Some structural and residual properties of 2-semilattices

Abstract

To each 2-semilattice, one can associate a digraph and a partial order. We analyze these two structures working toward two main goals: One goal is to give a structural dichotomy on minimal congruences of 2-semilattices. From this, we are able to deduce information about the tame-congruence-theoretic types that occur in 2-semilattices. In particular, we show that the type of a finite simple 2-semilattice is always either 3 or 5 and can be deduced immediately from its associated digraph. The other goal is to introduce and explore a property that some 2-semilattices have which we have named the “component-semilattice property”. We show that this property must hold in every algebra in a variety of 2-semilattices that is both locally finite and residually small. Hence, a finite 2-semilattice that lacks this property generates a residually large variety.