On Lattices Embeddable Into Lattices of Order-Convex Sets: Case of Trees

Abstract

We find a syntactic characterization of the class of lattices embeddable into convexity lattices of posets which are trees. The characterization implies that this class forms a finitely based variety.

Lattices of Order-Convex Sets of Forests

“…All the identities considered here (and their corresponding join irreducible interpretations) were introduced in [7], see also [10]. We denote the following identity with variables x, y, y 0 , y 1 , z by (S):…”

“…Identities we consider in this section were introduced in the paper [10]. The reader can consult the same paper for the proof that those identities hold on convexity lattices of forests.…”

“…[10]) We define binary relation on the set J a by putting x y for x, y ∈ J a , if one of the following cases occurs:…”

“…The following identity with variables x, x 0 , x 1 , y which we will refer to as (B 0 ) was introduced in [10].…”

“…The present paper is a continuation of [10], where lattices embeddable into lattices from Co(F ) [from Co(L), respectively] were described by a set of identities. Having in mind Birkhoff's HSP-theorem, nobody would even dare to hope that such an equational characterization might be obtained for Co(F ) or for Co(L) themselves.…”