On the representation of finite distributive lattices

Abstract

A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice P can be constructed from P by removing a particular family I L of its irreducible intervals.Applying this in the case that P is a product of a finite set C of chains, we get a one-to-one correspondence L → D P (L) between the sublattices of P and the preorders spanned by a canonical sublattice C ∞ of P.We then show that L is a tight sublattice of the product of chains P if and only if D P (L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of P and the posets spanned by its poset J(P) of non-zero join-irreducible elements.With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice L into products of chains, and the chain decompositions of its poset J(L) of non-zero join-irreducible elements.2010 Mathematics Subject Classification. 06D05.

Distributive Lattice Polymorphism on Reflexive Graphs

“…As Dilworth [6] observed in the case that C is a decomposition, we observed in [18] that e C is in fact a lattice embedding of D(P ) into the product of chains P C = d i=1 P i where P i is the chain 0 ≺ 1 ≺ • • • ≺ n i with one more element than C i . Thus by Birkoff's result from [2], every chain cover of J L gives an embedding e C of L as a sublattice of a product P C of chains.…”

“…Theorem 4.3 provides us a sub-digraph A of J L such that G ∼ = G(J L , A). As we mentioned above, we have from [18] that every chain cover C of J L yields an embedding e C : L ∼ = P C − V into a product of chains. By Corollary 5.4 it is induced if and only if red(A c ) is a subgraph of C. We can assure this by taking every arc of red(A c ) as a two element chain in C and then covering then rest of J L with one element chains.…”

“…As we mentioned above, it follows from [18] that V (G) = V (G) − V so we are done with the following claim.…”

“…Setup for the proof of Theorem 5.1. The first statement of Theorem 5.1 is acually from [18]. We explain, as we will have to build on this.…”

“…In (Corollary 6.6 of) [18] we showed that every embedding of L as a sublattice of a product of chains such that L contains the zero and unit of the product, is e C for some chain cover C of J L .…”