COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI SZEGED (HUNGARY), 1980.

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1 COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 33. CONTRIBUTIONS TO LATTICE THEORY SZEGED (HUNGARY), A SURVEY OF PRODUCTS OF LATTICE VARIETIES G. GRATZER - D. KELLY Let y and Wbe varieties of lattices. The ppoduat of y and ~,denoted by y 0 ~, consists of all lattices L for which there is a congruence e on L with every class of e in y and LIe in W. Observe that y 0 ~ contains both y and ~. Moreover, any simple lattice in VoW is already in y or w. This concept was studied extensively for varieties of groups by H. Neumann [30J and later generalized for quasivarieties of universal algebras by A. I. Malfcev [29J. The lattices in Figures 1, 2 and 4 are in!? 0 Q, where Q is the variety of distributive lattices. All the nontrivial classes of a congruence relation are shown in each of these diagrams. Clearly, each class is distributive, and so is the quotient. Let N S be the S-element nonmodular lattice and let N S be the variety it generates. It is easily shown that DoD contains all but two of the varieties convering ~5 (see B. Jonsson and I. Rival [25]). Verification of membership in a product of varie- This research was supported by the NSERC of Canada

2 ties can be simplified by using the results of G. Graczynska [11]. The reader is referred to our paper [15] for further details. Henceforth, this reference is to be understood when no other one is given. H. Neumann [30] showed that the product of two group varieties is again a variety. The same is true for Brouwerian algebras (see P. Kohler [26]). However, the product of two lattice varieties is not necessarily a variety. On the other hand, we show in [16] that gog is a variety. The class QOQ was utilized by S. V. Polin [31]. He did not show that it is a variety, but he did prove that the variety generated by QoQ is distinct from the variety ~ of all lattices. A. I. Mal'cev [29] showed that yo~ is always a quasivariety. In other words, it is closed under the formation of sublattices, direct products, and ultraproducts. Alternatively, it is defined by (equational) implications. Consequently, Yo~ is a variety iff it is closed under homomorphic images. We show that Yo~ is always closed under the formation of ideal lattices, a property not enjoyed by quasivarieties of lattices in general. We now consider a lattice construction that was introduced by R. P. Dilworth [7]. Let L be a lattice. If there is an ideal A of L and a dual ideal B of L such that~ L = AUB, and AnB + Q, then we say that L is obtained by gluing A and B over AnB. Observe that C = AnB is a dual ideal in A and an ideal in B. Moreover, the lattice L is uniquely determined by the lattices A and B and the embeddings of C in A and B. (This construction has been generalized by C. Herrmann [22])

3 For a class ~ of lattices, ~(~) is the class of homomorphic images of lattices in ~ and g(~) is the class of all lattices that are obtained (up to isomorphism) by gluing pairs of lattices in ~. A variety y is czosed under gzuing if ~(y) S y. Let G denote the class of all such varieties. R. P. Dilworth [7] showed that ~, the variety of all modular lattices, is in G. This result was later applied by M. Hall and R. P. Dilworth [21]. It is also known that D E G. For any lattice variety y, it is obvious that Consequently, if yo!? is a variety, then G(V) C VoD ,.., Thus, a necessary condition for yo Q to be a variety is that y contain every simple lattice in G(V). This condition is significant because it is easy to construct simple lattices that belong to G(y): gluing two simple lattices over a nontrivial lattice yields a simple lattice. Let y be a nondistributive variety that is generated by a finite lattice. We shall show that yo!? is not a variety. Suppose, to the contrary, that it is. The lattice of Figure 5 is in ~S o!? and has the lattice S Figure 6 as a must lie in y whenever N S E must contain a homomorphic image. Since S is simple, it y. It now follows that y simple lattice with more than two elements. By Jonsson's Lemma [24], there is a finite bound on the size of simple lattices in y. Let L be a of simple lattice of maximum size in y. Since y cannot contain the simple lattice obtained gluing L to itself over a prime interval, V 0 D is not a variety

4 Let y and Wbe nontrivial lattice varieties that do not contain the lattice S of Figure 6. If we also assume that N5" E y., then it. follows from the previos paragraph that VoW is not a variety" In particular, (Q (I Q) 0!? is not a variety" Let Wbe a nondistributive variety of modular lattices, and y = ~VN5" By J6nsson's Lemma, S ~ V. By chasing Was in K. A. Baker [1], we can construct continuumly many varieties y. for which Y(I!? is not a variety" the quasivarieties V 0 D are all distinct. To membership in of varieties, we make the followlng definitlon: for a lattice L and a quasivariety ~, let 8(L,~) be the (unique) smallest congruence e on L such that LIe E ~. The quotient LI8(L,y) is the maximum homomorphic image of L that lies in y.. In terms of y and W, there Is an easy test whether a lattice L is in yo LEy (I Wiff every class of B(L,!) is in y. (This definition and its application are due to A. I. Mal'cev [29]. ) The congruences.shown In 1, 2, 4 and 5 are all of the form B(L,Q). The infinite lattice of Figure 1 11es in Q(I!? and it is generated by 4 elements. Consequently, QoQ is not locally flnite. (An additlonal property of the lattice in Figure 1 is that it has only two elements in Its max Imum distributive homomorphic image.) We denote an n-element chain by Let L = 02'*C 2 ' the free product of with itself. Since L is in!? 0!?, L is also the D 0 product of with However, C *C is not in PO!? (See [13], (32), or [33) for 4 1 diagrams of' 02*02 and * 1,,) Let y and Nbe varieties with Y2 N. Let L be the free y-lattice by the set X" In symbols, L =

5 = FV(X). If e ;;; e(l,w), then L/e is isomorphic to FW(X). SimIlar results hold for free products and lattices~completely freely generated by posets. The correctness of Figure 2 follows from these observations. (This lattice has 19 elements in its maximum distributive homomorphic image.) Similarly, starting from the completely free lattice generated by the poset H of Figure 3 (given in I. Rival and R. Wille [32]), we obtain the completely Q 0 Q -free lattice gene.rated by H (see Figure 4). The maximum distributive homomorphic image of this lattice has 27 elements. We - - =!?n O!? The!?n,S are distinct by V. B. Lender [27]. We define the dimension of y as the maximum n for which y contains Qn. A. Day [4] has shown that ~ is the only inductively define the quasivariety D n as follows: DO = T, the class of trivial lattices;-d n + 1 = lattice variety that contains every Qn. Therefore, every variety y + ~ has a finite dimension. Consequently, every lattice variety is a product of indecomposable varieties. We show in [16] that, for y E G, yo > is a variety. A.Day [6] has generalized this result by replacing Q by any locally finite variety generated by lattices that are bounded in the sense of McKenzie [28]. These M-bounded lattices are characterized in Day [5J. Using a family of projective modular lattices of R. Freese [9], we show in [16] that G contains continuumly many modular varieties. (In an unpublished appendix to C. Herrmann and A. Huhn [23], some modular members of G are defined by identities.) Thus, applying a result of Lender [27J, there are continuumly many varieties of the form V 0 D. We have not found any variety y for which

6 VoD is a variety but Yf G. Let us call a class; of l~ttices subdirectly complete if every lattice in X can be embedded in a subdirectly irreducible lattice in ~. R. p. Dilworth and R. Freese [8] have shown that ~ is subdirectly complete. ~ is also known to be subdirectly complete. In [18], we show that VoW is subdirectly complete whenever y and ~ are nontrivial varieties. We show that pop contains continuumly many subvarieties in [17]. Consequently, the product of any two nontrivial lattice varieties contains continuumly many subvarieties. Therefore, one may conjecture that any nontrivial subdirectly complete variety of lattices contains continuumly many subvarieties. By Jonsson's Lemma, any subdirectly complete lattice variety is join-irreducible. Thus, we can easily find all solutions of the equation yo W= yv~. (See also [14].) The previous results imply the existence of continuumly many join-irreducible lattice varieties. J. Berman [2] has given a different proof of this statement. We have already observed that p Q is not locally finite. However, it is shown in [19] that!? o!? is generated by its finite members. Since the implications holding in!? o!? can be recursively enumerated (see [15]), it follows that the (free lattice) word problem for!? P is solvable. We use different techniques in [20] to give a primitve recursive basis for the identities of!? Q

7 Figure 1 A lattice in DoD

8 Figure 2 The DOD-free product of C4 and C

9 Figure 3 The poset H

10 AXIS OF ---_ SYMMETRY Figure 4 The completely Q 0 Q-free lattice generated by H

11 Figure 5 A lattice in ~5 o!?

12 Figure 6 A simple lattice

13 REFERENCES [1] K. A. Baker, tices, classes of modular lat J. Math., 28 (1969), [ J. Interval lattices and the [3) A. A, 12 (1981), , lattices all lattices, Universalis, 7 (1977), [4] A., in the classes of lattices, (1978), of all SI? Bull., 21 [5] [6] [7] A. Characterizations of finite lattices that are free lattices, Preprint. A. Private or sublatticesof R. P. The arithmetical of Birkhoff lattices, Duke Math. J., 8 (1941), [8] R. P. Dilworth - R. Generators of lattice varieties, A Universalis, 6 (1976), [9] R.Freese, as modular lattices, Trans. Amep. Math. Soa., 251 (1979), [101 A. M. W. Glass - W. C.Holland - S. H. The structure of varieties, A Uni-, 10 (1980), [11]E., On the sums of double system of lattices, in: Contributions to Universal ( 1975 ), Co l Math. Soa. Janos Bo, vol 17. North-Holland, New York, 1977,

14 [12] E. Graczynska - G. Gratzer, On double systems of lattices, Demonstratio Mathematica, 13 (1980), [13] G. Gratzer, GeneraL Lattice Theory, Pure and Applied Mathematics Series, Academic Press, New York, [14] G. Gratzer - D. Kelly, On a special type of subdirectly irreducible lattice with an application to products of varieties, C. R. Math. Rep. Acad. Sci. Canada, 2 (1980), [15] G. Gratzer - D. Kelly, The product of lattice varieties, to appear. [16] G. Gratzer - D. Kelly, Certain products of lattice varieties that are varieties, to appear. [17] G. Gratzer - D. Kelly, The lattice variety QoQ, to appear. [18] G. Gratzer - D. Kelly, Products of lattice varieties permit embedding into subdirectly irreducible members, to appear. [19] G. Gratzer - D. Kelly, A property of products of locally finite lattice varieties, to appear. [20] G. Gratzer - D. Kelly, Identities satisfied by certain products of lattice varieties, to appear. [21] M. Hall - R. P. Dilworth, The imbedding problem for modular lattices, Ann. of Math., 45 (1944), [22] C. Herrmann, S-verklebte Summen von Verbanden, Math. Z., 130 (1973),

15 [2 c. Herrmann - A. Huhn, Zum Charakteristik modularer (1975), der Math. Z., 144 [24] B. Jonsson, distributive, Math. whose congruence lattices are, 21 (1967), [25] B. Jonsson - I. Rival, Lattice varieties covering the smallest non-modular 82 (1979), J. Math., [26] P. Kohler, Varieties of Brouwerian, Mitt. Math. Bem. Giessen, Heft 116, Giessen [ V. B. Lender, The of of lattices (, Siberian Math. J., 16 (1975), in translation. [28] R. McKenzie, bases and nonmodular lattice varieties, Trans. Amer. Math. Soo., 174 (1972), J A. I. Mal'cev, braic (1967), Mathematios A (Russian), Amsterdam, 1971, of classes of Siberian th. J., 8 translation: The terns, [30] H. Varieties der Mathematik und ihrer, Band 37,, New York, [31] S. V. On identities in congruence lattices of universal Russian, Mat. Zametki, 22 (1977) I

16 [32] I. Rival - R. Wille, Lattices freely generated by partially ordered sets: which can be "drawn"? J. Reine Angew. Math., 310 (1979), [33] H. L. Rolf, The free lattice generated by a set of chains, Paaifia J. Math., 8 (1958), George Gratzer - David Kelly Department of Mathematics University of Manitoba Winnipeg, Manitoba R3T 2N2, Canada