algebraic tools for modal logic -

Algebraic Tools for Modal Logic

Mai Gehrke Yde Venema

ESSLLI’01

August 13 – 17, 2001Helsinki, Finland

Algebraic Tools for Modal Logic

Mai Gehrke�

Yde Venemay

General aim

There is a long and strong tradition in logic research of applying algebraic techniques in orderto deepen our understanding of logic. Such applications are possible because many logics

correspond to classes of algebras; typically, the consequence relation of the logic translates

into the (quasi-)equational theory of the corresponding class of algebras.

This correspondence between logic and algebra allows one, on a �rst level, to study the

algebras in order to understand the deductive system. But often, metalogical properties also

end up having algebraic counterparts. In modal logic, a striking example of this phenomenon

can be found using the duality theory between Kripke structures and Boolean Algebras with

Operators. For instance, a modal logic is complete if and only if its corresponding algebraic

variety is generated by the class of algebras that are dual to the Kripke frames of the logic.

A central tool in proving completeness for modal logics is the notion of canonicity, which

happens to have an equally important and interesting algebraic expression. For a long timethe algebraic and the logical strand of research on these issues have been carried out in relative

isolation, and one of the aims of this course is to strengthen the ties between the two research

lines.

More concretely, the purpose of the course is twofold. Apart from giving a general intro-

duction to the fundamental ideas and methods of applying algebra in logic, we also intend to

present recent developments from algebra as well as modal logic, in an integrated format. Our

intention is to illuminate and generalize existing results concerning the issues of completeness,

canonicity and correspondence for modal and generalized modal logics.

Course outline

In the �rst part of the course (consisting of the �rst two lectures) we give a general introduction

to the algebraic perspective on logic. As our running examples we take classical propositional

logic and modal logic. We show how the technical notion of a modal logic corresponds to the

algebraic one of a variety of Boolean algebras with operators, and we discuss the connection

between the logical and the algebraic angle on properties such as completeness and canonicity.

�Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA. E-mail:

[email protected] of Logic, Language and Computation, Univ. of Amsterdam, Plantage Muidergracht 24, 1018 TV

Amsterdam. E-mail: [email protected].

1

In the last three lectures we concentrate on the notion of canonicity, which plays an equally

fundamental role in the theory of modal logic as in the algebraic theory of Boolean algebras

with operators. The aim of this part of the course is to to present some recent results in the

�eld that generalize and illuminate the classical results.

Intended audience & prerequisites

The intended audience for this course consists of two groups of people, and of each group we

presuppose some background knowledge.

First and foremost, we have tailored our course towards students that have encountered

modal logic before and want to study its general theory in more detail. We presuppose that

these students are familiar with the canonical model method for proving completeness of a

modal logic, and with the notion of a boolean algebra. Furthermore, some experience with

mathematical argumentation is required. It would be useful if the student had encountered

some lattice theory as well, but this is not strictly needed.And second, the course could be of value for mathematics students that have an interest in

universal algebra, and more particular, in the theory of lattices and lattices with additional

operators. For these students, some familiarity with duality theory would be useful, but

knowledge of modal logic is not strictly required.

About these notes

These Notes consist of three parts.

1. Modal Logic and Their Algebras, by Patrick Blackburn, Maarten de Rijke and Yde

Venema. This is an excerpt from the recently published textbook Modal Logic by the

same three authors (Cambridge University Press, Cambridge, 2001). There are two

chapters, of which the �rst starts with a recollection of some basic facts concerningmodal logic and then gives a fairly detailed description of the canonical model method

for proving modal completeness results. The second chapter introduces the algebraic

approach towards modal logic, and ends with a discussion of the importance of the

notion of canonicity in logic and algebra.

2. The article On the Canonicity of Sahlqvist Identities by Bjarni J�onsson, (Studia Logica,

1994) provides an algebraic description of the notion of canonicity. It modernizes and

further develops the classical algebraic methodology for proving canonicity and leads

to an algebraic proof of an important result in modal logic stating that all so-calledSahlqvist formulas are canonical.

3. Monotone Bounded Distributive Lattice Expansions, by Mai Gehrke and Bjarni J�onsson,

(Mathematica Japonica, 2000). This paper treats canonical extensions of distributive

lattice based algebras from an algebraic point of view. It includes the abstract algebraic

characterization of canonical extensions, the algebraic de�nition of extensions of maps,

and algebraic canonicity results mainly of semantic nature.

2

Modal Logic and Their Algebras

Patrick Blackburn, Maarten de Rijke and Yde Venema

1

Modal Logic: Basic Concepts

In this chapter we recollect some basic facts concerning modal logic, concentratingon completeness theory.

1.1 Syntax

Definition 1.1 Thebasic modal languageis defined using a set ofproposition let-ters(or proposition symbolsor propositional variables) � whose elements are usu-ally denotedp, q, r, and so on, and a unary modal operator3 (‘diamond’). Thewell-formedformulas� of the basic modal language are given by the rule

� ::= p j ? j :� j _ � j 3�;

wherep ranges over elements of�.Just as the familiar first-order existential and universal quantifiers are duals to

each other (in the sense that8x�$ :9x:�), we have a dual operator (‘box’) forour diamond which is defined by�� := :3:�. We also make use of the classi-cal abbreviations for conjunction, implication, bi-implication and the constant true(‘top’): �^ := :(:�_: ), �! := :�_ , �$ := (�! )^( ! �)

and> := : ?. a

Although we generally assume that the set� of proposition letters is a countablyinfinite setfp0; p1; : : :g, occasionally we need to make other assumptions. This iswhy we take� as an explicit parameter when defining the set of modal formulas.That is the basic modal language. Let us now generalize it. There are two obviousways to do so. First, there seems no good reason to restrict ourselves to languageswith only one diamond. Second, there seems no good reason to restrict ourselvesto modalities that take only a single formula as argument. Thus the general modallanguages we will now define may contain many modalities, of arbitrary arities.

1

2 1 Modal Logic: Basic Concepts

Definition 1.2 A modal similarity typeis a pair� = (O; �) whereO is a non-empty set, and� is a functionO ! N. The elements ofO are calledmodal oper-ators; we useM (‘ triangle’), M0, M1, . . . , to denote elements ofO. The function�assigns to each operatorM 2 O a finitearity, indicating the number of argumentsM can be applied to.

In line with Definition 1.1, we often refer tounary triangles asdiamonds, anddenote them by3a or hai, wherea is taken from some index set. We often assumethat the arity of operators is known, and do not distinguish between� andO. a

Definition 1.3 A modal languageML(�; �) is built up using a modal similaritytype � = (O; �) and a set of proposition letters�. The setForm(�; �) of modalformulasover� and� is given by the rule

� := p j ? j :� j �1 _ �2 j M(�1; : : : ; ��(M));

wherep ranges over elements of�. a

Definition 1.4 We now define dual operators for non-nullary triangles. For eachM 2 O thedualO of M is defined asO(�1; : : : ; �n) := :M(:�1; : : : ;:�n). Thedual of a triangle of arity at least2 is called anabla. As in the basic modal language,the dual of a diamond is called abox, and is written2a or [a]. a

Example 1.5 (An Arrow Language)The type�! of arrow logic is a similaritytype with modal operators other than diamonds. The language of arrow logic isdesigned to talk about the objects in arrow structures (entities which can be picturedas arrows). The well-formed formulas� of the arrow language are given by the rule

� := p j ? j :� j � _ j � Æ j � j 1’ :

That is,1’ (‘identity’) is a nullary modality (a modal constant), the ‘converse’ oper-ator is a diamond, and the ‘composition’ operatorÆ is a dyadic operator. Possiblereadings of these operators are:

1’ identity ‘skip’ ;� converse ‘� conversely’;� Æ composition ‘first�, then ’ : a

1.2 Semantics

Basic modal language

We start by defining frames, models, and the satisfaction relation for the basicmodal language.

1.2 Semantics 3

Definition 1.6 A frame for the basic modal language is a pairF = (W;R) suchthatW is a non-empty set andR is a binary relation onW .

That is, a frame for the basic modal language is simply a relational structurebearing a single binary relation. Elements ofW will be referred to by variousnames such asnodes, worlds, or points.

A modelfor the basic modal language is a pairM = (F; V ), whereF is a framefor the basic modal language, andV is a function assigning to each propositionletter p in � a subsetV (p) of W . Formally, V is a map:� ! P(W ), whereP(W ) denotes the power set ofW . Informally we think ofV (p) as the set ofpoints in our model wherep is true. The functionV is called avaluation. Given amodelM = (F; V ), we say thatM is based onthe frameF, or thatF is the frameunderlyingM. a

Definition 1.7 Supposew is a state in a modelM = (W;R; V ). Then we induc-tively define the notion of a formula� beingsatisfied(or true) in M at statew asfollows:

M; w p iff w 2 V (p); wherep 2 �;

M; w ? never;

M; w :� iff not M; w �;

M; w � _ iff M; w � or M; w ;

M; w 3� iff for somev 2W with Rwv we haveM; v �: (1.1)

It follows from this definition thatM; w 2� if and only if for all v 2 W suchthatRwv, we haveM; v �. Finally, we say that aset� of formulas is true at astatew of a modelM, notation:M; w �, if all members of� are true atw. a

It is convenient to extend the valuationV from proposition letters to arbitraryformulas so thatV (�) always denotes the set of states at which� is true:

V (�) := fw jM; w �g:

Definition 1.8 A formula� is globallyor universally truein a modelM (notation:M �) if it is satisfied at all points inM (that is, ifM; w �, for all w 2 W ).A formula � is satisfiablein a modelM if there issomestate inM at which� istrue; a formula isfalsifiableor refutablein a model if its negation is satisfiable.

A set� of formulas is globally true (satisfiable, respectively) in a modelM ifM; w � for all statesw in M (some statew in M, respectively). a

Arbitrary modal languages

We now define frames, models and satisfaction for modal languages of arbitrarysimilarity type.


Definition 1.9 Let � be a modal similarity type. A� -frameis a tupleF consistingof the following ingredients:

(i) a non-empty setW ,(ii) for eachn � 0, and eachn-ary modal operatorM in the similarity type� ,

an (n+ 1)-ary relationRM.

We turn such a frame into a model in exactly the same way that we did forthe basic modal language: by adding a valuation. That is, a� -model is a pairM = (F; V ) whereF is a� -frame, andV is a valuation with domain� and rangeP(W ), whereW is the universe ofF.

The notion of a formula� beingsatisfied(or true) at a statew in a modelM =

(W; fRM j M 2 �g; V ) (notation:M; w �) is defined inductively. The clausesfor the atomic and boolean cases are the same as for the basic modal language (seeDefinition 1.7). As for the modal case, when�(M) > 0 we define

M; w M(�1; : : : ; �n) iff for somev1, . . . ,vn 2W with RMwv1 : : : vn

we have, for eachi,M; vi �i:

This is an obvious generalization of the way3 is handled in the basic modal lan-guage. Before going any further, the reader should formulate the satisfaction clausefor O(�1; : : : ; �n).

As before, we often writew � for M; w � whereM is clear from thecontext. The concept ofglobal truth (or universal truth) in a model is definedas for the basic modal language: it simply meanstruth at all states in the model.And, as before, we sometimes extend the valuationV supplied byM to arbitraryformulas. a

Example 1.10 (Arrow Models)Consider the similarity type�! of arrow logic,cf. Example 1.5. Arrow frames are structures of the formF = (W;C;R; I), whereC is a ternary relation interpretingÆ, R a binary relation for andI a unary rela-tion for 1’. An arrow modelis a structureM = (F; V ) such thatF = (W;C;R; I)

is an arrow frame andV is a valuation. Then:

M; a 1’ iff Ia;

M; a � iff M; b � for someb with Rab;

M; a � Æ iff M; b � andM; c for someb andc with Cabc:

Interesting examples of such structures are the two-dimensional ones arising asfollows. LetU be some set, and letU �U be the set of pairs overU ; suppose nowthat the relationsC,R andI onU � U are defined as follows:

Cabc iff a0 = b0; a1 = c1 andb1 = c0;

Rab iff a0 = b1 anda1 = b0;

1.3 Completeness 5

Ia iff a0 = a1:

Then we call the structure(U �U;C;R; I) the two-dimensional arrow frame overU , notation:SU . In such a structure, the semantics works out as follows.V nowmaps propositional variables to sets ofpairs overU ; that is, to binary relations.The truth definition can be rephrased as follows:

M; (a0; a1) 1’ iff a0 = a1;

M; (a0; a1) � iff M; (a1; a0) �;

M; (a0; a1) � Æ iff M; (a0; u) � andM; (u; a1) for someu 2 U:

a

Validity

Definition 1.11 A formula� is valid at a statew in a frameF (notation:F; w �)if � is true atw in every model(F; V ) based onF; � is valid in a frameF (notation:F �) if it is valid at every state inF. A formula� is valid on a class of framesF (notation:F �) if it is valid on every frameF in F; and it isvalid (notation: �) if it is valid on the class of all frames. The set of all formulas that are valid ina class of framesF is called thelogic of F (notation:�F). a

Our definition of the logic of a frame classF (as the set of ‘all’ formulas that arevalid onF) is underspecified: we did not say which collection of proposition letters� should be used to build formulas. But usually the precise form of this collectionis irrelevant for our purposes. Readers that are worried about this may define, givena set� of proposition letters,�F(�) to be the setf� 2 Form(�; �) j F �g.)

Note that the above is a fairly restricted viewpoint on what a logic is: in particu-lar, we do not take the semantics consequence relation into consideration.

1.3 Completeness

Normal Modal Logics

Suppose we are interested in a certain class of framesF: are there syntactic mech-anisms capable of generating�F, the formulas valid onF? One response to suchquestions is embodied in the concept of anormal modal logic. A normal modallogic is simply a set of formulas satisfying certain syntactic closure conditions. Inthis section we are working in a fixed countable set of proposition letters, and wefirst confine ourselves to the basic modal language.

Definition 1.12 (Normal Modal Logics)A normal modal logicin the basic modallanguage is a set� of modal formulas that contains, besides all propositional tau-tologies, the axioms


(K) 2(p! q)! (2p! 2q),

(Dual) 3p$ :2:p.

while it is closed under the rules

(modus ponens) if� 2 � and�! 2 � then 2 �,

(generalization) if� 2 � then2� 2 �, and

(uniform substitution) if� belongs to� then so do all of its substitution in-stances.

If � 2 � we say that� is a theoremof � and write`� �; if not, we write 6`� �. If�1 and�2 are modal logics such that�1 � �2, we say that�2 is anextensionof�1.

In what follows, we usually drop the words ‘normal modal’ and talk simply of‘logics.’ a

Remark 1.13 Apart from the inclusion of the Dual axiom – needed because wetook the diamond as the primitive operator in our language – the above definition ofa normal modal logic is probably the most popular way of stipulating what normallogics are. But it is not the only way. Here, for example, is a simple diamond-basedformulation of the concept, which will be useful in our algebraic work: a logic� isnormal if it contains (besides the propositional tautologies) the axioms3? $ ?

and3(p_q)$ 3p_3q, and is closed under the rules of modus ponens, universalsubstitution and: � �! implies`� 3�! 3 . This formulation is equivalentto Definition 1.12, as the interested reader may verify. a

Example 1.14 (i) The collection of all formulas is a logic, theinconsistentlogic.

(ii) The set of formulas valid in a given frame class is a logic.

(iii) If f�i j i 2 Ig is a collection of logics, thenTi2I �i is a logic.

(iv) If M is a class of models, then the set of formulas that are true throughoutany model inM neednotbe a logic since it may not be closed under uniformsubstitution. a

Example 1.14(iii) guarantees that there is a smallest normal modal logic containingany set of formulas.

Definition 1.15 The normal modal logicgeneratedor axiomatizedby� , notation:K�, is the smallest logic containing� ; the formulas in� are called theaxiomsofthis logic. The logic generated by the empty set is theminimal normal modal logicK. a

1.3 Completeness 7

This generative perspective, which is essentiallysyntactic, can be phrased in theterminology of Hilbert style axiomatizations, but since we are not interested in thedetails of proof systems we will refrain from doing so.

Defining a logic by stating which formulas generate it (that is, extending theminimal normal logicK with certain axioms of interest) is the usual way of syn-tactically specifying normal logics. Here are some of the better known axioms,together with their traditional names:

(4) 33p! 3p

(T) p! 3p

(B) p! 23p

(D) 2p! 3p

(.3) 3p ^3q! 3(p ^3q) _3(p ^ q) _3(q ^3p)

There is a convention for talking about the logics generated by such axioms: ifA1; : : : ;An are axioms thenKA 1 : : :An is the normal logic generated by A1, . . . ,An. But irregularities abound. Many historical names are firmly entrenched, thusmodal logicians talk ofT, S4, B, andS5 instead ofKT , KT4 , KB and KT4Brespectively.

Soundness and Completeness

Now that we know what normal modal logics are, we are ready to introduce the twofundamental concepts linking the syntactic and semantic perspectives:soundnessandcompleteness.

Definition 1.16 (Soundness and Completeness)Let F be a class of frames (ormodels, or general frames). A normal modal logic� is soundwith respect toF if� � �F. (Equivalently:� is soundwith respect toF if for all formulas�, and allstructuresF 2 F, `� � impliesF �.) If � is sound with respect toF we say thatF is a class of frames(or models, or general frames)for �.

Conversely,� is completewith respect toF if for any formula�, if �F � �; thatis, if F � then`� �. a

These notions, which only refer to the theorems of the logic and the set of va-lidities of a frame class, are often referred to as theweakversions of soundnessand completeness. The correspondingstrongnotions, which relate the derivabilityof formulas from assumptions to the semantic consequence relation over a frameclass, will not be discussed in these notes.Table 1.1 lists a number of well-known logics together with classes of frames forwhich they are sound. Recall that aright-unboundednessframe(W;R) is a framesuch that8x9yRxy. Also, a frame(W;R) satisfying8x8y8z (Rxy ^ Rxz !(Ryz _ y = z _Rzy)) is said to haveno branching to the right.


These completeness results are among the best known in modal logic, and wewill soon be able to prove them. Together with their soundness counterparts, theyconstitute perspicuous semantic characterizations of important logics.K4, for ex-ample, is not just the logic obtained by enrichingK with some particular axiom:it is precisely the set of formulas valid on all transitive frames. There is alwayssomething arbitrary about syntactic presentations; it is pleasant (and useful) to havethese semantic characterizations at our disposal.

K the class of all framesK4 the class of transitive framesT the class of reflexive framesB the class of symmetric framesKD the class of right-unbounded framesS4 the class of reflexive, transitive framesS5 the class of frames whose relation is an equivalence relationK4.3 the class of transitive frames with no branching to the rightS4.3 the class of reflexive, transitive frames with no branching to the rightKL the class of finite transitive trees (weakcompleteness only)

Table 1.1.Some Soundness and Completeness Results

These completeness results are among the best known in modal logic, and we willsoon be able to prove them. Together with their soundness counterparts, they con-stitute perspicuous semantic characterizations of important logics.K4, for exam-ple, is not just the logic obtained by enrichingK with some particular axiom: it isprecisely the set of formulas valid on all transitive frames. There is always some-thing arbitrary about syntactic presentations; it is pleasant (and useful) to havethese semantic characterizations at our disposal. Not all normal modal logics canbe semantically characterized as the set of formulas valid in a certain frame class.This fact makes the following definition meaningful.

Definition 1.17 A normal modal logic� iscompleteis it is the logic of some frameclassF, i.e., if� = �F. a

Arbitrary Modal Languages

To conclude this section, we extend the definition of normal modal logics to arbi-trary similarity types.

Definition 1.18 Assume we are working with a modal language of similarity type� . A modal logicin this language is a set of formulas containing all tautologies thatis closed under modus ponens and uniform substitution. A modal logic� is normal

1.4 Canonical Models 9

if for every operatorO it contains the axiom KiO

(for all i such that1 � i � �(O))

and the axiom DualO, and is closed under the generalization rules described below.The required axioms are obvious polyadic analogs of the earlier K and Dual

axioms:

(KiO

) O(r1; : : : ; p! q; : : : ; r�(O)) !

!�O(r1; : : : ; p; : : : ; r�(O))! O(r1; : : : ; q; : : : ; r�(O))

�:

(DualO) M(r1; : : : ; r�(O)) $ :O(:r1; : : : ;:r�(O)):

(Herep; q; r1; : : : ; r�(O) are distinct propositional variables, and the occurrencesKiO

of p and q occur in thei-th argument place ofO.) Finally, for a polyadicoperatorO, generalization takes the following form:

`� � implies `� O(?; : : : ; �; : : : ;?):

That is, ann-place operatorO is associated withn generalization rules, one foreach of itsn argument positions.

Note that these axioms and rules do not apply tonullary modalities. Nullarymodalities are rather like propositional variables and – as far as the minimal logicis concerned – they do not give rise to any axioms or rules. a

Definition 1.19 Let � be a modal similarity type. Given a set of� -formulas� ,we defineK��, the normal modal logicaxiomatizedor generatedby � , to be thesmallest normal modal� -logic containing all formulas in� . Formulas in� arecalledaxiomsof this logic, and� may be called anaxiomatizationof K��. Thenormal modal logic generated by the empty set is denoted byK� . a

1.4 Canonical Models

Soundness proofs are usually routine, but completeness of a logic is often hard toprove. The standard approach to show that a logic� is complete with respect to aframe classF uses contraposition: one proves that if a formula� doesnot belongto a logic, it is satisfiable in some model based on a frame inF. Thus completenesstheorems are essentially model existence theorems. In this section we discuss, in afair bit of detail, the most important method for proving completeness; it is calledthecanonical modelmethod because it constructs a single model in which all non-theorems of the logic can be refuted.

The idea behind the construction of this canonical model is of an elegant sim-plicity: let the points of the model form a coherently related collection ofsets offormulas, and make sure that any modal formula is true at such a point if and onlyif it is a member of it. In a slogan:

truth = membership.


Maximal Consistent Sets

First we need some auxiliary technical definitions. Fix a normal modal logic�.The points of the canonical model for� will be maximal�-consistent sets.

Definition 1.20 If � [ f�g is a set of formulas then� is deducible in� from �

(or: � is�-deducible from� ) if there are formulas 1,. . . , n 2 � such that

`� ( 1 ^ � � � ^ n)! �:

If this is the case we write� `� �, if not, � 6`� �. A set of formulas� is �-consistentif � 6`�?, and�-inconsistentotherwise. A formula� is�-consistent iff�g is�-consistent; otherwise� is�-inconsistent. A set of formulas� is maximal�-consistent, or: a�-MCS, if � is �-consistent, and any set of formulas properlycontaining� is�-inconsistent. It is easy to see that� is�-consistent iff:� 2 �.

a

Why useMCSs in completeness proofs? To see this, note that every pointw in everymodelM for a logic� is associated with a set of formulas, namelyf� j M; w

�g. It is easy to check (and the reader should do so) that this set of formulas isactually a�-MCS. That is: if� is true in some model for�, then� belongs to a�-MCS. The idea of the canonical model construction is to turn this observationaround.

We need to learn a little more aboutMCSs.

Proposition 1.21 (Properties ofMCSs) If � is a logic and� is a�-MCS then:

(i) � is closed under modus ponens: if�, �! 2 � , then 2 � ;(ii) � � � ;

(iii) for all formulas�: � 2 � or :� 2 � ;(iv) for all formulas�, : � _ 2 � iff � 2 � or 2 � .

Proof. Left to the reader. a

As MCSs are to be our building blocks, it is vital that we have enough of them. Infact, any consistent set of formulas can be extended to a maximal consistent one.

Lemma 1.22 (Lindenbaum’s Lemma) If � is a�-consistent set of formulas thenthere is a�-MCS�+ such that� � �+.

Proof. Let �0, �1, �2; : : : be an enumeration of the formulas of our language. Wedefine the set�+ as the union of a chain of�-consistent sets as follows:

�0 = �;

�n+1 =

��n [ f�ng; if this is�-consistent�n [ f:�ng; otherwise


�+ =Sn�0�n:

The proof of the following properties of�+ is left as an exercise to the reader:(i) �n is �-consistent, for alln; (ii) exactly one of� and:� is in �+, for everyformula�; (iii) if �+ `� �, then� 2 �+; and finally (iv)�+ is a�-MCS. a

The Canonical Model

In particular, Lindenbaum’s Lemma guarantees that every�-consistent formulabelongs to some�-MCS. Thus, if we take as the universe of our canonical modelthe collection ofall �-MCSs, we are sure that every�-consistent formula is anelement of some point in the model. Hence, if we can indeed prove a Truth Lemmastating that ‘truth = membership’, we have shown that every�-consistent formulais satisfiable in the canonical model for�.

Obviously, to make this work it is crucial to find the right definition of the canon-ical valuation and the canonical accessibility relations. The first is straightforward;for the latter, let us have look at how maximal consistent sets are related in a con-crete model. Ifw is related tow0 in some modelM, then it is clear that the in-formation embodied in theMCSs associated withw andw0 is ‘coherently related’.The idea behind the canonical model construction is to try and turn this observationaround: that is, to work backwards from collections of coherently relatedMCSs tothe desired model.

We are now ready to build models out ofMCSs, and in particular, to build thevery special models known as canonical models. With the help of these structureswe will be able to prove the Canonical Model Theorem, a universal completenessresult for normal logics. We first define canonical models and prove this result forthe basic modal language; at the end of the section we generalize our discussion tolanguages of arbitrary similarity type.

Definition 1.23 Thecanonical modelM� for a normal modal logic� (in the basiclanguage) is the triple(W�; R�; V �) where:

(i) W� is the set of all�-MCSs;(ii) R� is the binary relation onW� defined byR�wu if for all formulas ,

2 u implies3 2 w. R� is called thecanonical relation;(iii) V � is the valuation defined byV �(p) = fw 2W� j p 2 wg. V � is called

thecanonical(or natural) valuation.

The pairF� = (W�; R�) is called thecanonical framefor �. a

All three clauses deserve comment. First, the canonical valuation equates the truthof a propositional symbol atw with its membership inw. Our ultimate goal is to


prove a Truth Lemma which will lift this ‘truth = membership’ equation to arbitraryformulas.

Second, note that the states ofM� consist ofall �-consistentMCSs. The signif-icance of this is that, by Lindenbaum’s Lemma,any�-consistent set of formulasis a subset of some point inM� – hence, by the Truth Lemma proved below, any�-consistent set of formulas is true at some point in this model. In short, the sin-gle structureM� is a ‘universal model’ for the logic�, which is why it is called‘canonical.’

Finally, consider the canonical relation: a statew is related to a stateu preciselywhen for each formula in u, w contains the information3 . Intuitively, thiscaptures what we mean byMCSs being ‘coherently related.’ A dual formulation interms of boxes instead of diamonds is ac The following lemma, stating that an al-ternative formulation in terms of boxes instead of diamonds is actually equivalent,shows that we are getting things right:

Lemma 1.24 For any normal logic�, R�wv iff for all formulas , 2 2 w

implies 2 v.

Proof. For the left to right direction, supposeR�wv. Further suppose 62 v. Asvis anMCS, by Proposition 1.21: 2 v. AsR�wv,3: 2 w. Asw is consistent,:3: 62 w. That is,2 62 w and we have established the contrapositive. Weleave the right to left direction to the reader. a

In fact, the definition ofR� is exactly what we require; all that remains to bechecked is that enough ‘coherently related’MCSs exist for our purposes.

Lemma 1.25 (Existence Lemma)For any normal modal logic� and any statew 2W�, if 3� 2 w then there is a statev 2W� such thatR�wv and� 2 v.

Proof. Suppose3� 2 w. We will construct a statev such thatR�wv and� 2 v.Let v� bef�g [ f j 2 2 wg. Thenv� is consistent. For suppose not. Thenthere are 1, . . . , n such that � ( 1 ^ � � � ^ n) ! :�, and it follows by aneasy argument that� 2( 1 ^ � � � ^ n)! 2:�: As the reader should check, theformula (2 1 ^ � � � ^ 2 n) ! 2( 1 ^ � � � ^ n) is a theorem of every normalmodal logic, hence by propositional calculus,`� (2 1^� � �^2 n)! 2:�. Now,2 1 ^ � � � ^ 2 n 2 w (for 2 1, . . . ,2 n 2 w, andw is anMCS) thus it followsthat2:� 2 w. Using Dual, it follows that:3� 2 w. But this is impossible:w isanMCS containing3�. We conclude thatv� is consistent after all.

Let v be anyMCS extendingv�; such extensions exist by Lindenbaum’s Lemma.By construction� 2 v. Furthermore, for all formulas , 2 2 w implies 2 v.Hence by Lemma 1.24,R�wv. a

With this established, the rest is easy. First we lift the ‘truth = membership’ equa-tion to arbitrary formulas:


Lemma 1.26 (Truth Lemma) For any normal modal logic� and any formula�,M�; w � iff � 2 w.

Proof. By induction on the degree of�. The base case follows from the definitionof V �. The boolean cases follow from Proposition 1.21. It remains to deal with themodalities. The left to right direction is more or less immediate from the definitionof R�:

M�; w 3� iff 9v (R�wv ^ M�; v �)

iff 9v (R�wv ^ � 2 v) (Induction Hypothesis)only if 3� 2 w (DefinitionR�):

For the right to left direction, suppose3� 2 w. By the equivalences above, itsuffices to find anMCS v such thatR�wv and� 2 v – and this is precisely whatthe Existence Lemma guarantees. a

The Canonical Model Theorem

We now have all the material to prove the Canonical Model Theorem. Although itis not strictly necessary here, it is often useful to think of completeness proofs inthe following way.

Proposition 1.27 A logic� is complete with respect to a class of structures (mod-els or frames)S iff every�-consistent formula is satisfiable on someS 2 S.

Proof. To prove the right to left implication we argue by contraposition. Suppose� is not complete with respect toS. Thus there is a formula� such thatS � but� 62 �. But then:� is �-consistent, but not satisfiable on any structure inS. Theleft to right direction is left to the reader. a

The following Theorem is the main result concerning canonical models. It statesthat every normal modal logic is identical to the set of formulas that are throughat every state of its canonical model. (Comparing this to the remark in Exam-ple 1.14(iv), we see that the canonical model is rather special indeed.)

Theorem 1.28 (Canonical Model Theorem)Any normal modal logic is completewith respect to its canonical model. That is, for any formula�, we have

M� � iff � 2 �:

Proof. Soundness is left to the reader. For completeness, suppose� is a consistentformula of the normal modal logic�. By Lindenbaum’s Lemma there is a�-MCS

� to which� belongs. By the Truth Lemma,M�; � �, whence� is satisfiable inthe canonical model. Thus by the previous proposition,� is complete with respectto its canonical model. a


At first glance, the Canonical Model Theorem may seem rather abstract. It is acompleteness result with respect to asingle model, not a class of frames, and arather abstract model at that. (ThatK4 is complete with respect to the class of tran-sitive frames is interesting; that it is complete with respect to the singleton classcontaining only its canonical model seems rather dull.) But appearances are mis-leading: canonical models are by far the most important tool used in completenessproofs. For a start, the Canonical Model Theorem immediately yields the followingresult:

Theorem 1.29 Kis complete with respect to the class of all frames.

Proof. By Proposition 1.27, to prove this result it suffices to find, for anyK -consistent formula�, a modelM (based on any frame whatsoever) and a statew in M such thatM; w �. This is easy: simply chooseM to be(FK; V K), thecanonical model forK , and letw be anyK -MCS containing�. a

More importantly, it is often easy to get useful information about the structure ofcanonical frames. For example, as we will learn in the next section, the canonicalframe forK4 is transitive – and this immediately yields the (more interesting) re-sult thatK4 is complete with respect to the class of transitive frames. Even whena canonical model is not as cleanly structured as we would like, it still embod-ies a vast amount of information about its associated logic; one of the importantthemes pursued later in the chapter is how to make use of this information in-directly. Furthermore, canonical models are mathematically natural. As we willlearn in Chapter 2, from an algebraic perspective canonical models are not abstractoddities: indeed, they are precisely the structures one is lead to by considering theideas underlying the Stone Representation Theorem.

1.4.1 Arbitrary Modal Languages

To conclude this section we sketch the generalizations required to extend the resultsobtained so far to languages of arbitrary similarity types.

Definition 1.30 Let � be a modal similarity type, and� a normal modal logic inthe language over� . The canonical modelM� = (W�; R�

M; V �)M2� for � has

W� andV � as defined in Definition 1.23, while for ann-ary operatorM 2 � therelationR�

M� (W�)n+1 is defined byR�

Mwu1 : : : un if for all formulas 1 2 u1,

. . . , n 2 un we haveM( 1; : : : ; n) 2 w. a

There is an analog of Lemma 1.24.

Lemma 1.31 For any normal modal logic�, R�Mwu1 : : : un iff for all formulas

1.5 Applications 15

1; : : : ; n, O( 1; : : : ; n) 2 w implies that for somei such that1 � i � n, i 2 ui.

Proof. Left to the reader. a

Now for the crucial lemma – we must show that enough coherently relatedMCSsexist. This requires a more delicate approach than was needed for Lemma 1.25.

Lemma 1.32 (Existence Lemma)SupposeM( 1; : : : ; n) 2 w. Then there areu1, . . . ,un such that 1 2 u1, . . . , n 2 un andR�wu1 : : : un.

Proof. The proof of Lemma 1.25 establishes the result for any unary operators inthe language, so it only remains to prove the (trickier) case for modalities of higherarity. To keep matters simple, assume thatM is binary; this illustrates the key newidea needed.

So, supposeM( 1; 2) 2 w. Let �0, �1, . . . enumerate all formulas. We con-struct two sequences of sets of formulas

f 1g = �0 � �1 � � � � and f 2g = �0 � �1 � � � �

such that all�i and�i are finite and consistent,�i+1 is either�i [ f�ig or�i[f:�ig, and similarly for�i+1. Moreover, putting�i :=

V�i and�i :=

V�i,

we will have thatM(�i; �i) 2 w.The key step in the inductive construction is

M(�i; �i) 2 w ) M (�i ^ (�i _ :�i); �i ^ (�i _ :�i)) 2 w

) M ((�i ^ �i) _ (�i ^ :�i); (�i ^ �i) _ (�i ^ :�i)) 2 w

) one of the formulasM(�i ^ [:]�i; �i ^ [:]�i) is inw.

If, for example,M(�i ^ �i; �i ^ :�i) 2 w, we take�i+1 := �i [ f�ig, �i+1 :=

�i [ f:�ig. Under this definition, all�i and�i have the required properties.Finally, letu1 =

Si�i andu2 =

Si�i. It is easy to see thatu1, u2 are�-MCSs

andR�Mwu1u2, as required. a

With this lemma established, the real work has been done. The Truth Lemmaand the Canonical Model Theorem for general modal languages are now obviousanalogs of Lemma 1.26 and Theorem 1.28. The precise statements and proofs areleft to the reader.

1.5 Applications

In this section we put canonical models to work. First we show how to prove someof the frame completeness results noted in Table 1.1 using a simple and uniformmethod of argument. This leads us to isolate one of the most important conceptsof modal completeness theory:canonicity.


Suppose we suspect that a normal modal logic� is complete with respect to aclass of framesF; how should we go about proving it? Actually, there is no infal-lible strategy. (Indeed, many normal modal logics are not complete with respectto any class of frames whatsoever.) Nonetheless, a very simple technique worksin a large number of interesting cases: simply show that the canonical frame for� belongs toF. We call such proofscompleteness-via-canonicityarguments, forreasons which will soon become clear. Let us consider some examples.

Theorem 1.33 The logicK4 is complete with respect to the class of transitiveframes.

Proof. Given aK4-consistent formula�, it suffices to find a model(F; V ) and astatew in this model such that (1)(F; V ); w �, and (2)F is transitive. LetMK4 be the canonical model forK4 and let� be anyK4-MCS containing�. ByLemma 1.26,MK4; � � so step (1) is established. It remains to show thatFK4

is transitive. So supposew, v andu are points in this frame such thatRK4wv andRK4vu. We wish to show thatRK4wu. Suppose 2 u. AsRK4vu, 3 2 v, soasRK4wv, 33 2 w. Butw is aK4-MCS, hence it contains33 ! 3 , thusby modus ponens it contains3 . ThusRK4wu. a

In spite of its simplicity, the preceding result is well worth reflecting on. Twoimportant observations should be made.

First, the proof actually establishes something more general than the theoremclaims: namely, that the canonical frame ofanynormal logic� containing33p!3p is transitive. The proof works because allMCSs in the canonical frame containthe 4 axiom; it follows that the canonical frame of any extension ofK4 is transitive,for all such extensions contain the 4 axiom.

Second, the result suggests that there may be a connection between the structureof canonical frames and the frame correspondences studied in modal correspon-dence theory. It is well-known that the 4 axiomdefinestransitivity (in the sensethat a frameF is transitive iff F 4 – and now we know that it imposes thisproperty on canonical frames as well.

Theorem 1.34 T, KB andKD are complete with respect to the classes of reflexiveframes, of symmetric frames, and of right-unbounded frames, respectively.

Proof. For the first claim, it suffices to show that the canonical model forT isreflexive. Letw be a point in this model, and suppose� 2 w. Asw is aT-MCS,�! 3� 2 w, thus by modus ponens,3� 2 w. ThusRTww.

For the second claim, it suffices to show that the canonical model forKB issymmetric. Letw andv be points in this model such thatRKBwv, and supposethat� 2 w. Asw is aKB -MCS, �! 23� 2 w, thus by modus ponens23� 2 w.Hence by Lemma 1.24,3� 2 v. But this means thatRKBvw, as required.

1.6 Completeness and canonicity 17

For the third claim, it suffices to show that the canonical model forKD is right-unbounded. (This is slightly less obvious than the previous claims since it requiresan existence proof.) Letw be any point in the canonical model forKD. Wemust show that there exists av in this model such thatRKDwv. Asw is aKD-MCS it contains2p ! 3p, thus by closure under uniform substitution it contains2>! 3>. Moreover, as> belongs to all normal modal logics, by generalization2> does too; so2> belongs toKD, hence by modus ponens3> 2 w. Hence,by the Existence Lemma,w has anRKD successorv. a

Once again, these result hint at a link between definability and the structure ofcanonical frames: after all, T defines reflexivity, B defines symmetry, and D rightunboundedness. And yet again, the proofs actually establish something more gen-eral than the theorem states: the canonical frame ofany normal logic containingT is reflexive, the canonical frame ofanynormal logic containing B is symmetric,and the canonical frame ofanynormal logic containing D is right unbounded. Thisallows us to ‘add together’ our results. Here are two examples:

Theorem 1.35 S4is complete with respect to the class of reflexive, transitive frames.S5 is complete with respect to the class of frames whose relation is an equivalencerelation.

Proof. The proof of Theorem 1.33 shows that the canonical frame ofanynormallogic containing the 4 axiom is transitive, while the proof of the first clause ofTheorem 1.34 shows that the canonical frame ofanynormal logic containing theT axiom is reflexive. AsS4 contains both axioms, its canonical frame has bothproperties, thus the completeness result forS4follows.

As S5 contains both the 4 and the T axioms, it also has a reflexive, transitivecanonical frame. As it also contains the B axiom (which by the proof of the secondclause of Theorem 1.34 means that its canonical frame is symmetric), its canonicalrelation is an equivalence relation. The desired completeness result follows.a

1.6 Completeness and canonicity

As the above examples suggest, canonical models are an important tool for prov-ing frame completeness results. Moreover, their utility evidently hinges on somesort of connection between the properties of canonical frames and the frame cor-respondences studied earlier. Let us introduce some terminology to describe thisimportant phenomenon.

Definition 1.36 (Canonicity)A formula� is canonicalif, for any normal logic�,� 2 � implies that� is valid on the canonical frame for�. A normal logic� iscanonicalif its canonical frame is a frame for�. (That is,� is canonical if for all� such that � �, � is valid on the canonical frame for�.) a


Clearly 4, T, B and D axioms are all canonical formulas. For example, any normallogic� containing the 4 axiom has a transitive canonical frame, and the 4 axiom isvalid on transitive frames. Similarly, any modal logic containing the B axiom hasa symmetric canonical frame, and the B axiom is valid on symmetric frames.

MoreoverK4, T, KB , KD , S4 and S5 are all canonical logics. Our previouswork has established that all the axioms involved are valid on the relevant canon-ical frames. But modus ponens, uniform substitution, and generalization preserveframe validity. It follows thateveryformula in each of these logics is valid on thatlogic’s canonical frame. In general, to show thatKA1 : : :An is a canonical logicit suffices to show thatA1; : : : ; An are canonical formulas.The general importance of the notion of canonicity is given by the following The-orem.

Theorem 1.37 Every canonical logic is complete.

Proof. Let� be a canonical normal modal logic, and letF� be the class of framesfor �. Clearly then� is sound with respect toF�. For completeness, take a�-consistent formula�. It follows from the canonical model theorem that� is satis-fiable inF�, and since� is canonical, this frame actually belongs toF�. Thus thetheorem follows from Proposition 1.27. a

Of the various questions that naturally suggest themselves, we mention two clus-ters. First, which formulas are canonical? For instance, can we decide on the basisof the syntactic shape alone whether a formula is canonical? Unfortunately, the an-swer to this question is negative, but then perhaps we can find sufficient syntacticcriteria for canonicity? There are some interesting positive answers here, and thisissue will keep us occupied for a large part of the course. For instance, there is animportant collection of modal formulas, the so-calledSahlqvist formulasthat areall canonical. The precise definition will be given later on, but all of the formulasdiscussed above belong to the Sahlqvist fragment of modal logic.

The second cluster of questions circles around the relation between the notionsof correspondence and canonicity. All the above examples concern canonicalmodal formulas that areelementary, that is, they correspond to first order frameproperties like transitivity or reflexivity; in fact, all Sahlqvist formulas have thisproperty. Not all canonical modal formulas are elementary, but all modal formulasthat are known to be canonical, do correspond to a first order property when weconfine our attention to thecanonicalframes.

When it comes toproving the canonicity of a given modal formula, once weknow that the formula is elementary, the standard approach is to show that thecanonical frame satisfies the first order condition that corresponds to the formula.This was the approach that we took in the previous section. The aim of these

1.6 Completeness and canonicity 19

Notes however is to provide tools that are tailored towards more direct proofs.This approach is based on algebraic methods that we will develop next.

2

Algebraizing Modal Logic

In this chapter we develop analgebraicsemantics for modal logic. The basic ideais to extend the algebraic treatment of classical propositional logic (which usesboolean algebras) to modal logic. The algebras employed to do this are calledboolean algebras with operators(BAOs). The boolean part handles the underlyingpropositional logic, the additional operators handle the modalities.

But why algebraize modal logic? There are two main reasons. First, the alge-braic perspective allows us to bring powerful new techniques to bear on modal-logical problems. Second, the algebraic semantics turns out to be better-behavedthan frame-based semantics: we will be able to prove an algebraic completenessresult foreverynormal modal logic. No analogous result holds for frames.

In order to make the reader familiar with the basic ideas underlying algebraiclogic, we first discuss the algebraic approach towards propositional logic and onlythen turn to the algebraization of modal logic. In the third section we prove tworesults that underpin the whole approach, namely the Stone theorem for booleanalgebras and its exension, the J´onsson-Tarksi theorem for boolean algebras withoperators.

2.1 Logic as Algebra

What do algebra and logic have in common? And why bring algebra into the studyof logic? This section provides some preliminary answers: we show that algebraand logic share key ideas, and analyze classical propositional logic algebraically.Along the way we will meet a number of important concepts (notably formulaalgebras, the algebra of truth values, set algebras, abstract boolean algebras, andLindenbaum-Tarski algebras) and results (notably the Stone Representation Theo-rem), but far more important is the overall picture. Algebraic logic offers a naturalway of re-thinking many basic logical issues, but it is important not to miss thewood for the trees. The bird’s eye view offered here should help guide the readerthrough the more detailed modal investigations that follow.

20

2.1 Logic as Algebra 21

Algebra as logic

Most school children learn how to manipulate simple algebraic equations. Giventhe expression(x + 3)(x + 1), they learn how to multiply these factors to formx2+4x+3, and (somewhat later) study methods for doing the reverse (that is, fordecomposing quadratics into factors).

Such algebraic manipulations are essentially logical. For a start, we have a well-defined syntax: we manipulateequationsbetweenterms. This syntax is rarelyexplicitly stated, but most students (building on the analogy with basic arithmetic)swiftly learn how to build legitimate terms using numerals, variables such asx, yandz, and+, � and�. Moreover, they learn the rules which govern this symbolmanipulation process: replacing equals by equals, doing the same thing to bothsides of an equation, appealing to commutativity, associativity and distributivity tosimplify and rearrange expressions. High-school algebra is a form of proof theory.

But there is also asemanticperspective on basic algebra, though this usuallyonly becomes clear later. As students learn more about mathematics, they realizethat the familiar ‘laws’ do not hold for all mathematical objects: for example, ma-trix multiplication is not commutative. Gradually the student grasps that variablesneed not be viewed as standing for numbers: they can be viewed as standing forother objects as well. Eventually the semantic perspective comes into focus: thereare various kinds ofalgebras(that is, sets equipped with collections of functions,or operations, which satisfy certain properties), andterms denote elements in al-gebras. Moreover, an equation such asx � y = y � x is not a sacrosanct law: it issimply a property that holds for some algebras and not for others.

So algebra has a syntactic dimension (terms and equations) and a semantic di-mension (sets equipped with a collection of operations). And in fact there is atight connection between the proof theory algebra offers and its semantics. InAppendix A we give a standard derivation system forequationallogic (that is, astandard set of rules for manipulating equations) and state a fundamental result dueto Birkhoff: the system is strongly sound and complete with respect to the standardalgebraic semantics. Algebra really can be viewed as logic.

But logic can also be viewed as algebra. We will now illustrate this by examin-ing classical propositional logic algebraically. Our discussion is based around threemain ideas: the algebraization of propositional semantics in the class of set alge-bras; the algebraization of propositional axiomatics in the class of abstract booleanalgebras; and how the Stone Representation Theorem links these approaches.

Algebraizing propositional semantics

Consider any propositional formula, say(p_ q)^ (p_ r). The most striking thingabout propositional formulas (as opposed to first-order formulas) is their syntacticsimplicity. In particular, there is no variable binding – all we have is a collection of

22 2 Algebraizing Modal Logic

atomic symbols (p, q, r, and so on) that are combined into more complex expres-sions using the symbols?, >, :, _ and^. Recall that we take?, : and_ as theprimitive symbols, treating the others as abbreviations.

Now, as the terminology ‘proposition letters’ suggests, we think ofp, q, andras symbols denoting entities called propositions, abstract bearers of information.So what do?, >, :, _ and^ denote? Fairly obviously: ways of combiningpropositions, or operations on propositions. More precisely,_ and^ must denotebinary operations on propositions (let us call these operations+ and� respectively),: must denote a unary operation on propositions (let us call it�), while? and>denote special nullary operations on propositions (that is, they are the names oftwo special propositions: let us call them 0 and 1 respectively). In short, we haveworked our way towards the idea thatformulas can be seen as terms denotingpropositions.

But which kinds of algebras are relevant? Here is a first step towards an answer.

Definition 2.1 Let Bool be the algebraic similarity type having one constant (ornullary function symbol)?, one unary function symbol:, and one binary functionsymbol_. Given a set of propositional variables�, Form(�) is the set ofBool -terms in�; this set is identical to the collection of propositional formulas in�.

Algebras of typeBool are usually presented as 4-tuplesA = (A;+;�; 0). Wemake heavy use of the standard abbreviations� and 1. That is,a � b is shorthand for�(�a+�b), and1 is shorthand for�0. a

But this only takes us part of the way. There are many different algebras of thissimilarity type – and we are only interested in algebras which can plausibly beviewed as algebras of propositions. So let us design such an algebra. Propositionallogic is about truth and falsehood, so let us take the set2 = f0; 1g as the setAunderlying the algebra; we think of ‘0’ as the truth valuefalse, and ‘1’ as the valuetrue. But we also need to define suitable operations over these truth values, and wewant these operations to provide a natural interpretation for the logical connectives.Which operations are appropriate?

Well, the terms we are working with are just propositional formulas. So howwould we go about evaluating a formula� in the truth value algebra? Obviouslywe would have to know whether the proposition letters in� are true or false, but letus suppose that this has been taken care of by a function� : �! 2 mapping the set� of proposition letters to the set2 of truth values. Given such a� (logicians willcall � a valuation, algebraists will call it an assignment) it is clear what we have to


do: compute~�(�) according to the following rules:

~�(p) = �(p); for all p 2 �;~�(?) = 0;~�(:�) = 1� ~�(�);

~�(� _ ) = max(~�(�); ~�( )):

(2.1)

Clearly the operations used here are the relevant ones; they simply restate the fa-miliar truth table definitions. This motivates the following definition:

Definition 2.2 Thealgebra of truth valuesis 2 = (f0; 1g;+;�; 0)), where� and+ are defined by�a = 1� a anda+ b = max(a; b), respectively. a

Let us sum up our discussion so far. The crucial observations are that formulas canbe viewed as terms, that valuations can be identified with algebraic assignmentsin the algebra2, and that evaluating the truth of a formula under such a valua-tion/assignment is exactly the same as determining the meaning of the term in thealgebra2 under the assignment/valuation.

So let us move on. We have viewed meaning as a map~� from the setForm(�)

to the setf0; 1g – but it is useful to consider thismeaning functionin more math-ematical detail. Note the ‘shape’ of the conditions on~� in (2.1): the resemblanceto the defining condition of ahomomorphismis too blatant to miss. But since ho-momorphisms are the fundamental maps between algebras (see Appendix A) whynot try and impose algebraic structure on thedomainof such meaning functions(that is, on the set of formulas/terms) so that meaning functions really are homo-morphisms? This is exactly what we are about to do. We first define the neededalgebraic structure on the set of formulas.

Definition 2.3 Let � be a set of proposition letters. The propositionalformulaalgebraover� is the algebra

Form(�) = (Form(�);+;�;?);

where� is the collection of propositional formulas over�, and� and+ are theoperations defined by�� := :� and�+ := � _ , respectively. a

In other words, the carrier of this algebra is the collection of propositional formulasover the set of proposition letters�, and the operations� and+ give us a simplemathematical picture of the dynamics of formula construction.

Proposition 2.4 Let � be some set of proposition letters. Given any assignment� : � ! 2, the function~� : Form(�) ! 2 assigning to each formula its meaningunder this valuation, is a homomorphism fromForm(�) to 2.


Proof. A precise definition of homomorphism is given in Appendix A. Essentially,homomorphisms between algebras map elements in the source algebra to elementsin the target algebra in an operation preserving way – and this is precisely what theconditions in~� in (2.1) express. a

The idea of viewing formulas as terms, and meaning as a homomorphism, is fun-damental to algebraic logic.

Another point is worth stressing. As the reader will have noticed, sometimes wecall a sequence of symbols likep _ q a formula, and sometimes we call it a term.This is intentional. Any propositional formula can be viewed as – simplyis – analgebraic term. The one-to-one correspondence involved is so obvious that it is notworth talking about ‘translating’ formulas to terms or vice-versa; they are simplytwo ways of looking at the same thing. We simply choose whichever terminologyseems most appropriate to the issue under discussion.

But let us move on. As is clear from high-school algebra, algebraic reasoning isessentiallyequational. So a genuinelyalgebraic logic of propositions should giveus a way of determining when two propositions are equal. For example, such alogic should be capable of determining that the formulasp _ (q ^ p) andp denotethe same proposition. How does the algebraic approach to propositional semanticshandle this? As follows: an equations � t is valid in an algebraA if for everyassignment to the variables occurring in the terms,s andt have the same meaningin A (see Appendix A for further details). Hence, an algebraic way of saying thata formula� is a classical tautology (notation:j=C �) is to say that the equation� � > is valid in the algebra of truth values.

Now, an attractive feature of propositional logic (a feature which extends tomodal logic) is that not only terms, butequationscorrespond to formulas. Thereis nothing mysterious about this: we can define the bi-implication connective$ inclassical propositional logic, and viewed as an operation on propositions,$ assertsthat both terms have the same meaning:

~�(�$ ) =

�1 if ~�(�) = ~�( );

0 otherwise:

So to speak, propositional logic is intrinsically equational.Theorem 2.5 neatly summarizes our discussion so far: it shows how easily we

can move from a logical to an algebraic perspective and back again.

Theorem 2.5 (2 Algebraizes Classical Validity) Let � and be propositionalformulas/terms. Then

j=C � iff 2 j= � � >: (2.2)

2 j= � � iff j=C �$ : (2.3)

j=C � $ (�$ >). (2.4)


Proof. Immediate from the definitions. a

Remark 2.6 The reader may wonder about the presence of (2.3) and in particular,of (2.4) in the Theorem. The point is that for a proper, ‘full’, algebraization of alogic, one has to establish not only that the membership of some formula� in thelogic can be rendered algebraically as the validity of some equation�� in some(class of) algebra(s). One also has to show that conversely, there is a translationof equations to formulas such that the equation holds in the class of algebras ifand only if its translation belongs to the logic. And finally, one has to prove thattranslating a formula� to an equation��, and then translating this equation backto a formula, one obtains a formula�0 that isequivalentto the original formula�.The fact that our particular translations satisfy these requirements is stated by (2.3)and (2.4), respectively.

Since we will not go far enough into the theory of algebraic logic to use these‘full’ algebraizations, in the sequel we will only mention the first kind of equiva-lence when we algebraize a logic. Nevertheless, in all the cases that we consider,the second and third requirements are met as well. a

Set algebras

Propositional formulas/terms and equations may be interpreted in any algebra oftype Bool . Most algebras of this type are uninteresting as far as the semanticsof propositional logic is concerned – but other algebras besides2 are relevant. Aparticularly important example is the class ofset algebras. As we will now see,set algebras provide us with a second algebraic perspective on the semantics ofpropositional logic. And as we will see in the following section, the perspectivethey provide extends neatly to modal logic.

Definition 2.7 (Set Algebras)Let A be a set. As usual, we denote thepower setof A (the set of all subsets ofA) by P(A). The power set algebraP(A) is thestructure

P(A) = (P(A);[;�;?);

where? denotes theemptyset,� is the operation of taking thecomplementofa set relative toA, and[ that of taking theunion of two sets. From these basicoperations we define in the standard way the operation\ of taking theintersectionof two sets, and the special elementA, thetop setof the algebra.

A set algebraor field of setsis a subalgebra of a power set algebra. That is, a setalgebra (onA) is a collection of subsets ofA that contains? and is closed under[ and� (so any set algebra containsA and is closed under\ as well). The classof all set algebras is calledSet. a


Set algebras provide us with a simple concrete picture of propositions and the waythey are combined – moreover, it is a picture that even at this stage contains anumber of traditional modal ideas. Think ofA as a set of worlds (or situations,or states) and think of a proposition as a subset ofA. And think of a propositionas a set of worlds – the worlds that make it true. So viewed,? is a very specialproposition: it is the proposition that is false in every situation, which is clearly agood way of thinking about the meaning of?. Similarly,A is the proposition truein all situations, which is a suitable meaning for>. It should also be clear that[is a way of combining propositions that mirrors the role of_. After all, in whatworlds isp _ q true? In precisely those worlds that makep true orq true. Finally,complementation mirrors negation, for:p is true in precisely those worlds wherep is not true.

As we will now show, set algebras and the algebra2 make precisely the sameequations true. We will prove this algebraically by showing that the class of setalgebras coincides (modulo isomorphism) to the class of subalgebras of powers of2. The crucial result needed is the following:

Proposition 2.8 Every power set algebra is isomorphic to a power of2, and con-versely.

Proof. Let A be an arbitrary set, and consider the following function� mappingelements ofP(A) to 2-valued maps onA:

�(X)(a) =

�1 if a 2 X;0 otherwise:

In other words,�(X) is thecharacteristic functionof X. The reader should verifythat � is an isomorphism betweenP(A) and2A, where the latter algebra is asdefined in Definition A.6.

Conversely, to show that every power of2 is isomorphic to some power setalgebra, let2I be some power of2. Consider the map� : 2I ! P(I) defined by

�(f) = fi 2 I j f(i) = 1g:

Again, we leave it for the reader to verify that� is the required isomorphism be-tween2I andP(I). a

Theorem 2.9 (Set algebraizes classical validity)Let � and be propositionalformulas/terms. Then

j=C � iff Set j= � � >: (2.5)

Proof. It is not difficult to show from first principles that the validity of equations ispreserved under taking direct products (and hence powers) and subalgebras. Thus,with the aid of Theorem 2.5 and Proposition 2.8, the result follows. a


Algebraizing propositional axiomatics

We now have two equational perspectives on the semantics of propositional logic:one via the algebra2, the other via set algebras. But what about the syntacticaspects of propositional logic? It is time to see how the equational perspectivehandles such notions as theoremhood and provable equivalence.

Assume we are working in some fixed (sound and complete) proof system forclassical propositional logic. LetC �mean that� is a theorem of this system, andcall two propositional formulas� and provably equivalent(notation: �C )if the formula� $ is a theorem. Theorem 2.11 is a syntactic analog of Theo-rem 2.9: it is the fundamental result concerning the algebraization of propositionalaxiomatics. Its statement and proof makes use ofboolean algebras, so let us definethese important entities right away.

Definition 2.10 (Boolean Algebras)Let A = (A;+;�; 0) be an algebra of theboolean similarity type. ThenA is called aboolean algebraiff it satisfies thefollowing identities (recall thatx � y and1 are shorthand for�(�x+�y) and�0,respectively):

(B0) x+ y = y + x x � y = y � x

(B1) x+ (y + z) = (x+ y) + z x � (y � z) = (x � y) � z

(B2) x+ 0 = x x � 1 = 1

(B3) x+ (�x) = 1 x � (�x) = 0

(B4) x+ (y � z) = (x+ y) � (x+ z) x � (y + z) = (x � y) + (x � z)

The operations+ and� are calledjoin andmeet, respectively, and the elements1and0 are referred to as thetop andbottomelements. Weorder the elements of aboolean algebra by defininga � b if a+ b = b (or equivalently, ifa � b = a). Givena boolean algebraA = (A;+;�; 0), the setA is called itscarrier set. We call theclass of boolean algebrasBA. a

By a famous result of Birkhoff (discussed in Appendix A) a class of algebras de-fined by a collection of equations can be structurally characterized as avariety.Thus in what follows we sometimes speak of the variety of boolean algebras, ratherthan the class of boolean algebras.

If you have not encountered boolean algebras before, you should check that thealgebra2 and the set algebras defined earlier are both examples of boolean algebras(that is, check that these algebras satisfy the listed identities). In fact, set algebrasare what are known asconcreteboolean algebras. As we will see when we dis-cuss the Stone Representation Theorem, the relationship between abstract booleanalgebras (that is, any algebraic structure satisfying the previous definition) and setalgebras lies at the heart of the algebraic perspective on propositional soundnessand completeness.


But this is jumping ahead: our immediate task is to state the syntactic analog ofTheorem 2.9 promised above.

Theorem 2.11 (BA Algebraizes Classical Theoremhood)Let� and be propo-sitional formulas/terms. Then

`C � iff BA j= � � >: (2.6)

Proof. Soundness (the direction from left to right in (2.6) can be proved by astraightforward inductive argument on the length of propositional proofs. Com-pleteness will follow from the Propositions 2.14 and 2.15 below. a

How are we to prove this completeness result? Obviously we have to show that ev-ery non-theorem of classical propositional logic can be falsified on some booleanalgebra (falsified in the sense that there is some assignment under which the for-mula does not evaluate to the top element of the algebra). So the key question is:how do we build falsifying algebras? Our earlier work on relational completenesssuggests an answer. In our completeness proof we made use ofcanonicalmodels:that is, we manufactured models out of syntactical ingredients (sets of formulas)taking care to hardwire in all the crucial facts about the logic. So the obviousquestion is: can we construct algebras from (sets of) formulas in a way that buildsin all the propositional logic we require? Yes, we can. Such algebras are calledLindenbaum-Tarski algebras. In essence, they are ‘canonical algebras.’

First, some preliminary work. The observation underpinning what follows isthat the relation of provable equivalence is acongruenceon the formula algebra. Acongruence on an algebra is essentially an equivalence relation on the algebra thatrespects the operations (a precise definition is given in Appendix A) and it is nothard to see that provable equivalence is such a relation.

Proposition 2.12 The relation�C is a congruence on the propositional formulaalgebra.

Proof. We have to prove that�C is an equivalence relation satisfying

� �C only if :� �C : (2.7)

and

�0 �C 0 and�1 �C 1 only if (�0 _ �1) �C ( 0 _ 1): (2.8)

In order to prove that�C is reflexive, we have to show that for any formula�, theformula� $ � is a theorem of the proof system. The reader is invited to provethis in his or her favorite proof system for proposition calculus. The properties ofsymmetry and transitivity are also left to the reader.


But we want to prove that�C is not merely an equivalence relation but a congru-ence. We deal with the case for negation, leaving (2.8) to the reader. Suppose that� �C , that is,`C � $ . Again, given that we are working with a sound andcomplete proof system for propositional calculus, this implies that`C :�$ : .Given this, (2.7) is immediate. a

The equivalence classes under�C are the building blocks for what follows. As anysuch class is a maximal set of mutually equivalent formulas, we can think of suchclasses as propositions. And as�C is a congruence, we can define a natural al-gebraic structure on these propositions. Doing so gives rise to Lindenbaum-Tarskialgebras.

Definition 2.13 (Lindenbaum-Tarski Algebra) Given a set of proposition letters�, let Form(�)=�C be the set of equivalence classes that�C induces on the setof formulas, and for any formula� let [�] denote the equivalence class containing�. Then theLindenbaum-Tarski algebra(for this language) is the structure

LC(�) := (Form(�)=�C ;+;�; 0);

where+, � and 0 are defined by:[�] + [ ] := [� _ ], �[�] := [:�] and0 :=

[?]. Strictly speaking, we should write[�]� instead of[�], for �’s congruenceclass depends on the set� of proposition letters. But unless there is potential forconfusion, we usually will not bother to do so. a

Lindenbaum-Tarski algebras are easy to work with. For instance, it is easy to seethat the meet operation in such an algebra is given by[�] � [ ] := [� ^ ], whilethe top element1 is [>]. As another example, we show thata + (�a) = 1 forall elementsa of LC(�). The first observation is thata, just like any element ofLC(�), is of the form[�] for some formula�. But then we have

a+ (�a) = [�] + (�[�]) = [�] + [:�] = [� _ (:�)] = [>] = 1; (2.9)

where the fourth equality holds because`C (� _ :�)$ >.It is fairly obvious that the structure of a Lindenbaum-Tarski algebra only de-

pends on the cardinality of the set� of proposition letters; the reader is asked toprove this in Exercise 2.1.4.

We need two results concerning Lindenbaum-Tarski algebras. First, we have toshow that they are indeed an ‘algebraic canonical model’ – that is, that they give usa counterexample foreverynon-theorem of propositional logic. Second, we haveto show that they are counterexamples of the right kind: that is, we need to provethat any Lindenbaum-Tarski algebra is a boolean algebra.

Proposition 2.14 Let� be some propositional formula, and� a set of proposition


letters of size not smaller than the number of proposition letters occurring in�.Then

`C � iff LC(�) j= � � >: (2.10)

Proof. We may and will assume that� actually contains all variables occurring in�, cf. Exercise 2.1.4. We first prove the easy direction from right to left. Assumethat � is not a theorem of classical propositional logic. This implies that� and> are not provably equivalent, whence we have[�] 6= [>]. We have to find anassignment onLC(�) that forms a counterexample to the validity of�. There isone obvious candidate, namely the assignment� given by�(p) = [p]. It can easilybe verified (by a straightforward formula induction) that with this definition weobtain~�( ) = [ ] for all formulas that use variables from the set�. But then byour assumption on� we find that

~�(�) = [�] 6= [>] = 1;

as required.For the other direction we have to work a bit harder. If`C � then it is obvious

that~�(�) = [�] = [>] = 1, but only looking at� is not sufficient now. We have toshow that~�(�) = [>] for all assignments�.

So let � be an arbitrary assignment. That is,� assigns an equivalence class(under�C) to each proposition letter. For each variablep, take a representingformula �(p) in the equivalence class�(p); that is, we have�(p) = [�(p)]. Wemay view� as afunctionmapping proposition letters to formulas; in other words,� is asubstitution. Let�( ) denote the effect of performing this substitution on theformula . It can be proved by an easy formula induction that, for any formula ,we have

~�( ) = [�( )]: (2.11)

Now, the collection of propositional theorems is closed under uniform substitution(depending on the formulation of your favorite sound and complete proof system,this is either something that is hardwired in or can be shown to hold). This closureproperty implies that the formula�(�) is a theorem, and hence that�(�) �C >, orequivalently,[�(�)] = [>]. But then it follows from (2.11) that

~�(�) = [>];

which is precisely what we need to show thatLC(�) j= �. a

Thus it only remains to check thatLC(�) is the right kind of algebra.

Proposition 2.15 For any set� of proposition letters,LC(�) is a boolean algebra.


Proof. Fix a set�. The proof of this Proposition boils down to proving that all theidentities B0–4 hold inLC(�). In (2.9) we proved that the first part of B3 holds;we leave the reader to verify that the other identities hold as well. a

Summarizing, we have seen that the axiomatics of propositional logic can be al-gebraized in a class of algebras, namely the variety of boolean algebras. We havealso seen that Lindenbaum-Tarski algebras act as canonical representatives of theclass of boolean algebras. (For readers with some background in universal algebra,we remark that Lindenbaum-Tarski algebras are in fact thefreeboolean algebras.)

Weak completeness via Stone

It is time to put our findings together, and to take one final step. This step is moreimportant than any taken so far.

Theorem 2.9 captured tautologies as equations valid in set algebras:

j=C � iff Set j= � � >:

On the other hand, in Theorem 2.11 we found an algebraic semantics for the notionof classical theoremhood:

`C � iff BA j= � � >:

But there is a fundamentallogical connection betweenj=C and`C : the soundnessand completeness theorem for propositional logic tells us that they are identical.Does this crucial connection show up algebraically? That is, is there an algebraicanalog of the soundness and completeness result for classical propositional logic?There is: it is called the Stone Representation Theorem.

Theorem 2.16 (Stone Representation Theorem)Any boolean algebra is isomor-phic to a set algebra.

Proof. We will make a more detailed statement of this result, and prove it, in Sec-tion 2.3. a

(Incidentally, this immediately tells us that any boolean algebra is isomorphic to asubalgebra of a power of2 – for Proposition 2.8 tells us that any power set algebrais isomorphic to a subalgebra of a power of2.) But what really interests us hereis the logical content of Stone’s Theorem. In essence, it is the key to the weakcompleteness of classical propositional logic.

Corollary 2.17 (Soundness and Weak Completeness)For any formula�, � isvalid iff it is a theorem.

Proof. Immediate from the equations above, since by the Stone RepresentationTheorem, the equations valid inSet must coincide with those valid inBA. a


The relation between Theorem 2.11 and Corollary 2.17 is the key to much of ourlater work. Note that from a logical perspective, Corollary 2.17 is the interesting re-sult: it establishes the soundness and completeness of classical propositional logicwith respect to the standard semantics. So why is Theorem 2.11 important? Afterall, as it proves completeness with respect to an abstractly defined class of booleanalgebras, it does not have the same independent logical interest. This is true, butgiven that the abstract algebraic counterexamples it provides can be represented asstandard counterexamples – and this is precisely what Stone’s Theorem guarantees– it enables us to prove the standard completeness result for propositional logic.

To put it another way, the algebraic approach to completeness factors the algebrabuilding process into two steps. We first prove completeness with respect to anabstract algebraic semantics by building an abstract algebraic model. It is easy todo this – we just use Lindenbaum-Tarski algebras. We then try andrepresenttheabstract algebras in the concrete form required by the standard semantics; that is,in terms of set algebras or of the algebra2.

In the next two sections we extend this approach to modal logic. Algebraizingmodal logic is more demanding than algebraizing propositional logic. For a start,there is not just one logic to deal with – we want to be able to handle any normalmodal logic whatsoever. Moreover, the standard semantics for modal logic is givenin terms of frame-based models – so we are going to need a representation resultthat tells us how to represent algebras as relational structures.

But all this can be done. In the following section we will generalize booleanalgebras to boolean algebras with operators; these are theabstractalgebras we willbe dealing with throughout the chapter. We also generalize set algebras to com-plex algebras; these are theconcretealgebras which model the idea of set-basedalgebras of propositions for modal languages. We then define the Lindenbaum-Tarski algebras we need – and every normal modal logic will give rise to its ownLindenbaum-Tarski algebra. This is all a fairly straightforward extension of ideaswe have just discussed. We then turn, in Section 2.3, to the crucial representationresult: the J´onsson-Tarski Theorem. This is an extension of Stone’s Representa-tion Theorem that tells us how to represent a boolean algebra with operators as anordinary modal model. It is an elegant result in its own right, but for our purposesits importance is the bridge it provides between completeness in the universe ofalgebras and completeness in the universe of relational structures.

Exercises for Section 2.12.1.1 Let A andB be two sets, andf : A ! B some map. Show thatf�1 : P(B) !P(A) given byf�1(Y ) = fa 2 A j f(a) 2 Y g is ahomomorphismfrom the power setalgebra ofB to that ofA.

2.1.2 Prove that every power set algebra is isomorphic to a power of the algebra2, and

2.2 Algebraizing Modal Logic 33

that conversely, every power of2 is isomorphic to a power set algebra. That is, fill in thedetails of the proof of Theorem 2.8.

2.1.3 Here is a standard set of axioms for propositional calculus:p ! (q ! p), (p !(q ! r)) ! ((p ! q) ! (p ! r)), and(:p ! :q) ! (q ! p). Show that all threeaxioms are valid on any set algebra. That is, show that whatever subset is used to interpretthe proposition letters, these formulas are true in all worlds. Furthermore, show that modusponens and uniform substitution preserve validity.

2.1.4 Let� and be two sets of proposition letters.

(a) Prove thatForm(�) is a subalgebra ofForm() iff � � .(b) Prove thatLC(�) can beembeddedin LC() iff j�j � j j.(c) Prove thatLC(�) andLC() are isomorphic iffj�j = j j.(d) Does� � imply thatLC(�) is a subalgebra ofLC()?

2.2 Algebraizing Modal Logic

Let us adapt the ideas introduced in the previous section to modal logic. The mostbasic principle of algebraic logic is that formulas of a logical language can beviewed as terms of an algebraic language, so let us first get clear about the algebraiclanguages we will use in the remainder of this chapter:

Definition 2.18 Let � be a modal similarity type. Thecorresponding algebraicsimilarity typeF� contains as function symbols all modal operators, together withthe boolean symbols_ (binary),: (unary), and? (constant). For a set� of vari-ables, we letTer � (�) denote the collection ofF� -terms over�. a

The algebraic similarity typeF� can be seen as the union of the modal similaritytype� and the boolean typeBool . In practice we often identify� andF� , speakingof � -terms instead ofF� -terms. The previous definition takes the formulas-as-terms paradigm quite literally: by our definitions

Form(�; �) = Ter � (�):

Just as boolean algebras were the key to the algebraization of classical proposi-tional logic, in modal logic we are interested inboolean algebras with operatorsorBAOs. Let us first defineBAOs abstractly; we will discuss concreteBAOs shortly.

Definition 2.19 (Boolean Algebras with Operators) Let � be a modal similaritytype. Aboolean algebra with� -operatorsis an algebra

A = (A;+;�; 0; fM)M2�

such that(A;+;�; 0) is a boolean algebra and everyfM is anoperator of arity�(M); that is,fM is an operation satisfying


(normality) fM(a1; : : : ; a�(M)) = 0 wheneverai = 0 for somei (0 < i � �(M)).(additivity) for all i (such that0 < i � �(M)),

fM(a1; : : : ; ai + a0i; : : : ; a�(M)) =

fM(a1; : : : ; ai; : : : ; a�(M)) + fM(a1; : : : ; a0i; : : : ; a�(M)):

If we abstract from the particular modal similarity type� , or if � is known fromcontext, we simply speak ofboolean algebras with operators, or BAOs. a

Now, the boolean structure is obviously there to handle the propositional connec-tives, but what is the meaning of the normality and additivity conditions on thefM?Consider a unary operatorf . In this case these conditions boil down to:

f(0) = 0;

f(x+ y) = fx+ fy:

But these equations correspond to the following modal formulas:

3? $ ?;

3(p _ q) $ 3p _3q;

both of which formulas are modal validities. Indeed (as we noted in Remark 1.13)they can be even be used to axiomatize the minimal normal logicK . Thus, evenat this stage, it should be clear that our algebraicoperatorsare well named: theirdefining properties are modally crucial.

Furthermore, note that all operators have the property ofmonotonicity. An oper-ationg on a boolean algebra ismonotonicif a � b impliesga � gb. (Here� refersto the ordering on boolean algebra given in Definition 2.10:a � b iff a � b = a

iff a + b = b.) Operators are monotonic, because ifa � b, thena + b = b, sofa+ fb = f(a+ b) = fb, and sofa � fb. Once again there is an obvious modalanalog, namely the rule of proof mentioned in Remark 1.13: if`� p ! q then`� 3p! 3q.

Example 2.20 Consider the collection of binary relations over a given setU . Thiscollection forms a set algebra on which we can define the operationsj (compo-sition), (�)�1 (inverse) andId (the identity relation); these are binary, unary andnullary operations respectively. It is easy to verify that these operations are actu-ally operators; to give a taste of the kind of argumentation required, we show thatcomposition is additive in its second argument:

(x; y) 2 R j (S [ T )

iff there is az with (x; z) 2 R and(z; y) 2 S [ T

iff there is az with (x; z) 2 R and(z; y) 2 S or (z; y) 2 T

iff there is az with (x; z) 2 R and(z; y) 2 S,


or there is az with (x; z) 2 R and(z; y) 2 T

iff (x; y) 2 R j S or (x; y) 2 R j T

iff (x; y) 2 R j S [R j T:

The reader should check the remaining cases. a

Algebraizing modal semantics

However it is the next type ofBAO that is destined to play the leading role: complexalgebras. These structures make crucial use of operationsmR that we define next.

Let us first consider the basic modal similarity type with one diamond. Given aframeF = (W;R), letmR be the following operation on the power set ofW :

mR(X) = fw 2W j Rwx for somex 2 X g:

Think of mR(X) as the set of states that ‘see’ a state inX. This operation corre-sponds to the diamond in the sense that for any valuationV and any formula� wehaveV (3�) = mR(V (�)).

Moving to the general case, we obtain the following definition.

Definition 2.21 Given an(n+1)-ary relationR on a setW , we define the follow-ing n-ary operationmR on the power setP(W ) of W :

mR(X1; : : : ;Xn) =

fw 2W j Rww1 : : : wn for somew1 2 X1; : : : ; wn 2 Xng: a

Example 2.22 Let be the converse operator of arrow logic, and recall that weuse the letterR to denote the accessibility relation for. Thus on asquareframeSU , by the rather special nature ofR, we have that, for any subsetX of U2:

mR(X) = f(a0; a1) 2 U2 j a0 = x1 anda1 = x0 for some(x0; x1) 2 X g

= f(x1; x0) 2 U2 j (x0; x1) 2 Xg:

In other words,mR(X) is nothing but theconverseof the binary relationX. a

Definition 2.23 (Complex Algebras) Let � be a modal similarity type, andF =

(W;RM)M2� a � -frame. The(full) complex algebra ofF (notation: F+), is theexpansion of the power set algebraP(W ) with operationsmRM for every operatorM in � . A complex algebrais a subalgebra of a full complex algebra. IfK is aclass of frames, then we denote the class of full complex algebras of frames inK

byCmK. a


It is important that you fully understand this definition. For a start, note that com-plex algebras are set algebras (that is, concrete propositional algebras) to whichmR operations have been added. Recall that for abinary relationR, the unaryoperationmR yields the set of all states which ‘see’ a state in a given subsetX ofthe universe:

mR(X) = fy 2W j there is anx 2 X such thatRyxg:

For a relation of arityn+1, then-ary operationmR maps ann-tuple of subsets ofthe universe to the set of all points which ‘see’ ann-tuple of states each of whichbelongs to the corresponding subset. It easily follows that if we have some modelin mind and denote with~V (�) the set of states where� is true, then

~V (M(�1; : : : ; �n)) = mRM( ~V (�1; : : : ; �n)):

Thus it should be clear that complex algebras are intrinsically modal. In the previ-ous section we said that set algebras model propositions as sets of possible worlds.By adding themR operations, we have modeled the idea that one world may beable to access the information in another. In short, we have defined a class of con-crete algebras which capture the modal notion of access between states in a naturalway.

How are complex algebras connected with abstractBAOs? One link is obvious:

Proposition 2.24 Let � be a modal similarity type, andF = (W;RM)M2� a � -frame. ThenF+ is a boolean algebra with� -operators.

Proof. We have to show that operations of the formmR are normal and additive.This rather easy proof is left to the reader; see Exercise 2.2.2. a

The other link is deeper. As we will learn in the following section (Theo-rem 2.45), complex algebras are toBAOs what set algebras are to boolean algebras:every abstract boolean algebra with operators has a concrete set theoretic repre-sentation, for every boolean algebra with operators is isomorphic to a complexalgebra.

But we have a lot to do before we are ready to prove this – let us continue ouralgebraization of the semantics of modal logic. We will now define the interpreta-tion of � -terms and equations in arbitrary boolean algebras with� -operators. Aswe saw for propositional logic, the basic idea is very simple: given an assignmentthat tells us what the variables stand for, we can inductively define the meaning ofany term.

Definition 2.25 Assume that� is a modal similarity type and that� is a set ofvariables. Assume further thatA = (A;+;�; 0; fM)M2� is a boolean algebra with


� -operators. Anassignmentfor � is a function� : � ! A. We can extend�uniquely to a meaning function~�: Ter � (�)! A satisfying:

~�(p) = �(p); for all p 2 �;~�(?) = 0;

~�(:s) = �~�(s);

~�(s _ t) = ~�(s) + ~�(t);

~�(M(s1; : : : ; sn)) = fM(~�(s1); : : : ; ~�(sn)):

Now lets � t be a� -equation. We say thats � t is true inA (notation:A j= s � t)if for every assignment�: ~�(s) = ~�(t). a

But now consider what happens whenA is acomplexalgebraF+. Since elementsof F+ aresubsetsof the power setP(W ) of the universeW of F, assignments� aresimply ordinary modal valuations! The ramifications of this observation are listedin the following proposition:

Proposition 2.26 Let � be a modal similarity type,� a � -formula,F a � -frame,�an assignment (or valuation) andw a point inF. Then

(F; �); w � iff w 2 ~�(�); (2.12)

F � iff F+ j= � � >; (2.13)

F+ j= � � iff F �$ : (2.14)

Proof. We will only prove the first part of the proposition (for the basic modalsimilarity type); the second and third part follow immediately from this and thedefinitions.

Let �, F and� be as in the statement of the theorem. We will prove (2.12) (forall w) by induction on the complexity of�. The only interesting part is the modalcase of the inductive step. Assume that� is of the form3 . The key observationis that

~�(3 ) = mR3(~�( )): (2.15)

We now have:

(F; �); w 3 iff there is av such thatR3wv and(F; �); v iff there is av such thatR3wv andv 2 ~�( )

iff w 2 mR3(~�( ))

iff w 2 ~�(3 ):

Here the second equivalence is by the inductive hypothesis, and the last one by(2.15). This proves (2.12). a


The previous proposition is easily lifted to the level of classes of frames and com-plex algebras. The resulting theorem is a fundamental one: it tells us that classesof complex algebras algebraize modal semantics. It is the modal analog of Theo-rem 2.9.

Theorem 2.27 Let � be a modal similarity type,� and � -formulas, andK aclass of� -frames. Then

K � iff CmK j= � � >; (2.16)

CmK j= � � iff K �$ : (2.17)

Proof. Immediate by Proposition 2.26. a

This proposition allows us to identify themodal logic�K of a class of framesK(that is, the set of formulas that are valid in eachF 2 Kg) with the equationaltheoryof the classCmK of complex algebras of frames inK (that is, the set ofequationsfs � t j F+ j= s � t, for all F 2 Kg).

Let us summarize what we have learned so far. We have developed an algebraicapproach to the semantics of modal logic in terms of complex algebras. Thesecomplex algebras,concreteboolean algebras with operators, generalize to modallanguages the idea of algebras of propositions provided by set algebras. And mostimportant of all, we have learned that complex algebras embodyall the informationabout normal modal logics that frames do. Thus, mathematically speaking, we candispense with frames and instead work with complex algebras.

Algebraizing modal axiomatics

Turning to the algebraization of modal axiomatics, we encounter a situation similarto that of the previous section. Once again, we will see that the algebraic counter-part of a logic is an equational class of algebras. To give a precise formulation weneed the following definition.

Definition 2.28 Given a formula�, let�� be the equation� � >. Now let� be amodal similarity type. For a set� of � -formulas, we defineV� to be the class ofthose boolean algebras with� -operators in which the set�� = f�� j � 2 �g isvalid. a

We now state the algebraic completeness theorem for modal logic. It is the obviousanalog of Theorem 2.11.

Theorem 2.29 (Algebraic Completeness)Let � be a modal similarity type, and


� a set of� -formulas. ThenK�� (the normal modal� -logic axiomatized by�) issound and complete with respect toV�. That is, for all formulas� we have

`K�� iff V� j= ��:

Proof. We leave the soundness direction as an exercise to the reader. Completenessis an immediate corollary of Theorems 2.34 and 2.35. a

As a corollary to the soundness direction of Theorem 2.29, we have thatVK�� =

V�, for any set� of formulas. In the sequel this will allow us to forget aboutaxiom sets and work with logics instead.

To prove the completeness direction of Theorem 2.29, we need a modal versionof the basic tool used to prove algebraic completeness results: Lindenbaum-Tarskialgebras. As in the the case of propositional languages, we will build an algebra ontop of the set of formulas in such a way that the relation of provable equivalencebetween two formulas is a congruence relation. The key difference is that we donot have just one relation of provable equivalence, but many: we want to define thenotion of Lindenbaum-Tarski algebras for arbitrary normal modal logics.

Definition 2.30 Let � be an algebraic similarity type, and� a set of propositionletters. Theformula algebraof � over� is the algebraForm(�; �) = (Form(�; �);

+; �; ?; fM)M2� where+, � and? are given as in Definition 2.3, while for eachmodal operatorM, the operationfM is given by

fM(t1; : : : ; tn) = M(t1; : : : ; tn): a

Notice the double role ofM in this definition: on the right-hand side of the equation,M is a ‘static’ part of thetermM(t1; : : : ; tn), whereas in the-left hand side we havea more ‘dynamic’ perspective on theinterpretationfM of the operation symbolM.

Definition 2.31 Let � be a modal similarity type,� a set of propositional variables,and� a normal modal� -logic. We define�� as a binary relation between� -formulas (in�) by

� �� iff `� �$ :

If � �� , we say that� and areequivalent modulo�. a

Proposition 2.32 Let � be a modal similarity type,� a set of proposition lettersand� a normal modal� -logic. Then�� is a congruence relation onForm(�; �).

Proof. We confine ourselves to proving the proposition for the basic modal simi-larity type. First, we have to show that�� is an equivalence relation; this is easy,and we leave the details to the reader. Next, we must show that�� is acongruence


relation on the formula algebra; that is, we have to demonstrate that�� has thefollowing properties:

�0 �� 0 and�1 �� 1 imply �0 _ �1 �� 0 _ 1;

� �� implies :� �� : ;

� �� implies 3� �� 3 :

(2.18)

We leave the proofs as exercises to the reader. a

Proposition 2.32 tells us that the following are correct definitions of functions onthe setForm(�; �)=�� of equivalence classes under��:

[�] + [ ] := [� _ ];

�[�] := [:�];

fM([�1]; : : : ; [�n]) := [M(�1; : : : ; �n)]:

(2.19)

For unary diamonds, the last clause boils down to:f3[�] := [3�].Given Proposition 2.32, the way is open to define the Lindenbaum-Tarski algebra

for any normal modal logic�: we simply define it to be thequotient algebraof theformula algebra over the congruence relation��.

Definition 2.33 (Lindenbaum-Tarski Algebras) Let � be a modal similaritytype,� a set of proposition letters, and� a normal modal� -logic in this language.TheLindenbaum-Tarski algebra of� over the set of generators� is the structure

L�(�) := (Form(�; �)=��;+;�; fM);

where the operations+,� andfM are defined as in (2.19). a

As with propositional logic, we need two results about Lindenbaum-Tarski alge-bras. First, we must show that modal Lindenbaum-Tarski algebras are booleanalgebras with operators; indeed, we need to show that the Lindenbaum-Tarski al-gebra of any normal modal logic� belongs toV�. Second, we need to prove thatLindenbaum-Tarski algebras provide canonical counterexamples to the validity ofnon-theorems of� in V�. The second point is easily dealt with:

Theorem 2.34 Let � be a modal similarity type, and� a normal modal� -logic.Let� be some propositional formula, and� a set of proposition letters of size notsmaller than the number of proposition letters occurring in�. Then

`� � iff L�(�) j= ��: (2.20)

Proof. This proof is completely analogous to that of Proposition 2.14 and is left tothe reader. a

So let us verify that Lindenbaum-Tarski algebras are canonical algebraic models ofthe right kind:


Theorem 2.35 Let� be a modal similarity type, and� be a normal modal� -logic.Then for any set� of proposition letters,L�(�) belongs toV�.

Proof. Once we have shown thatL�(�) is a boolean algebra with� -operators, thetheorem immediately follows from Theorem 2.34. Now, thatL�(�) is a booleanalgebra is clear, so the only thing that remains to be done is to show that the modal-ities really give rise to� -operators.

As an example, assume that� contains a diamond3; let us prove additivity off3. We have to show that

f3(a+ b) = f3a+ f3b;

for arbitrary elementsa andb of L�(�). Leta andb be such elements; by definitionthere are formulas� and such thata = [�] andb = [ ]. Then

f3(a+ b) = f3([�] + [ ]) = f3([� _ ]) = [3(� _ )]

while

f3a+ f3b = f3([�]) + f3([ ]) = [3�] + [3 ] = [3� _3 ]:

It is easy to check that

`� 3(� _ )$ (3� _3 );

whence it follows that[3(� _ )] = [3� _3 ]. We leave it for the reader to fillin the remaining details of this proof as Exercise 2.2.4. a

As an immediate corollary we have the following result: modal logics are alwayscomplete with respect to the variety of boolean algebras with operators where theiraxioms are valid. This is in sharp contrast to the situation in relational semantics,where modal logics neednot be complete with respect to the class of frames thatthey define.

This is an interesting result, but it is not what we really want, for it proves com-pleteness with respect to abstractBAOs rather than complex algebras. Not only arecomplex algebras concrete algebras of propositions, we also know (recall Proposi-tion 2.26) that complex algebras embody all the information of relevance to framevalidity – so we really should be aiming for completeness results with respect toclasses of complex algebras.

And that is why the long-promised J´onsson-Tarski Theorem, which we stateand prove in the following section, is so important. This tells us thateverybooleanalgebra with operators is isomorphic to a complex algebra, and thus guarantees thatwe can represent the Lindenbaum-Tarski algebras of any normal modal logics�

as a complex algebra. In effect, it will convert Theorem 2.34 into a completenessresult with respect to complex algebras. Moreover, because of the link between


complex algebras and relational semantics, it will open the door to exploring framecompleteness algebraically.

Exercises for Section 2.22.2.1 LetA be a boolean algebra. Prove that� is an operator. How about+?

2.2.2 Show that every complex algebra is a boolean algebra with operators (that is, proveProposition 2.24).

2.2.3 LetA be the collection of finite and co-finite subsets ofN. Definef : A! A by

f(X) =

�fy 2 N j y + 1 2 Xg if X is finite;N if X is co-finite:

Prove that(A;[;�;?; f) is a boolean algebra with operators.

2.2.4 Let � be a normal modal logic. Prove that the Lindenbaum-Tarski algebraL� is aboolean algebra with� -operators (that is, fill in the missing proof details in Theorem 2.35).

2.2.5 Let� be a set of� -formulas. Prove that for any formula�, `K�� impliesV� j=��. That is, prove the soundness direction of Theorem 2.29.

2.2.6 Call a varietyV of BAOs completeif it is generated by a class of complex algebras,i.e., if V = HSPCmK for some frame classK. Prove that a logic� is complete iff thevarietyV� is complete.

2.2.7 Let A be a boolean algebra. In this exercise we assume familiarity with the notionof an infinite sum (supremum). An operationf : A! A is calledcompletely additiveif itdistributes over infinite sums (in each of its arguments).

(a) Show that every operation of the formmR is completely additive.(b) Give an example of an operation that is additive, but not completely additive. (Hint:

as the boolean algebra, take the set of finite and co-finite subsets of some frame.)

2.3 The Jonsson-Tarski Theorem

We already know how to construct aBAO from a frame: simply form the frame’scomplex algebra. We will now learn how to construct a frame from aBAO by form-ing theultrafilter frameof the algebra. As we will see, this operation generalizestwo constructions that we have met before: taking the ultrafilter extension of amodel, and forming the canonical frame associated with a normal modal logic.

Our new construction will lead us to the desired representation theorem: by tak-ing the complex algebra of the ultrafilter frame of aBAO, we obtain thecanonicalembedding algebraof the original BAO. The fundamental result of this section(and, indeed, of the entire chapter) is thatevery boolean algebra with operatorscan be isomorphically embedded in its canonical embedding algebra. We willprove this result and along the way discuss a number of other important issues,

2.3 The Jonsson-Tarski Theorem 43

such as the algebraic status of canonical models and ultrafilter extensions, and theimportance of canonical varieties ofBAOs for modal completeness theory.

Let us consider the problem of (isomorphically) embedding an arbitraryBAO

A in a complex algebra. Obviously, the first question to ask is: what should bethe underlyingframe of the complex algebra? To keep our notation simple, letus assume for the moment that we are working in a similarity type with just oneunary modality, and thatA = (A;+;�; 0; f) is a boolean algebra with one unaryoperatorf . Thus we have to find a universeW and a binary relationR on Wsuch thatA can be embedded in the complex algebra of the frame(W;R). Stone’sRepresentation Theorem 2.16 gives us half the answer, for it tells us how to embedthe boolean part ofA in the power set algebra of the setUfA of ultrafilters of A.Let us take a closer look at this fundamental result.

Stone’s Representation Theorem

A central role in the representation theory of boolean algebras is played by filtersand ultrafilters.

Definition 2.36 A filter of a boolean algebraA = (A;+;�; 0) is a subsetF � A

satisfying

(F1) 1 2 F ,(F2) F is closed under taking meets; that is, ifa; b 2 F thena � b 2 F ,(F3) F is upward closed; that is, ifa 2 F anda � b thenb 2 F .

A filter is proper if it does not contain the smallest element0, or, equivalently, ifF 6= A. An ultrafilter is a proper filter satisfying

(F4) For everya 2 A, eithera or�a belongs toF .

The collection of ultrafilters ofA is calledUfA. a

Note the difference in terminology: an (ultra)filterover the setW is an (ultra)filterof the power set algebraP(W ).

Example 2.37 For any elementa of a boolean algebraA, the seta" = fb 2 A j

a � bg is a filter. In the field of finite and co-finite subsets of a countable setW ,the collection of co-finite subsets ofW forms an ultrafilter. a

Example 2.38 Since the collection of filters of a boolean algebra is closed undertaking intersections, we may speak of thesmallestfilter FD containing a given setD � A. This filter can also be defined as the following set:

fa 2 A j there ared0; : : : ; dn 2 D such thatd1 � : : : � dn � ag (2.21)


which explains why we will also refer toFD as the filtergenerated byD. Thisfilter is proper ifD has the so-calledfinite meet property; that is, if there is no finitesubsetfd0; : : : ; dng of D such thatd1 � : : : � dn = 0. a

For future reference, we gather some properties of ultrafilters; the proof of the nextproposition is left to the reader.

Proposition 2.39 LetA = (A;+;�; 0) be a boolean algebra. Then

(i) For any ultrafilteru of A and for every pair of elementsa; b 2 A we havethata+ b 2 u iff a 2 u or b 2 u.

(ii) UfA coincides with the set of maximal proper filters onA (‘maximal’ isunderstood with respect to set inclusion).

The main result that we need in the proof of Stone’s Theorem is the UltrafilterTheorem: this guarantees that there are enough ultrafilters for our purposes.

Proposition 2.40 (Ultrafilter Theorem) LetA be a boolean algebra,a an elementofA, andF a proper filter ofA that does not containa. Then there is an ultrafilterextendingF that does not containa.

Proof. We first prove that every proper filter can be extended to an ultrafilter. LetG be a proper filter ofA, and consider the setX of all proper filtersH extendingG. Suppose thatY is achain in X; that is,Y is a nonempty subset ofX of whichthe elements are pairwise ordered by set inclusion. We leave it to the reader toverify that

SY is a proper filter; obviously,

SY extendsG; so

SY belongs toX

itself. This shows thatX is closed under taking unions of chains, whence it followsfrom Zorn’s Lemma thatX contains a maximal elementu. We claim thatu is anultrafilter.

For suppose otherwise. Then there is ab 2 A such that neitherb nor�b belongstou. Consider the filtersH andH 0 generated byu[fbg andu[f�bg, respectively.Since neither of these can belong toX, both must be improper; that is,0 2 H and0 2 H 0. But then by definition there are elementsu1; : : : ; un, u01; : : : ; u

0m in u such

that

u1 � : : : � un � b � 0 and u01 � : : : � u0m � �b � 0:

From this it easily follows that

u1 � : : : � un � u01 � ::: � u

0m = 0;

contradicting the fact thatu is a proper filter.Now suppose thata andF are as in the statement of the proposition. It is not

hard to show thatF [ f�ag is a set with the finite meet property. In Example 2.38we saw that there is a proper filterG extendingF and containing�a. Now we use


the first part of the proof to find an ultrafilteru extendingG. But if u extendsG italso extendsF , and if it contains�a it cannot containa. a

It follows from Proposition 2.40 and the facts mentioned in Example 2.38 that anysubset of a boolean algebra can be extended to an ultrafilter provided that it hasthe finite meet property. We now have all the necessary material to prove Stone’sTheorem.

Theorem 2.16 (Stone Representation Theorem)Any boolean algebra is iso-morphic to a field of sets, and hence, to a subalgebra of a power of2. As aconsequence, the variety of boolean algebras is generated by the algebra2:

BA = V(f2g):

Proof. Fix a boolean algebraA = (A;+;�; 0). We will embedA in the power setof UfA. Consider the map� : A! P(Uf A) defined as follows:

�(a) = fu 2 UfA j a 2 ug:

We first show that� is a homomorphism. As an example we treat the join operation:

�(a+ b) = fu 2 UfA j a+ b 2 ug

= fu 2 UfA j a 2 u or b 2 ug

= fu 2 UfA j a 2 ug [ fu 2 UfA j b 2 ug

= �(a) [ �(b):

Note that the crucial second equality follows from Proposition 2.39.It remains to prove that� is injective. Suppose thata andb are distinct elements

of A. We may derive from this that eithera 6� b or b 6� a. Without loss ofgenerality we may assume the second. But ifb 6� a thena does not belong to thefilter b" generated byfbg, so by Proposition 2.40 there is some ultrafilteru suchthatb" � u anda 62 u. Obviously,b" � u implies thatb 2 u. But then we havethatu 2 �(b) andu 62 �(a).

This shows thatA is isomorphic to a field of sets; it then follows by Propo-sition 2.8 thatA is isomorphic to a subalgebra of a power of2. From this it isimmediate thatBA is the variety generated by the algebra2. a

Remark 2.41 That every boolean algebra is isomorphic to a subalgebra of a powerof the algebra2 can be proved more directly by observing that there is a one-to-one correspondence between ultrafilters ofA and homomorphisms fromA onto2.Given an ultrafilteru of A, define�u : A! 2 by

�u(a) =

�1 if a 2 u;0 otherwise:


And conversely, given a homomorphism� : A! 2, define the ultrafilteru� by

u� = ��1(1) (= fa 2 A j �(a) = 1g):

We leave further details to the reader. a

Ultrafilter frames

Now that we have a candidate for the universe of the ultrafilter frame of a givenBAO A, let us see how to define a relationR on ultrafilters such that we can embedA in the algebra(Uf A; R)+. To motivate the definition ofR, we will view theelements of the algebra aspropositions, and imagine thatr(a) (the representationmapr applied to propositiona) yields the set of states wherea is true accordingto some valuation. Hence, readingfa as3a, it seems natural that a stateu shouldbe in r(fa) if and only if there is av with Ruv andv 2 r(a). So, in order todecide whetherRuv should hold for two arbitrary states (ultrafilters)u andv, weshould look at all the propositionsa holding atv (that is, all elementsa 2 v) andcheck whetherfa holds atu (that is, whetherfa 2 u). Putting it more formally,the natural, ‘canonical’ choice forR seems to be the relationQf given by

Qfuv iff fa 2 u for all a 2 v:

The reader should compare this definition with the definition of the canonical re-lation given in Definition 1.23. Although one is couched in terms of ultrafilters,and the other in terms of maximal consistent sets (MCSs), both clearly trade on thesame idea. As we will shortly learn (and as the above identification of ‘ultrafilters’and ‘maximal sets of propositions’ already suggests), this is no accident.

In the general case, we use the following definition (an obvious analog of Defi-nition 1.30).

Definition 2.42 Given ann-ary operatorf on a boolean algebra(A;+;�; 0), wedefine the(n+ 1)-ary relationQf on the set of ultrafilters of the algebra by

Qfuu1 : : : un iff f(a1; : : : ; an) 2 u for all a1 2 u1, . . . ,an 2 un:

Let A = (A;+;�; 0; fM)M2� be a boolean algebra with operators. Theultrafilterframeof A, notation:A+, is the structure(Uf A; QfM)M2� . The complex algebra(A+)

+ is called the(canonical) embedding algebraof A (notation:EmA). a

We leave it to the reader to verify that theultrafilter extensionue F of a frameF is nothing but the ultrafilter frame of the complex algebra ofF, in symbols:ue F = (F+)+.

For later reference, we state the following proposition (an obvious analog ofLemma 1.31) which shows that we could have given an alternative but equivalentdefinition of the relationQf .


Proposition 2.43 Let f be ann-ary operator on the boolean algebraA, andu,u1, . . . ,un an (n+ 1)-tuple of ultrafilters ofA. Then

Qfuu1 : : : un iff �f(�a1; : : : ;�an) 2 u implies that for somei, ai 2 ui.

Proof. We only prove the direction from left to right. Suppose thatQfuu1 : : : un,and that�f(�a1; : : : ;�an) 2 u. To arrive at a contradiction, suppose that there isno i such thatai 2 ui. But asQfuu1 : : : un, it follows thatf(�a1; : : : ;�an) 2 u.But this contradicts the fact that�f(�a1; : : : ;�an) 2 u. a

As the above sequence of analogous definitions and results suggest, we have al-ready encountered a kind of frame which is very much like an ultrafilter frame,namely thecanonical frameof a normal modal logic (see Definition 1.23). Thebasic idea should be clear now: the states of the canonical frame are theMCSs ofthe logic, and an ultrafilter is nothing but an abstract version of anMCS. But this isno mere analogy: the canonical frame of a logic is actuallyisomorphicto the ultra-filter frame of its Lindenbaum-Tarski algebra, and the mapping involved is simpleand intuitive. When making this connection, the reader should keep in mind thatwhen we defined ‘the’ canonical frame in the previous Chapter, we always had afixed, countable set� of proposition letters in mind.

Theorem 2.44 Let� be a modal similarity type,� a normal modal� -logic, and�the set of proposition letters used to define the canonical frameF�. Then

F� �= (L�(�))+:

Proof. We leave it to the reader to show that the function� defined by

�(� ) = f[�] j � 2 �g;

mapping a maximal�-consistent set� to the set of equivalence classes of its mem-bers, is the required isomorphism betweenF� and(L�(�))+. a

The Jonsson-Tarski Theorem

We are ready to prove the J´onsson-Tarski Theorem: every boolean algebra withoperators is embeddable in the full complex algebra of its ultrafilter frame.

Theorem 2.45 (Jonsson-Tarski Theorem)Let � be a modal similarity type, andA = (A;+;�; 0; fM)M2� be a boolean algebra with� -operators. Then the repre-sentation functionr : A! P(Uf A)) given by

r(a) = fu 2 UfA j a 2 ug

is an embedding ofA into EmA.


Proof. To simplify our notation a bit, we work in a similarity type with a singlen-ary modal operator, assuming thatA = (A;+;�; 0; f) is a boolean algebra witha singlen-ary operatorf . By Stone’s Representation Theorem, the mapr : A !

P(Uf (A)) given by

r(x) = fu 2 Uf (A) j x 2 ug

is a boolean embedding. So, it suffices to show thatr is also amodalhomomor-phism; that is, that

r(f(a1; : : : ; an)) = mQf(r(a1); : : : ; r(an)): (2.22)

We will first prove (2.22) for unaryf . In other words, we have to prove that

r(fa) = mQf(r(a)):

We start with the inclusion from right to left: assumeu 2 mQf(r(a)). Then by

definition ofmQf, there is an ultrafilteru1 with u1 2 r(a) (that is,a 2 u1) and

Qfuu1. By definition ofQf this impliesfa 2 u, or u 2 r(fa).For the other inclusion, letu be an ultrafilter inr(fa), that is,fa 2 u. To

prove thatu 2 mQf(r(a)), it suffices to find an ultrafilteru1 such thatQfuu1 and

u1 2 r(a), or a 2 u1. The basic idea of the proof is that we first pick out thoseelements ofA (other thana) that we cannot avoid putting inu1. These elements aregiven by the conditionQfuu1. By Proposition 2.43 we have that for every elementof the form�f(�y) in u, y has to be inu1; therefore, we define

F := fy 2 A j �f(�y) 2 ug:

We will now show that there is an ultrafilteru1 � F containinga. First, an easyproof (using the additivity off ), shows thatF is closed under taking meets. Sec-ond, we prove that

F 0 := fa � y j y 2 Fg

has the finite meet property. AsF is closed under taking meets, it is sufficient toshow thata � y 6= 0 whenevery 2 F . To arrive at a contradiction, suppose thata � y = 0. Thena � �y, so by the monotonicity off , fa � f(�y); therefore,f(�y) 2 u, contradictingy 2 F .

By Proposition 2.40 there is an ultrafilteru1 � F 0. Note thata 2 u1, as1 2 F .Finally,Qfuu1 holds by definition ofF : if �f(�y) 2 u theny 2 F � u1.

We now prove (2.22) for arbitraryn � 1 by induction on the arityn of f . Wehave just proved the base case. So, assume that the induction hypothesis holds forn. We only treat the direction from left to right, since the other direction can beproved as in the base case. Letf be a normal and additive function of rankn+ 1,and suppose thata1; : : : ; an+1 are elements ofA such thatf(a1; : : : ; an+1) 2 u.


We have to find ultrafiltersu1; : : : ; un+1 of A such that (i)ai 2 ui for all i with1 � i � n + 1, and (ii)Qfuu1 : : : un+1. Our strategy will be to let the inductionhypothesis take care ofu1; : : : ; un and then to search forun+1.

Let f 0 : An ! A be the function given by

f 0(x1; : : : ; xn) = f(x1; : : : ; xn; an+1):

That is, for the time being we fixan+1. It is easy to see thatf 0 is normal andadditive, so we may apply the induction hypothesis. Sincef 0(a1; : : : ; an) 2 u, thisyields ultrafiltersu1; : : : ; un such thatai 2 ui for all i with 1 � i � n, and

f(x1; : : : ; xn; an+1) 2 u; wheneverxi 2 ui (1 � i � n). (2.23)

Now we will define an ultrafilterun+1 such thatan+1 2 un+1 andQfuu1 : : : un+1.This second condition can be rewritten as follows (we abbreviate ‘x1 2 u1, . . . ,xn 2 un’ by ‘ ~x 2 ~u’):

Qfuu1 : : : un+1

iff for all ~x; y: if ~x 2 ~u, theny 2 un+1 impliesf(~x; y) 2 u

iff for all ~x; y: if ~x 2 ~u, thenf(~x; y) 62 u impliesy 62 un+1

iff for all ~x; y: if ~x 2 ~u, then�f(~x; y) 2 u implies�y 2 un+1

iff for all ~x; z: if ~x 2 ~u, then�f(~x;�z) 2 u impliesz 2 un+1.

This provides us with a minimal set of elements thatun+1 should contain; put

F := fz 2 A j 9~x 2 ~u (�f(~x;�z) 2 u)g:

If �f(~x;�z) 2 u, we say that~x drivesz into F . We now take the first conditioninto account as well, definingF 0 := fan+1g [ F .

Our aim is to prove the existence of an ultrafilterun+1 containingF 0. It will beclear that this is sufficient to prove the theorem (note thatan+1 2 F

0 as1 2 F ). Tobe able to apply the Ultrafilter Theorem 2.40, we will show thatF 0 has the finitemeet property. We first need the following fact:

F is closed under taking meets. (2.24)

Let z0; z00 be inF ; assume thatz0 andz00 are driven intoF by ~x0 and~x00, respec-tively. We will now see that~x := (x01 � x

001 ; : : : ; x

0n � x

00n) drivesz := z0 � z00 into F ,

that is, that�f(~x;�z) 2 u.Sincef is monotonic, we havef(~x;�z0) � f(~x0;�z0), and hence we find that

�f(~x0;�z0) � �f(~x;�z0). As u is upward closed and�f(~x0;�z0) 2 u byour ‘driving assumption’, this gives�f(~x;�z0) 2 u. In the same way we find�f(~x;�z00) 2 u. Now

f(~x;�z) = f(~x;�(z0 � z00)) = f(~x; (�z0) + (�z00)) = f(~x;�z0) + f(~x;�z00);


whence

�f(~x;�z) = [�f(~x;�z0)] � [�f(~x;�z00)]:

Therefore,�f(~x;�z) 2 u, sinceu is closed under taking meets. This proves(2.24).

We can now finish the proof and show that indeed

F 0 has the finite meet property. (2.25)

By (2.24) it suffices to show thatan+1�z 6= 0 for all z 2 F . To prove this, we reasonby contraposition: suppose thatz 2 F andan+1 � z = 0. Let ~x 2 ~u be a sequencethat drivesz into F , that is,�f(~x;�z) 2 u. Froman+1 � z = 0 it follows thatan+1 � �z, so by monotonicity off we get�f(~x;�z) � �f(~x; an+1). But then�f(~x; an+1) 2 u, which contradicts (2.23). This proves that indeedan+1 � z 6= 0

and hence we have shown (2.25) and thus, Theorem 2.45. a

Exercises for Section 2.32.3.1 Let� be a normal modal logic. Give a detailed proof that the canonical frameF� isisomorphic to the ultrafilter frame ofL� (over a countable set of proposition letters).

2.3.2 Let A denote the collection of setsX of integers satisfying one of the followingfour conditions: (i)X is finite, (ii) X is co-finite, (iii) X � E is finite, (iv) X � E isco-finite. HereE denotes the set of all even integers, and� denotes symmetric difference:X �E = (X nE)[ (E nX). Consider the following algebraA = (A;[;�;?; f) wherethe operationf is given by

f(X) =

�fx� 1 j x 2 Xg if X is of type (i) or (iii);Z if X is of type (ii) or (iv):

(a) Show thatA is a boolean algebra with operators.(b) DescribeA+.

2.3.3 Let W be the setZ[ f�1;1g and letS be the successor relation onZ, that is,S = f(z; z + 1) j z 2 Zg.

(a) Give aBAO whose ultrafilter frame is isomorphic to the frameF = (W;R) withR = S [ f(�1;�1); (1;1)g.

(b) Give aBAO whose ultrafilter frame is isomorphic to the frameF = (W;R) withR = S [ (W � f�1;1g).

(c) Give aBAO whose ultrafilter frame is isomorphic to the frameF = (W;R) withR = S [ f(�1;�1)g [ (W � f1g).

2.3.4 An operation on a boolean algebra is called2-additiveif it satisfies

f(x+ y + z) = f(x+ y) + f(x+ z) + f(y + z):

Prove an analog of the J´onsson-Tarski Theorem for boolean algebras augmented with 2-additive operations.

2.4 Canonicity: the algebraic perspective 51

2.4 Canonicity: the algebraic perspective

To conclude this chapter, we discuss the algebraic perspective on canonicity. Clearlythe Jonsson-Tarski Theorem guarantees that we can represent the Lindenbaum-Tarski algebras of normal modal logics as complex algebras, so it immediatelyconverts Theorem 2.34 into a completeness result with respect to complex alge-bras.

But modal logicians want more: because of the link between complex algebrasand relational semantics, it seems to offer a plausible algebraic handle on framecompleteness. And in fact it does – but we need to be careful. As should be clearfrom our work in the previous Chapter, even with the J´onsson-Tarski Theorem atour disposal, one more hurdle remains to be cleared. In Exercise 2.2.6 we definedthe notion of acompletevariety ofBAOs: a varietyV is complete if there is a frameclassK that generatesV in the sense thatV = HSPCmK. The exercise askedthe reader to show that any logic� is complete if and only ifV� is a completevariety. Now does the J´onsson-Tarski Theorem establish such a thing? Not really– it doesshow that every algebraA is a complex algebra oversomeframe, thusproving that for any logic� we have thatV� � CmK for some frame classK. So,this certainly givesV� � HSPCmK. However, in order to prove completeness,we have to establish an equality instead of an inclusion. One way to prove this isto show that the complex algebras that we have found form a subclass ofV�. ByProposition 2.26 it would suffice to show that for any algebraA in the varietyV�,the frameA+ is a frame for the logic�. This requirement gives us an algebraichandle on the notion of canonicity.

Let us examine a concrete example. Recall thatK4 is the normal logic gen-erated by the 4 axiom,33p ! 3p. We know from Theorem 1.33 thatK4 iscomplete with respect to the class of transitive frames. How can we prove thisresult algebraically?

A little thought reveals that the following is required: we have to show that theLindenbaum-Tarski algebras forK4 are embeddable in full complex algebras oftransitiveframes. Recall that the 4 axiomcharacterizesthe transitive frames, thusin our proposed completeness proof, we would have to show that4 is valid in theultrafilter frame(LK4(�))+ of LK4(�), or equivalently, that((LK4(�))+)+ be-longs to the varietyV4. Note that by Theorem 2.34 we already know thatLK4(�)

belongs toV4.As this example suggests, proving frame completeness results for extensions of

K algebraically leads directly to the following question:which varieties ofBAOsare closed under taking canonical embedding algebras?In fact, this is the requiredalgebraic handle on canonicity and motivates the following definition.

Definition 2.46 Let � be a modal similarity type, andC a class of boolean algebraswith � -operators.C is canonicalif it is closed under taking canonical embedding


algebras; that is, if for all algebrasA, EmA is in C wheneverA is in C. Likewise,an equation iscanonicalif its validity is preserved when moving from aBAO to itscanonical embedding algebra. a

Thus we now have two notions of canonicity, namely the logical one of Defini-tion 1.36 and the algebraic one just defined. Using Theorem 2.34, we show thatthese two concepts are closely related.

Proposition 2.47 Let � be a modal similarity type, and� a set of� -formulas. IfV� is a canonical variety, then� is canonical.

Proof. Assume that the varietyV� is canonical, and let� be the fixed countable setof proposition letters that we use to define canonical frames. By Theorem 2.34, theLindenbaum-Tarski algebraLK�(�) is in V�; then, by assumption, its canonicalembedding algebraEmLK� is inV�. However, from Theorem 2.44 it follows thatthis algebra is isomorphic to the complex algebra of the canonical frame ofK�:

EmLK�(�) = ((LK�(�))+)+ �= (FK�)+:

Now the fact that(FK�)+ is inV� means thatFK� � by Proposition 2.26. Butthis implies that� is canonical. a

An obvious question is whether the converse of Proposition 2.47 holds as well;that is, whether a varietyV� is canonical if� is a canonical set of modal formu-las. However, note that canonicity of� only implies that oneparticular booleanalgebra with operators has its embedding algebra inV� , namely the Lindenbaum-Tarski algebra over acountably infinite number of generators. This is becausethroughout the completeness chapter we were working in a fixed,countableset ofproposition letters. In fact, we are facing an open problem here:

Open Problem 1 Let � be a modal similarity type, and� a canonical set of� -formulas. IsV� a canonical variety?

Equivalently, suppose thatE is a set of equations such that for allcountableboolean algebras with� -operators we have the following implication

if A j= E thenEmA j= E: (2.26)

Is VE a canonical variety? In other words, does (2.26) hold forall boolean alge-bras with� -operators?

This is an interesting problem. However, arguably the restriction of the notion ofcanonicity to countable languages that we adopted in the previous Chapter was notmathematically natural. Thus, let us redefine the logical notion of canonicity sothat it refers to languages of arbitrary size. The definition of canonical frames and

2.4 Canonicity: the algebraic perspective 53

models can easily be parametrized by a set of proposition letters: the maximal con-sistent sets are supposed to be maximal within the induced set of formulas. We nowsimply define a logic� to becanonicalif it is valid on eachof its canonical frames.With this definition we can indeed establish equivalence between the logical andalgebraic notions of canonicity. (An alternative, and mathematically quite interest-ing alternative, would be to introduce, both logically and algebraically, ahierarchyof canonicity notions, parametrized by cardinal numbers. Such an approach hasindeed been studied in the literature, but this option will not be pursued here.)

Regardless of our approach towards this issue, the algebraic notion of canonicitycan do a lot of work for us. The important point is that it offers a genuinely newperspective on what canonicity is, a perspective that will allow us to use algebraicarguments. This will be demonstrated in the two papers forming part of theseLecture Notes.

Finally, in the course we will concentrate on canonicity results of a syntacticnature, proving that terms of various syntactic shape have various properties. Butthere is an interesting result, and an intriguing open problem, concerning a orestructural side of canonicity as well; both of these have to do with the relationbetween canonical varieties and frame classes that are closed under the formationof ultraproducts (such as elementary frame classes).

First we need the following definition.

Definition 2.48 Let � be modal similarity type, andK be a class of� -frames. Thevariety generated byK (notation:VK) is the classHSPCmK. a

Theorem 2.49 Let� be modal similarity type, andK be a class of� -frames whichis closed under ultraproducts. Then the varietyVK is canonical.

Example 2.50 Consider the modal similarity typefÆ;; 1’g of arrow logic, whereÆ is binary, is unary and1’ is a constant. The standard interpretation of thislanguage is given in terms of thesquares(cf. Example 1.10). Recall that the squareSU = (W;C;R; I) is defined as follows:

W = U � U;

C((u; v); (w; x); (y; z)) iff u = w and v = z and x = y;

R((u; v); (w; x)) iff u = x and v = w;

I(u; v) iff u = v:

It may be shown that the classSQ of (isomorphic copies of) squares is first-orderdefinable in the frame language with predicatesC, R and I. Therefore, Theo-rem 2.49 implies that the variety generated bySQ is canonical. This variety is wellknown in the literature on algebraic logic as the varietyRRA of Representable Re-lation Algebras. a


Rephrased in terminology from modal logic, Theorem 2.49 boils down to the fol-lowing result.

Corollary 2.51 Let � be a modal similarity type, andK be a class of� -frameswhich is closed under ultraproducts. Then the modal theory ofK is a canonicallogic.

We conclude the section with the foremost open problem in this area: does theconverse of Theorem 2.49 holds as well?

Open Problem 2 Let � be modal similarity type, andV a canonical variety ofboolean algebras with� -operators. Is there a classK of � -frames, closed undertaking ultraproducts, such thatV is generated byK?

Notes

The main aim of algebraic logic is to gain a better understanding of logic by treat-ing it in universal algebraic terms – in fact, the theory of universal algebra wasdeveloped in tandem with that of algebraic logic. Given a logic, algebraic logi-cians try to find a class of algebras that algebraizes it in a natural way. When alogic is algebraizable, natural properties of a logic will correspond to natural prop-erties of the associated class of algebras, and the apparatus of universal algebra canbe applied to solve logical problems. For instance, we have seen that represen-tation theorems are the algebraic counterpart of completeness theorems in modallogic. The algebraic approach has had a profound influence on the development oflogic, especially non-classical logic; readers interested in the general methodologyof algebraic logic should consult Blok and Pigozzi’s [7] or Andr´ekaet al. [1].

The field has a long and strong tradition dating back to the nineteenth century.In fact, nineteenth century mathematical logicwasalgebraic logic: to use the ter-minology of Section 2.1, propositions were represented as algebraic terms, notlogical formulas. Boole is generally taken as the founding father of both proposi-tional logic and modern algebra – the latter because, in his work, terms for the firsttime refer to objects other than numbers, and operations very different from thearithmetical ones are considered. The work of Boole was taken up by de Morgan,Peirce, Schr¨oder and others; their contributions to the theory of binary relationsformed the basis of Tarski’s development of relational algebra. In the HistoricalOverview in Chapter 1 we mentioned MacColl, the first logician in this tradition totreat modal logic. A discussion of the nineteenth century roots of algebraic logic isgiven by Anellis and Houser in [4].

However, when the quantificational approach to logic became firmly establishedin the early twentieth century, interest in algebraic logic waned, and it was onlythe influence of a relatively small number of researchers such as Birkhoff, Stone,

Notes to Chapter 2 55

Tarski, and Rasiowa and Sikorski [43, 44], that ensured that the tradition waspassed on to the present day. The method of basing an algebra on a collectionof formulas (or equivalence classes of formulas), due to Lindenbaum and Tarski,proved to be an essential research tool. This period also saw the distinction betweenlogical languages and their semantics being sharpened; an algebraic semantics fornon-classical logics was provided by Tarski’s matrix algebras. But the great suc-cess story of algebraic logic was its treatment of classical propositional logic inthe framework of boolean algebras, which we sketched in Section 2.1. Here, thework of Stone [48] was a milestone: not only did he prove the representation the-orem for boolean algebras (our Theorem 2.16), he also recognized the importanceof topological notions for the area (something we did not discuss in the text). Thisenabled him to prove a duality theorem permitting boolean algebras to be viewedas essentially the same objects as certain topologies (now called Stone spaces).Stone’s work has influenced many fields of mathematics, as is witnessed by John-stone [25].

McKinsey and Tarski [40] drew on Stone’s work in order to prove a representa-tion theorem for so-called closure algebras (that is,S4-algebras); this result signif-icantly extended McKinsey’s [39] which dealt with finite closure algebras. How-ever, when it comes to the algebraization of modal logics, the reader is now in aposition to appreciate the full significance of the work of Jónsson and Tarski [29].Although modal logic is not mentioned in their paper, the authors simultaneouslyinvented relational semantics, and showed (via their representation theorem) howthis new relational world related to the algebraic one. Both Theorem 2.45 fromthe present chapter and important results on canonicity are proved here. It is alsoobvious from their terminology (for instance, the use of the words ‘closed’ and‘open’ for certain elements of the canonical embedding algebra) that hidden be-neath the surface of the paper lies a duality theory that extends Stone’s result tocover operators on boolean algebras.

For many years after the publication of the Jónsson-Tarski paper, research inmodal logic and inBAO theory pretty much took place in parallel universes. Inalgebraic circles, the work of Jónsson and Tarski was certainly not neglected. Thepaper fitted well with a line of work onrelation algebras. These were introducedby Tarski [49] to be to binary relations what boolean algebras are to unary ones;the concrete, so-called representable relation algebras have, besides the booleanrepertoire, operations for taking the converse of a relation and the compositionof two relations, and as a distinguished element, the identity relation. (From theperspective of our book, the classRRA of representable relation algebras is nothingbut the variety generated by the complex algebras of the two-dimensional arrowframes; see Example 2.50. In fact, one of the motivations behind the introduction ofarrow logic was to give a modal account of the theory of relation algebras.) Muchattention was devoted to finding an analog of Stone’s result for boolean algebras:


that is, a nice equational characterization of the representable relation algebras.But this nut turned out to be hard to crack. Lyndon [34] showed that the axiomsthat Tarski had proposed did not suffice, and Monk [41] proved that the varietydoes not even have a finite first-order axiomatization. Later work showed thatequational axiomatizations will be very complex (Andréka [2]) and not in Sahlqvistform (Hodkinson [23] and Venema [53]), although Tarski proved thatRRA is acanonical variety. As a positive result, a nice game-theoretical characterizationwas given by Hirsch and Hodkinson [22]. But these Notes can only provide a lop-sided account of one aspect of the theory of relation algebras; for more, the readeris referred to Jónsson [26, 27], Maddux [35] or Hirsch and Hodkinson [24]. Onelast remark on relation algebras: it is averypowerful theory. In fact, as Tarski andGivant show in [50], one can formalize all of set theory in it.

Tarski and his students developed other branches of algebraic logic as well: forexample, the theory of cylindric algebras. The standard reference here is Henkin,Monk and Tarski [21]. Cylindric algebras (and also the polyadic algebras of Hal-mos [20]), are boolean algebras with operators that were studied as algebraic coun-terparts of first-order logic. For an introductory survey of these and other algebrasof relations we refer the reader to Németi [42]; modal logic versions of these alge-bras are discussed in Marx and Venema [38].

But in modal circles, the status of algebraic methods was very different. Indeed,with the advent of relational semantics for modal logic in the 1960s, it seemed thatalgebraic methods were to be swept away: model theoretic tools seemed to betheroute to a brave new modal world. (Bull’s work was probably the most impor-tant exception to this trend. For example, his theorem that all normal extensionsof S4.3are characterized by classes of finitemodelswas proved usingalgebraicarguments.) Indicative of the spirit of the times is the following remark made byLemmon in his two part paper on algebraic semantics for modal logic. After thank-ing Dana Scott for ideas and stimulus, he remarks:

. . . I alone am responsible for the ugly algebraic form into which I have castsome of his elegant semantics. [33, page 191]

Such attitudes only seemed reasonable because Jónsson and Tarski’s work had beenoverlooked by the leading modal logicians, and neither Jónsson nor Tarski haddrawn attention to its modal significance. Only when the frame incompletenessresults (which began to appear around 1972) showed that not all normal modallogics could be characterized in terms of frames were modal logicians forced toreappraise the utility of algebraic methods.

The work of Thomason and Goldblatt forms the next major milestone in thestory: Thomason [52] not only contains the first incompleteness results and usesBAOs, it also introduced general frames (though similar, language dependent, struc-tures had been used in earlier work by Makinson [36] and Fine [11]). Thomason

Notes to Chapter 2 57

showed that general frames can be regarded as simple set theoretic representationsof BAOs, and notes the connection between general frames and Henkin models forsecond-order logic. In [51], Thomason developed a duality between the categoriesof frames with bounded morphisms and that of complete and atomic modal alge-bras with homomorphisms that preserve infinite meets. It was Goldblatt, however,who did the most influential work: in [17] the full duality between the categories ofmodal algebras with homomorphisms and descriptive general frames with boundedmorphisms is proved, a result extending Stone’s. Independently, Esakia [10] cameup with such a duality for closure algebras. Goldblatt generalized his duality to ar-bitrary similarity types in [18]; a more explicitly topological version can be foundin Sambin and Vaccaro [46]. Ever since their introduction in the seventies, gen-eral frames, a nice compromise between algebraic and the relational semantics,have occupied a central place in the theory of modal logic. Kracht [31, 32] devel-oped an interesting calculus of internal descriptions which connects the algebraicand first-order side of general frames. Zakharyaschev gave an extensive analysisof transitive general frames in his work on canonical formulas – see for instanceChagrov and Zakharyaschev [8]

The first proof of the canonicity of Sahlqvist formulas, for the basic modal sim-ilarity type, was given by Sahlqvist [45], although many particular examples andless general classes of Sahlqvist axioms were known to be canonical. In particu-lar, Jonsson and Tarski proved canonicity, not only for certain equations, but alsofor various boolean combinations of suitable equations. This result overlaps withSahlqvist’s in the sense that canonicity for simple Sahlqvist formulas follows fromit, but on the other hand, Sahlqvist formulas allowing properly boxed atoms inthe antecedent do not seem to fall under the scope of the results in Jónsson andTarski [29]. Incidentally, Sahlqvist’s original proof is well worth consulting: it isnon-algebraic, and very different from the one given in the text. Our proof of theSahlqvist Completeness Theorem is partly based on the one given in Sambin andVaccaro [47].

Recent years have seen a revived interest in the notion of canonicity. De Rijkeand Venema [9] defined the notion of a Sahlqvistequationand generalized thetheory to arbitrary similarity types. Jónsson became active in the field again; in [28]he gave a proof of the canonicity of Sahlqvist equations by purely algebraic means,building on the techniques of his original paper with Tarski. Subsequent work ofGehrke, Jónsson and Harding (see for instance [14, 13]) generalized the notioneven further by weakening the boolean base of the algebra to that of a (distributive)lattice. Ghilardi and Meloni [15] prove canonicity of a wide class of formulasusing an entirely different representation of the canonical extension of algebraswith operators. Whereas the latter lines of research tend to separate the canonicityof formulas from correspondence theoretic issues, a reverse trend is visible in thework of Kracht (already mentioned) and Venema [54]. The latter work shows that


for some non-Sahlqvist formulas there is still an algorithm generating a first-orderformula which is now not equivalent to the modal one, but does define a propertyfor which the modal formula is canonical.

The applications of universal algebraic techniques in modal logic go much fur-ther than we could indicate in this chapter. Various properties of modal logics havebeen successfully studied from an algebraic perspective; of the many examples weonly mention the work of Maksimova [37] connecting interpolation with amalga-mation properties, and the work by Goldblatt and Thomason [19] relating modaldefinability to Birkhoff’s Theorem [6] identifying varieties with equational classes.Also, most work by Blok, Kracht, Wolter and Zakharyaschev on mapping the lat-tice of modal logics makes essential use of algebraic concepts such as splittingalgebras; a good starting point for information on this line of research would beKracht [32].

Finally, the proof of Theorem 2.49 is a generalization and algebraization byGoldblatt [18] of results due to Fine [12] and van Benthem [5]. The Open Prob-lems 1 and 2 seem to have been formulated first in Fine [12]; readers who wouldlike to try and solve the second one should definitely consult Goldblatt [16].

Appendix A

An Algebraic Toolkit

In this appendix we review some basic (universal) algebraic notions. The first partdeals with algebras and operations on (classes of) algebras, the second part is aboutalgebraic model theory, and in the third part we discuss equational logic. Birkhoff’sfundamental theorems are stated without proof.

Universal Algebra

An algebra is a set together with a collection of functions over the set; these func-tions are usually calledoperations. Algebras come in varioussimilarity types,determined by the number and arity of the operations.

Definition A.1 (Similarity Type) An algebraic similarity typeis an ordered pairF = (F; �) whereF is a non-empty set and� is a functionF ! N. Elementsof F are calledfunction symbols; the function� assigns to each operatorf 2 F afinite arity or rank, indicating the number of arguments thatf can be applied to.Function symbols of rank zero are calledconstants. We will usually be sloppy inour notation and terminology and writef 2 F instead off 2 F . a

Given a similarity type, it is obvious what an algebra of this type should be.

Definition A.2 (Algebras) Let A be some set, andn a natural number; ann-aryoperationonA is a function fromAn toA.

Let F be an algebraic similarity type. Analgebra of type F is a pairA =

(A; I) whereA is a non-empty set called thecarrier of the algebra, andI is aninterpretation, a function assigning, for everyn, ann-ary operationfA on A toeach function symbolf of rankn. We often use the notationA = (A; fA)f2F forsuch an algebra. When no confusion is likely to arise, we omit the subscripts onthe operations. a

We now define the standard constructions for forming new algebras from old. Firstwe define the natural notion of structure preserving maps between algebras.

59

60 A An Algebraic Toolkit

Definition A.3 (Homomorphisms) Let A = (A; fA)f2F andB = (A; fB)f2Fbe two algebras of the same similarity type. A map� : A! B is a homomorphismif for all f 2 F , and alla1, . . . ,an 2 A (wheren is the rank off ):

�(fA(a1; : : : ; an)) = fB(�a1; : : : ; �an): (A.1)

(Here�ai is shorthand�(ai).) The special case for constantsc is

�(cA) = cB:

Thekernelof a homomorphismf : A! B is the relationker f = f(a; a0) 2 A2 j

f(a) = f(a0)g. We say thatB is ahomomorphic imageof A (notation:A� B),if there is a surjective homomorphism fromA ontoA0. Given a classC of algebras,HC is the class of homomorphic images of algebras inC. a

Definition A.4 (Isomorphisms) A bijective homomorphism is called anisomor-phism. We say that two algebras areisomorphicif there is an isomorphism betweenthem. Usually we do not distinguish isomorphic algebras, but if we do, we writeIC for the class of isomorphic copies of algebras inC. a

The second way of making new algebras from old is to find a small algebra insidea larger one.

Definition A.5 (Subalgebras)Let A be an algebra, andB a subset of the carrierA. If B is closed under every operationfA, then we callB = (B; fA �B)f2F asubalgebraof A. We say thatC is embeddablein A (notation: C � A), if C isisomorphic to a subalgebra ofA; the isomorphism is called anembedding. Givena classC of algebras,SC denotes the class of isomorphic copies of subalgebras ofalgebras inC. a

A third way of forming new algebras is to make a big algebra out of a collection ofsmall ones.

Definition A.6 (Products) Let (Aj)j2J be a family of algebras. We define theproduct

Qj2J Aj of this family as the algebraA = (A; fA)f2F whereA is the

cartesian productQj2J Aj of the carriersAj, and the operationfA is defined co-

ordinatewise; that is, for elementsa1; : : : ; an 2Qj2J Aj, fA(a1; : : : ; an) is the

element ofQj2J Aj given by:

fA(a1; : : : ; an)(j) = fAj(a1(j); : : : ; an(j)):

When all the algebrasAj are the same, sayA, then we callQj2J A a powerof

A, and writeAJ rather thanQj2J A. Given a classC of algebras,PC denotes the

class of isomorphic copies of products of algebras inC. a

An Algebraic Toolkit 61

Suppose you are working with a class of algebras from which you cannot obtainnew algebras by the three operations defined above. Intuitively, such a class is‘complete’, for it is closed under the natural algebra-forming operations. Suchclasses play an important role in universal algebra. They are calledvarieties:

Definition A.7 (Varieties) A class of algebras is called avariety if it is closedunder taking subalgebras, homomorphic images, and products. Given a classC

of algebras,VC denotes the variety generated byC; that is, the smallest varietycontainingC. a

A well-known result in universal algebra states thatVC = HSPC. That is, in orderto obtain the variety generated byC, you can start by taking products of algebrasin C, then go on to take subalgebras, and finish off by forming homomorphic im-ages. You do not need to do anything else: subsequent applications of any of theseoperations will not produce anything new.

Homomorphisms and Congruences

Homomorphisms on an algebraA are closely related to special equivalence rela-tions on the carrier ofA.

Definition A.8 (Congruences)Let A be an algebra for the similarity typeF . Anequivalence relation� onA (that is, a reflexive, symmetric and transitive relation)is acongruenceif it satisfies, for allf 2 F

if a1 � b1 & : : : & an � bn; thenfA(a1; : : : ; an) � fA(b1; : : : ; bn); (A.2)

wheren is the rank off . a

The standard examples of congruences are the ‘modulo’ relations on the integers.Consider the algebraZ = (Z;+; �; 0; 1) of the integers under addition and multi-plication, and, for a positive integern, let the relation�n be defined byz �n z0

if n dividesz � z0. We leave it to the reader to verify that these relations are allcongruences.

The importance of congruences is that they are precisely the kind of equivalencerelations that allow a natural algebraic structure to be defined on the collection ofequivalence classes.

Definition A.9 (Quotient Algebras) LetA be anF-algebra, and� a congruenceonA. Thequotient algebra ofA by� is the algebraA=� whose carrier is the set

A=� = f[a] j a 2 Ag


of equivalence classes ofA under�, and whose operations are defined by

fA=�([a1]; : : : ; [an]) = [fA(a1; : : : ; an)]:

(This is well-defined by (A.2).) The function� taking an elementa 2 A to itsequivalence class[a] is called thenatural mapassociated with the congruence.a

As an example, taking the quotient ofZ under the relation�n makes the algebraZn of arithmetic modulon.

The close connection between homomorphisms and congruences is given by thefollowing proposition (the proof of which we leave to the reader).

Proposition A.10 (Homomorphisms and Congruences)LetA be anF-algebra.Then

(i) If f : A! B is a homomorphism, its kernel is a congruence onA.

(ii) Conversely, if� is a congruence onA, its associated natural map is asurjective homomorphism fromA ontoA=�.

Algebraic Model Theory

Universal algebra can be seen as a branch of model theory in which one is onlyinterested in structures where all relations are functions. The standard languagefor talking about such structures isequational, where an equation is a statementasserting that twotermsdenote the same element.

Definition A.11 (Terms and Equations)Given an algebraic similarity typeF anda setX of elements calledvariables, we define the setTerF (X) of F-terms overX inductively: it is the smallest setT containing all constants and all variables inX such thatf(t1; : : : ; tn) is in T whenevert1, . . . ,tn are inT andf is a functionsymbol of rankn.

An equationis a pair of terms(s; t); the notations � t is usually used. a

Having defined the algebraic language, we now consider the way it is interpretedin algebras. Obviously terms refer to elements of algebras, but in order to calculatethe meaning of a term we need to know what the variables in the term stand for.This information is provided by an assignment.

Definition A.12 (Algebraic Semantics)LetF be an algebraic similarity type,Xa set of variables, andA anF-algebra. AnassignmentonA is a function� : X !

A associating an element ofA with each variable inX. Given such an assignment�, we can calculate themeaning~�(t) of a termt in TerF (X) as follows:


~�(x) = �(x);~�(c) = cA;

~�(f(t1; : : : ; tn)) = fA(~�(t1); : : : ; ~�(tn)): a

The last equality bears an obvious resemblance to the condition (A.1) defininghomomorphisms. In fact, we can turn the meaning function into a genuine homo-morphism by imposing a natural algebraic structure on the setTerF (X) of terms:

Definition A.13 (Term Algebras) Let F be an algebraic similarity type, andXa set of variables. Theterm algebra ofF over X is the algebraTerF (X) =

(TerF (X); I) where every function symbolf is interpreted as the operationI(f)onTerF (X) given by

I(f)(t1; : : : ; tn) = f(t1; : : : ; tn): (A.3)

a

In other words, the carrier of the term algebra overF is the set ofF-terms over theset of variablesX, and the operationI(f) or fTerF (X) maps ann-tuple t1; : : : ; tnof terms to the termf(t1; : : : ; tn). Note the double role off in (A.3): on the right-hand side,f denotes a ‘static’part of the syntacticterm f(t1; : : : ; tn), while onthe left-hand sideI(f) denotes a ‘dynamic’interpretationof f as anoperationonterms.

The perspective onF-terms as constituting anF-algebra is extremely useful.For example, we can view the operation ofsubstitutingterms for variables in termsas anendomorphismon the term algebra, that is, a homomorphism from an algebrato itself.

Definition A.14 LetF be a similarity type, andX a set of variables. Asubstitutionis a map� : X ! TerF (X) mapping variables to terms. Such a substitution canbe extended to a map~� : TerF (X) ! TerF (X) by the following inductivedefinition:

~�(x) := �(x);

~�(f(t1; : : : ; tn)) := f(~�(t1); : : : ; ~�(tn)):

We sometimes use the word ‘substitution’ for a function mapping terms to termsthat satisfies the second of the conditions above (that is, we sometimes call~� asubstitution). a

Proposition A.15 Let � : TerF (X) ! TerF (X) be a substitution. Then� :

TerF (X) ! TerF (X) is a homomorphism.


Moreover, themeaning functionassociated with an assignment� is now a homo-morphism:

Proposition A.16 Given any assignment� of variablesX to elements of an alge-bra A, the corresponding meaning function~� is a homomorphism fromTerF (X)

toA.

The standard way of making statements about algebras is to compare the meaningof two terms under the same valuation – that is, to useequations.

Definition A.17 (Truth and Validity) An equations � t is true or holds in analgebraA (notation:A j= s � t), if for all assignments�, ~�(s) = ~�(t).

A setE of equations holds in an algebraA (notation:A j= E), if each equationin E holds inA. If A j= s � t or A j= E we will also say thatA is amodelfors � t, or forE, respectively.

An equations � t is asemantic consequenceof a setE of equations (notation:E j= s � t), if every model forE is a model fors � t. a

Algebraists are often interested in specific classes of algebras such as groups andboolean algebras. Such classes are usually defined by sets of equations.

Definition A.18 (Equational Class)A classC of algebras is equationally defin-able, or anequational class, if there is a setE of equations such thatC containsprecisely the models forE. a

The following theorem, due to Birkhoff, is one of the most fundamental results ofuniversal algebra:

Theorem A.19 (Birkhoff) A class of algebras is equationally definable if and onlyif it is a variety.

Unfortunately, we do not have the space to prove this theorem here. The readeris advised to try proving the easy direction (that is, to show that any equationallydefinable class is closed under taking homomorphic images, subalgebras and directproducts) for him- or herself.

Equational Logic

Equational logic arises when we formalize the rules that enable us to deduce newequations from old. Although we do not make direct use of equational logic in thetext, it will be helpful if the reader is acquainted with it. Here is a fairly standardsystem.


Definition A.20 (Equational Logic) LetF be an algebraic similarity type, andEa set of equations. The set of equations that arederivablefrom E is inductivelydefined by the following schema:

(axioms) The equations inE are derivable fromE; they are calledaxioms.(reflexivity) Every equationt � t is derivable fromE.(symmetry) If t1 � t2 is derivable fromE, then so ist2 � t1.(transitivity) If the equationst1 � t2 andt2 � t3 are derivable fromE, then so is

t1 � t3.(congruence) Suppose that all equationst1 � u1, . . . ,tn � un are derivable from

E, and thatf is a function symbol of rankn. Then the equationf(t1; : : : ; tn) � f(u1; : : : ; un) is derivable fromE as well. Thisschema is sometimes calledreplacement.

(substitution) If t1 � t2 is derivable fromE, then so is the equation�t1 � �t2,for every substitution�.

The notationE ` t1 � t2 means that the equationt1 � t2 is derivable fromE.A derivation is a list of equations such that every element is either an axiom, or

has the formt � t, or can be obtained from earlier elements of the list using thesymmetry, transitivity, congruence/replacement, or substitution rules. a

A fundamental completeness result, also due to Birkhoff, links this deductive ap-paratus to the semantic consequence relation defined earlier.

Theorem A.21 Let E be a set of equations for the algebraic similarity typeF .Then for all equationss � t,E j= s � t iff E ` s � t.

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Stanford, 2000.

On the Canonicity of Sahlqvist Idendities

Bjarni Jonsson

Monotone Bounded Distributive LatticeExpansions

Mai Gehrke and Bjarni J´onsson

Monotone Bounded Distributive Lattice

Expansions

Mai Gehrke and Bjarni J�onsson

Abstract

Monotone bounded distributive lattice expansions (DLMs) are boundeddistributive lattices augmented by �nitary operations that are isotone orantitone in each coordinate. Such algebras encompass most algebras withbounded distributive lattice reducts that arise from logic, and general-ize bounded distributive lattices with operators. We de�ne the canonicalextension for DLMs and show that this extension is functorial.

A class of compatible DLMs is said to be canonical provided it isclosed under canonical extensions. The main theorems are two criteriafor canonicity: (1) If K is a canonical class of compatible DLMs closedunder ultraproducts, then the variety generated by K is also canonical; inparticular every �nitely generated variety of DLMs is canonical. (2) Anyvariety of DLMs, for which the canonical extension of each basic operationof each DLM in the class is both continuous and dually continuous, iscanonical. Both of these preservation theorems encompass varieties notencompassed by previous results in the literature even in the setting ofBoolean algebras with additional operations.

1 Introduction

Birkho�'s Representation Theorem for �nite distributive lattices states thatany �nite distributive lattice is isomorphic to the ring of order ideals of a �nitepartially ordered set. This makes �nite distributive lattices particularly simpleto work with as this enables one to work with the underlying ordered set andthe set theoretic operations of intersection and union. The Stone and Priest-ley dualities [?, ?] generalize this representation theorem to arbitrary boundeddistributive lattices. Underlying these dualities, as well as Birkho�'s represen-tation theorem is the embedding D ! P(P ) of a bounded distributive latticein the power set of its prime spectrum. In the �nite case the image of this mapis completely determined by the order on P , namely it is the set O(P ) of orderideals of P . In the in�nite case the image is not all of O(P ), and more structureis necessary. The dualities show that one way of characterizing this embeddingis by generating a topology on P using the image of the map. However, inworking with the dualities one has to leave the realm of algebra and orderedsets as the lattice is identi�ed topologically. That is, algebraic questions must

1

be translated to this topological setting in order to be studied there. In [?], weabstractly characterize the embedding D ! O(P ) within the realm of completelattices. This allows one to work with the useful set representation while keepingan algebraic and order theoretic point of view. Moreover, the resulting canoni-cal extension, is a complete lattice that is completely distributive and generatedby its completely join irreducible elements. The (complete lattice) theory ofsuch lattices is essentially the same as the theory of �nite distributive lattices,thus allowing the embedding of an arbitrary bounded distributive lattice in a�nite-acting one.

A setting in which the canonical extension is likely to o�er an advantageouspoint of view is in the presence of additional operations on the bounded dis-tributive lattice. In order to make use of the extensions, the question is �rst ofall how to extend the additional operations, and then in turn whether the ex-tended algebras stay within the variety from which the original algebras come.The canonical extension was �rst de�ned and put to use in exactly this wayby B. J�onsson and A. Tarski's [?] for Boolean Algebras with Operators. Thereal strength of their result is that they showed that any positive equationalproperty, that is, one not involving the negation, is satis�ed in the extension if(and only if) it is satis�ed in the original algebra. Thus this theorem suppliesrepresentation/extension theorems for all varieties of Boolean algebras with op-erators given by a set of positive equations. Due to the fact that the algebraiccounterparts of many logics are varieties (or quasi-varieties) of DLMs, thesetypes of algebras are very actively investigated. In particular, the tight rela-tionship between Kripke semantics for such logics and either the topologicalduals or the canonical extensions of the corresponding algebras, gives the rep-resentation/extension results a central role. For varieties of DLMs that areBAOs, where the canonical extension and preservation of positive identities re-sults have been available, these results have been used by some researchers, e.g.[?, ?, ?, ?, ?]. Alternatively, one can use the topological dualities augmentedby additional relational structure. This approach has been used to some extentfor BAOs but especially for more general types of DLMs where the results andmethods of [?] do not apply, for example [?, ?, ?, ?, ?, ?, ?]. Of strong relationto our work are the recent e�orts to work out the right Kripke semantics fornon-classical logics, e.g. [?, ?, ?]. There are some general papers investigatingthe relationship between varieties or quasivarieties of DLMs and their represen-tations/extensions in topological or algebraic terms [?, ?, ?, ?, ?, ?, ?, ?]. Ourcurrent work, including the results herein, is a further e�ort in this direction.

This paper contains results completed by the Spring of 1998. Since thenseveral conceptual advancements, including the introduction of two topologieson canonical extensions as well as a new de�nition of extensions for maps haveallowed us to simplify proofs, and generalize and add to our results. In addition,we are working on a comprehensive exposition of the consequences of theseresults. Due to the enormous work involved in this endeavor, we o�er these�rst results in published form to make at least some part of the ideas involvedavailable at this time.

In section two we brie y state the results on canonical extensions of bounded

2

distributive lattices from [?] and give the de�nition of the most general type ofalgebraic system we will consider, that is what we call monotone bounded dis-tributive lattice expansions (DLMs). These are bounded distributive latticeswith additional operations whose restrictions to any one coordinate are eitherisotone or antitone. In section three we make a detailed study of the interactionbetween canonical extension, dualization, and composition for isotone maps cul-minating in Theorem 17 which states that the canonical extension is functorialwhen the maps are taken to be algebra homomorphisms for the appropriate typeof algebras. Section four contains the main theorems on the preservation of iden-tities. Theorem 21, which is an algebraic counterpart to Goldblatt's Theorem3.6.7 in [?], states that if a class of compatible DLMs is closed under ultra-products, then the variety generated by the class is also closed under canonicalextensions. As a consequence, any �nitely generated variety of DLMs is closedunder canonical extensions. Theorem 23 states that any DLM for which thecanonical extension of each basic operation is both continuous and dually con-tinuous has the same equational theory as its canonical extension. Neither ofthese theorems implies or is implied by the preservation theorems in [?] and [?].However, they allow applications to structures involving negation, implication,and other such basic operations that these other theorems do not address.

2 Canonical Extensions of Bounded Distribu-

tive Lattices

All the lattices considered in this paper will be bounded and distributive, with0 and 1 regarded as distinguished elements. For brevity, a bounded distributivelattice will be referred to as a DL. The following results, with proofs can befound in [?]. They are listed here for easy reference.

Theorem 2.1 Let A be a bounded distributive lattice (DL), then there is aunique (up to isomorphism) DL A � satisfying:

1. A � is a DL-extension of A ;

2. A � is complete, completely distributive, and join generated by the setJ(A � ) of all completely join irreducible elements of A � ;

3. J(A � ) � K(A � ), where K(A � ) is the complete meet closure of A in A � ;

4. for all subsets K;L � A, ifWK �

VL in A � , then there are �nite subsets

K 0 � K, and L0 � L so thatWK 0 �

VL0.

Theorem 2.2 For a complete DL L, the following conditions are equivalent:

1. L is completely distributive and join generated by J(L);

2. L is algebraic and generated by J(L);

3. L �=O(J(L)) (the order ideals of J(L));

3

4. L �=O(P;�) for some partially ordered set (P;�);

5. L is doubly algebraic.

Notice that since the last condition clearly is self-dual, the dual of each ofthe other conditions also are equivalent to these. In particular, it follows thatA � also is meet generated by M(A � ), the set of completely meet irreducibleelements in A � . We call the elements of K(A � ), of O(A � ), and of A the closedelements, the open elements, and the clopen elements of A � , respectively. Givena lattice A , the lattice obtained from A by reversing the order is called the duallattice of A and we denote it by A � .

Corollary 2.3 Given a DL A , we have that (A � )� = (A � )�.

Proof It is clear that (A � )� satis�es conditions 1,2, and 4 of Theorem 1for A � . To see that condition 3 is also satis�ed let q; q0 2 M(A � ) with q � q0.Then there is p 2 J(A � ) with p � q but p � q0. Now since J(A � ) � K(A � )we have

Vfa 2 A : p � ag = p � q. Since q is completely meet irreducible,

there is a 2 A with p � a � q, but p � q0 implies a � q0. It follows that everycompletely meet irreducible is the join of the elements of A below it, that is,M(A � ) � O(A � ).

Corollary 2.4 Given DLs A and B , we have that A ��B � = (A �B )� , K(A � )�K(B� ) = K ((A � B )� ), and O(A � )�O(B � ) = O ((A � B )� ).

Proof It is straight forward to verify that A � � B� satis�es the conditionsof Theorem 1 for A � B .

We are interested in proving an extension theorem for DLs with additionaloperations that ensures the preservation of identities as far as this is possible.We want to make our class of DL expansions broad enough that it encompassesthose obtained by considering weakened negations, implications and such. Thismeans that we need to allow our maps to be antitone, or order reversing, insome coordinates. We work with the following class of objects as our broadestscope:

De�nition 2.5 A DLM (monotone DL expansion) is an algebra A = (A 0; (fi)i2I)so that

1. A 0 is a DL;

2. fi : A "i1 � : : : � A "

ini ! A is isotone for the appropriate choices of "ij =�

1�

for each j, 1 � j � ni; where ni is the arity of fi.

A class of DLMs is said to be a class of compatible algebras provided allthe algebras in the class are similar and the indexed family

�"iji2I

is the samefor all algebras in the class. A compatibility type is a similarity type along

4

with the "ij 's. Notice that the DLMs of a particular compatibility type form asubvariety of the class of all algebras of the corresponding similarity type.

If all the basic operations of a DLM are operators, that is, they preserve�nite joins in each coordinate (when viewed without the turning upside downprovided by the "ij 's), then it is called a DLO. A DLM (DLO) augmented by aBoolean complementation of the underlying lattice is denoted by BLE (BAO).

3 Canonical Extensions of Isotone Maps

Let A and B be DL's, f : A ! B an isotone (or order preserving) map. Wewant to de�ne f� : A � ! B� that extends f . Recall that A � is join generatedby J(A � ), and that J(A � ) lies in K(A � ), the complete meet closure of A . Forx 2 K(A � ), we let

f�(x) =^ff(a) : x � a 2 Ag.

Notice that :

� The mapping f� � K(A � ) is isotone and extends f ;

� The mapping f� � K(A � ) maps into K(B� );

� It is the largest possible isotone extension of f to K(A � ).

Now we extend this map to all of A � by de�ning, for u 2 A � ,

f�(u) =_ff�(x) : u � x 2 K(A � )g.

Here again we have:

� The mapping f� is isotone and extends f ;

� It is the least possible isotone extension of f� � K(A � ) to all of A � .

Since we will be dealing with functions of the form fi : A "i1 � : : :� A "

in ! A ,

and since we will need to compose these, it will be useful to use f�i sometimesinstead of fi. So the question arises whether for isotone f : A ! B it holds that(f�)

�= (f�)

�. Of course (f�)

�is the same function as f�, just considered

with the order on its domain and codomain turned upside down. The function(f�)

�however, is obtained by using the above de�nition but on the dual lattice.

The extension thus de�ned we call f�. We spell out the details:For y 2 O(A � ), we let

f�(y) =_ff(a) : y � a 2 Ag.

Notice that :

� The mapping f� � O(A � ) is isotone and extends f ;

5

� The mapping f� � O(A � ) maps into O(B � );

� It is the least possible isotone extension of f to O(A � ).

Now we extend this map to all of A � by de�ning, for u 2 A � ,

f�(u) =^ff�(y) : u � y 2 O(A � )g.

Here again we have:

� The mapping f� is isotone and extends f ;

� It is the largest possible isotone extension of f� � O(A � ) to all of A � .

These two functions do not agree in general. We have the following relation-ship:

Theorem 3.1 Let A and B be DL's, f : A ! B an isotone map, then f� �f� with equality holding for open as well as for closed elements (consequently,we will at times denote both extensions by f when acting on closed or openelements).

Proof Let x 2 K(A � ), then f�(x) � f�(x) since f� is the largest extensionof f to K(A � ). We need to show that f�(x) � f�(x) =

Vff�(y) : x � y 2

O(A � )g. So let x � y 2 O(A � ); then since x is closed and y is open, there isa 2 A with x � a � y. Thus we have f�(x) � f�(a) = f(a) = f�(a) � f�(y),and we have shown that f� and f� agree on closed elements. By duality theymust agree on the open elements as well. Finally, it now follows that f� � f�

holds on A � since f� is the least isotone extension of f� � K(A � ) to all of A � .

Remark 3.2 Notice that since f� maps closed elements to closed elements, itfollows from this theorem that the same is true for f�. Similarly we concludethat f� maps open elements to open elements.

De�nition 3.3 An isotone map f : A ! B between DL's is said to be smoothprovided f� = f�.

De�nition 3.4 A map f : K ! L between complete lattices is said to be (Scott)continuous provided it preserves (up-)directed joins. The dual notion, that is, amap that preserves down-directed meets is said to be dually continuous.

Theorem 3.5 If f� is continuous, or f� is dually continuous, then f is smooth.

Proof For each u 2 A � , by de�nition f�(u) =Wff�(x) : u � x 2 K(A � )g.

Since the set D = fx : u � x 2 K(A � )g is directed, it follows that if f� iscontinuous, then f�(u) = f�(

WD) =

Wff�(x) : u � x 2 K(A � )g. But f� = f�

on closed elements, so f�(u) =Wff�(x) : u � x 2 K(A � )g = f�(u).

6

Theorem 3.6 If f is join preserving, then f� is completely join preserving,and thus continuous. Dually, if f is meet preserving, then f� is completelymeet preserving, and therefore dually continuous. In either case f is smooth.

Proof Let U be any subset of A � , withWU = u0, then

Wu2U f

�(u) � f�(u0)as f� is isotone. Now using the de�nition of f� and complete distributivity ofA � we get

_u2U

f�(u) =_u2U

0BB@ ^y2O(A� )u�y

f�(y)

1CCA=

^ : U!O(A� )

(u)�u

_u2U

f�( (u))

!

So we need to show that for each : U ! O(A � ) with (u) � u we haveWu2U f

�( (u)) � f�(u0). We haveWu2U f

�( (u)) =Wff(a) : a 2 A and

a � (u) for some u 2 Ug by the de�nition of f� on open elements. We nowclaim that the fact that f preserves joins implies that this last join is in fact equaltoWff(a) : a 2 A and a �

Wu2U (u)g. It is clear that

Wff(a) : a 2 A and

a � (u) for some u 2 Ug �Wff(a) : a 2 A and a �

Wu2U (u)g. For the other

inequality, take a0 2 A, a0 �Wu2U (u) =

Wfa 2 A : a � (u) for some u 2 Ug.

Since a0 is clopen, there are a1; a2; a3; : : : ; an 2 A and u1;u2; u3; : : : ; un 2 Uwith ai � (ui) and a0 � a1 _ a2 _ a3 _ : : : _ an . Now f being join preservingimplies

f (a0) � f (a1 _ a2 _ a3 _ : : : _ an)

= f(a1) _ f(a2) _ f(a3) _ : : : _ f(an)

�_ff(a) : a 2 A and a � (u) for some u 2 Ug

thus proving the other inequality. So_u2U

f�( (u)) =_ff(a) : a 2 A and a �

_u2U

(u)g

= f�(_u2U

(u))

� f�(u0)

and we can conclude thatWu2U f

�(u) � f�(u0), and consequently equalityholds, thus proving that f� is completely join-preserving.

As we proved in [?], the weaker condition of f being an operator impliesthat f� is completely join preserving in each coordinate, and thus, as it hasonly �nitely many coordinates, f� is continuous. However, f being an operatoris not enough to guarantee smoothness.

7

Example 3.7 This is an example of a binary operation on a DL which is bothan operator and a dual operator but f� 6= f�. Let A = f 1

n;� 1

n: n 2 Ng as

ordered in Q. De�ne f : A � A ! A by

f(a; b) =

8>><>>:1 if b > 0a if a > �b > 0b if 0 < a � �b�1 if a; b < 0

In order to check that f is meet and join preserving in both coordinates, we justneed to check that f is isotone in both coordinates since A is a chain. This canbe done by checking in all the possible cases for each coordinate. The canonicalextension A � of A is the chain obtained when adding two elements x and y toA ; both between the positive and the negative pieces of A and with y < x. It canbe checked that f�(x; y) = y < x = f�(x; y).

We now turn to the question of composability of � and �. This will play animportant role in determining whether or not equational properties are preservedby taking canonical extensions.

Theorem 3.8 Let g : A ! B , f : B ! C be isotone maps on DL's. Then

(fg)�� f�g� �

�f�g�

f�g�

�� f�g� � (fg)

�

with equality holding on closed elements and on open elements of A � .

Proof If we just prove that (fg)�� f�g� holds, then the rest follows by

Theorem 6 and duality. Furthermore, in order to prove that (fg)�� f�g� holds

on all of A � it is enough to show that it holds on K(A � ) since (fg)� is the leastisotone extension of (fg)� from K(A � ) to A � . For x 2 K(A � )

(fg)�(x) =

^ffg(a) : x � a 2 Ag

f�g�(x) =^ff(b) : g�(x) � b 2 Bg

Let b 2 B with g�(x) =Vfg(a) : x � a 2 Ag � b. Then, since b is clopen, there

are a1; : : : ; an 2 A, all greater than or equal to x, with g (a1) ^ : : : ^ g (an) �b. Now we let a = a1 ^ : : : ^ an, and we have a 2 A, x � a, and g (a) =g (a1 ^ : : : ^ an) � g (a1) ^ : : : ^ g (an) � b, and we conclude that (fg)

�(x) �

f�g�(x).

Example 3.9 We give an example to show that we don't always have equality inthe above theorem. Let B be a Boolean algebra, let f : B ! B be the function thatsends everything to zero, except 1 which gets sent to 1. Also, let g : B � B ! Bdenote the join operation in B . Then it is straight forward to check that (fg)� <f�g�, for example, (fg)

�(x; x0) = 0 < 1 = f�g�(x; x0) as soon as x is not in B .

Notice that f is meet preserving, and that g is both join preserving and a dualoperator.

8

Proposition 3.10 Let g : A ! B , f : B ! C be isotone maps on DL's. Iff� is continuous, then (fg)

�= f�g�. Dually, if f� is dually continuous, then

(fg)� = f�g�.

Proof Recall that

(fg)�(u) =

_f(fg)

�(x) : u � x 2 K(A � )g

=_ff�g�(x) : u � x 2 K(A � )g

and notice that D = fg�(x) : u � x 2 K(A � )g is a directed set in B� withWD = g�(u). Using the fact that f� is continuous, we get:

(fg)� (u) =_ff�(d) : d 2 Dg

= f��_

D�

= f� (g�(u))

As mentioned above, if f is an operator, then f� is continuous and then� is compositional. This is the essence of the proof of the canonical extensiontheorem for DLO's in [?]. However, a new result is that we also can guaranteecompositionality if the �rst (or inner) map is 'nice' { it needs to be very nicethough:

Theorem 3.11 Let g : A ! B , f : B ! C be isotone maps on DL's. If gis meet preserving, then (fg)

�= f�g�. Dually, if g is join preserving, then

(fg)� = f�g�.

Proof We know that (fg)�� f�g� with equality on closed elements and on

open elements holds in general. For u 2 A �

(fg)�(u) =

_ff�g�(x) : u � x 2 K(A � )g

f�g�(u) =_ff�(s) : g�(u) � s 2 K(B� )g

We show that for each s 2 K(B� )\ # g�(u), there is x 2 K(A � )\ # u withs � g�(x). Let s 2 K(B� )\ # g�(u), then

s � g�(u) = g�(u) =^fg�(y) : u � y 2 O(A � )g

That is, for each y 2 O(A � )\ " u, s � g�(y) =Wfg(a) : y � a 2 Ag and since

s is closed, there are a1; : : : ; an 2 A\ # y with s � g (a1) _ : : : _ g (an). We letay = a1 _ : : : _ an 2 A\ # y, then s � g (a1) _ : : : _ g (an) � g (a1 _ : : : _ an) =g (ay), so s �

Vfg(ay) : y 2 O(A � )\ " ug = g� (

Vfay : y 2 O(A � )\ " ug)

since g is meet preserving, making g� completely meet preserving. So we �nallylet x =

Vfay : y 2 O(A � )\ " ug �

Vfy : y 2 O(A � )\ " ug = u then

x 2 K(A � )\ # u and s � g�(x).For the proof of the next theorem we need a de�nition:

9

De�nition 3.12 Let fi : A i ! B i for i = 1; : : : ; n. Then we de�ne the superpo-sition of f1; : : : ; fn to be the function hf1; : : : ; fni : A 1�: : :�A n ! B 1�: : :�Bnde�ned by hf1; : : : ; fni (a1; : : : ; an) = (f1(a1); : : : ; fn(an)).

Remark 3.13 It is straightforward to check that hf1; : : : ; fni� = hf�1 ; : : : ; f

�n i

whenever the fi's are isotone.

Theorem 3.14 Let � be a compatibility type for DLMs, and let A be the cat-egory of all DLMs of compatibility type � with algebra homomorphisms of thecorresponding type, then � : A �! A is a functor. Moreover, � preserves andre ects injections and surjections.

Proof On objects � : Ob(A) �! Ob(A) is de�ned by �(A 0 ; (fi)i2I) =(A �0 ; (f

�i )i2I ), and on maps � : Hom(A ; B ) �! Hom(A � ; B� ) is de�ned by

�(h) = h�. Since the homomorphisms are meet preserving, � is compositional.The only thing to check is whether h� is a DLM homomorphism. Let fA :A "10 �: : :�A

"n0 ! A 0 and f

B : B "10 �: : :�B"n0 ! B 0 be corresponding operations of

A and B , respectively. Suppose h : A 0 ! B 0 with h Æ fA = fB Æ hh"1 ; : : : ; h"ni.Then h� Æ fA

�

= h� Æ�fA��

=�h Æ fA

��since h is join preserving, so that

h� is continuous. Thus h� Æ fA�

=�h Æ fA

��=�fB Æ hh"1 ; : : : ; h"ni

��. Now

hh"1 ; : : : ; h"ni : A "10 � : : :� A "n0 ! B "10 � : : :� B "n0 is a homomorphism since h isa homomorphism and since the dual of a homomorphism is again a homomor-phism. So hh"1 ; : : : ; h"ni is meet preserving and thus, by the previous theorem,�fB Æ hh"1 ; : : : ; h"ni

��=�fB��Æ (hh"1 ; : : : ; h"ni)

�= fB

�

Æ h(h"1)�; : : : ; (h"n)

�i.

Finally, since h"i is a homomorphism, it is smooth, and (h"i)�= (h�)

"i for eachi 2 f1; : : : ; ng. We have shown that h� Æ fA

�

= fB�

Æ h(h�)"1 ; : : : ; (h�)

"ni, thatis, h� is a DLM homomorphism.

We now need to show that � preserves injections and surjections. Thisis independent of any consideration involving the additional operations andwe consider h : A ! B , a DL homomorphism. First suppose h is injective.Let s; x 2 K(A � ) and suppose that h�(x) = h�(s). Let a 2 A\ " x, thenh(a) � h�(x) = h�(s) =

Vfh(b) : s � b 2 Ag. Since h(a) is clopen there

are b1; : : : ; bn 2 A\ " s with h(a) � h(b1) ^ : : : ^ h(bn) � h(b1 ^ : : : ^ bn).So b = b1 ^ : : : ^ bn � s and a � b since h is injective. That is, s �

Vfa :

x � a 2 Ag = x. By symmetry, s = x and h� is injective on the closedelements. Now let u; v 2 A � with h�(u) = h�(v). Let s 2 K(A � )\ # u,then h�(s) � h�(u) = h�(v) = h�(v) =

Vfh�(y) : v � y 2 O(A � )g. Thus,

for each y 2 O(A � )\ " v we have h�(s) � h�(y) =Wfh(a) : y � a 2 Ag.

Since s is closed, so is h�(s), and thus there are a1; : : : ; an 2 A\ # y withh�(s) � h(a1) _ : : : _ h(an) � h(a1 _ : : : _ an). For each y 2 O(A � )\ " v welet ay = a1 _ : : : _ an, then ay � y and h�(s) � h(ay). So h�(s) �

Vfh(ay) :

y 2 O(A � )\ " vg = h�(Vfay : y 2 O(A � )\ " vg) since h� is completely meet

preserving. Finally we let x =Vfay : y 2 O(A � )\ " vg 2 K(A � ), and x �

Vfy :

y 2 O(A � )\ " vg = v while h�(s) � h�(x). Now, since h� is injective on closedelements, it follows that s � x � v. That is, u =

Wfs : u � s 2 K(A � )g � v

and by symmetry u = v. We have shown that h� is injective.

10

We show that if h : A ! B is surjective, then so is h� . Let s 2 K(B� ), thens =

Vfb : s � b 2 Bg. For each b 2 B\ " s, there is ab 2 A with h(ab) = b.

Let x =Vfab : s � b 2 Bg. Since h is meet preserving, h� is completely meet-

preserving and thus h�(x) =Vfh(ab) : s � b 2 Bg =

Vfb : s � b 2 Bg = s, so

h� is onto K(B� ). Now let v 2 B� , then v =Wfs : v � s 2 K(B� )g. Since h�

is onto K(B� ), we have that for each s 2 K(B� )\ # v there is xs 2 K(A � ) withh�(xs) = s. Let u =

Wfxs : s 2 K(B� )\ # vg, then as h� is completely join

preserving, h�(u) =Wfh(xs) : s 2 K(B� )\ # vg =

Wfs : s 2 K(B� )\ # vg = v,

and h� is surjective.The fact that � re ects injections is clear; the fact that � re ects surjections

is also quite easy to show as there being a u 2 A � with h�(u) = b 2 B easilyis seen to imply that there also must be a 2 A with h(a) = b by compactnessarguments similar to the ones given above.

Remark 3.15 Notice that the image actually falls in the category DL+ of alldoubly algebraic distributive lattices (or equivalently, complete, completely dis-tributive lattices, join generated by their completely join irreducible elements)with complete homomorphisms. It is however not clear exactly what the imageobjects are. In particular, the underlying distributive lattices are exactly thoselattices O(P ) for which P is a spectral set, that is, a poset that is order isomor-phic to the set of prime ideals or �lters of a DL ordered by inclusion. No ordertheoretic description of these posets is known.

Our goal is to determine when properties, such as identities, are preservedwhen going to the canonical extension. The above theorem gives an immediatecorollary in this direction. We need the following terminology:

De�nition 3.16 A class K of (compatible) DLMs is canonical provided it isclosed under �. A single DLM A is canonical provided V(A ), the variety gen-erated by A , is canonical. A set of identities , or more generally a property, iscanonical provided its class of models is canonical.

Corollary 3.17 If K is a class of compatible DLMs that is closed under prod-ucts and that is canonical, then V(K) is canonical.

In the following section we explore the preservation of identities further.

4 Preservation of identities

We �rst sharpen the above corollary by using the fact that any product can beviewed as a Boolean product of ultraproducts. We �rst sketch this fact: LetB =

Qi2I B i , and let X be the Stone space of the Boolean algebra P(I). Then

each � 2 X is an ultra�lter of P(I). We let B� =Q

i2I B i=�, the ultraproductof fB igi2I with respect to the ultra�lter �. Then it is straight forward to checkthat B ,!

Q�2X B� where x 7! (x=�)�2X is a Boolean product.

We also need the following theorem which appeared in its topological formin [?].

11

Theorem 4.1 Let fA xgx2X be a family of compatible DLMs, X a Booleanspace, and A �

Qx2X A x a Boolean product of fA xgx2X . Then A �=

Qx2X A �x .

Proof We show that the DL reduct ofQ

x2X A �x satis�es the conditions forbeing the canonical extension of the DL reduct of A . We use the same notationforQ

x2X A �x , A and their DL reducts in order to simplify the notation.It is clear that

Qx2X A �x is a DL extension of A . It is also clear that

Qx2X A �x

is complete, completely distributive, and generated by J(Q

x2X A �x ) (which con-sists of those X-tuples that have a completely join irreducible in one coordinate,and 0's in all the others).

Let p; q 2 J(Q

x2X A �x ) with p � q. Then there are x; y 2 X with pz = 0for each z 6= x, and qz = 0 for each z 6= y. If x 6= y then there is a clopen setV � X with x =2 V and y 2 V . Since 0 = (0x)x2X and 1 = (1x)x2X belongto A , a = 0 � V { [ 1 � V also belongs to A by the patching property. It isnow clear that a satis�es p � a while q � a. On the other hand, if x = y thenpx; qx 2 J(A �x ) and px � qx, so there is ax 2 A x with px � ax and qx � ax.Since A is a subdirect product of fA xgx2X , there is a 2 A with ax = ax, andwe have p � a but q � a.

Finally we need to check the generalized compactness property. This isthe part that requires that the subdirect product be Boolean. Let K;L � Aand suppose

WK �

VL in

Qx2X A �x . Then for each x 2 X , there are �nite

subsets K 0x � K, L0x � L so that (

WK 0x)x � (

VL0x)x. For each x 2 X , let

Vx = fy 2 X : (WK 0x)y ^ (

VL0x)y = (

VL0x)yg then the Vx's are open and they

cover X . Since X is compact, it follows that there are x1; : : : ; xn 2 X so thatVx1 ; : : : ; Vxn cover X . Let K 0 = K 0

x1[ : : :[K 0

xnand L0 = L0x1 [ : : :[L

0xn, then

(WK 0)y � (

VL0)y for all y 2 X , that is,

WK 0 �

VL0 as desired.

In addition, for an n-ary order preserving operation f on A we must showthat for all u 2 A �n , f�(u) = (f�x (ux))x2X . We have

f�(u) =_f^ff(a) : s � a 2 Ang : u � s 2 K(A � )ng

whereas for x 2 X

f�x (ux) =_f^ff(b) : t � b 2 An

xg : ux � t 2 K(A �x )ng.

We show that for x 2 X and u 2 A � ,

ft 2 K(A �x ) : ux � tg = fsx : u � s 2 K(A � )g

and for x 2 X and s 2 K(A � ),

fb 2 Ax : sx � bg = fax : s � a 2 Ag.

Once this has been showed it follows that f� = (f�x )x2X . Let x 2 X andu 2 A � . Clearly if u � s 2 K(A � ), then ux � sx 2 K(A �x ). For the reverseinclusion, let ux � t 2 K(A �x ). Let s 2

Qy2X A �y be given by sx = t and

sy = 0 for y 6= x. Then clearly u � s, and we show that s 2 K(A � ). This

12

follows since s =Vfa � V

S0 � V { : a 2 A with ax � t and V is a clopen

neighborhood of x in Xg. Finally let x 2 X and s 2 K(A � ). If s � a 2 Athen certainly sx � ax 2 Ax. For the reverse inclusion, let sx � b 2 Ax, thenVfax : s � a 2 Ag = sx � b. So by compactness in A �x , there is a 2 A with

s � a and ax � b. Also there is c 2 A with cx = b. Now take d = a _ c. Thend 2 A and s � d and dx = b.

Theorem 4.2 If K is a class of compatible DLMs that is closed under ultra-products and canonical, then V(K) is also canonical.

Corollary 4.3 Any �nitely generated variety of compatible DLMs is canonical.

This applies to quite a few varieties, such as for example

Corollary 4.4 All the proper subvarieties of the variety of pseudo-complementedDL's are canonical.

Corollary 4.5 Varieties of de Morgan algebras are all canonical.

The second main result we present here is based on knowledge of the smooth-ness, continuity and dual continuity of the operations in the DL expansions.

For a single DLM, A , preservation of identities when going to the canonicalextension can be phrased in terms of the relationship between the two kernels:

Clon(A )% j

T(X) j& #

Clon(A � )

where X = fx1; : : : ; xng. The identities that hold in A , Id(A ), is exactly thekernel of the surjection T(X) ! Clon(A ), and the identities that hold in A � ,Id(A � ), is exactly the kernel of the surjection T(X) ! Clon(A � ). So sayingthat Id(A ) � Id(A � ) is equivalent to showing that there is a homomorphism

� : Clon(A ) ! Clon(A � )

making the diagram commute (on generators). In order to show this for a givenmap �, we need:

1. For each j = 1; : : : ; n, the map � commutes on the jth projection:��A

j

��=

�A�

j ;

2. For each i 2 I , and for h1; : : : ; hni 2 Clon(A ) we have�fAi Æ [h1; : : : ; hni ]

��=

fA�

i Æ [h�1 ; : : : ; h�ni]

13

Here [h1; : : : ; hni ] : An ! A ni is de�ned by

[h1; : : : ; hni ] (a) = (h1 (a) ; : : : ; hni (a)) :

On the isotone part of the clone we have a map �; namely our extension � (or� if appropriate). It is easy to check that the �rst condition above holds. Wealso have

�fA

i

��= fA

�

i for all the basic operations: it is true for the additionaloperations in the DL expansion by de�nition, and it is straight forward to checkfor meet and join. So, as long as the whole clone is isotone, we just need toknow that � is compositional, i.e.,

�fA

i Æ [h1; : : : ; hni ]��

=�fA

i

��Æ [h1; : : : ; hni ]

�

for each i 2 I and for h1; : : : ; hni 2 Clon(A ).Here we see how the proof works for DLO's: the whole clone is isotone, the

compositionality works because fA

i is an operator, that is,�fA

i

��is continuous

for each i 2 I . We can also use our knowledge about � and � on isotone mapsto get a theorem that encompasses some DLMs with maps that are antitone insome coordinates:

Theorem 4.6 If A is a DLM for which the extension of each basic operationis both continuous and dually continuous, then A is canonical.

To simplify the discussion we give this hypothesis on maps a name:

De�nition 4.7 An isotone map f : A ! B is said to be �-doubly continuousprovided f� is both continuous and dually continuous.

Notice that if f� is �-doubly continuous then f is smooth. Also, notice thatfor an isotone map to be �-doubly continuous it is suÆcient that one of thefollowing conditions hold:

1. f is an operator and is meet preserving;

2. f is a dual operator and is join preserving.

This is of course not at all necessary. It does however show that the meet andjoin operations of a DL always are �-doubly continuous, so that the hypothesisof the theorem does not restrict the class of DL reducts allowed. In addition,this clearly holds for operations that are either lattice homomorphisms from aproduct of copies of the underlying lattice and its dual to the underlying lattice.This holds, for example, for all Ockham algebras. Thus

Corollary 4.8 All varieties of Ockham algebras are canonical.

In addition, it can be shown that �-double continuity of the basic operationsis preserved by taking subalgebras, homomorphic images, and Boolean products.Thus, the basic operations of any DLM that lies in the variety generated by a�nite DLM (as in Corollary 21) are �-doubly continuous.

For the proof of the theorem we need two lemmas, the �rst of which followsfrom our results in the previous section.

14

Lemma 4.9 Let g : A ! B and f : B ! C be isotone maps that are �-doublycontinuous then the same holds true for the composition fg. Also, superpositionpreserves �-double continuity.

Proof This follows since continuity of f� implies (fg)�= f�g�, dual con-

tinuity of f� = f� implies(fg)�= f�g�, and continuity and dual continuity

is preserved by composition. The statement for superposition is clear sinceeverything is done coordinate-wise.

The additional complication in this setting is that not all of Clon(A ) maybe considered as isotone even with the use of dualization of coordinates. Forexample, if � is the pseudo-complementation in a pseudo-complemented DL,x 7! x _ x� is in general neither isotone nor antitone. However, every functionin Clon(A ) can be obtained in the form fA Æ

��A

i1; : : : ; �A

ik

�where f is obtained

from the basic operations using superposition and composition, and �A

ijis the

ijth projection of An onto A;where ij 2 f1; : : : ; ng for each j 2 f1; : : : ; kg (given

a term t, obtaining the term f corresponds to replacing repeated variables bydistinct, previously unused, variables).

The commutativity of the diagram corresponding to Id(A ) �Id(A � ) tells usthat we must extend � to Clon(A ) by letting

�fA Æ

��A

i1; : : : ; �A

ik

��= fA

�

Æ��A

�

i1; : : : ; �A

�

ik

�. But since fA is the composition of �-doubly continuous op-

erations that can all be viewed as isotone with matching domains and ranges,fA

�

=�fA��, so fA

�

Æ��A

�

i1; : : : ; �A

�

ik

�=�fA��Æ��A

�

i1; : : : ; �A

�

ik

�. What remains

to be seen is that this last step yields a well-de�ned extension of � to Clon(A ).For this purpose we need the following lemma:

Lemma 4.10 Let f : A "11 � : : : � A "

1k ! A and g : A "

21 � : : : � A "

2l ! A be

isotone maps, and suppose that for each a 2 An

f��A

i1(a) ; : : : ; �A

ik(a)�� g

��A

j1(a) ; : : : ; �A

jl(a)�,

where 1 � i1; : : : ; ik; j1; : : : ; jl � n

then for each u 2 (A � )n

f��A

�

i1(u) ; : : : ; �A

�

ik(u)�� g�

��A

�

j1(u) ; : : : ; �A

�

jl(u)�.

Before we prove this lemma, we see how this proves that � as extended toClon(A ) above is well-de�ned. Let t be an n-ary term, and suppose that forall a 2 An tA (a) = fA

��A

i1(a) ; : : : ; �A

ik(a)�= gA

��A

j1(a) ; : : : ; �A

jl(a)�where f

and g are terms without repeated variables. Then fA and gA can be consideredas isotone and by lemma 26 we have for all u 2 (A � )n

(fA )��A

�

i1(u) ; : : : ; �A

�

ik(u)�

� (gA )��A

�

j1(u) ; : : : ; �A

�

jl(u)�

and

(gA )��A

�

j1(u) ; : : : ; �A

�

jl(u)�

� (fA )��A

�

i1(u) ; : : : ; �A

�

ik(u)�.

15

Now since fA and gA are �-doubly continuous, it follows that

f (A� )��A

�

i1(u) ; : : : ; �A

�

ik(u)�= g(A

� )��A

�

j1(u) ; : : : ; �A

�

jl(u)�

for all u 2 (A � )n and the last step of the extension of � is well-de�ned. Now forthe proof of the lemma:

Proof Suppose the conclusion not to hold. That is, there are u 2 (A � )n andp 2 J(A � ) with

p � f��A

�

i1(u) ; : : : ; �A

�

ik(u)�

but

p � g��A

�

j1(u) ; : : : ; �A

�

jl(u)�

Suppose (WLOG) that "11; : : : ; "1k1

= 1, and "1k1+1; : : : ; "1k = �, that is, f is

isotone in the �rst k1 variables, and antitone in the last k2 = k � k1 variables.Similarly for g with l1 + l2 = l. Given a k-tuple ( l-tuple) we write it asv = (v1; v2) where vi is the ki-tuple (li-tuple) consisting of the �rst, or last ki(li) entries of v for i = 1; 2. There should be no confusion as to whether we aretalking about k-tuples or l-tuples as this will depend on whether we are talkingabout f or g.

For v 2 (A � )k we have

f�(v) = f�(v1; v2)

=_ff�(x; y) : v1 � x 2 K(A � )k1 ; v2 � y 2 O(A � )k2g

Now p � f��A

�

i1(u) ; : : : ; �A

�

ik(u)�means that there exist x 2 K(A � )k1 and

y 2 O(A � )k2 with p � f�(x; y) and x � (�A�

i1(u) ; : : : ; �A

�

ik1(u)), and y �

(�A�

ik1+1(u) ; : : : ; �A

�

ik(u)). Similarly for v 2 (A � )l we have

g�(v) = g�(v1; v2)

=^fg�(y; x) : v1 � y 2 O(A � )l1 ; v2 � x 2 K(A � )l2g

�A�

Now p � g��A

�

j1(u) ; : : : ; �A

�

jl(u)�means that there exist y0 2 O(A � )l1

and x0 2 K(A � )l2 with p � g�(y0; x0) and y0 � (�A�

j1(u) ; : : : ; �A

�

jl1(u)), and

x0 � (�A�

jl1+1(u) ; : : : ; �A

�

jl(u)). For each m between 1 and n we let

bxm =�_

fx� : i� = m; 1 � � � k1g�

_�_

fx0� : jl1+� = m; 1 � � � l2g�

and

bym =�^

fy� : ik1+� = m; 1 � � � k2g�

^�^

fy0� : j� = m; 1 � � � l1g�

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Then since x� � �A�

i�(u) and x0� � �A

�

jl1+�(u) we get bxm � um. Similarly

um � bym. Let x0 2 K(A � )k1 where (x0)� = bxm if and only if i� = m for1 � � � k1. Then

x � x0 � (�A�

i1(u) ; : : : ; �A

�

ik1(u))(1)

Similarly, we let y0 2 O(A � )k2 where (y0)� = bym if and only if i�+k1 = m for1 � � � k2. Then

y � y0 � (�A�

ik1+1(u) ; : : : ; �A

�

ik(u)).(2)

We go through the same constructions for g, y0, and x0 thus getting y00 2 O(A � )l1

where (y00)� = bym if and only if j� = m for 1 � � � l1 so that

y0 � y00 � (�A�

j1(u) ; : : : ; �A

�

jl1(u))(1')

and x00 2 K(A � )l2 where (x00)� = bxm if and only if j�+l1 = m for 1 � � � l2 sothat

x0 � x00 � (�A�

jl1+1(u) ; : : : ; �A

�

jl(u)).(2')

>From (1) and (2) we get f�(x; y) � f�(x0; y0) so that p � f�(x0; y0). From(1') and (2') we get g�(y0; x0) � g�(y00; x

00) so that p � g�(y00; x

00). Now for each

m between 1 and n, since bxm � um � bym, there is am 2 A with bxm � am � bym.We let a = (a1; : : : ; an), then by the de�nitions of x0, y0, y

00, x

00, and a we get

x0 � (�A

i1(a) ; : : : ; �A

ik1(a))

y0 � (�A

ik1+1(a) ; : : : ; �A

ik(a))

y00 � (�A

j1(a) ; : : : ; �A

jl1(a))

x00 � (�A

j11+1(a) ; : : : ; �A

jl(a))

so that

p � f�(x0; y0) � f(�A

i1(a0) ; : : : ; �

A

ik(a0))

and

p � g�(y00; x00) � g(�A

j1(a0) ; : : : ; �

A

jl(a0))

thus contradicting our hypothesis that for each a 2 An we have

f��A

i1(a) ; : : : ; �A

ik(a)�� g

��A

j1(a) ; : : : ; �A

jl(a)�.

5 Further work

This paper gives an idea of the types of general canonicity results one canobtain for distributive lattices with additional operations that are either order

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reversing or preserving in each coordinate. In ongoing work we have generalizedthe de�nition of canonical extensions of maps to encompass all maps on DLs,and through the introduction of two topologies on canonical extensions of DLs,we are able to greatly simplify the proofs of the results given here as well asgeneralize them. For example, one can show that all �nitely generated varietiesof algebras with a DL reduct are canonical. A comprehensive paper detailingthese new concepts and generalized results and giving applications to manyspeci�c situations of interest, such as residuation in one and two variables,quasi-identities etcetera is currently in preparation.

References

[1] Andr�eka, A., Givant, S., N�emeti, I., Perfect extensions and derived algebras,J. Symbolic Logic 60 (1995), no. 3, 775-796.

[2] Blok, W., J., The lattice of modal logics: an algebraic investigation, J.Symbolic Logic 45 (1980), 221-236.

[3] Blyth, T. S., Varlet, J. C., Ockham Algebras, Oxford University Press, 1994.

[4] Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S., Lukasiewicz-MoisilAlgebras, Annals of Discrete Mathematics 49, North-Holland, 1991.

[5] Celani, S., Jansana, R., A new semantic for positive modal logic, NotreDame J. of Formal Logic, 38 (1997), 1-18.

[6] Cignoli, R., Quanti�ers on distributive lattices, Discrete Mathematics 96(1991), 183-197.

[7] de Rijke, M., Venema, Y., Sahlqvist's theorem for Boolean algebras withoperators with an application to cylindric algebras, Studia Logica 54 (1995),no. 1, 61-78.

[8] Dunn, J. Michael, Positive modal logic, Studia Logica 55 (1995), no. 2,301{317.

[9] Gehrke, M., The order structure of Stone spaces and the TD-separationaxiom, Zeitschr. f. math. Logik und Grundlagen d. Math. 37 (1991), 5-15.

[10] Gehrke, M., J�onsson, B., Bounded distributive lattices with operators, Math.Japonica 40, no. 2 (1994), 207-215.

[11] Ghilardi, S., Meloni, G., Constructive canonicity in non-classical logics,Annals of Pure and Applied Logic, 86 (1997), 1-32.

[12] Goldblatt, R., Varieties of complex algebras, Annals of pure and AppliedLogic 44 (1989), 173-242.

[13] Goldblatt, R., The McKinsey axiom is not canonical, J. of Symbolic Logic,56 (1991), 554-562.

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[14] Jonsson, B., On the canonicity of Sahlquist identities, Studia Logica 53,no. 4,(1994), 473-491.

[15] J�onsson, B., Tarski, A., Boolean algebras with operators, I, Amer. J. Math.73 (1951), 891-939.

[16] Nilsson, J. Fischer, A Concept Object Algebra CA+�[:], in H. Kangassalo,J. Jaakkola, K. Hori, T. Kitahashi, eds., Information Modeling and Knowl-edge Bases IV, IOS Press, Amsterdam, 1993.

[17] Priestley, H., A., Representations of distributive lattices by means of orderedtopological spaces, Bull. London Math. Soc. 2 (1970), 186-190.

[18] Priestley, H. A., Natural dualities, Lattice theory and its applications(Darmstadt, 1991), 185{209, Res. Exp. Math., 23, Heldermann, Lemgo,1995.

[19] Priestley, H. A., Natural dualities for varieties of distributive lattices witha quanti�er, Algebraic methods in logic and computer science (Warsaw,1991), 291-310, Banach Center Publ., 28, Polish Acad. Sci., Warsaw, 1993.

[20] Priestley, H. A., Varieties of distributive lattices with unary operations. I.,J. Austral. Math. Soc. Ser. A 63 (1997), no. 2, 165{207.

[21] Priestley, H. A., Santos, R., Varieties of distributive lattices with unaryoperations. II. Portugal. Math. 55 (1998), no. 2, 135{166.

[22] Sahlqvist, H, Completeness and correspondence in the �rst and second ordersemantics for modal logic, Proceedings of the Third Scandinavian LogicSymposium (Univ. Uppsala, Uppsala 1973), 110-143, Stud. Logic Found.Math., 82, North-Holland, Amsterdam, 1975.

[23] Sofronie-Stokkermans, V., Duality and canonical extensions of bounded dis-tributive lattices with operators, and applications to the semantics of non-classical logics, I, II, to appear in Studia Logica.

[24] Stone, M., H., Topological representations of distributive lattices and Brow-erian logics, Casopis pest. Mat. 67 (1937), 1-25.

[25] Urquhart, A., Equational classes of distributive double p-algebras, AlgebraUniversalis, 14 (1982) 235-243.

[26] Venema, Y., Atom structures and Sahlqvist equations, Alg. Univ., 38

(1997), 185-199.

Mai Gehrke,New Mexico State University,Department of Mathematical Sciences,Las Cruces, NM88003,USA

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Work phone 505-646-4218Work fax 505-646-1064e-mail: [email protected]

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algebraic tools for modal logic -

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