Externalization of lattices

Let r be a type of algebras. An identity s = t of type r is said to be externally compatible, or simply external, if the terms s and t are either the same variable or both start with the same operation symbol fj of the type. A variety is called external if all of its identities are external. For any variety V , there is a least external variety E(V ) containing V , the variety determined by the set of all external identities of V . External identities and varieties have been studied by [4], [5] and [2], and a general characterization of the algebras in E(V ) has been given in [3]. In this paper we study the algebras of the variety E(V ) where V is the type (2, 2) variety L of lattices. Algebras in L may also be described as ordered sets, and we give an ordered set description of the algebras in E(L). We show that on any algebra in E(L) there is a natural quasiorder having an additional property called externality, and that any set with such a quasiorder can be given the structure of an algebra in E(L). We also characterize algebras in E(L) by an inflation construction.