Sublattices of Lattices of Order-convex Sets, I. the Main Representation Theorem

For a partially ordered set P , we denote by Co(P ) the lattice of order-convex subsets of P . We find three new lattice identities, (S), (U), and (B), such that the following result holds: Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P ) iff L satisfies (S), (U), and (B). Furthermore, if L has an embedding into some Co(P ), then it has such an embedding that preserves the existing bounds. Furthermore, if L is finite, then one can take P finite, with |P | ≤ 2|J(L)| − 5|J(L)|+ 4, where J(L) denotes the set of all join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.