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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Інститут прикладної математики і механіки НАН України
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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 |
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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. |
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Skowronek-Kaziow, J. n-distributivity and n-modularity in lattices Український математичний вісник |
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Skowronek-Kaziow, J. |
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Skowronek-Kaziow, J. |
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n-distributivity and n-modularity in lattices |
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n-distributivity and n-modularity in lattices |
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n-distributivity and n-modularity in lattices |
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n-distributivity and n-modularity in lattices |
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n-distributivity and n-modularity in lattices |
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n-distributivity and n-modularity in lattices |
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Інститут прикладної математики і механіки НАН України |
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2004 |
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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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Український математичний вісник |
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AT skowronekkaziowj ndistributivityandnmodularityinlattices |
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2025-07-09T01:44:11Z |
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2025-07-09T01:44:11Z |
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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.
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