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 $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals. Applying this in the case that $\mathcal{P}$ is a product of a finite set $\mathcal{C}$ of chains, we get a one-to-one correspondence $L \mapsto \mathcal{D}_\mathcal{P}(L)$ between the sublattices of $\mathcal{P}$ and the preorders spanned by a canonical sublattice $\mathcal{C}^\infty$ of $\mathcal{P}$. We then show that $L$ is a tight sublattice of the product of chains $\mathcal{P}$ if and only if $\mathcal{D}_\mathcal{P}(L)$ is asymmetric. This yields a one-to-one correspondence between the tight sublattices of $\mathcal{P}$ and the posets spanned by its poset $J(\mathcal{P})$ 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.