Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order

2011 ◽  
Vol 54 (2) ◽  
pp. 277-282
Author(s):  
Jonathan David Farley
Keyword(s):  
Distributive Lattice ◽  
Maximal Chain ◽  
Distributive Lattices ◽  
Finite Distributive Lattice ◽  
Finite Distributive Lattices

AbstractLet L be a finite distributive lattice. Let Sub0(L) be the lattice﹛S | S is a sublattice of L﹜ [ ﹛∅﹜ and let ℓ*[Sub0(L)] be the length of the shortest maximal chain in Sub0(L). It is proved that if K and L are non-trivial finite distributive lattices, then ℓ*[Sub0(K × L)] = ℓ*[Sub0(K)] + ℓ[Sub0(L)]. A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.

scholarly journals Covering in the lattice of subuniverses of a finite distributive lattice

1998 ◽  
Vol 65 (3) ◽  
pp. 333-353 ◽  
Author(s):  
Zsolt Lengvárszky ◽  
George F. McNulty
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Finite Distributive Lattice ◽  
Covering Relation ◽  
Finite Distributive Lattices

AbstractThe covering relation in the lattice of subuniverses of a finite distributive lattices is characterized in terms of how new elements in a covering sublattice fit with the sublattice covered. In general, although the lattice of subuniverses of a finite distributive lattice will not be modular, nevertheless we are able to show that certain instances of Dedekind's Transposition Principle still hold. Weakly independent maps play a key role in our arguments.

Reimer's Inequality on a Finite Distributive Lattice

2013 ◽  
Vol 22 (4) ◽  
pp. 612-621
Author(s):  
CLIFFORD SMYTH
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Finite Distributive Lattice ◽  
Finite Distributive Lattices

We generalize Reimer's Inequality [6] (a.k.a. the BKR Inequality or the van den Berg–Kesten Conjecture [1]) to the setting of finite distributive lattices.

scholarly journals Congruences on a Distributive Lattice

1954 ◽  
Vol 10 (2) ◽  
pp. 76-77
Author(s):  
H. A. Thueston
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Distributive Lattices ◽  
The Subject ◽  
The Many ◽  
Finite Distributive Lattices

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

scholarly journals On the relation of a distributive lattice to its lattice of ideals

1972 ◽  
Vol 7 (3) ◽  
pp. 377-385 ◽  
Author(s):  
Herbert S. Gaskill
Keyword(s):  
Distributive Lattice ◽  
Related Result ◽  
Distributive Lattices ◽  
Relationship Of ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
Lattice Of Ideals

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

scholarly journals On Decompositions of Matrices over Distributive Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yizhi Chen ◽  
Xianzhong Zhao
Keyword(s):  
Distributive Lattice ◽  
Direct Product ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
The Matrix ◽  
Nilpotent Matrices ◽  
Decomposition Theorems ◽  
Finite Distributive Lattices ◽  
Matrix Semigroup

LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1m‍Liof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1m‍Mn,q(Li)(semiring∏i=1m‍Mn(Li), resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.

Homomorphisms of Distributive Lattices as Restrictions of Congruences

1986 ◽  
Vol 38 (5) ◽  
pp. 1122-1134 ◽  
Author(s):  
George Grätzer ◽  
Harry Lakser
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Classical Result ◽  
Finite Lattice ◽  
Distributive Lattices ◽  
Lattice Homomorphism ◽  
Finite Distributive Lattice ◽  
Full Generality ◽  
Ideal Lattices

Given a lattice L and a convex sublattice K of L, it is well-known that the map Con L → Con K from the congruence lattice of L to that of K determined by restriction is a lattice homomorphism preserving 0 and 1. It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. Although it has been conjectured that any algebraic distributive lattice is the congruence lattice of some lattice, this has not yet been proved in its full generality. The best result is in [4]. The conjecture is true for ideal lattices of lattices with 0; see also [3].

scholarly journals Dumont's Statistic on Words

10.37236/1555 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Mark Skandera
Keyword(s):  
Symmetric Group ◽  
Distributive Lattice ◽  
Distributive Lattices ◽  
Finite Distributive Lattices

We define Dumont's statistic on the symmetric group $S_n$ to be the function dmc: $S_n \rightarrow {\bf N}$ which maps a permutation $\sigma$ to the number of distinct nonzero letters in code$( \sigma )$. Dumont showed that this statistic is Eulerian. Naturally extending Dumont's statistic to the rearrangement classes of arbitrary words, we create a generalized statistic which is again Eulerian. As a consequence, we show that for each distributive lattice $J(P)$ which is a product of chains, there is a poset $Q$ such that the $f$-vector of $Q$ is the $h$-vector of $J(P)$. This strengthens for products of chains a result of Stanley concerning the flag $h$-vectors of Cohen-Macaulay complexes. We conjecture that the result holds for all finite distributive lattices.

scholarly journals The Central Decomposition of FD01(n)

Order ◽  
2021 ◽  
Author(s):  
Peter Köhler
Keyword(s):  
Distributive Lattices ◽  
Finitely Generated ◽  
Finite Distributive Lattices

AbstractThe paper presents a method of composing finite distributive lattices from smaller pieces and applies this to construct the finitely generated free distributive lattices from appropriate Boolean parts.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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