C.R. Math. Rep. Acad. Sci. Canada - Vol. IX, No.6, December 1987 decembre

Memoir

The Amalgamation Property in LatticeTheory

G. Gratzer FRSC

1. Introduction. The AmalgamationProperty migrated to lattice theory from grouptheory via universal algebra. In this paper. Iwill try to introduce the general reader to thebasic ideas related to the Amalgamation Prop­erty in lattices. I will not attempt to survey allthe lattice theoretic results.

This paper is based on the invited ad­dress I gave on August 2. 1986 at the UniversalAlgebra and Lattice Theory Conference at theNational Institute of Health in Bethesda. Mary­land. Only the results with E. Fried concerningpasting are of more recent origin.

Special thanks are due to J. Berman. E.Fried. R. Padmanabhan. and G. H. Wenzel fortheir cr1t1cal comments on the draft of thispaper.

The research for this paper was supported bythe NSERC of Canada.

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274 G. Gratzer

2. Basic concepts. I shall assume that the reader Is famll1ar withthe basic concepts of lattice theory: lattices. diagrams., identities(equations). varieties (equational classes), see G. GrAtzer U) and (31.

In particular, we need two lattices:The lattice of Figure 1 will be denoted byN5. It is closely related to the modular

identity. A lattice L is called modular ifffor all x, y. z e L. z S x implies thatx 1\ (y V z) =Ix 1\ y) v z. A lattice is modulariff it does not contain Ns as a sublattice,

The class of all modular lattices will bedenoted M.

Figure 1

The lattice of Figure 2 win be denotedby M3. It Is closely related to thedistribut(ve identity.. A lattice L IsdtstrtbuUve iff for all x. y. z e L.x 1\ (y V z) = (x A y) vex A z). The class

of all distributive lattices will be de·noted byD. M3 is the smallest mod.·ular nondistdbut1ve lattice. A modularlattice L belongs to D iff it does notcontain M3 as a sublattlce.

Figure 2

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3. Definition. There are two ways to define the AmalgamationProperty: informally with "potatoes" and formally with a categorical di­agram.

Informal deflnttion:

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Figure 3

Let A and B be lattices. and let S bea common sublattice of A and B asshown on Figure 3 (A () B = Scan

be assumed without loss of gene­rality). The lattice L amalgamatesA and B over S iff L containssublattices A'. B', and S' such that Ais isomorphic to A', B to B', S to S',and S' holds the same position inA' (resp., B') as S in A (resp. B).

Figure 4

Figure 4 shows the lat­tice L as it amalgamatesA and B over S; notethat, in general, the in­tersection of A' and B'may be larger than S'. Ifthere is no danger ofconfusion. A', B'. S' willalso be denoted by A, B,S.

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The categorical definition follows:

S "A

1B

Figure 5

S "A

1 1B .. L

Figure 6

We are given two embeddlngs of the lat­tice S Into the lattices A and B. as In Fig­ure 5.

The lattice L amalgamates A and B over SIf there are embeddlngs of A and B Into Lsuch that the diagram of Figure 6 iscommutative.

We say that a class K of lattices has the Amalgamation Property

Iff every pair of lattices can be amalgamated over a cornmon sublattice.The class L of all lattices and the variety D have the AmalgamationProperty, but as we shall see below, M does not.

The Strong Amalgamation Property re-

quires that. In L. the sublattlces A and B Inter- <>sect In S. L has the Strong AmalgamationProperty. D does not. Indeed. consider thelattice of Figure 7. Let this lattice be both Aand B. and let S be the three element sublat-

tlce consisting of the elements marked by Figure 7solid circles.M3 shows that there Is a strong amalgamation of A and B over S In L. It

Is easily seen that there Is none In D: the middle filled element musthave more than one relative complement In any strong amalgamation,and this Is Impossible In a distributive lattice.

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4. Gluing and pasting. If L amalgamates A and Bover S. so does every lattice that contains L.

However. in general. there may be many lat-tices amalgamating A and B over S. none containingany of the other ones as sublattices. Consider thetriVial example of Figure 8. Let A and B be the latticeof Figure 8. S is the two-element lattice of black­filled elements. There are four ways to amalgamateA and B over S. as shown in Figure 9.

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Figure 8

Note that no lattice of Figure 9 can be embedded into anotherlattice of Figure 9 over S (a black-filled element has to be mapped intoa black-filled element).

Figure 9

Using some well-known results (see Section VI.2 of G. GrAtzer[3]), it is easy to show that there are finite lattices A and B. and acommon sublattlce S, such that there are countably infinite amalgama-

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tions of A and B over Stand no can be embedded Intoanother amalgamation preserving S.

Figure 10 Hlustrates one of theoldest forms of (due to R. P.Dilworth): S Is a dual idealo! A {If x, yeS.then x 1\ yeS and If xeS, yeA, and x S y,then y e SI and S is an ideal of B (if X, ye S.then .x v yeS and if xeS, y e B, and y S x,then ye $); then L = Au B andB over S. The partial order m Lis the naturalone: S retains its meaning in A and B: fora e A and b eB, as b iff a S sm A and s S b inB for some s e S.

10

If L glues A and B over S. then L A and B over S; infact. if any other lattice K amalgamates A and B over S. then L isfh""rni'\1"nhf" to} a sublattlce of K. This is the "most onecan get out of the Anlall~an!1ationPrt\npri\1

There are otherinstances where we getthis type of uniqueness.Let A and B be given asin Figure 11; in bothlattices S is the two~

element chain of solieelements. There isonly one minimal wayto amalgamate A and Bover S. as shown in

12.

11 Figure 12

Figure 13

Figure 14

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Again, any amalgamation containsthe unique lattice we obtain from L :: A u B.

This last example lllustrates another specialtype of amalgamation which we shall callpasting: Let L be a lattice. Let A, B, and S besublattices of L. A () B :: S, A u B :: L, see

Figure 13. Then L pastes A and B togetherover S, if every amalgamation of A and Bover S contains L as a sublattice. This con­cept was investigated in A. Slavik U J: seealso G. GrAtzer {3}, Exercise 12 of SectionVA. (A. Slavik used the term "A-decom­posable".)

In a more categorical version: Let L be alattice. Let A. B, S be sublattlces of L. A () B =S,A u B = L. Let fA and fa be the embeddings of A

and B, respectively, Into L. Then L pastes Aand B together over S If whenever ~ and ga

are embeddings of A and B into a lattice K sat­isfying x~ :: xga for all XES, then there Is anembedding) h of L Into K Satisfying fAh = gAand fah:: ga, see Figure 14.

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5. Varieties with the Amalgamation Property. When the Amal­gamation Property was first Introduced, three lattice varieties wereknown to have it: T (the trivial variety of one element lattices), D (thevariety of all distributive lattices), and L (the variety of all lattices).

It Is worth recalling why the Amalgamation Property holds for L.Let A and B be given with the common sublattice S as In Figure 3,A () B = S. By B. J6nsson {I}. we can define on p:: A u B a partial order­Ing relation S as follows:

(I) For x, yEA (and for x, y e B), x S y in P Iff x S y In A (resp..x Syin B).

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(ti) For x e A and ye B, x S y In P iff there ex1sts an s e S withx s s in A and sS y in B: and dually, for y S x .

The MacNeille completion pc of P (the lattice of an subsets X ofPwith the property that X contains all the elements that are lowerbounds of the set of all upper bounds of Xl is a lattice containing P as asubposet (p e P Is identified with {x I x e P and x S pH and all joins andmeets that exist in P are preserved In pc (see, e.g., G. Gratzer [SJ).Thus Pc amalgamates A and B over S.

Since the early fifties. many attempts have been made to findlattice varieties having the Property, or to prove thatthere are nOne other than T. D. and L. In particular. there was greatinterest whether M (the varl.ety of all modular lattices) has theAmalgamation Property. Lists of unsolved problems invariably includedone or more problems relating to the Amalgamation Property (see.e.g., B. J6nsson(3) and G. Gratzer IS)).

The problem was finally resolved the following

Theorem. The only lattice varieties having the Ama1samationPr0eel"!Y are T, D, and L.

The solution came in two steps, in 1972 and 1983.In 1972. G. Gratzer, B. J.6nsson. and H. Lakser made the follow­

ingsimple observation: let V be a variety of modular lattices having theAmalgamation Property. V::> D. Then any lattice. L. In V can be em­bedded into a projective geometry (that is. into the lattice of all sub­spaces of a projective space).

Indeed. to start wUh. we can assume that L has a zero and a unit.Now recall from Sectlon2 that Ms must be in V. The crucial step isthe fonowing: We embed L into a lattice L' in V such that every a e L' Iscontained In a sublattlce of L' isomorphic to Ms.

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To accomplish this. take anya e L that is not contained in a sublat­tice isomo,rphic to MS. AmalgamateMS with L over the three-elementchain in V as shown in Figure 15 {thisfigure shows the A. B, and S of theamalgamation, not the resulting lat­tice of V. Proceed like this with allthe elements of L; by forming a directlimit we obtain a lattice Ll In V Inwhich every element of L Is containedIn a sublattice Isomorphic to MS. Figure 15Repeating this process (l) times, andagain forming a direct limit. we obtain the lattice L'.

This L' Is a simple complemented modular lattice. so by a resultof O. Frink UI. L' can be embedded In a projective geometry. It nowimmediately fonows that M: does not have the Amalgamation Property:indeed. In this projective geometry. the lattice theoretic version ofDesargues' Theorem, the arguestan fdenttty. holds, while there aremodular lattices L in which it does not.

It takes some more work to get our fun result: T and D are theonly varieties of modular lattices haVing the Amalgamation Property.For the complete proof. the reader is referred to our paper.

It took another dozen years or so for A. Day and J. Jezek to re­solve the nonmodular case. They built on earlier attempts of J. Jezekand A. Slavik (IJ and of A. Slavik UI who proved that any nonmodularlattice variety haVing the Amalgamation Property has to contain a largeclass of lattices (the "primitive lattices"). They used the concept ofpasting which proved to be cruCial.

Their argument uses an earlier result of A. Day (I}. Let L be a lat­tice. and let I = (a. bl be an interval of L. We split the interval I in thelattice L to obtain L(IJ. The lattice L(t) consists of L - t and I x C2 (C2 isthe two-element chain with elements 0 and 1) With the natural partialordering.

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282G. Gratzer

To visualIZe ft. think ·of L splitinto four parts as in Figure 16:U is the set of elements in L -I with an upper bound in (:V

Is the set of elements in L - I

with a lower bound in I: X IsL (I u U u V). Then LIt] is asdepicted In Figure 17. 10 is({l. 0) liE n and It Is{(t. 1) I f Iii n. In LfII. x S (i, 0)iff x S i in L: (:I. 1) S Yiff I S Y In

Figure 16 L 17

Now result is the fonowing: if we start with the free dts~

trtbutive lattice Fn(3)on3 generators and we keep intervalsso that every interval eventuaUygets split.. the inverse Umit of thissequence oflaWces cont.a1ns the free lattice FLf3} on 3 generators as asublatUce. So if V Is a variety closed under the of an interval.then V is generated by FL(3), and so V= L.

If e Is a congruence relaUon of L under which I is a con~

gruenceclass. then obviously LUJ is a subdtrect of Land(L/9)(I/91, where 1/6 Is a Thus If L Is a lattice in a non~

modular lattice varlety V and LIe Is C2 x C2. then (L/eml611s Ns , soLlI) is in V.

For the remainder of the argument: For a nonmodular latticevartety V having the Amalgamation Property,an Ingenious Induction onthe number of elements tn Lconcludes that Lm Is in V withoutto assume that I Is a congruence class. The reader is referred to thepaper and JeZek for the details.

6. The Amalgamation Class. In G. GrAtzer and H. Lakser Ill.. weInvestigate which va.rtetles of pseudocomplemented dlstrtbutive lat~

Uces have the Amalgamation Property. The result Is ratherpollntlng: only the three smaDest varteties and the one have the

283Memoir

Amalgamation Property. It seems more appropriate to investigate towhat extent a variety has the Amalgamation Property. We do this byintroducing the concept of the amalgamation class:

For a variety V. the amalgamation class oIv. Amal(V), is the classof all those members 5 of V for which for any A. B e V. if 5. A. B are asin Figure 3. then there is an amalgamation L of A and B over 5 in V. asin Figure 4.

Obviously. V has the Amalgamation Property iff Amal(V) = V. InG. GrAtzer and H. Lakser (l]. we have given a pretty effective deSCrip­tion of the finite members of Amal(V) for any proper variety V of pseu­docomplemented distributive lattices.

For lattice varieties. we have beenmuch less successful in saying anythingabout Amal(V). In G. GrAtzer. B. Jonsson.and H. Lakser (I]. we have proved that thetwo-element chain. C2. is not in Amal(M).The proof is very geometric: wereC2 e Amal(M). it would follow that everymodular lattice L is a sublattice of a modularlattice L' in which L lies in an edge of an M3

as shown in Figure 18. But then we wouldhave the extra "two dimensions" to imitate Figure 18the proof of Desargues' Theorem in L' in itslattice theoretic form (the arguesian identi-ty). a contradiction.

Obviously. the one-element lattice is· in Amal(V) for any latticevariety V. We have been unable to find any other member of Amal(M).50 it was a great surprise to us when M. Yasuhara (1] proved that forany variety V. Amal(V) is coflnal with V (that is. every lattice in V canbe embedded into a lattice in Amal(V)). Unfortunately. the proof doesnot reveal how to find a member of Amal(M); we still have not seen anynontrivial member of Amal(M).

In general. not much is known about Amal(V). In G. GrAtzer (21.I point out that if C2 e Amal(V). then V is a Join-irreducible element of

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the lattice of lattice varieties, J, Berm.an (I) constructs a nonmodularva.riety V with C2 E Amal(V): V is by a sequence of latticespa!ttelme~d after Figure 12, a,Jonsson (41 describes all finite membersof Amal(V) for finitely lattice varieties; in he ob~

tains a nice description of all the finite members of Amal(N:s l. whereNs is the variety generated by No, .

7. The concept of pasting (see Sections 4 and 5) hasbeen in A. Sla.vi'k Ill. A. and J, JeZek UI. and in twopapers E. Fried and G. Gratzer ({3] and (4)). Here is the main char-acterizaUon th.eorem of pasting (E, Fried and G. Gratzer (4)):

In the finite case, (11) can be the condition (A. Dayand J. Je2eklll):

(U') For s E S, an the covers ofs in L are in A or all are in B: and

Check (1) and (U') in thelattlces Sand 10.

The main of the characterization is the result (E.Fried and G. Gratzer (4]): M and D are closed under pasting.

8. Concluding comments. There are some interesting connec­tions between the Amalgamation Property and products of varieties.

If V andW are lattice varieties. their VoW consists ofall lattices L for Which there is a congru.ence relation e (0 alla-classes of L are in V; on Lla is in W. In VoW is not a vari-

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ety; however. R(V 0 W) (the class of all homomorphic images of mem­bers ofVoW) always is.

In G. GrAtzer and D. Kelly [21. we prove that if V is closed undergluing then V 0 D is a variety. In fact. based on a result of R. Freese [1I.we present contlnuumly many modular lattice varieties closed undergluing.

R. N. McKenzie (in R. McKenzie and D. Hobby [1n proves thatDoW and MoW are always varieties. A very deep result of T. Harrison

(1) proves that there are no nonmodular varieties (other than L) forwhich a similar result holds: if V is a nonmodular lattice variety suchthat V 0 D is a variety. then V =L; in fact. he proves that if VoW is a

variety for some nontrivial modular lattice variety W. then V =L;.Since Harrison's result uses only that V is closed under finite

gluing. by the result of Gra.tzer and Kelly. we obtain the followingsharp form of the result of Day and JeZek: If V is a nonmodular latticevariety closed under finite gluing. then V = L.

If VoW is not a variety. then R(V 0 W) is; however. the membersof R(V 0 W) cannot be described by congruences so neatly. R. N.

McKenzie conjectured that an "amalgamated" version of congruences.the so called tolerance relations. will serve to describe R(V 0 W). A

tolerance relation T on a lattice L Is defined as a reflexive and sym­metric binary relation having the substitution property. A maximal T­connected subset of L Is a T-block. The quotient lattice LIT consists ofthe T-blocks with the natural ordering. R. N. McKenzie conjecturedthat a lattice K belongs to the variety generated by VoW iff there is a

tolerance relation T on K satisfying (I) all T-classes of L are in V: (li)

LIT is in W. The following result is proved in E. Fried and G. Gratzer[11:

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286G. Gratzer

is in D.

Figure 19In E.Fried and G. Gratzer (2J. we introduced a of

the concept of congruence relation. called general.i.Zed congruence.With this new concept. the of McKenzie's conJecture can be

Lattices fonn an important of congruence distributiveuniversal algebras: many results mentioned in this paper have ana~

logues in the theory of congruence distributive varieties. Many valuablecontributions have been made to this field by c .. Bergman. B. M.B. J6nsson. and others. The interested reader may start With C.Belrs:!Itlan III as an introduction to this field.

Final note: I have just received two manuscripts containing fe~

suits relevant to this survey. The first is by C. Bergman proving thatAmal(Malls not axiomatizable. The second is A. Day and Ch. Herr~mann in which the last theorem of this section is for Ns 0 D.

Memoir

References

C.Bergman[l) Amalgamation classes of some distributive varieties. AlgebraUniversalis 20 (1985), 143 - 146.

J.Berman[1) Interval lattices and the amalgamation property. Algebra Uni­versalis 12 (1981), 360 - 375.

A Day[1) CharacteriZation of finite lattices that are bounded homomor­phiC images of sublattices of free lattices. Canadian J. Math. 3 1

(1979), 69 - 78.A Day and J.Jezek

[1) The amalgamation property for varieties of lattices. Mathe­matics Report # 1 - 83, Lakehead University.

R. Freese[I) Projective geometries as projective modular lattices. Trans.Arner. Math. Soc. 251 (1979). 329 - 342.

E. Fried and G. GrAtzer[1) Notes on Tolerance Relations of Lattices: A Conjecture of R. N.McKenzie. Manuscript submitted to FestschrY£. Journal of Pureand Applied Algebra..(2) Generalized Congruences and Products of Lattice Varieties.Manuscript submitted to Acta Sci. Math. (SzegedJ.(3) Pasting and modular lattices. Abstract: Notices Amer. Math.SOC.87f-06-209. Manuscript.(4) Pasting infinite lattices. Abstract. Notices Amer. Math. Soc.87f-06-193. Manuscript submitted to Journal ofAutralian Math.Soc.

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288G. Graber

E. Flied. G. GriUzer. and H. LakserII} Projective geometries as cover preservingsublattices. Manu~

script submitted to Algebra UniversalistO. Frink

{I) Complemented modular lattices and projective spaces of infi~

nUe dimension. 1Tans. Amer. Math. Soc. 60 (1946), 452 ~ 467.G, Grlitzer

(1) Lattice Theory: First Concepts a.M Distributive Lattices. W. H,Freeman and Company. San Francisco. Calif.• 1971.(2] On the Amalgamation Property. Notices Amer:. Math. Soc, 23(1976), A-268.(3) General Lattice Pure and Applied Mathematics series.,AcademiC Press. New York. N.Y.• 1978: Birkhauser Verlag. Mate~

matische Relhe. Band 52. Basel, 1978.G, Gratzer. B. Jonsson. and H. Lakser

(1) The amalgamation property in equational classes of modularI.attices. Pac(ftc J, Math. 46 (1913), 507 - 524.

G. Gratzer and D.III A survey of products of lattice varieties. ColloqUia Mathemati.caSocietatis Jdnos Bolyat.. 33. Universal Algebra. {Hun­gary}, 1980. 457 - 412.(21 The lattice variety DoD. Acta Sct. Math, (Szeged). 61 (1987).

131 Products of lattice varieties. Algebra Untversalis 21 (1985).33 - 45.

G. Gratzer and H, LakserIl) The structure of pseudocomplemented distributive lattices. n.Congruence extension and amalgamation. 1Tans. Amer. Math. Soc.156 (1971). 343 358.

G, Gratzer and G. H. Wenzel(11 Notes on Tolerance Relations of Lattices: Some Basic Observa­tions. Manuscript submitted to Acta Set. Math. (S2;eg<~d).

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T. A. Harrison(1) P a nontrivial modular lattice variety and V a nonmodular lat­tice va,riety such that V 0 P is a variety implies V is the variety of

alliattlces. Ph.D. Thesis, University of Hawatt, 1985.J. Jezek and A. Slavik

(1) Primitive lattices. Czech. Math. Journal 29 (104) (1979).595 - 634.

B.Jonsson(1) Universal relational systems. Math. Scand. 4 (1956), 193 - 208.(2) Extensions of relational structures. Theory of Models(Proceedings 1963 Intemat. Sympos. Berkeley), 146 - 157. NorthHolland. Amsterdam. 1965.(3) Varieties of lattices: Some open problems. ColloqUia Mathema­tlea Societatis Janos B6lyai. 29. Universal Algebra, Esztergom(Hungary). 1977. 421 - 436.(4) Amalgamation in small varieties of lattices. Festschrift. Journal

ofPure and Applied Algebra. to appear.

R. McKenzie and D. Hobby(1) The structure of finite algebras (tame congruence theory).Manuscript, 1985.

A. Slavik(1) A note on the amalgamation property in lattice varieties.Comm. Math. Univ. Carolinae 21 (1980),473 - 478.

M. Yasuhara(1) The Amalgamation Property, the Universal-HomogeneousModels, and the Generic Models. Math. Scand. 34 (1974), 5 - 36.

Received Sept. 15. 1987 Department of MathematicsUniversity of ManitobaWinnipeg, ManitobaR3T 2N2