n-distributivity and n-modularity in lattices

In this paper we consider some forbidden sublattices for n-distributive, but non-modular lattices. We define the new notion of n-modularity (weaker than n-distributivity). We also consider some forbidden sublattice for an n-modular lattice. We prove that n-modularity implies (n + 1)-modularity. The...

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Автор: Skowronek-Kaziow, J.
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Мова:English
Опубліковано: Інститут прикладної математики і механіки НАН України 2004
Назва видання:Український математичний вісник
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Цитувати:n-distributivity and n-modularity in lattices / J. Skowronek-Kaziow // Український математичний вісник. — 2004. — Т. 1, № 2. — С. 273-278. — Бібліогр.: 8 назв. — англ.

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spelling irk-123456789-1246202017-10-01T03:02:53Z n-distributivity and n-modularity in lattices Skowronek-Kaziow, J. In this paper we consider some forbidden sublattices for n-distributive, but non-modular lattices. We define the new notion of n-modularity (weaker than n-distributivity). We also consider some forbidden sublattice for an n-modular lattice. We prove that n-modularity implies (n + 1)-modularity. The counter-examples for the inverse implication are shown. 2004 Article n-distributivity and n-modularity in lattices / J. Skowronek-Kaziow // Український математичний вісник. — 2004. — Т. 1, № 2. — С. 273-278. — Бібліогр.: 8 назв. — англ. 1810-3200 2000 MSC. 06C99, 06D99, 06D50. http://dspace.nbuv.gov.ua/handle/123456789/124620 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description In this paper we consider some forbidden sublattices for n-distributive, but non-modular lattices. We define the new notion of n-modularity (weaker than n-distributivity). We also consider some forbidden sublattice for an n-modular lattice. We prove that n-modularity implies (n + 1)-modularity. The counter-examples for the inverse implication are shown.
format Article
author Skowronek-Kaziow, J.
spellingShingle Skowronek-Kaziow, J.
n-distributivity and n-modularity in lattices
Український математичний вісник
author_facet Skowronek-Kaziow, J.
author_sort Skowronek-Kaziow, J.
title n-distributivity and n-modularity in lattices
title_short n-distributivity and n-modularity in lattices
title_full n-distributivity and n-modularity in lattices
title_fullStr n-distributivity and n-modularity in lattices
title_full_unstemmed n-distributivity and n-modularity in lattices
title_sort n-distributivity and n-modularity in lattices
publisher Інститут прикладної математики і механіки НАН України
publishDate 2004
url http://dspace.nbuv.gov.ua/handle/123456789/124620
citation_txt n-distributivity and n-modularity in lattices / J. Skowronek-Kaziow // Український математичний вісник. — 2004. — Т. 1, № 2. — С. 273-278. — Бібліогр.: 8 назв. — англ.
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fulltext Український математичний вiсник Том 1 (2004), № 2, 273 – 278 n-distributivity and n-modularity in lattices Joanna Skowronek-Kaziów (Presented by Usenko V. M.) Abstract. In this paper we consider some forbidden sublattices for n-distributive, but non-modular lattices. We define the new notion of n-modularity (weaker than n-distributivity). We also consider some for- bidden sublattice for an n-modular lattice. We prove that n-modularity implies (n + 1)-modularity. The counter-examples for the inverse impli- cation are shown. 2000 MSC. 06C99, 06D99, 06D50. Key words and phrases. n-distributive lattice, n-modular lattice, dually n-distributive lattice, dually n-modular lattice, Boolean lattice. 1. Introduction We recall the n-distributivity notion, which was introduced by G. M. Bergman (in [1]) and A. P. Huhn (in [4]) as a generalization of the ordinary distributivity (for n = 1), for modular lattices: A lattice (L,∨,∧) is n-distributive if for every x, y0, . . . , yn ∈ L the condition is satisfied: (Dn) x ∧∨n i=0 yi = ∨n j=0(x ∧∨n i=0;j 6=i yi). A lattice L is dually n-distributive if for every x, y0, . . . , yn ∈ L the following equality is satisfied: x ∨∧n i=0 yi = ∧n j=0(x ∨∧n i=0;i6=j yi). A lattice L is modular, if for every x, y, z ∈ L, x ≤ y implies x ∧ (y ∨ z) = x ∨ (x ∧ z). The condition (Dn) is equivalent to the dual n-distributivity condi- tion iff a lattice L is modular (see [4]). It is easy to show that every n-distributive lattice (dually n-distribu- tive) is also (n+1)-distributive (dually (n+1)-distributive, respectively). For standard terminology, see [3]. We introduce two notions weaker than notion of n-distributivity and dual n-distributivity, respectively: ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України 274 n-distributivity and n-modularity in lattices 1) A lattice (L,∨,∧) is n-modular if for every x, y0, . . . , yn ∈ L the following implication is true: [ ∨n−1 i=0 yi ≤ x] ⇒ [x ∧∨n i=0 yi = ( ∨n−1 i=0 yi) ∨ ∨n−1 j=0 (x ∧∨n i=0;i6=j yi)]. 2) A lattice (L,∨,∧) is dually n-modular if for every x, y0, . . . , yn ∈ L the implication: [ ∧n−1 i=0 yi ≥ x] ⇒ [x ∨∧n i=0 yi = ( ∧n−1 i=0 yi) ∧ ∧n−1 j=0 (x ∨∧n i=0;i6=j yi)] is valid. The 1-modular lattices and dually 1-modular lattices are exactly modular. If P is a poset and for a, b, c ∈ P the conditions a < b, a ≤ c ≤ b imply c = a or c = b, then we say, that b covers a in the set P (or a is covered by b). 2. Some properties for n-distributive and n-modular lat- tices; Characterization of an n-modular lattice by the forbidden sublattice In 1972 A. P. Huhn (see [4]) proved that a modular lattice L is not n- distributive iff it contains a sublattice B isomorphic to the 2n+1-element Boolean lattice and an element x such that x ∧ a = ∧ B, x ∨ a = ∨ B, for every atom a of B. For n = 1, it is the well-known criterion of distributivity. The following proposition without the modularity assumption is some partial generalization for the above Huhn’s result. Proposition 1. A lattice (L,∨,∧) is not n-distributive whenever it con- tains a sublattice B isomorphic to the 2n+1-element Boolean lattice and an element x such that x > b, for some b ∈ B and ∨ B is the only element in B, which covers x in L. Proof. Let {y0, . . . , yn} be the set of atoms in the algebra B. Then x ∧∨n i=0 yi = x ∧∨B = x. According to the assumption there is an element b0 ∈ B such that x covers b0 in the poset B ∪ {x}. Hence, x ∧ ∨n i=0;j 6=i yi ≤ b0 < x, for 0 ≤ j ≤ n and ∨n j=0(x ∧ ∨n i=0;j 6=i yi) ≤ b0 < x, which contradicts the n-distributivity. Corollary 1. A lattice (L,∨,∧) is not dually n-distributive whenever it contains a sublattice B isomorphic to the 2n+1-element Boolean lattice and an element x covering ∧ B in L such that x < b0, for some b0 ∈ B. The inverse implication in the above theorem seems true, but it is still an open problem. J. Skowronek-Kaziów 275 Proposition 2. A lattice (L,∨,∧) is n-modular iff for comparable ele- ments x and ∨n−1 i=0 yi the following equality is satisfied: (Mn) x ∧ [(x ∧∨n−1 i=0 yi) ∨ yn] = ∨n j=0(x ∧∨n i=0;i6=j yi). Proof. Assuming ∨n−1 i=0 yi ≤ x in (Mn) we get x ∧∨n i=0 yi = ( ∨n−1 i=0 yi) ∨∨n−1 j=0 (x ∧ ∨n i=0;i6=j yi), what gives n-modularity. Let ∨n−1 i=0 yi > x, then we get n-modularity applying the absorbtion laws. Now, let ∨n−1 i=0 yi ≤ x and assume that (Mn) fails, for some x, y0, . . . , yn ∈ L. Then x ∧∨n i=0 yi = x ∧ [(x ∧∨n−1 i=0 yi) ∨ yn] 6= ∨n j=0(x ∧∨n i=0; i6=j yi) = = ( ∨n−1 i=0 yi) ∨ ∨n−1 j=0 (x ∧∨n i=0; i6=j yi), which contradicts the n-modularity. Corollary 2. A lattice (L,∨,∧) is dually n-modular iff for comparable elements x and ∧n−1 i=0 yi the following equality is valid: x ∨ [(x ∨∧n−1 i=0 yi) ∧ yn] = ∧n j=0(x ∨∧n i=0;i6=j yi). Proposition 3. Let n ≥ 1. Then: (i) Every n-distributive (dually n-distributive) lattice is n-modular (du- ally n-modular, respectively). (ii) Every n-modular (dually n-modular) lattice is (n+ 1)-modular (du- ally (n+ 1)-modular, respectively). Proof. First implication is obvious. Now we prove that the usual mod- ularity implies n-modularity for n > 1. Let ∨n−1 i=0 yi ≤ x. Then using modularity, we get x ∧∨n i=0 yi = x ∧ ( ∨n−1 i=0 yi ∨ ∨n i=0;j 6=i yi) = ∨n−1 i=0 yi ∨ ( x ∧∨n i=0;j 6=i yi ) , for every 0 ≤ j ≤ n − 1. Hence, x ∧ ∨n i=0 yi = ( ∨n−1 i=0 yi) ∨ ∨n−1 j=0 (x ∧∨n i=0;i6=j yi), what gives n-modularity. Now, let x, y0, . . . , yn, yn+1 ∈ L,∨n i=0 yi ≤ x and let 0 ≤ l, k ≤ n be fixed indices. Then assuming n- modularity and treating yl ∨ yk as a single element we get the equality: x ∧∨n+1 i=0 yi = ∨n i=0 yi ∨ [ ∨n j=0;j 6=l,k(x ∧∨n+1 i=0;i6=j yi)] ∨ (x ∧∨n+1 i=0;i6=l,k yi). The supremum over all 0 ≤ l, k ≤ n of the right-hand side of this equal- ity is exactly equal to ∨n i=0 yi ∨ ∨n j=0(x ∧ ∨n+1 i=0;i6=j yi). Hence we get (n+ 1)-modularity. Analogously, inverting operations we prove the dual theorem. Remark. The inverse implications in the Proposition 3 are not always true! The lattices L1, L2, L3 (see Figure 1) are not modular; L2, L3 are not 2-distributive, but they are 2-modular; L1 is not 2-distributive and not 2-modular. 276 n-distributivity and n-modularity in lattices x x x Figure 1. Proposition 4. A lattice (L,∨,∧) is not n-modular whenever it contains a sublattice B isomorphic to the 2n+1-element Boolean lattice and an element x such that c0 < x < ∨ B, for some coatom c0 of B. A lattice L is not 2-modular if and only if it contains an isomorphic copy of L1 as a poset (see Figure 1). Proof. Let A = {y0, y1, . . . , yn} ⊆ B be the set of atoms of B. Since c0 = ∨n−1 i=0 yi < x, hence x ∧ ∨n i=0 yi = x. An element ∨n i=0;j 6=i yi is a coatom of B, for 0 ≤ j ≤ n. Hence x ∧ ∨n i=0;j 6=i yi ≤ c0 < x, for every 0 ≤ j ≤ n and ( ∨n−1 i=0 yi)∨ ( ∨n−1 j=0 (x∧∨n i=0;j 6=i yi)) = ∨n−1 i=0 yi = c0, which contradicts the n-modularity. Now, we prove the inverse implication, in the case n = 2. Assume that L is not 2-modular. Then for some x, y1, y2, y3 ∈ L, y1∨y2 ≤ x we get the inequality (*) x∧ (y1∨y2∨y3) > (y1∨y2)∨ [x∧ (y1∨y3)]∨ [x∧ (y2∨y3)]. Notice that comparability of every pair of elements y1, y2, y3 contra- dicts this inequality. Now, let y1∨y2 = y1∨y3. Then (y1∨y2) ≥ (y2∨y3), x∧ (y1 ∨ y2 ∨ y3) = y1 ∨ y2 and (y1 ∨ y2)∨ [x∧ (y1 ∨ y3)]∨ [x∧ (y2 ∨ y3)] = (y1 ∨ y2) ∨ [x ∧ (y2 ∨ y3)] = (y1 ∨ y2). If (y1 ∨ y3) = (y2 ∨ y3), then (y1 ∨ y2) ≤ (y2 ∨ y3), x∧ (y1 ∨ y2 ∨ y3) = x ∧ (y1 ∨ y3) and (y1 ∨ y2) ∨ [x ∧ (y1 ∨ y3)] ∨ [x ∧ (y2 ∨ y3)] = (y1 ∨ y2) ∨ [x ∧ (y1 ∨ y3)] = x ∧ (y1 ∨ y3). Hence, elements y1 ∨ y2, y1 ∨ y3, y2 ∨ y3 must be different. Similarly, if any two of the following elements y1 ∨ y2, y1 ∨ y3, y2 ∨ y3 are comparable, then it contradicts the inequality (*). Three incomparable elements y1 ∨ y2, y1 ∨ y3, y2 ∨ y3 generate a lattice isomorphic to the 23-element Boolean lattice (see. [3], p. 48). Hence, L must contain L1. Corollary 3. A lattice (L,∨,∧) is not dually n-modular whenever it contains a sublattice B isomorphic to the 2n+1-element Boolean lattice and an element x such that ∧ B < x < a0, for some atom a0 of B (the inverse implication is true for n = 1 and n = 2). J. Skowronek-Kaziów 277 The inverse implication of Proposition 4 seems true also for n > 2, but it is still an open problem (for n = 1 it is the well-known criterion of modularity). A. P. Huhn proved, that for a modular lattice L the equality:∧n+1 j=0 ∨n+1 i=0;i6=j yi = ∨n+1 k=0 ∧n+1 j=0;j 6=k ∨n+1 i=0;i6=j,k yi, for y0, ..., yn+1 ∈ L is equivalent to (Dn) (see [4], [5]). The next proposition gives the equality condition implying (Dn) without modularity assumption: Proposition 5. A latticeL is n-distributive whenever for every y0, ..., yn+1 ∈ L the following equality is satisfied:∧n+1 j=0 ∨n+1 i=0;i6=j yi = ( ∧n i=0 yi) ∨ ∨n j=0(yn+1 ∧ ∨n i=0;j 6=i yi). Proof. Denote the left-hand side of the above equality by a, and the right-hand one by b. Assuming ∨n−1 i=0 yi ≤ yn+1 in a = b and using the absorbtion laws we get yn+1 ∧ ∨n i=0 yi = ( ∨n−1 i=0 yi) ∨ ( ∨n−1 j=0 (yn+1 ∧∨n i=0;j 6=i yi)), what gives n-modularity. Notice, that yn+1 ∧ a = yn+1 ∧ ( ∧n+1 j=0 ∨n+1 i=0;i6=j yi) = = yn+1 ∧ ∨n i=0 yi ∧ ∧n j=0(yn+1 ∨ ∨n i=0;j 6=i yi) = yn+1 ∧ ∨n i=0 yi. Since L is n-modular and ∨n j=0(yn+1 ∧ ∨n i=0;j 6=i yi) ≤ yn+1, hence yn+1 ∧ b = yn+1 ∧ [( ∧n i=0 yi) ∨ ∨n j=0(yn+1 ∧ ∨n i=0;j 6=i yi)] = = [ ∨n j=0(yn+1∧ ∨n i=0;j 6=i yi)]∨ ∨n j=0{yn+1∧[( ∧n i=0 yi)∨(yn+1∧ ∨n i=0;i6=j yi)]}. Because of the inequality {yn+1 ∧ [( ∧n i=0 yi) ∨ (yn+1 ∧ ∨n i=0;i6=j yi)]} ≤ (yn+1 ∧ ∨n i=0;i6=j yi), 0 ≤ j ≤ n, which is valid for an arbitrary lattice, we deduce yn+1 ∧ b = ∨n j=0(yn+1 ∧ ∨n i=0;i6=j yi). The equality yn+1 ∧ a = yn+1 ∧ b gives n-distributivity. There are some useful applications for (Dn) condition in lattices of closed sets with respect to a given closure operator. For example, the n-distributivity property can be asocciated to the Carathéodory number, which is some parameter describing a closure operator on a given set (see [2], [6]–[8]). References [1] G. M. Bergman, On 2-firs (weak Bézout rings) with distributive divisors lattices, preprint, 1969. [2] K. G lazek, J. Grytczuk and J. Skowronek-Kaziów, Remarks on an algebraic realiza- tion of closure operators // Contributions to General Algebra 11 (1999), 85–100. [3] G. Grätzer, General Lattice Theory, (the second edition). Birkhäuser Verlag, 1998. [4] A. P. Huhn, Schwach distributive Verbände. // I, Acta Sci. Math. (Szeged) 33 (1972), 297–305. [5] A. P. Huhn, On n-distributive system of elements of a modular lattice. // Publ. Math. (Debrecen) 27 (1980), 107–115. 278 n-distributivity and n-modularity in lattices [6] L. Libkin, n-Distributivity, dimension and Carathéodory’s theorem. // Algebra Universalis. 34 (1995), 72–95. [7] J. Skowronek-Kaziów, Matroids and incidence structures, p. 193–202 in: General Algebra and applications, Shaker Verlag, Aachen 2000. [8] J. Skowronek-Kaziów, Closure operators, related contexts and algebras. // Miscel- lanea Algebraicae. 5 (2001), no. 2, 73–80. Contact information J. Skowronek- Kaziów Institute of Mathematics, University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Poland.