For finite S every E-variety is the intersection of themodel classes of the free algebras related to the system of all S-subsetsderived from Y.Let S be a finite set of sorts, let K ⊆ PAlg(Σ) be any class of partial algebras, and letY be as in Definition 8.37. Then the following statements are equivalent:

(i) K is an E-variety, i.e. K = Mod EeqK.

(ii) K =⋂S′⊆S Mod EeqY|S′F (Y|S′ ,K) .

Remark 8.45 Observe that one does not need the finiteness of S for the directionfrom (i) to (ii). If S is infinite, one has to replace (ii) by

(ii)’ K =⋂Mod EeqY|S′F (Y|S′ ,K) | S ′ ⊆ S and S ′ is finite ,

i.e. by another finiteness condition. And then the argument would have to involvereduced products. We postpone this until the next section, where reduced productshave to be used for all the kinds of quasivarieties showing up there, anyway.

Proof (ii) implies (i): This is quite obvious, since it is easy to realize that (ii)implies that K = Mod (

⋃S′⊆S EeqY|S′F (Y|S′ ,K) , and this shows that K is indeed an

E-variety, since all the terms involved only use a finite set of variables; and for theindication, which phyla have to be non-empty in order that the E-equation has anon-trivial effect, only finitely many extra variables have to be used.

(i) implies (ii): Let K be any E-variety. By Theorem 8.41 K is primitive andtherefore also quasiprimitive. Hence, by Corollary 8.28, K contains all universal solu-tions, and therefore F (Y|S′ ,K) | S ′ ⊆ S ⊆ K . Since |= induces (kind of) a Galoiscorrespondence between Eeq and PAlg(Σ) one gets91

K = Mod Eeq K ⊆ Mod Eeq F (Y|S′ ,K) | S ′ ⊆ S .91Cf. Definition 6.26 and Lemma 6.27.

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In order to show the other inclusion consider any B ∈ Mod Eeq F (Y|S′ ,K) | S ′ ⊆ S .We claim that B is the homomorphic image of F (B,K) induced by i := idB as amapping out of F (B,K) onto B. Since K is primitive, this would imply B ∈ K.

In order to prove the claim, we use the Tacking Lemma 8.36 in which we choose asM the set of all finite subsets F ofB with92 supp SF = supp SB. In the rest of the proofwe shall abbreviate supp SD by SD for any S-set D.93 Let F ∈M. Then there is aninjective mapping, say f : F → Y|SF — set X := f(F ), and by Lemma 8.40, F (F,K)is isomorphic to the subalgebra CF (Y|SF ,K)f(M) of F (Y|SF , K). By Lemma 8.43 and

our assumptions on M, F is K-independent in F (B,K), and F (B,K) is isomorphic toCF (B,K)F . Set v1 := f−1 : X → B and v2 := f−1 : X → F (B,Σ). Since B |= EeqK,we get

EeqX K = charXK = ker v2∼ ⊆ ker v1

∼ ,

and this implies the existence of a homomorphism iF : CF (B,K)F → B extendingi. Thus the assumptions of the Tacking Lemma 8.36 are satisfied, and we have ahomomorphism i : F (B,K) → B extending i; and therefore i is surjective, sincei already has this property. Since K is an E-variety, it contains all homomorphicimages, and therefore B ∈ K, what was to be shown.

Corollary 8.46 If K is any class of homogeneous partial algebra of some signatureΣ, and if M is any at least countably infinite set, then

indF (M,K)M = HSPK ∪ Ø = HSPoK = HSPoKF (M,K).

8.8 Preservation of formulas by operators

Definition 8.47 of the preservation of formulas by operators: Let Y be any ofthe operators introduced in Definition 7.31, let (X;Phi) be any formula from L(Y,Σ),and let K ⊆ PAlg(Σ). We say that the operator Y preserves the formula (X; Φ),iff

K |= (X; Φ) always implies that YK |= (X; Φ) .

Theorem 8.48 (i) Weak homomorphic images (i.e. the operator Hw) pre-serve E-equations:

For every surjective homomorphism f : A→ B and every E-equation (X; te≈ t′)

one has:A |= (X; t

e≈ t′) implies B |= (X; t

e≈ t′) .

(ii) Closed homomorphic images (i.e. the operator Hc) preserve ECE-equations — and therefore also E-equations:

92This is the place, where we use the finiteness of S.93Cf. the Notation 8.37.

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For every closed and surjective homomorphism f : A → B and every ECE-

equation (X;∧ni=1Dti ⇒ t

e≈ t′) one has:

A |= (X;n∧i=1

Dti ⇒ te≈ t′) implies B |= (X;

n∧i=1

Dti ⇒ te≈ t′) .

(iii) Subalgebras (i.e. the operator Sc) preserve QE-equations — and there-fore also E- and ECE-equations:For every closed and injective homomorphism i : B → A and every QE-equation

(X;∧ni=1 tn

e≈ t′i ⇒ t

e≈ t′) one has:

A |= (X;n∧i=1

tie≈ t′i ⇒ t

e≈ t′) implies B |= (X;

n∧i=1

tie≈ t′i ⇒ t

e≈ t′) .

(iv) Products (i.e. the operator P) preserve QE-equations — and thereforealso E- and ECE-equations:

For every family (Ai)i∈I and for every QE-equation (X;∧ni=1 tn

e≈ t′i ⇒ t

e≈ t′)

one has:

Ai |= (X;n∧i=1

tie≈ t′i ⇒ t

e≈ t′) (i ∈ I) implies

∏i∈I

Ai |= (X;n∧i=1

tie≈ t′i ⇒ t

e≈ t′) .

Theorem 8.49 (i) Let K be an arbitrary class of partial algebras. ThenModEeqX K = HSP K ∪ Ø,ModEeq K = HSP K.

(ii) For any set M ∈ Ø, X and for any set G ⊆ EeqM the following statementsare equivalent:

(A) G = EeqM ModG.

(B) G is a closed and fully invariant congruence relation on the relative subal-gebra PG := dom G of F (M,TAlg(τ)) such that PG is freely generated byX.

(iii) For any sets G ⊆ EeqX and G0 ⊆ Eeqø the following statements are equivalent:

(a) (G,G0) = EeqMod (G,G0).

(b) G = EeqXModG, and (G0 = Ø or else G0 = G ∩ F (Ø,TAlg(τ))2)(where one considers F (Ø,TAlg(τ)) as a subalgebra of F (X; TAlg(τ))).

Thus the primitive classes are exactly the existence equationally definable classes.However, if all existence equations are related to the infinite set X of variables,

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then always Ø has to belong in addition to the existence equationally definable (i.e.primitive) classes providing in them the so-called initial object. Since in theoreticalcomputer science one wants in general to specify a non-empty initial object, one haseither to choose the a little bit more complex description of primitive classes by pairsof sets of existence equations or one has to forbid the empty partial algebra as amodel (what may only cause trouble in general, if in a partial algebra no constantexist, since then the empty set will generate no subalgebra).

We now have similar to the total case (see [Mar58]) the following

8.9 A characterization of free partial algebras

Theorem 8.50 Characterization Theorem for Free Partial Algebras Let Abe a partial algebra generated by a set M such that the carrier set A has at least twoelements in each phylum. Then the following statements are equivalent:

(i) A is A-freely generated by M .

(ii) A = F (M,K) for some non-trivial class K of partial algebras.

(iii) A is isomorphic to the partial algebra FM(A) of all total term operations ofarity M on A. This isomorphism is induced by eMm 7→ m(m ∈ M), whereeMm (a) := a(m) for all a ∈ AM , i.e. eMm is the m-th projection from AM into A.

(iv) For every a ∈ A there exists a term ta ∈ F (M,TAlg(τ)) such that a = tAa (idM).And for any two terms t, t′ ∈ F (M,TAlg(τ)) the fact that idM belongs todom tA∩dom tA implies that tA and t′A are total. Moreover, tA(idM) = t′A(idM)implies already tA = t′A (what means that two total term operations of A arealready identical, when they coincide on idM).

(v) There exists a free total algebra B, freely generated by M , such that A isan M-generated relative subalgebra of B, and such that for every term t ∈F (M,TAlg(τ)) the fact that tB(idM) exists and belongs to A implies that therestriction of tB to AM is a total term operation on A:

tB|AM = tA ∈ FM(A).

(vi) Let F = F (M,TAlg(τ)). Consider β : M → A, β(m) := m for m ∈ M , asa mapping out of F into A, and let β∼ := CF×Aβ be the subalgebra of F × Agenerated by the graph of β. Then β∼ is the graph of a closed and surjectivehomomorphism β∼ : F ⊇ dom β∼ → A, kerβ∼ is a (closed and) fully invariantcongruence relation, and dom β∼ is freely generated by M .

Proof (i) implies (ii): Choose K := indAM . This is primitive by Proposition 8.4.And by assumption on A one has A ∼= F (M,K) ∈ K. Hence K is no-trivial, since itcontains the non-trivial partial algebra A.

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8.10 Characterizations of primitive classes — finitely manysorts

This yields the following characterization theorem for primitive classes of homogeneouspartial algebras of the same signature. For heterogeneous algebras the situation is alittle bit more complicated, since there the E-varieties (which are primitive classes,see Theorem 8.41) can in general not be generated w.r.t. the operator HSP by asingle algebra, as we can infer from Proposition 8.44.

Theorem 8.51 For a class K of homogeneous partial algebras of type τ and for anyat least countably infinite set M the following statements are equivalent:

(i) K = HSPK ∪ Ø,

(ii) K = indF (M,K)M .

Moreover, the following statements are equivalent:

(a) K = HSPK,

(b) K = indF (M,K)M ∩ indF (ø,K)Ø.

Lemma 8.52 Let M and N be any sets with N ⊆ M , and let K be any class ofpartial algebras. Then

charNK ⊆ charMK,

and if N 6= Ø, then

charNK = charMK ∩ (CF (M,TPAlg(Σ))

N)2.

Observing the description of F (M ; K) as quotient algebra (dom charMK)/charMK

and the above results one may realize that, for any (at least) countably infinite setM one has:

Theorem 8.53 Characterization Theorem for Primitive Classes — homo-geneous case Let M be any at least countably infinite set. Then each primitive classK is characterized by a pair (θM , θ0), where θ0 and θM are closed and fully invariantcongruence relations on their respective domains, which are relative subalgebras ofF (M,TPAlg(Σ)) freely generated by Ø and M , respectively, such that θ0 ⊆ θM and

K = ind(dom θ0)/θ0

Ø ∩ ind(dom θM )/θM

M,

and each such pair θ0 ⊆ θM characterizes a primitive class in this way.Moreover, the pair (θM , θ0) characterizing the primitive class K in such a way is

uniquely determined by K.

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The reader knowing the equational theory of total algebras may realize the closerelationship of the above results in the homogeneous case with the Birkhoff-Theoremand the characterization of equational theories for total homogeneous algebras. Theonly difference here is that charMK (for an infinite set M) only characterizes primitiveclasses containing Ø, while for the characterization of arbitrary primitive classes K

one also needs charøK.

8.11 The E-equational theories

We start with a more general result for QE-equations in general.

Lemma 8.54 Key Lemma for closed sets of QE-equations:Let K ⊆ PAlg(Σ) be any class of partial algebras, let Xi be any sets, let R1, R2 ⊆T (Xi,Σ)2 be any relations, and define, for i ∈ 1, 2 ,

Di :=↓suppRi , Θi :=⋂ ker w | K ∈ K , w : Di → K , Ri ⊆ ker w , Ei := supp Θi .

Then:

(a) The induced homomorphisms rRi,Θi : Di/ConDiRi → Ei/Θi are K-universalsolutions of D/ConDiRi .

(b) Every homomorphism g : D1 → E2 satisfying (g × g)(R1) ⊆ Θ2 has a homo-morphic extension g : E1 → E2 such that (g × g)(Θ1) ⊆ Θ2 .

Proof Observe that rRi,Θi exists, since, by definition, Ri ⊆ Θi and therefore Di ⊆ Eiand ConDiRi ⊆ Θi, and we can apply the First Diagram Completion Lemma 6.11.Moreover, rRi,Θi is always an epimorphism, since idDi,Ei : Di → Ei, natConDiRi

andnatΘi are epimorphism by definition (apply the Diagram Completion Lemma 8.17 forEpimorphisms). Moreover,

Θi = (rRi,Θi natConDiRi)∼ , (31)

since Θi is a closed congruence relation on Ei, which is a Di- and Xi-generated relativesubalgebra of T (Xi,Σ). If K is empty, then the statement is trivial — observe thatthen Ei = T (Xi,Σ) and Θi = T (Xi,Σ)2.

Ad (a): rRi,Θi is K-extendable: Let K ∈ K and let f : Di/ConDiRi be anyhomomorphism. Then f natConDiRi

is a homomorphism from Di into K such thatRi ⊆ ker(f natConDiRi

)∼. Hence Θi ⊆ ker(f natConDiRi)∼, and we have a homomor-

phism f : Ei/Θi → K such that

(f rRi,Θi natConDiRi=) f natΘi idDi,Ei = f natConDiRi

(since natΘi idDi,Ei = rRi,Θi natConDiRi). And since natConDiRi

is an epimorphism(even surjective), we get

f rRi,Θi = f .

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rRi,Θi is K-universal: Let e : Di/ConDiRi → C be any K-extendable epimor-phism. Then, for every f : Di/ConDiRi → K ∈ K, one has a homomorphism, sayf ∗ : C → K, such that f ∗ e = f . This means by Lemma 8.17 that ker e∼ ⊆ ker f∼

(in F (Di/ConDiRi,TAlg(Σ))). — Since every homomorphism h : Di → K ∈ K from

Di into some K-algebra such that Ri ⊆ ker h factors through natConDiRi, we obtain

ker(e natConDiRi)∼ ⊆ ker(f natConDi

)∼ for all homomorphisms h : Di → K ∈ K from

Di into some K-algebra such that Ri ⊆ ker h. Hence we have

ker(e natConDiRi)∼ ⊆ Θi .

Since e natConDiRiis an epimorphism, Lemma 8.17 implies the existence of a homo-

morphism, say e∗ : C → Ei/Θi such that

e∗ e natConDiRi= natΘi idDi,Ei = rRi,Θi natConDiRi

.

Since natConDiRiis an epimorphism, this implies that

e∗ e = rRi,Θi .

This ends the proof that rRi,Θi is K-universal.Ad (ii): Consider g : D1 → E2 satisfying (g × g)(R1) ⊆ Θ2 . Then R1 ⊆

ker(natΘ2g. This induces by Lemma 6.11 a homomorphism, say g∗ : D1/ConDiR1 →

E2/Θ2 such that g∗ natConD1R1 = natΘ2

g . From the K-universality of rRi,Θi (i =1, 2) proved in (i), we obtain by Lemma 8.31 also their IScP(K)-universality, andfrom Theorem 8.27.(ii) and part (i) of this theorem we obtain Ei/Θi ∈ IScP(K)(i = 1, 2). This implies that g∗ induces a homomorphism g∗∗ : E1/Θ1 → E2/Θ2 suchthat

natΘ2g = g∗∗ rR1,Θ1

natConD1R1 = g∗∗ natΘ1

idD1,E1 .

Since natΘ2 is a closed homomorphism, we obtain D1 ⊆ E1 ⊆ ker g, and thereforewe obtain g := g|E1 : E1 → E2 as a homomorphic extension of g. Because ofnatΘ2

g = g∗∗ natΘ1 we can immediately conclude that

(g × g)(Θ1) ⊆ Θ2 ,

what was to be shown.

Remark 8.55 Note that Lemma 8.54.(i) gives still another proof of the existenceand characterization of K-universal solutions, this time referring to its description bya closed congruence relation in an appropriate term algebra, when the partial algebrais described as a factor algebra of a relative subalgebra of this term algebra w.r.t. asuitable “presentation”. Such a “presentation” of a given partial algebra A is alwayspossible by choosing X := A and R := ker(idA)∼.

Moreover, the above lemma is a generalization of full invariance to all kinds ofQE-equations. In particular we have the following corollary.

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Corollary 8.56 Consequences for K-free algebras:If, with the assumptions and notation of the above lemma, Ri = ∆Xi , then everymapping f : X1 → supp charX2K extends to a homomorphism f : supp charX1K →supp charX2K , such that (f × f)(charX1K) ⊆ charX1K.

Theorem 8.57 Characterization of E-equational theories: Let E :=⋃X∈Pfin(Y )X ×

QX ⊆ Eeq be any set of E-equations. Then the following statements are equivalent:

(i) E = Eeq Mod(E) .

(ii) 1. QX is a closed and fully invariant congruence relation for each X ∈Pfin(Y ).

2. For any two finite sets X, X ′ ∈ Pfin(Y ) of variables and for every mappingw : X → suppQX′ there exists a homomorphic extension f : suppQX →suppQX′ such that (f × f)(QX) ⊆ QX′.

(iii) 1. QX = charXMod(E) (is a closed and fully invariant congruence relation)for each X ∈ Pfin(Y ).

2. For any two finite sets X, X ′ ∈ Pfin(Y ) of variables and for every map-ping w : X → supp charX′Mod(E) there exists a homomorphic extension f :supp charXMod(E)→ supp charX′Mod(E) such that (f×f)(charXMod(E)) ⊆charX′Mod(E).

Remark 8.58 Here we list characteristic subsets K0 of IScP(K) for a class K ⊆PAlg(Σ) such that the E-, ECE- or QE-equational theory of K0 is the same as the oneof K — recall that Y is an S-set of variables with each phylum countably infinite:

(i) For EeqK:

(a) K ⊆ TAlg(Σ), |S| = 1, i.e. total and homogeneous case:

K0 := F (Y,K) .

(b) K ⊆ PAlg(Σ), |S| = 1, i.e. partial and homogeneous case:

K0 := F (Ø,K), F (Y,K) .

(c) K ⊆ PAlg(Σ), |S| > 1 finite, i.e. partial and heterogeneous case with finitelymany sorts:

K0 := F (Y|S′ ,K) | S ′ ⊆ S .

(d) K ⊆ PAlg(Σ), |S| > 1 infinite, i.e. partial and heterogeneous case withinfinitely many sorts:

K0 := F (Y|S′ ,K) | S ′ ⊆ S , S ′ finite .

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(ii) For ECEeqK:

K0 := F (D,K) | D a finite relative subalgebra of T (Y,Σ) generated by D∩Y .

(iii) For QEeqK:

K0 := F (D/θ,K) | D ∈ SrT (Y,Σ) finite , D = CD(D ∩ Y ) , θ ∈ ConD .

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9 Quasivarieties of partial algebras 196

9 Quasivarieties of partial algebras

9.1 About the first order language for partial algebras basedon existence equations

Based on existence equations as atomic formulas one can now build the syntax of ausual first order language.

Definition 9.1 Let X be any set of variables, τ : Ω → N0 a given similarity typesuch that X ∩Ω = Ø. A first order language L(X, τ) is now defined in the usual way:

(FX1) te≈ t′ is an atomic formula and hence a formula of L(X, τ) for any t, t′ ∈

F (X,TAlg(τ)).

(FX2) If F and F ′ are formulas of L(X, τ) then ¬F , (F ∧ F ′), (F ∨ F ′), (F ⇒ F ′),and (F ⇔ F ′) are formulas of L(X, τ).

(FX3) If F is a formula of L(X, τ), and if x ∈ X is a variable, then (∀x)F and(∃x)F are formulas of L(X, τ).

Before we can also define L(Ø, τ), let us define the functions

fvar : L(X, τ)→ P(X)

assigning to each formula the set of its free variables, and

var : F (X,TAlg(τ))→ P(X),

assigning to each term the set of variables on which it really depends:

– var(x) := x for every variable x ∈ X,

– var(ωt1 . . . tτ(ω)) :=⋃τ(ω)i=1 var(ti) for all ω ∈ Ω and t1, . . . , tτ(ω) ∈ F (X,TAlg(τ)),

for which var has already been defined, e.g. var(ω) := Ø, if τ(ω) = 0.

Once we know the function var, we have

– fvar(t1e≈ t2) := var(t1) ∪ var(t2) for all t1, t2 ∈ F (X,TAlg(τ)).

– If fvar is defined for F and F ′ of L(X, τ), then we have

fvar(¬F ) := fvar(F ),

fvar((F ∧ F ′)) := fvar((F ∨ F ′)) :=

fvar((F ⇒ F ′)) := fvar((F ⇔ F ′)) := fvar(F ) ∪ fvar(F ′).

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– If fvar is defined for F , and if x ∈ X, then

fvar((∀x)F ) := fvar((∃x)F )) := fvar(F ) \ x.

In the same way one can define the set ovar(F ) of variables occuring in F withthe only difference that

ovar((∀x)F ) := ovar((∃x)F ) := ovar(F ) ∪ x.

We then have L(Ø, τ) defined by the rules (FX1) and (FX2) for X = Ø and

(Fø3∗) If F ∈ L(X, τ) and fvar(F ) = Ø, i.e. if F is a sentence of L(X, τ) thenFø ∈ L(Ø, τ), where Fø is the same formula as F , where only the indexindicates that it is now considered as a formula of L(Ø, τ).

We set L(τ) := L(X, τ) ∪ L(Ø, τ). For M ∈ Ø, X we define the semantics ofexistence equations as before, and for F, F ′ ∈ L(M, τ), A a partial algebra of type τand v : M → A we define:A |= (F ∧ F ′)[v] iff A |= F [v] and A |= F ′[v],A |= (F ∨ F ′)[v] iff A |= F [v] or A |= F ′[v],A |= (F ⇒ F ′)[v] iff (if A |= F [v] then A |= F ′[v]),A |= (F ⇔ F ′)[v] iff A |= ((F [v]⇒ F ′) ∧ (F ′ ⇒ F ))[v],

A |= (¬F )[v] iff it is not true that A |= F [v].

If x is any variable, and if F ∈ L(M, τ), then

A |= (∀x)F [v] iff for all a ∈ A and for all valuations vxa : M ∪ x → A with

vxa(y) :=

v(y), if y ∈M \ xa, if y = x

one has A |= F [vxa ],

A |= (∃x)F [v] iff there exists an element a ∈ A and a valuation vxa : M∪x → A(like above) such that A |= F [vxa ].

Finally, we say that a formula F ∈ L(M, τ) holds in a partial algebra, if and onlyif A |= F [v] for all valuations v : M → A.

A complete and correct system of rules w.r.t. satisfaction of formulas from L(X, τ)in partial algebras has been given in [B82]. Since in what follows we shall restrictour considerations to (mainly positive) universal Horn formulas, we do not go hereinto more details about the general language. Let us only observe again what wealready have mentioned earlier: If we do not forbid the empty partial algebra, thefirst order language only gains its full expressive power, if it also refers to valuationsof the empty set of variables. Moreover, the category theoretical translation below ofelementary implications also supports the above approach.

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9.2 Elementary implications and their translation into epi-morphisms

Definition 9.2 For a given similarity type τ we define an elementary implicationι to be a formula (we omit brackets in the usual way)

ι ≡ (∧i∈I

tie≈ t′i ⇒

∧j∈J

t∗je≈ t∗j

′).

“∧i∈I ti

e≈ t′i” stands for the formation of an “arbitrarily long” conjunction in gener-

alization of “∧”. Observe that we have

A |= ι iff, for every valuation v : fvar(ι)→ A , one has

(ti, t′i) | i ∈ I ⊆ ker v∼ implies (t∗j , t∗j ′) | j ∈ J ⊆ ker v∼ .

This means that ι holds in A iff, for every (partial) interpretation v∼ : dom v∼ →A, one has:Whenever v∼ interprets all the terms occurring in the premise and if it interprets thoseocurring in the same E-equation by the same element of A, then it also interprets allthe terms occurring in the conclusion, and it interprets those in the same E-equationby the same element of A.

Particular elementary implications are

– ECE-equations (i.e. existentially conditioned existence equations)∧i∈I

tie≈ ti ⇒ t

e≈ t′,

– QE-equations (i.e. quasi existence equations)∧i∈I

tie≈ t′i ⇒ t

e≈ t′,

where the conclusion consists of one existence equation only.

In principle in what follows I and J may be arbitrary sets, while w.r.t. L(τ) theyhave to be finite.

Special elementary implications occur in connection with three further equationalconcepts, two of which are also frequently used as axioms for the description of (classesof) partial algebras, while the third one has been used by T.Evans in [Ev51] asmentioned in the “Motivation”:

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– Weak equations tw≈ t′ (for t, t′ ∈ F (X,TAlg(τ))) are in our approach special

ECE-equations94:

tw≈ t′ := (t

e≈ t ∧ t′

e≈ t′ ⇒ t

e≈ t′).

– Strong equations (or Kleene equations) ts≈ t′ (for t, t′ ∈ F (X,TAlg(τ)))

are conjunctions of special ECE-equations:

ts≈ t′ := ((t

e≈ t⇒ t

e≈ t′) ∧ (t′

e≈ t′ ⇒ t

e≈ t′)).

– Evans equations tE≈ t′ are even more complicated:

tE≈ t′ := (((t

e≈ t ∧

∧t′′∈↓t′,t′′ 6=t′

t′′e≈ t′′)⇒ t

e≈ t′)∧

((t′e≈ t′ ∧

∧t′′∈↓t,t′′ 6=t

t′′e≈ t′′)⇒ t

e≈ t′)).

Their very special implicational form makes it understandable that their theoryis not so easily describable as the one of the existence equations (see e.g. S.C.Kleene[Kl52], R.Kerkhoff [Ke70], H.Hoft [Ho70] and [Ho73], R.John [J75] and [J78], L.Rudak[Ru83], and W.Craig [Cr89]).

For ECE-equations and QE-equations in general we still have “nice” Birkhoff typetheorems. In what follows X designates a countably infinite set of variables, and wedenote for an arbitrary class K of partial algebras of type τ and for a set M ∈ Ø, Xof variables by

– ECEeqM(K) the set of all ECE-equations in L(M, τ), which are valid in allK-algebras,

– QEeqM(K) the set of all QE-equations in L(M, τ), which are valid in all K-algebras.

Observe that now all implications under consideration are finite.

Theorem 9.3 Let K be any class of partial algebras of type τ . In each instance belowthe statements (a) and (b) are equivalent:

(i) (a) K = Mod ECEeqX(K),

(b) K = HcSPr(K) ∪ Ø.94We want to mention in this connection that P.Kosiuczenko — see [Kos94] — has recently used a

combination of E-equations and weak equations in order to characterize axiomatic classes of partialalgebras, in which each partial algebra has a permutable respectively distributive lattice of closedcongruence relations.

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(ii) (a) K = Mod (ECEeqX(K) ∪ ECEeqø(K)),

(b) K = HcSPr(K).

(iii) (a) K = Mod QEeqX(K),

(b) K = ISPr(K) ∪ Ø.

(iv) (a) K = Mod (QEeqX(K) ∪QEeqø(K)),

(b) K = ISPr(K).

As an example for ECE-varieties let us list the axioms for the class of all smallcategories considered as homogeneous partial algebras of type

(Ω, τ) = (D,C, , (D, 1), (C, 1), (, 2))

satisfying the axioms (cf. subsection 1.1 for the properties (C 1) through (C 4)formulated there).

1. xDxe≈ x (this implies Dx

e≈ Dx, i.e. D has to be

total),

2. Cxxe≈ x (this implies Cx

e≈ Cx, i.e. C has to be total),

3. yxe≈ yx⇒ Cx

e≈ Dy ∧ C yx

e≈ Cy ∧D yx

e≈ Dx,

4. yxe≈ yx ∧ zy

e≈ zy ⇒ z yx

e≈ zyx.

One might have expected in addition the QE-equation

5. Dye≈ Cx⇒ yx

e≈ yx,

but this can be proved from the other axioms, showing that the class of all smallcategories is really an ECE-variety. We briefly sketch the proof of 5.:

Let K be any small category, f, g ∈ K such that DKg = CKf . By axioms 1and 2 we have the existence of g K DKg(= g) and CKf K f(= f). Because of theassumption DKg = CKf the premise of axiom 4 is satisfied: g K CKf and CKf K fexist, and therefore (g K CKf) K f and g K (CKf K f) exist and are equal. ButCKf K f = f , i.e. g K f exists, and this was to be proved.

Another example may be the specification of an interval Zlk := [−l, k] of integers(l, k ∈ N). We choose the similarity type

(Ω, τ) = (0, s, p, (0, 0), (s, 1), (p, 1)).

Observe that the algebra (Z; 0Z, sZ, pZ) of all integers can be specified as the initialobject F (Ø,K) of the model class K of the axioms

0e≈ 0, psx

e≈ x, spx

e≈ x.

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Zlk can be specified as the initial object F (Ø,Klk) of the model class Klk of theaxioms (sn(x) stands as abbreviation for s . . . s︸︷︷︸

n times

x) :

sk(0)e≈ sk(0),

pl(0)e≈ pl(0),

sxe≈ sx⇒ psx

e≈ x,

pxe≈ px⇒ spx

e≈ x.

The reader is asked (as an exercise) to find a specification of Zlk as F (Ø,K′) ofan E-variety K′. He will realize that the above ECE-equational one is much simpler.Recall the implementation of integers by TURBO PASCAL already mentioned in the“Motivation”, where an interval [−32768, 32767] of Z is implemented in such a waythat s(32767) = −32768, while the value is not defined in the (initial) data typebelonging to our specification.

Elementary implications are of special interest, since the classes defined by themhave still a fairly simple description as seen above and they have free algebras and inmore generality universal solutions of any partial algebra of the same type.

Moreover, they allow a relatively simple translation into a category theoreticallanguage95:

Definition 9.4 of the epimorphism induced by an elementary implication:Let again

ι ≡ (∧i∈I

tie≈ t′i ⇒

∧j∈J

t∗je≈ t∗j

′).

be an elementary implication. Then we assign to ι a homomorphism eι : Pι → Cι —which is indeed an epimorphism — as follows: Let

P0 := var(ι)∪ ↓ ti, t′i | i ∈ I,

where for any set T of terms in F (X,TAlg(τ)), ↓ T designates the set of all subtermsof terms occurring in T , and

C0 := P0∪ ↓ t∗j , (t′j)∗ | j ∈ J.

Moreover, let P 0 and C0 be the relative subalgebras of F (X,TAlg(τ)) with carriersets P0 and C0, respectively. Let

θP 0:= ConP 0

(ti, t′i) | i ∈ I,95See however [AN79], where such a translation has been carried through for all first order formulas.

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P ι := P 0/θP 0Cι := C0/θC0

-eι ∈ Epi

P 0 C0--

idP0C0

??

natθP0

??

natθC0

var(ι)disc. --idvar(ι)P0

Figure 25: eι is the epimorphism encoding the elementary implication ι.

P ι := P 0/θP 0,

θC0:= ConC0

((ti, t′i) | i ∈ I ∪ (t∗j , (t′j)∗) | j ∈ J), and

Cι := C0/θC0.

Then eι : P ι → Cι is the homomorphism induced by the inclusion mapping fromP 0 into C0 (see Figure 25).

Lemma 9.5 Elementary imlications and epimorphisms are equivalent toolsfor the description of classes of partial algebras.

(i) For a given elementary implication ι the encoding homomorphism eι : P ι →Cι is an epimorphism. Moreover, ι holds in a partial algebra A iff for everyhomomorphism f : P ι → A there exists a (unique) homomorphism g : Cι → Asuch that g eι = f , i.e. iff eι is an A-extendable epimorphism, iff A isinjective w.r.t. eι.

(ii) Every epimorphism e : P → C encodes a — possibly infinitary — implication;namely, if X is a generating subset of P and β : X → P the inclusion mapping,R a generating subset of ker β∼, and S a generating subset of ker (eβ)∼, then

ιe := (∧

(t,t′)∈R

te≈ t′ ⇒

∧(s,s′)∈S

se≈ s′)

is an elementary implication encoded by e. If the sets X, R and S can be chosento be finite, then ιe ∈ L(τ).

Definition 9.6 If E is a class of epimorphisms, then we define

Inj(E) := A ∈ Alg(Σ) | each e ∈ E is A−extendable .

Thus Inj(E) = Modιe | e ∈ E, when we extend the concept of models also toinfinitary elementary implications.

Observe that existence equations ι ≡ te≈ t′ are special kinds of elementary implica-

tions, where the premise is empty; however for the encoding epimorphism ιe : P ι → Cι

one has P ι := fvar(ι), which is a discrete partial algebra. P ι only allows a homo-morphism into the empty partial algebra, if it is empty itself; else ι trivially holds inØ and Ø ∈ Inj(eι) is also true.

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For the description of closed sets of ECE- and QE-equations another representa-tion is useful, by which one can also include the one of E-equations:

Definition 9.7 Let

ι := (n∧i=1

tie≈ t′i ⇒ t

e≈ t′)

be any QE-equation. Then ι may be set theoretically represented by an ordered pair

((ti, t′i) | 1 ≤ i ≤ n, (t, t′)) ∈ Pfin(F × F )× (F × F ),

where F := F (X,TAlg(Σ)) and for any set S, Pfin(S) designates the set of all fi-nite subsets of S. If ι is an ECE-equation, then the corresponding pair belongs toPfin((t, t) | t ∈ F) × (F × F ), and if ι is an E-equation, then the correspondingpair belongs to Pfin((x, x) | x ∈ X)× (F × F ). Since we want all these axioms tocorrespond to epimorphisms, we define

PremE := Pfin((x, x) | x ∈ X),

PremECE := Pfin((t, t) | t ∈ F), and

PremQE := Pfin((t, t′) | t, t′ ∈ F).And for Prem ∈ PremE,PremECE,PremQE, we considerQ ⊆

⋃P∈Prem(P × F (fvar(P ),TAlg(Σ))2) to be any set of set theoretically encoded

elementary implications of the corresponding type. For P ∈ Prem define

Q(P ) := (t, t′) | (P, (t, t′)) ∈ Q.

For any class K of partial algebras define

ImpPrem(K) := (P, (t, t′)) | P ∈ Prem, t, t′ ∈ F (fvar(P ),TAlg(Σ)),K |= (∧

(p,p′)∈P pe≈

p′ ⇒ te≈ t′) and set ↓E to be the relative subalgebra of F = F (X,TAlg(Σ))

consisting of all subterms of terms occurring in E ⊆ F × F , and let suppE (i.e.the support of E) be the set of all terms occurring as at least one component of apair in E.

With the above notation one has the following description of closed sets of ele-mentary implications of one of the three kinds of Prem:

Theorem 9.8 96 (Characterization of theories of E-, ECE- and QE-equations)Let Prem ∈ PremE,PremECE,PremQE, and let Q ⊆

⋃P∈Prem(P×F (fvar(P ),TAlg(Σ))2)

be any set representing elementary implications connected with Prem.

96The proof of this theorem can be found first — formulated for QE-equations — in [ABN81] (andin another form in [AN83]). Later it appeared in [B86] and, without proof, in [B93]. Yet in all threecases (I1) contained an error, since we there refer to ↓Q(P ) rather than to suppQ(P ), and ↓Q(P ) istrivially generated by fvar(P ), i.e. then (I1) does not contain any non-trivial condition.

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(a) Then the following statements are equivalent:

(i) Q = ImpPrem(Mod(Q)).

(ii) Q has the following properties (I1) through (I4) for any P, P ′ ∈ Prem:

(I1) suppQ(P ) is an fvar(P )-generated relative subalgebra of F (fvar(P ),TAlg(Σ))— in particular one has suppQ(P ) =↓ Q(P ).

(I2) Q(P ) is a closed congruence relation on suppQ(P ).

(I3) P ⊆ Q(P ).

(I4) For every homomorphism f : ↓P → suppQ(P ′) which satisfies (f ×f)(P ) ⊆ Q(P ′), there exists a homomorphic extension fPP ′ : suppQ(P )→suppQ(P ′), which satisfies (fPP ′ × fPP ′)(Q(P )) ⊆ Q(P ′).

(b) If Q = ImpPrem(Mod(Q)), and P ∈ Prem, then

Q(P ) =⋂ker f∼ | f : ↓P → A, A ∈Mod(Q) and P ⊆ ker f∼.

From the above theorem one can easily derive a result similar to the one forequational theories of total algebras97 (observe that the above theorem yields thecorresponding results for total algebras, if one replaces (I1) above by(I1)t suppQ(P ) = F (fvar(P ),TAlg(Σ)) ,and that in this case one also gets the following theorem for the case of total algebras.)

Theorem 9.9 The sets of theories of E-, ECE- or QE-equations, respec-tively, ordered by set theoretical inclusion, form algebraic lattices.More precisely: For Prem ∈ PremE,PremECE,PremQE, the ordered set

(Q ⊆⋃

P∈Prem

(P × F (fvar(P ),TAlg(Σ))2) | Q = ImpPrem(Mod(Q)) , ⊆)

is an algebraic closure system on⋃P∈Prem(P × F (fvar(P ),TAlg(Σ))2) .

Proof Set LPrem :=⋃P∈Prem(P × F (fvar(P ),TAlg(Σ))2) to be the “language”

under consideration.First let Q be a set of theories. We have to show that Q0 :=

⋂Q is again a

theory, i.e. that Q0 satisfies the axioms (I1) through (I4). For this purpose considerany P, P ′ ∈ Prem:

• Let us first observe that, for each P ∈ Prem, Q0(P ) =⋂Q∈QQ(P ) , by the

definition of Q0.

97According to H.-J.Hoehnke the following result was first communicated to him by I.Sain about1985.

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• (I1), (I2) and (I3) are obvious, too, since the intersection of fvar(P )-initial closedcongruence relations of F (fvar(P ),TAlg(Σ)) containing P is again such a con-gruence.

• Let f0 : ↓P → suppQ0(P ′) be any homomorphism satisfying (f0 × f0)(P ) ⊆Q0(P ′) . Then we have, for eachQ ∈ Q, a homomorphism fQ : ↓P → suppQ(P ′)satisfying (fQ × fQ)(P ) ⊆ Q(P ′) — since Q0(P ′) ⊆ Q(P ′) — and having thesame graph as f0. Each of them is the restriction to ↓P of the homomor-

phic extension f0 : F (fvar(P ),Σ) → F (fvar(P ′),Σ) of f0. Since each Q ∈ Q

is a theory, each fQ has a homomorphic extension fQ,PP ′ : suppQ(P ) →suppQ(P ′), which satisfies (fQ,PP ′ × fQ,PP ′)(Q(P )) ⊆ Q(P ′), and therefore,in particular, (fQ,PP ′ × fQ,PP ′)(Q0(P )) ⊆ Q(P ′). Then consider fQ0,PP ′ :=

f0|Q0(P ) : suppQ0(P ) → F (fvar(P ′),Σ). Its graph can also be obtained as thegraph of the restriction to Q0(P ) of any fQ,PP ′ , for any Q ∈ Q. Therefore(fQ0,PP ′ × fQ0,PP ′)(Q0(P )) ⊆

⋂Q∈QQ(P ′) = Q0(P ′) .

This shows that the set Q ⊆ LPrem | Q = ImpPrem(Mod(Q)) of all theories underconsideration is a closure system on LPrem. Next we have to show that it is algebraic.

Therefore assume now that Q is an upward directed set of theories under con-sideration. We have to show that Q∗ :=

⋃Q∈QQ is again such a theory. The proof

runs similar to the case of the intersection, where again P, P ′ ∈ Prem are arbitrarypremises:

• Let us first observe that, for each P ∈ Prem, Q∗(P ) =⋃Q∈QQ(P ) , by the

definition of Q∗, and that, for Q ⊆ Q′ in Q, one has Q(P ) ⊆ Q′(P ), for everyP ∈ Prem.

• Next observe that the union of a directed set of closed and initial congruencerelations of F (fvar(P ),Σ) containing P has again these properties (cf. 8.23).Therefore Q∗ satisfies (I1), (I2) and (I3).

• In order to verify (I4) for Q∗, let f ∗ : ↓P → suppQ∗(P ′) be any homomorphismsatisfying (f ∗ × f ∗)(P ) ⊆ Q∗(P ′) . Since ↓ P is finite, and since Q is directed,there exists a Q′ ∈ Q such that f ∗(↓ P ) ⊆ suppQ′(P ′) . Choose such a Q′ andset

Q′ := Q ∈ Q | Q′ ⊆ Q .

Then, by the directedness of Q, one has⋃

Q′ = Q∗ , too. For each Q ∈ Q′,let fQ : ↓P → suppQ(P ′) be the homomorphism into suppQ(P ′) with the samegraph as f ∗. By assumption, fQ then has a homomorphic extension fQ,PP ′ :suppQ(P )→ suppQ(P ′), which satisfies (fQ,PP ′ × fQ,PP ′)(Q(P )) ⊆ Q(P ′) . Asin the argument concerning the intersection, we realize that each fQ,PP ′ is the

restriction to Q(P ) of the global homomorphic extension f ∗ : F (fvar(P ),Σ) →

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F (fvar(P ′),Σ) of f ∗. It is then easy to realize that

f ∗(suppQ∗(P )) =⋃Q∈Q′

f ∗(suppQ(P )) ⊆⋃Q∈Q′

suppQ(P ′) = suppQ∗(P ′) .

Therefore, we get a homomorphism fQ∗,PP ′ : suppQ∗(P ) → suppQ∗(P ′) as the

restriction of f ∗ to suppQ∗(P ) . Moreover, one then has

(fQ∗,PP ′×fQ∗,PP ′)(Q∗(P )) =⋃Q∈Q′

(fQ,PP ′×fQ,PP ′)(Q(P )) ⊆⋃Q∈Q′

Q(P ′) = Q∗(P ′) .

These arguments show that the set of all theories of E-, ECE- and QE-equations,respectively, is an algebraic lattice.

9.3 Preservation and reflection of formulas revisited

In subsection 3.2 we have already briefly treated the concepts of preservation andreflection of formulas. We add here some additional observations:

Let us recall from category theory that in the category Alg(τ) a homomorphismf : A → B is called initial, if for every partial algebra C and for every mappingg : C → A, g is a homomorphism from C into A if and only if f g is a homomorphismfrom C into B. The dual concept is called a final homomorphism. Thus we get thefollowing examples:

Proposition 9.10 In Table 1 some properties of homomorphisms are listed togetherwith the sets of formulas, the reflection of which characterizes them. Different vari-ables are assumed to be distinct. If an operation symbol ϕ ∈ Ω occurs, then thereflection of this kind of formulas for all ϕ ∈ Ω is meant. If some kind of TE-

statement te≈ t or E-equation t

e≈ t′ is mentioned, this means reflection of all such

axioms. Observe that “injective and initial” is equivalent to “injective and full”.

It may attract attention that such important properties like epimorphic, surjec-tive, full and surjective (quotient), etc. do not occur in this table. Yet they havealready been characterized in subsection 6.5 as “partners in factorization systems”of classes defined by the reflection of some existence equations. In order to get theircharacterization we need the concept of a factorization system, which is discussed inthe next subsection 3.5. However, let us first add some more facts about reflectionand preservation of formulas.

Observe that full homomorphisms cannot be defined by reflection of formulas,since their composition need not to be full. One could define some form of “weak re-flection” in order to characterize full homomorphisms (see [B86]), Observation 9.2.16),

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Property of homomorphisms Kind of reflected formulas

injective xe≈ y

closed ϕx1 . . . xτ(ϕ)

e≈ ϕx1 . . . xτ(ϕ)

closed te≈ t

initial ϕx1 . . . xτ(ϕ)

e≈ y

injective and closed xe≈ y, ϕx1 . . . xτ(ϕ)

e≈ ϕx1 . . . xτ(ϕ)

injective and closed te≈ t′

injective and initial xe≈ y, ϕx1 . . . xτ(ϕ)

e≈ z

Table 1: Properties of homomorphisms defined by reflection of formulas (seeProp. 9.10)

A B-f

P ι Cι-eι ∈ Epi

?

p

?

q 7→

A B-f

P ι Cι-eι ∈ Epi

?

p

?

qd

Figure 26: Reflection by f of the el. implication ι encoded by the epi. eι

but main applications of fullness are in connection with injectivity, where it is equiv-alent to “initial and injective” (as we have seen), or in connection with surjectivity,where they are isomorphic to quotient homomorphisms, which have been character-ized in subsection 9.4 below.

9.4 Factorization systems revisited

In subsection 6.5 we have already introduced the concept of a factorization system.Let us investigate here the category theoretical interpretation of the reflection ofelementary implications ι encoded by the epimorphism eι : P ι → Cι. Thus, letf : A → B be any homomorphism. If A does not satisfy the premise of ι, then ftrivially reflects ι. Hence we should assume that A satisfies the premise of ι withrespect to some valuation v : fvar(ι) → A. But this is equivalent to the fact that vinduces a homomorphism, say p : P ι → A, while the fact ”B |= ι[f v]” is equivalentto the existence of a homomorphism q : Cι → B such that f p = q eι. Then thereflection of ι by f is expressed by the existence of a homomorphism d : Cι → A suchthat d eι = p (see Figure 26).

Observe that d is unique and also satisfies f d = q, since eι is an epimorphism.

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This shows that one has Difip(eι, f).

Proposition 9.11 One always has for the operator Λ (and similarly for Λop) definedin subsection 6.5 and for E , E ′ ⊆ Hom:

(i) Iso ⊆ Λ(E),

(ii) Λ(E) Λ(E) ⊆ Λ(E),

(iii) Λ(E) ∩ E ⊆ Iso,

(iv) E ⊆ E ′ implies Λ(E) ⊇ Λ(E ′), ΛΛopΛ(E) = Λ(E) and E ⊆ ΛopΛ(E),

(v) g f ∈ Λ(E) and g ∈Mono imply f ∈ Λ(E);

g f ∈ Λop(E) and f ∈ Epi imply g ∈ Λop(E).

For those who know a little bit more about category theory we mention that Λis preserved by multiple pullbacks, products and induced product morphisms, whileΛop is preserved by multiple pushouts, coproducts and induced coproduct morphisms(for more details see G.E.Strecker [S72], or see [B86], section 10).

Factorization systems are abundant in Alg(τ), since we have the

Theorem 9.12 Let E ⊆ Epi and M⊆Mono in Alg(τ), then

(ΛopΛ(E),Λ(E)) as well as (Λop(M),ΛΛop(M))

are factorization systems in Alg(τ).

Particular examples are described in

Proposition 9.13 In Alg(τ) we have among others the factorization systems (E ,M)shown in Table 2, where e : P → C in E and m : A→ B in M.

The first factorization system in Table 2 has already been considered in Theo-rem 8.20, while the third one has been considered in Lemma 6.13.

Observe that final homomorphisms are full homomorphisms which totally inducethe structure on their image but need not be epimorphisms (since outside of the imagethe structure is just discrete). On the other hand closed homomorphisms as well asinitial homomorphisms are in general not injective, i.e. no monomorphisms.

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E is the class of all M is the class of allhomomorphisms e which are homomorphisms m which are

TAlg(τ)-extendable epimorphisms closedepimorphisms closed and injectivefull and surjective (= quotients) injectivesurjective initial and injective

(=full and injective)surjective, and initialc ∈ C \

⋃ϕ∈Ω ϕ

C(Cτ(ϕ)) implies #e−1(c) = 1

final bijective

Table 2: Some interesting factorization systems (see Prop. 9.13)

Proposition 9.13 shows us that the most interesting properties of homomorphismsthat have shown up so far are either definable by the reflection of existence equa-tions or are their partners in a factorization system (representing all the reflectedepimorphisms). The only exception here from this “rule” are final, bijective andfull homomorphisms, respectively, where at least final homomorphisms and bijectivehomomorphisms are “partners” in a factorization system, too.

In particular, the fact that the classes Ext of all TAlg(τ)-extendable epimorphismsand Mc of all closed homomorphisms form a factorization system shows that ourclosed initial homomorphic extensions are definable within the category Alg(τ) with-out using partial mappings between partial algebras: for a homomorphism f : A→ Bthe pair (idA dom f∼ , f

∼) is just its (Ext,Mc)-factorization (up to isomorphism).

9.5 A Meta Birkhoff Theorem by Andreka, Nemeti and Sain

Now we have almost all the tools available, which are needed for the formulation(and the proof) of a (still quite restricted version of a) result by H.Andreka, I.Nemetiand I.Sain (see [AN82] and [NSa82]) which yields many Birkhoff-type theorems forpartial algebras w.r.t. very different kinds of implications. However, we still have togeneralize our category theoretical description of formulas:

Definition 9.14 (i) A family c := (ei : P → Ci)i∈I of epimorphisms will be calleda cone in what follows.

(ii) We say that a cone c holds in a partial algebra A , or that A is injective w.r.t.c, in symbols A |= c or A ∈ Inj(c), if and only if for all homomorphismsf : P → A there are k ∈ I and a homomorphism g : Ck → A such that f = gek(see Figure 27). If I = Ø, then c = (P ), and we have A |= (P ) if and only ifthere is no homomorphims f : P → A.

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P Ck-ek

∀f

@@@@@@@@@@@RA?

(∃k ∈ I)(∃g)g ek = f

Ci

ei*

Cj

ejHHHHHj

·

·

·

·

·

·iff A |= (ei : P −→ Ci)i∈I

Figure 27: Validity of cones

(iii) If K is a class of cones, we say that A ∈ InjK if and only if A |= c for allc ∈ K, i.e.

InjK := A ∈ Alg(τ) | A |= c for all c ∈ K.

Observe that in a model theoretic interpretation a cone c represents an infinitaryimplication (for I 6= Ø) of the form

ιc ≡ (∧j∈J

tje≈ t′j ⇒

∨i∈I

∧k∈Ki

tike≈ t′ik)

and the injectivity of A w.r.t. c just means that ιc holds in A. The cone (P ) forI = Ø corresponds to the formula

¬∧j∈J

tje≈ t′j.

In what follows we shall use the lettersH and S both for classes of homomorphismsand for special operators induced by them; H for “H-homomorphic images” and Sfor “S-subobjects”, i.e. for K ⊆ Alg(τ) we define

H(K) := B ∈ Alg(τ) | there are A ∈ K and f : A→ B in H,

S(K) := A ∈ Alg(τ) | there are B ∈ K and f : A→ B in S.Recall that one has the following concept dual to injectivity:

Definition 9.15 Let H be a class of homomorphisms. A partial algebra P is calledH-projective if and only if for every homomorphism h : A → B from H and forevery homomorphism p : P → B there is a homomorphism f : P → A such thath f = p (see Figure 28).

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A B-h ∈ H

P

?

p 7→

A B-h ∈ H

f

P

?

p

Figure 28: P is H-projective

As the last preparation of the following theorem we have to specify different kindsof cones.

Definition 9.16 Let c = (ei : P → Ci)i∈I be a cone (of epimorphisms) then we saythat

(i) c is an H-cone (for H-images), if and only if P is H-projective;

(ii) c is an S-cone (for S-subobjects), if and only if ei ∈∧op(S) for all i ∈ I;

(iii) c is a P-cone (for products), if and only if #I = 1;

(iv) c is a P+-cone (for products with non-empty index sets), if and only if #I ≤ 1;

(v) c is a Pr-cone (for reduced products), if and only if #I = 1 and P is totallyfinite (respectively strongly small or finitely presented)98;

(vi) c is a Pr+-cone (for reduced products with non-empty index sets), if and only if#I ≤ 1 and P is totally finite (respectively strongly small);

(vii) c is an e-cone (for the empty product), if and only if #I ≥ 1;

(viii) c is a Pu-cone (for ultraproducts), if and only if I is finite and P and all Ci

(i ∈ I) are totally finite (respectively strongly small).

98 Recall that in Alg(τ) a partial algebra A is called totally finite, iff the set A as well as thedisjoint union of all graphs of fundamental operations of AU are finite. This is a special case of thecategory theoretical concept of strong smallness (see e.g. [B86], subsection 11.2) for the categoryAlg(τ) of all partial algebras of some finitary type τ with all homomorphisms as morphisms. In thecategory of all total algebras of finitary type τ with all homomorphisms as morphisms a total algebraA is called strongly small (or totally finite or finitely presented), iff it is the quotient of a finitelygenerated term algebra with respect to a finitely generated congruence relation. In an arbitrarysubcategory of Alg(τ) the description may be quite different and depends on the “factorizabilitythrough an arbitrary directed system” of a homomorphism into the colimit object of the directedsystem, where these homomorphisms start from such a partial or total algebra.

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(ix) If X , Y and Z are operators like in the preceding cases, then c is an XYZ-cone,iff c satisfies the conditions of an X -, a Y- as well as of a Z-cone.

From the results of H.Andreka, I.Nemeti and I.Sain in [AN82] and [NSa82] onecan extract the following

Theorem 9.17 Meta Birkhoff Theorem Let (O,M) be a category of partial al-gebras with O ⊆ Alg(Σ) and the class M⊆ Hom of homomorphisms such that

– (O,M) has products and direct limits,

– every A ∈ O is the direct limit (in (O,M)) of totally finite (respectively stronglysmall) partial algebras belonging to O.

Moreover, let H,S ⊆M be classes of morphisms such that:

(1) Each A ∈ O is the H-image of an H-projective P ∈ O.

(2) Every H-projective object P ∈ O is the direct limit of totally finite (respectivelystrongly small) H-projective partial algebras from O.

(3) (∧op(S),S) is a factorization system in (O,M).

(4) If g f ∈ H and f ∈∧op(S), then g ∈ H.

(5) From each A ∈ O there starts up to isomorphism only a set of∧op(S)-morphisms.

(6) S =∧e : P → C | e ∈

∧op(S) and P and C are totally finite (respectivelystrongly small).

Let K ⊆ O be any subclass of partial algebras and let F be one of the operators P,P+, Pr, Pr+, e or Pu. Then

InjKHSF(K) = HSF(K),

where KHSF designates the class of all HSF-cones, which hold in K.

For applications of this theorem we only consider the category Alg(τ) of all partialalgebras of type τ and of all homomorphisms between them. However, some ECE-varieties might also do. Moreover, let in this category

I be the class of all isomorphisms,

Sw be the class of all injective homomorphisms,

Hw be the class of all surjective homomorphisms,

Mc be the class of all closed homomorphisms,

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H \ S Si Sw Mi Sc Mc

I + + + + +Hc + + + + +Hf + + + + +Hw + + + + +Hi + + +Hb + + +

Table 3: Compatible pairs (H,S) (see Proposition 9.18)

Mi be the class of all initial homomorphisms,

Hf be the class of all full and surjective homomorphisms,

Hi :=Mi ∩Hw be the class of all initial and surjective homomorphisms,

Hb := Sw ∩Hw be the class of all bijective homomorphisms,

Hc :=Mc ∩Hw be the class of all closed and surjective homomorphisms,

Si :=Mi ∩ Sw be the class of all initial and injective homomorphisms,

Sc :=Mc ∩ Sw be the class of all closed and injective homomorphisms.

Then we get the

Proposition 9.18 In Table 3 it is indicated by + in a row for a class of homomor-phisms chosen for H and in the column for a class of homomorphisms chosen for S,when it is known that this pair (H,S) satisfies the assumptions of Theorem 9.17. Amissing entry means that the corresponding pair has not yet been investigated99.

This yields already 156 different Birkhoff type theorems, since we have 6 possibili-ties for the operator F in Theorem 9.17. For F = Pr, S = Sc, and for H being one ofthe classes I,Hc or Hw, we get the three results from Theorem 9.3, when we observein addition the influence of the conditions (1) through (6) from Theorem 9.17 on theimplications under consideration100. Some descriptions of premises and conclusions— derived from these conditions — for special operators are collected in Table 4(equality “=” here really means that the terms have to be identical, while X is theset of (free) variables under consideration).

99It seems that a student of W.Bartol at Warsaw has worked on them recently, and that twoentries are positive, two are negative.

100Observe that in the first case the operators HScPr and HScP are identical, since reducedproducts are weak homomorphic images of direct products.

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Restrictions on the premise∧i∈I ti

e≈ t′i in case of H as:

I no restrictionsHw ti = t′i is a variable (i ∈ I)Hf ti = t′i = ϕix1i . . . xτ(ϕi)i (i ∈ I, ϕi ∈ Ω),

and for (k, i) 6= (k′, i′) the variables xki and xk′i′ are distinctHc ti = t′i, arbitrary term (i ∈ I)

Restrictions on the conclusion te≈ t′ with respect to the premise∧

i∈I tie≈ t′i for S as:

Sw t, t′ ∈↓ ti, t′i | i ∈ I ∪XSi t arbitrary term, t′ ∈↓ ti, t′i | i ∈M ∪XSc t, t′ arbitrary termsMi t arbitrary term, t′ ∈↓ ti, t′i | i ∈M ∪X,

and not both of t, t′ are variablesMc t = t′ arbitrary term

Table 4: Premises and conclusions for some special operators

It should be observed that one consequence of Theorem 9.17 is the fact that thequasi-primitive classes K = IScP(K) are exactly the classes definable by elementaryimplications with no restrictions on the lengths of premise or conclusion.

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10 Some additions 215

10 Some additions

10.1 Some general remarks concerning the use of differentkinds of equations

There are quite different approaches to the concept of equations for partial algebras,and the one of E-equations extended to ECE-equations is not the only one. As wehave already indicated in the introduction and in connection with the definition of E-and ECE-equations, there are also in particular the concepts of strong equation orKleene-equation and the one of weak equation. Since both concepts do not havethe full expressive power of ECE-equations, they are often combined with a defined-ness predicate, say D, which corresponds to our term existence statement: Forany term t ∈ T (X,Σ), any partial algebra A of signature Σ and for any valuation

v : X → A the statement A |= D(X : t)[v] is equivalent to A |= (X : te≈ t)[v] (and

our TE-statement (X : ∃ t) is just an abbreviation for (X : te≈ t)), if we denote

it by (X : Dt) then our notation gets closer to the one in computer science, but wewant to stress that the definability of (X : Dt) from existence equations has a lotof technical advantages. In what follows we want to indicate, however, that another“trick” can give to the corresponding generalization of strong equality (almost) thesame expressive power as the one for ECE-equations — and if we would forbid theempty partial algebra of signature Σ, the concepts would even be equivalent withoutany restrictions.

Let us give some arguments, why we prefer (and suggest) to base a first orderlanguage for partial algebras on E-equations:

(e1) We do it because of their close connection with the description of free partialalgebras, which was actually long ago our first motivation to consider them.

(e2) They are exactly those axioms preserved by ordinary homomorphisms (see be-low, why we prefer homomorphisms to e.g. closed homomorphisms).

(e3) One does not need an additional definedness predicate.

(e4) The concept of ECE-equations based on the one of E-equations reflects theusual way of formulating axioms for partial algebras in the meta-language: Ifsome terms are interpreted, then others are interpreted and are equal (this canalso be achieved by the combination of the definedness statement with strong orweak equations, but here it is an intrinsic property fitting neatly into the generalconcept and the corresponding syntactical and semantical theories, see also thenext item). The (almost) equivalent concept of generalized Kleene-equationdoes not allow to formulate this in a similar suggestive way.

(e5) In contrast to weak and strong equations they have a “nice” syntactical as wellas semantical theory, and the definedness concept is subsumed in a very natural

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way, while the basic concepts of weak or strong equations, respectively, have nosuch “nice” theories (neither on the semantical nor on the syntactical side).

(e6) If one wants to consider the solution sets of equations between terms using someset of fundamental operations as domains of other fundamental operations, asit is done e.g. in the definition of the domain of the composition operation in(small) categories

dom C := (g, f) ∈ Mor(C) | DomCg = CodCf ,

or, more generally in the so-called hep-varieties of H. Reichel, then the abstractequational concept behind it is the one of existence equations and not the oneof strong equations (which would allow a pair (g, f) above to be a solution of

Domgs≈ Codf in C, when neither DomCg nor CodCf are defined).

(e7) The hierarchy E-equations — ECE-equations — QE-equationscorresponds “nicely” to the encoding of elementary implications by epimor-phisms e : P → C and to the corresponding Meta Birkhoff Theorem by Andreka,Nemeti and Sain, where one can classify the (finitely presentable) “premise alge-bra” P as discrete — any partial Peano algebra — any partial algebra,respectively.

(e8) A last but not the least reason for us is the close connection with the habits inthe metalanguage: If one states there that ϕA(a1, . . . , am) = ψA(b1, . . . , bn) forelements a1, . . . , am, b1, . . . , bn of A, where ϕ and ψ are any operation symbolsor terms, then one usually means that “both sides of the equation exist ans areequal”. Yet this is just the correspondence to existence equations on the levelof the metalanguage.

Actually we can only find very few reasons to prefer e.g. strong equality:

(s1) The “strong equational theory” of some class K of partial algebras of signatureΣ w.r.t. some fixed set of variables always forms a congruence relation on thewhole term algebra (yet here full invariance gets lost, which is still true forexistence equations).

(s2) If (X; ts≈ t′) holds in a partial algebra A, then this (also) means equality of

the induced term operations, However, the induced term operations need not

be total as it would then be the case w.r.t. an existence equation (X ; te≈ t′).

Moreover, let us briefly discuss our preference of plain homomorphisms to, say,closed homomorphisms or to any “meaningful” concept of morphisms based on partialmappings as basic morphisms for the category of all partial algebras of some givensignature:

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(h1) According to investigations by different authors the (plain) homomorphismsprovide the only concept of morphisms between partial algebras known so far(to us), which yields for every signature a complete and cocomplete category.

(h2) With respect to (h1) one can say that taking all homomorphisms as morphismsis the only way to get the concept of direct products (needed for the descriptionof E-varieties as being exactly the primitive classes) — and with them the one ofreduced products (needed for the description of ECE-varieties and QE-varieties)— as “natural” category theoretical concepts (without any restrictions to thesignature).

(h3) The category PAlg(Σ) based on homomorphisms is a very rich category with awealth of factorization systems allowing to formulate all (“usual”) model theo-retic concepts in it (without too much effort). In particular, it is the only oneknown to us (based on all partial algebras of a given signature) which satisfies allassumptions of the Meta Birkhoff Theorem by Andreka, Nemeti and Sain (whenwe do not want to have any restrictions w.r.t. the signature). And since thistheorem provides many useful model theoretic results on the description of spe-cial universal Horn formulas (and even a wider class of universal implications)it is good to have a suitable framework for it.

(h4) Within the categories PAlg(Σ) based on homomorphisms one can model theother interesting categories based on all partial algebras of signature Σ by cat-egory theoretical methods (cf. e.g. section 22 in the appendix of [B86]).

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10.2 Some remarks on term operations and clones

In connection with term operations there were in 1996 the following additions, whichwere not treated in 1998 and 2000. We present here the formulations for many-sortedpartial algebras.

• Let A be an S-set, and let w = s1 . . . sn ∈ S∗ be any word on the set S of sorts.By

ew,Ai : As1 × . . .× Asi × . . .× Asn → Asi ( (a1, . . . , an) 7→ ai )

we denote the i-th n-ary unit operation (or projection operation of inputtype w. These unit operations are always total; since in the case, when thecarriers of their output sorts are empty then this is also the carrier of an inputsort, and therefore the domain is then empty, too.

We point out that in generalization of the one-sorted definitions in section 1.1we denote by

POw,s(A) =⋃

D⊆As1×...×Asn

ADs

the set of all n-ary partial operations with input specification w and output sorts. In particular we denote by

POw(A) := (PO(w,s(A))s∈S

the S-set of all partial operations on A with the same input specification w.Finally we introduce

PO(A) :=⋃w∈S∗

POw(A)

as the S-set of all finitary partial operations on A (in the total or in the one-sorted case the definitions above become much simpler).

PO(A) can be considered as total algebra with S∗ × S as set of sorts, if S hasonly one element then we obtain a heterogeneous total algebra with N as set ofsorts). It has the family

(ew,Ai | 1 ≤ i ≤ n)(w,si)∈S∗×S

as nullary constants and the so-called superposition operations: For w :=s1 . . . sn, w

′ := s′1 . . . s′m ∈ S∗ and s ∈ S set

supposw,sw′ := supposs1...sn,ss′1...s′m

: POw,s(A)×(POw′,s1(A)×. . .POw′,sn(A))→ POw′,s(A)

defined bysupposw,sw′ (f, g1, . . . gn) := f [g1, . . . , gn] ,

(a1, . . . , am) 7→ f(g1(a1, . . . , am), . . . , gn(a1, . . . , am)) .

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A subset A of PO(A), which contains all unit operations and is closed w.r.t. allsuperposition operations is then called a (concrete) p-clone (more precisely:a (concrete) clone of partial operations on A.

If A is a partial algebra of signature Σ, and if w ∈ S∗, then POw(A) becomes apartial algebra of signature Σ in the following way:

For ϕ ∈ Ω with (η, σ)(ϕ) = (s1 . . . sn, s) =: (v, s) und gi ∈ POw,si(A) we set

ϕPOw(A)(g1, . . . , gn) := supposv,sw (ϕPOw

(A), g1, . . . , gn) = ϕPOw(A)[g1, . . . , gn] .

In what follows let, Y be an S-set of variables, where Ys =: ysn | n ∈ N isalways a countably infinite set. Consider w = s1 . . . sn, and set

Yw := ysii | 1 ≤ i ≤ n .

Then we set

pTO(A) :=⋃

(w,s)∈S∗×S

(Yw; t)A | t ∈ T (Yw,Σ)s

to be the set of all finitary (partial) term operations on A (w.r.t. to representa-tive sets of variables). In particular we have

pTOw(A) :=⋃s∈S

(Yw; t)A | t ∈ T (Yw,Σ)s

as set of all induced term operations on A.

Observe that we always have (Yw;xi)A = ew,Ai ∈ pTOw(A) , for 1 ≤ i ≤ n .

Let us denote by tTO(A) and tTOw(A), the sets of all total term operationsand of all total term operations of input specification w, respectively.

While in the case of total and partial (many-sorted) algebras A the set tTO(A)is always a clone (exercise), this is not true in general in the case of partialalgebras, as one may see from the following example:

Consider the one-sorted similarity type τ := (2, 1, 0) corresponding to Ω :=(ϕ, ν, γ), and consider the class K of all partial algebras of type τ defined by

the axioms (x, y ;ϕxye≈ ϕxy) (i.e. ϕ is required to be total but underlies

no further restrictions), and (x ; νγe≈ γ) (i.e. in the K-free partial K-algebra

FM := F (MK) K-freely generated by any set M we have dom νFM = γFM ,γFM is not contained in M and not a value w.r.t. ϕFM , and ϕFM is total,injective and has no value in M , as can easily be seen (exercise)). Then, say,for M := x, y,

g := e2,FM1 [e2,FM

1 , νFM [e2,FM2

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has domain FM × γFM , and g(a, γFM ) = a for all a ∈ FM . By its definitionone realizes that g belongs to the p-clone on FM generated by pTO(FM) but gdoes not belong to pTO(FM) itself, since the term operations on FM are eithertotal projections or “start” with ν and then take value γ or they “start” withϕ, and then they cannot behave like a partial projection.

In order to avoid this effect, W.Craig has proposed to introduce logical pro-jections. This will be explained below in more detail.

• (Introduction of “generalized terms”): This means that one should extendthe given signature in the way introduced below. Yet first we introduce thefollowing convention:

X = (Xs)s∈S will designate some — usually finite — fixed S-subset of Y , andif not stated differently, xs will always stand for ys0, the first variable in Ys. Inparticular we define the following two-element S-subsets101

xs, xu :=

ys0, yu0 , if s 6= u, ys0, ys1 , if s = u.

(32)

Ω := ϕ | ϕ ∈ Ω will be a fixed set of operation symbols , Ωe := Ω ∪ εsu |s, u ∈ S the corresponding extended set of operation symbols, where εsu /∈ Ωand εsu 6= εs

′u′ for s, u, s′, u′ ∈ S and (s, u) 6= (s′, u′). In the homogeneousor one-sorted case, i.e. when S = s has only one element s, then we haveonly εss, and we shall omit the index “ss” and only write ε instead of εss. Ifτ : Ω→ N0 is any (homogeneous) arity function (homogeneous similarity type),then we define τe : Ωe → N0, where

τe(ϕ) :=

τ(ϕ) , if ϕ ∈ Ω,2 , if ϕ = εsu, for some s, u ∈ S. (33)

Next, let (η, σ) : Ω → S∗ × S be a many-sorted (heterogeneous) sort speci-fication mapping assigning to each ϕ ∈ Ω a “sort specification” (η, σ)(ϕ) :=(sϕ1 . . . s

ϕτ(ϕ), s

ϕ), and as above we define the corresponding “extended sort spec-

ification” (ηe, σe) : Ωe → S∗ × S as

(ηe, σe)(ϕ) :=

(η, σ)(ϕ) , if ϕ ∈ Ω,(su, s) , if ϕ = εsu, for some s, u ∈ S. (34)

Finally, let Σ = (S,Ω, τ, (η, σ)) be the related heterogeneous signature, thenΣe := (S,Ωe, τe, (ηe, σe)) designates the corresponding extended (heterogeneous)signature.

101Observe that this convention does not apply to two-element sequences.

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Heterogeneous partial algebras A = ((As)s∈S, (ϕA)ϕ∈Ω), with fundamental par-

tial operations

ϕA : Asϕ1 × . . .× Asϕτ(ϕ)⊇ dom ϕA → Asϕ for ϕ ∈ Ω,

of signature Σ are defined as usual, while similar to the homogeneous case,which we have already mentioned in the introduction, the additional operationsεsu,Ae (s, u ∈ S) in the corresponding partial algebra Ae := ((As)s∈S, (ϕ

Ae)ϕ∈Ω∪(εsu,Ae)s,u∈S) — with ϕAe := ϕA for all ϕ ∈ Ω — of the extended signature Σe

will always be interpreted as total binary first projections102:

dom εsu,Ae := dom (xs, xu : εsu(xs, xu))Ae := As × Au and

εsu,Ae(a, b) := (xs, xu : εsu(xs, xu))Ae(a, b) := a (35)

for all heterogeneous partial algebras Ae of signature Σe,

and for all a ∈ As, b ∈ Au.

It should, however, be observed that (xs, xu : εsu(xs, xu))Ae is an empty term

operation on Ae, if at least one of the phyla (i.e. carrier sets) of sort s or u isempty in Ae (i.e. if As = Ø or Au = Ø). Recall in this connection that in theheterogeneous case one has to specify for each term operation and for each firstorder formula an information about the (free) variables to be involved in orderto make the language powerful enough for the intended purposes.

For convenience, we shall introduce for all non-empty words w ∈ S+, w :=s1 . . . sn, as abbreviations the general first projections εw as terms

εs1(x1) := εs1s1(x1, x1) (36)

εs1...sn(x1, . . . , xn) := εs1s2(x1, εs2s3(x2, . . . , ε

sn−1sn(xn−1, xn) . . .)), (37)

for xi ∈ Xsi , 1 ≤ i ≤ n. And in the homogeneous case this will just be desig-nated by εn. Observe that one can then derive also all k-th n-ary projectionswith a given pattern s1 . . . sn of input sorts as

εs1...sn,k(x1, . . . , xn) := εsksk+1...sns1...sk−1(xk, xk+1, . . . , xn, x1, . . . , xk−1) (38)

for 1 ≤ k ≤ n, which in the homogeneous case will then be denoted by

εnk(x1, . . . , xn) := εn(xk, xk+1, . . . , xn, x1, . . . , xk−1). (39)

We shall use these abbreviations freely in what follows.

102See the convention in (32) above.

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• (Consequences of the use of the extended signature):

(A) The partial algebras of signature Σe are the same as those of signature Σ,“when seen from the outside”, yet it is not difficult to realize that pTO(Ae)is always a p-clone of partial operations, namely the p-clone generated bypTO(A) w.r.t. the superposition operations.

(B) When one uses the extended signature Σe, then the use of “strong equa-tions” as axioms has the same expressive power as the use of ECE-equationsw.r.t. the “usual terms” and the class PAlg(Σ) \ Ø (i.e. if one forbidsthe empty partial algebra, where the carriers of all sorts are empty). Yetthe expressive power of ECE-equations as axioms is not enlarged by usingthe extended signature. — Observe that for “usual terms” strong equalityhas a quite difficult strong-equational theory, which has been revealed onlyquite recently.

(C) For each term t ∈ T (Yw,Σe)s there exist “usual terms” ti ∈ T (Yw,Σ)ui(1 ≤ i ≤ m) such that

(Yw ; t)Ae = (Yw ; εu1...um(t1, . . . , tm))Ae

for each partial algebra A of signature Σ.

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CONTENTS 223

Contents

References 2

0 Introduction 80.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80.2 Motivation for the consideration of of partial algebras . . . . . . . . . 80.3 Some remarks about set theory . . . . . . . . . . . . . . . . . . . . . 14

0.3.1 Classes and sets in general . . . . . . . . . . . . . . . . . . . . 140.3.2 Axioms of set theory . . . . . . . . . . . . . . . . . . . . . . . 190.3.3 Some useful set theoretical definitions and principles . . . . . 20

1 Similarity types, partial algebras, first visits to closed subsets, ho-momorphisms and direct products 241.1 Homogeneous partial algebras . . . . . . . . . . . . . . . . . . . . . . 241.2 Heterogeneous partial algebras . . . . . . . . . . . . . . . . . . . . . . 311.3 Closed subsets, generation and algebraic (structural) induction (1st visit) 361.4 A first visit to homomorphisms and direct products . . . . . . . . . . 41

2 Partial Peano algebras, terms and term operations 452.1 Partial Peano algebras and recursion theorems . . . . . . . . . . . . . 452.2 More on terms, term operations . . . . . . . . . . . . . . . . . . . . . 55

3 Some basic logical and model theoretic concepts, homomorphismsrevisited 613.1 Existence equations, ECE- and QE-equations . . . . . . . . . . . . . 613.2 Preservation and reflection of formulas . . . . . . . . . . . . . . . . . 693.3 Homomorphisms, closed homomorphisms and isomorphisms (second

visit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Examples 754.1 “Classical” one-sorted algebraic structures and modules . . . . . . . . 764.2 Some examples of partial algebras . . . . . . . . . . . . . . . . . . . . 884.3 A different view at heterogeneous (partial) algebras . . . . . . . . . . 91

5 Substructures, generation and homomorphisms revisited 965.1 Substructures and generation revisited . . . . . . . . . . . . . . . . . 965.2 On some properties of homomorphisms . . . . . . . . . . . . . . . . . 1005.3 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . 104

6 Congruence relations, factor algebras, diagram completion I 1086.1 Congruence relations and closed congruence relations . . . . . . . . . 1086.2 Factor algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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CONTENTS 224

6.3 Diagram Completion Lemma for mappings and full and surjective ho-momorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4 Isomorphism theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.5 Factorization systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Some constructions of partial algebras 1307.1 Products and reduced products . . . . . . . . . . . . . . . . . . . . . 1317.2 Coproducts and directed colimits . . . . . . . . . . . . . . . . . . . . 1397.3 Some operators derived from the constructions . . . . . . . . . . . . . 152

8 Free partial algebras, universal solutions and E-equations 1598.1 Some descriptive principles of the universal algebraic language . . . . 1598.2 Free partial algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.3 Free completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.4 Diagram completion II, the Epimorphism Theorem . . . . . . . . . . 1688.5 On the existence of universal solutions . . . . . . . . . . . . . . . . . 1748.6 Fully invariant congruence relations . . . . . . . . . . . . . . . . . . . 1798.7 E-varieties and the characterization of primitive classes and of free

partial algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.8 Preservation of formulas by operators . . . . . . . . . . . . . . . . . . 1888.9 A characterization of free partial algebras . . . . . . . . . . . . . . . . 1908.10 Characterizations of primitive classes — finitely many sorts . . . . . . 1918.11 The E-equational theories . . . . . . . . . . . . . . . . . . . . . . . . 192

9 Quasivarieties of partial algebras 1969.1 About the first order language for partial algebras based on existence

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1969.2 Elementary implications and their translation into epimorphisms . . . 1989.3 Preservation and reflection of formulas revisited . . . . . . . . . . . . 2069.4 Factorization systems revisited . . . . . . . . . . . . . . . . . . . . . . 2079.5 A Meta Birkhoff Theorem by Andreka, Nemeti and Sain . . . . . . . 209

10 Some additions 21510.1 Some general remarks concerning the use of different kinds of equations 21510.2 Some remarks on term operations and clones . . . . . . . . . . . . . . 218

Observe that starting from the middle of page 168 (in subsection 8.5) what isincluded into these notes is just a part of the survey [B93] on homogeneous partialalgebras which has not yet been fully included into these notes nor even adopted tomany-sorted algebras. Moreover, proofs and auxiliary results needed for the proofsas well as additional remarks are missing in that part. For more details concerningthe topics treated in these subsections the reader is still referred to [B86].

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CONTENTS 225

The following index is just the collection of the keywords handled in the form asLATEX supports it without requiring great efforts. This is still far from the final indexas we would like to provide it. In particular, the index starts with keywords startingwith some expression in mathmode, for what reason it could not be automaticallyput into the right place by LATEX.

17th July 2002

Index

I-th (heterogeneous) power, 32S-set, 31S-sorted set, 31S-subset, 32S-support, 184X-initial homomorphic extension, 57X-initial segment, 57inf-semilattice, 81sup-semilattice, 81i-th input sort, 33n-ary operation symbol, 24n-ary partial operation, 24n-ary relation, 17n-th power, 17H-projective, 210K-extendable, 166K-free subset, 160K-freely generated by, 160K-independent subset, 160K-universal, 166K-universal epimorphism, 166K-universal solution, 166(absolutely) free completion, 163, 167(canonical) injection, 139(canonical) projection, 43(closed) subalgebra, 36, 40(closed) subalgebras, 120, 153(commutative) field, 84(concrete) clone of partial operations,

219(concrete) p-clone, 219(fundamental) operation symbols, 24,

33(global partial) term operation, 58(kommutativer) Korper, 84(nullary) constant, 24(partial) interpretation of terms, 62

(relatively) free partial algebra, 160(weak) homomorphic image, 119(weak) homomorphic images, 120, 153Kleene-equations, 61Kuratowski-pair, 46

abelian group, 78absorption laws, 80abstract data type, 177Algebra, 86algebraic, 97algebraic induction, 36, 39algebraic quasi-order, 55, 56alphabet, 33antisymmetry, 20arity, 24, 33assignment, 58, 62associative, 77assoziativ, 77atomic formula, 63Axiom of Choice, 18, 117axiomatic class, 159axioms, 159

Baire-class, 99basis, 159, 160beschrankter Verband oder 0-1-Verband,

80bijection, 18bijective, 18binary operation symbol, 24binary relation, 17Boolean lattice, 81Boolescher Verband, 81bounded lattice or 0-1-lattice, 80

cancellative semigroups, 78canonical sort mapping, 91cardinal number, 135carrier set, 26

226

INDEX 227

carrier set of sort s of A, 31Cartesian product, 16category, 28, 35category theoretic reduced product, 152Characterization of monomorphisms, 104choice function, 18classes, 14closed, 42, 92closed congruence relation, 108closed homomorphic image, 120closed homomorphic images, 120, 153closed homomorphism, 42closed subset, 36, 38, 40closure operator, 38, 39closure system, 37, 39colimiting cocone, 142comma category, 93commutative, 77commutative (or abelian) semigroup, 77commutative monoid, 78commutative ring, 84compact, 97comparison, 100compatible, 145compatible family, 142complementation, 81complete lattice, 83complete lower semilattice, 83complete upper semilattice, 83concrete category, 101cone, 209confinal, 144conglomerates, 15congruence relation, 108conjunction, 64coproduct, 139

definable, 67definedness predicate, 215dense, 105diagonal, 109Diagram Completion Lemma for full and

surjective homomorphisms, 115

difference, 16difference class, 16difference of S-sets, 32direct limit, 142direct product, 16, 18, 43direct product of S-sets, 32direct sum, 141directed, 142directed colimit, 142directed ordered set, 142directed system, 142discrete (partial) operation, 25discrete heterogeneous partial algebra,

34discrete partial algebras, 29disjoint, 19disjoint union, 19disjunction, 64distributive 0-1-lattice, 80distributive lattice, 80distributive laws, 80distributiver Verband, 80division ring, 84Divisionsring, 84domain, 17, 24double, 22

E-equation, 62, 215E-variety, 67, 184ECE-equation, 215ECE-equations, 66, 198ECE-variety, 67eine Linksnull, 76eine Loop, 78eine Rechtsnull, 76elementary implication, 198elementary implications, 65empty (partial) operation, 25empty sequence, 25epimorphism, 104Epimorphism Theorem, 170epireflective subcategory, 177equalizer, 107, 138

17th July 2002

INDEX 228

equivalence, 64error handling, 164error value, 51evaluable, 58Evans equations, 199exhaustion, 41existence equation, 62existence equations, 61existence variety, 184existentially conditioned existence equa-

tion, 66existentially conditioned existence equa-

tions, 198

factor algebra, 115factor set, 114Factorization Lemma for full and sur-

jective homomorphisms and monomor-phisms, 119

factorization system, 124, 125Factorization Theorem for Epimorphisms

and Closed Monomorphisms, 171filter, 134final, 118, 206final homomorphism, 117final structure, 118finitely presentable, 147finitely presented, 211First Diagram Completion Lemma, 115first order axioms, 159first order language, 159forgetful functor, 26formal concept, 128formal context, 128formula, 63free, 159free partial algebra, 160free variable, 64free variables, 196freely generated by, 160freeness, 159full, 101full homomorphic image, 119

full homomorphic images, 120, 153fully invariant congruence relation, 179functor, 142fundamental (partial) operation, 26

Galois connection, 127, 128General Homomorphism Theorem, 170generalized Frechet filters, 135generalized Kleene-equation, 215generalized Peano axioms, 45generate, 39generating subset, 39generation, 36, 159graph, 17, 25group, 78groupoid, 77Gruppe, 78Gruppoid, 77

Halbgruppe, 77Halbring, 83Halbverband, 80Halbverband mit Eins(element), 80Halbverband mit Null, 80heterogeneous, 33heterogeneous (partial) algebra, 34hold, 65holds, 63homomorphic extension, 42homomorphic image, 43homomorphism, 41Homomorphism Theorem, 118hyperkomplexes System, 86

idempotent, 76, 80idempotente kommutative Halbgruppe,

80immediate predecessor, 55implication, 64independence class, 160induced product morphism, 131induction on the structure of terms, 36inductive, 20

17th July 2002

INDEX 229

inductive limit, 142infimum, 83infinitary ECE-equation, 66infinitary QE-euation, 66initial, 90, 96, 101, 118, 206initial algebra semantics, 177initial algebras, 177initial morphism, 101initial object, 159initial structure, 118injection, 18injective, 18, 71injective w.r.t. f , 166input sorts, 33interpretation, 26intersection, 15intersection of S-sets, 32intersection of substructures, 97inverse, 76inverse of the relation, 109Inverses, 76inversion, 76is generated, 39is valid, 63, 65isomorphic copies, 120, 153isomorphism, 42, 71

J.Schmidt-kernel, 168join, 15

kernel, 60, 111Kleene equations, 66, 199Kleene-equation, 215kommutativ, 77kommutative (oder abelsche) Halbgruppe,

77kommutativer Ring, 84kommutatives Monoid, 78

lattice, 80left cancellative semigroup, 77left inverse, 76left inversion, 76

left neutral, a left unit, 76left vectorspace, 85left zero (element), 76Lie rings, 86limit, 138linksinverse Operation, 76Linksinverses von, 76linkskurzbare Halbgruppe, 77linksneutral, eine Linkseins, 76logical projections, 220loop, 78lower bound, 83lower semilattice, 81

Magma, 77many-sorted (partial) algebra, 34many-sorted set, 31many-sorted similarity type, 33mapping from, 18minimal, 164minimal element principle, 20model theoretic homomorphism, 71model theoretic reduced product, 152Monoid, 78monoid, 78monoid ring, 87monomorphism, 104morphism, 35

natural homomorphism, 115natural mapping, 114natural projection, 115negation, 64neutral, a unit, 76neutral, ein Einselement, 76normal, 164normal one point (per sort) completion

of, 51nullary operation symbol, 24Nullelement, 76

object, 35onto, 18

17th July 2002

INDEX 230

open formula, 64operator preserves formula, 188ordered pair, 16, 21ordered set, 81ordered triple, 16output sort, 33

pairwise disjoint, 19partial algebra, 25partial interpretation of terms induced

by v, 55partial mapping, 17partial Peano algebra with Peano basis,

45Peano algebra, 45permutation, 79phylum, 91phylum of A of sort s, 31polynomial ring, 87power class, 16, 21power set, 82preservation of a formula, 69prime field, 178primitive, 154principal filter, 135principal ideal, 111process of approximation, 41process of generation from below, 99process of generation of structures, 40process of restriction of structure, 40product, 131products, 153products of non-empty families, 153projection operation, 218proper filter, 134pullback, 137

QE-equations, 66, 198QE-variety, 67quasi existence equations, 66, 198quasigroup, 78Quasigruppe, 78quasiprimitive, 154

quotient algebra, 115quotient of congruences, 121quotient set, 114

Rechtsinverses, 76rechtsneutral, eine Rechtseins, 76reduced product, 136reduced products, 135, 153reduced products of non-empty fami-

lies, 153reduct, 139reflection of a formula, 69reflexive and transitive closure, 109reflexive hull, 56reflexivity, 20regular semigroup, 77regulare Halbgruppe, 77relative subalgebra, 40relative subalgebras, 120, 153representative, 135restriction, 36, 121right cancellative semigroups, 78right inverse, 76right inversion, 76right neutral, a right unit, 76right zero (element), 76Ring, 83ring, 83ring with unit, 84

S-kernel, 168

satisfies the E-equation (X; te≈ t′) w.r.t.

the valuation, 62saturation, 121Schiefkorper, 84semi-ring, 83semi-ring with zero, 83semigroup, 77semilattice (or: idempotent commuta-

tive semigroup), 80semilattice with unit, 80semilattice with zero, 80Semiring, 83

17th July 2002

INDEX 231

sentence, 197sequence, 18set, 14, 15set of variables, 30signature, 33, 159similar, 34similarity type, 24skew field, 84small categories, 28sort algebra, 91sorts, 31, 33sorts of entries, 33specification, 177specification algebra, 91start object, 25strict, 56strong equation, 215Strong equations, 66, 199strong homomorphic image, 119strong homomorphism, 42strong smallness, 211strongly small, 147, 211structural induction, 36, 39subobject, 107substitution property, 179subterm, 55, 56superposition operation, 218support, 17, 203supremum, 83surjection, 18surjective, 18symmetric group, 79

Tacking Lemma, 182target object, 25TE-statement, 63, 68term, 55term algebra, 45term existence statement, 63, 215term reflecting, 71term reflecting homomorphism, 42terminal object, 132

terms, 30total, 18total (universal) algebras, 29total heterogeneous algebra, 34total heterogeneous term algebra on X

of signature Σ, 48total operation, 25total projection, 58total restriction, 40totality, 20totally finite, 147, 211TR-homomorphism, 42transformation monoid, 79transitive closure, 109transitive hull, 56transitivity, 20type, 24

ultrafilter, 135ultraproducts, 153unary operation symbol, 24unary relation, 17union, 15union of S-sets, 32unique-diagonal-fill-in-property, 125unit operation, 218unitarer Ring, 84universal formula, 64universal solution, 141, 160upper bound, 83upper semilattice, 81

valuation, 58, 62value, 18Verband, 80Verschmelzungsgesetze, 80

weak equation, 215Weak equations, 66, 199weak relative subalgebra, 41weak relative subalgebras, 120, 153weak subalgebra, 41well-ordering, 20

17th July 2002

INDEX 232

word algebra, 45

zero element, 76

17th July 2002

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