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G. Grätzer and J. Sichler, Free Decompositions of a Lattice

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G. Grätzer and J. Sichler, Free Decompositions of a Lattice Can. J. Math., VoL XXVII, No.2, 1975, pp. 276-285 FREE DECOMPOSITIONS OF A LATTICE G. GRATZER AND J. SICHLER 1. Introduction. Two basic questions have been raised for free products of lattices: 1, Do any two free products have a common refinement? 2. Can every lattice be decomposed into a free product of freely indecom­ posable lattices? Both questions have been around for some time and attempts at solving them were made especially after the Structure Theorem for Free Products was discovered (see G. Gratzer, H. Lasker, and C. R. Platt (3]). Partial answer to question one was supplied in A. Kostinsky (7]. In this paper we answer both questions. Our basic observation is that the proper framework for these results is the theory of free K-products, that is, free products in an arbitrary equational class K of lattices. This approach has the advantage that the answers are supplied for all equational classes of lattices. It is especially simple to answer Question 1 for equational classes having a special property (J) (see §2) describing certain sublattices of free products. We also show that many equational classes fail to have (J). 2. Results. An equational class K of lattices is called trivial if it is theclass of aU one element lattices; otherwise it is nontrivial. THEOREM 1, Let K be a nontrivial equational class of lattices. For any L in K, any two representations of L as a free K-product have a common refinement. It is easy to state what the common refinement is. To simplify our notation, we agree that we use the "internal" definition of free K-product, that is, the free K-factors are considered as sublattices of the free K-products. To further simplify our notation, let us agree that if L is a lattice and (A tli E I) is a family of subsets of L where each A t is either a sublattice or the empty set, then we say that L is a free K-product of (A iii E I) if and only if L is a free K-product of (A iii E I and At r£ 0). THEOREM I'. Let L be a free K-product of (A iii E I) and of (BAj E J). Then L is a free K-product of (A i II B iIi E I, j E J) and, for i E I, A i is a free K-product of (A ill Bjjj E J), and, for j E J, B j is a free K-product of (A t II Bjli E I). Theorem I' has many important consequences. Received August 20, 1973. This research was supported by the National Research Council of Canada. 276 FREE DECOMPOSITIONS 277 COROLLARY 1. If L is a free K-product of A and B, and also of A and C, then B = C. Corollary 1 is the Cancellation Property for Free Products. Observe that in Corollary 1 we assumed that A, B, and Care sublattices of L and we concluded the unexpectedly strong B = C. It is natural to ask whether the isomorphism of the free K-products would imply B isomorphic to C? To state it more precisely, let L 1, i = 1,2, be the free K-products of A 1 and B 1; does L 1 ::: L 2 and A 1 ::: A 2 imply that B 1 ::: B 2? It was remarked by S. Comer that no result of this sort can hold for an equational class K; we can always choose L 1 ::: L 2 ::: A 1 """ A 2 ::: FKOC O) , B 1 = FK(1), and B 2 = FK(2). (FK(m) is the free lattice over K on m generators). S. Comer raised the question what happens if A 1 is finitely generated. COROLLARY 2. LetA be afinitely generated lattice in K. If L 1, i = 1,2, is a free K-product ofA and B 1, and L 1 ::: L 2, then B 1 ::: B 2• Call a lattice L in K freely K-indecomposable if and only if L cannot be represented as a free K-product of two lattices. A result, related to Corollary 2, is the following: COROLLARY 3. Let A be a freely K-indecomposable lattice in K. If L 1, i = 1,2, is a free K-product of A and B 1, and L 1 ::: L 2, then B 1 ::: B 2• The standard consequences of Theorem 1 are as follows: COROLLARY 4. If L is a free K-product of A and B, and of A 1 and Bl, then A C A 1 implies that BB 1• COROLLARY 5. Let L be the free K-product of (A 11i E I) and of (BJlj E I). Assume that all A hiE I, and B 11 j E I, are freely K-indecomposable. Then there is a bijection tp between I and I such that A 1 = B 1'P for all i E I. In other words, decomposition into freely K-indecomposable components is absolutely unique. COROLLARY 6. Let L be the free K-product of (A 1li E I) and of (BAj E I). Assume that each A 1, i E I, is freely K-indecomposable. Then there is a partition (IAj E I) of I into nonvoid blocks such that, for each j E I, B; is the free K­ product of (A 1li E I;). The next result shows that there are many lattices to which Corollaries 5 and 6 do not apply. THEOREM 2. Let K be a nontrivial equational class of lattices. Then for each infinite cardinal m, there are 2m pairwise nonisomorphic lattices of cardinality m in K that cannot be represented as a free K-product of freely K-indecomposable lattices. 278 G. GRATZER AND J. SICHLER Another way of stating Theorem 2 is that the Common Refinement Property does not apply to infinitely many decompositions. The proof of Theorem l' is especially simple if the equational class K satis­ fies the following property: (]) If L is a free K-product of the (L.li E 1), A ( is a sublattice of L( for i E I, and A is the sublattice of L generated by U (A (Ii E I), then A is a free K-product of (A (Ii E I). Observe that Theorem l' is a special case of (]); it requires (J) to hold provided that A is a free K-factor. (]) was proved by B. J6nsson [5] for any equational class K having the Amalgamation Property. One can ask whether (]) holds for all equational classes. The following result shows that it is not the case; in fact, it provides 2No equational classes failing (]). Recall that an equational class is arguesian (see [4]) if it satisfies a special identity in six variables which reflect the Desargues Theorem for the lattice of all sublattices of a projective space. THEOREM 3. Let K be an equational class of modular lattices. If K satisfies property (]), then K is arguesian. Theorem 3 yields 2N o equational classes of modular lattices failing (]). A related result can be found in B. J6nsson [6] in which it is proved that if K is an equational class of modular lattices with the property that every sub­ directly irreducible member of K has dimension at most n for some fixed integer n, the K fails (]). 3. Proof of Theorem 1. Let L be a free K-product of (A (Ii E I). We assume that the A tare sublattices of L. We denote by LO the lattice obtained from L by adjoining a new zero, denoted by O. Observe that if L E K, then so is LO. So for every i E I we can consider the homomorphisms 'Pt determined by a'Pt = a for a EAt; a'P( = 0 for a E A j , j ~ i, j E I. For a E L we will use the notation: and call aA i the lower cover of a in A t. It follows from the definition that and if aAi E A (, then aAi is the largest element of A t below a (see B. J6nsson [6]). Now let L be also the free K-product of (B jlj E J). Take a E A (. We claim that FREE DECOMPOSITIONS 279 Since L is generated by U (Bjlj E J) we can represent a in the form where p is an (nl + + nk)-ary lattice polynomial, jl, ... , jk E J, and bj;,m E B j; for i = 1, , k, 1 ;;;i m ;;;i nl' Computing the lower AI-covers and observing that aA; = a we obtain a = p((bjl,l)A;, ... , (bjk,nk)A;)' We can assume, without any loss of generality, that j = jl. Forming lower B rcovers we get aBj = P(bjl,l' ... , bj"nl> 0, ... , 0, ... ,0), and aBj = (aA;)Bj = p(((bjl,l)A;)Bj, ... , ((bjk,nk)A;)B;)' Observe, however, that for any r ~ 1 and 1 ;;;i t ;;;i nn so aBj = P(bjl,l' ... ,bjl,nl> 0, ... ,0) = p(((bjl,l)A;)Bj, ... , ((bil,nl)A;)Bj, 0, ... , 0). Since p is isotone and for all 1 ;;;i t ;;;i nl bj,,1 ~ (bjl,I)A; ~ ((bji,l)A;)Bj, we obtain aBj = P(bjl,l, . .. ,bj"nl> 0, ... ,0) ~ p((bjl,l)A;,' .. , (bjl,n,)A;, 0, ... ,0) and so aBj = p((bjl,l)A;,"" (bjl,nl)A;, 0, ... ,0) E Al U {Ol. By definition, aBj E BjU {OJ, hence aBj E (A I n B j) U {Ol, as claimed.
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