On the Representations of Finite Distributive Lattices

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 bar right arrow D-P (L) between the sublattices of P and the preorders spanned by a canonical sublattice C-lozenge 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.