Sublattices of lattices of order-convex sets, I. The main representation theorem

Abstract

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. If L is finite, then one can take P finite, withwhere 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.

Sublattices of Lattices of Order-Convex Sets, Iii: The Case of Totally Ordered Sets

“…As observed in [12], (S) implies both join-semidistributivity and dual 2-distributivity. Therefore, we obtain the following consequence.…”

“…In the present paper, we extend these results to sublattices of products of lattices of convex subsets of chains (i.e., totally ordered sets), thus solving a problem of [12]. More specifically, we denote by SUB(LO) (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form i∈I Co(T i ), where T i | i ∈ I is a family of chains (resp., chains with at most n elements).…”

“…In particular, the identities (S), (U), and (B), together with their join-irreducible translations (S j ), (U j ), and (B j ), and tools such as Stirlitz tracks or the Udav-Bond partition, are defined in [12]. The identities (H n ) and (H m,n ), their join-irreducible translations, and bi-Stirlitz tracks are defined in [13].…”

“…(ii)⇒(iii) As in [12,13], we embed L into the filter lattice L of L, partially ordered by reverse inclusion. This embedding preserves the existing bounds and atoms.…”

Lattices of algebraic subsets

“…A great achievement was made in [31]- [33] in description of lattices embeddable into Co(P ). In particular, it was proved that the class of lattices embeddable into finite Co(P ) is a pseudovariety defined in the class of all finite lattices by three identities.…”