On the Representation of Finite Distributive Lattices

Classical results of Birkhoff and of Dilworth yield an extremely useful representation of a finite distributive lattice L as the downset lattice D(JL) of poset JL of join-irreducible elements of L. Through this they get a correspondence between the chain decompositions of JL and the tight embeddings of L into a product of chains. We use a simple but elegant result of Rival which characterizes sublattices L in terms of the possible make-up of the sets L′ − L, to give an alternate interpretation of a chain decomposition of JL as a catalog of the irreducible intervals removed from a product of chains to get L. This interpretation allows us to extend the correspondence of Birkhoff and Dilworth to non-tight embeddings of L into products of chains. In the non-tight case, JL is replaced with a digraph D, which is no longer a poset, and no longer canonical, and L is represented as the lattice of terminal sets of D. Taking a quotient of this D yields a one-to-one correspondence between the embeddings of L into products of chains, and what we call loose chain covers of a a canonical supergraph J∞ L of JL. There is another useful representation, from results of Dilworth and of Koh, of a finite distributive lattice L as the lattices of maximal antichains A(P ) of various posets P . Through the correspondence of Birkhoff and Dilworth this relates chain decompositions of various posets P to tight embeddings of L into products of chains. But it is not a correspondence. We similarly extend the representation theory of Dilworth and Koh to nontight embeddings of L into products of chains. This allows us to correspond any chain decomposition of any finite poset to an embedding of a finite distributive lattice into a product of chains; in a way, completing the correspondence of Dilworth and Koh. The development of our two representations is essentially the same, and we move between them via a simple construction based on a construction of Koh.