Semilattice sums of algebras and Mal’tsev products of varieties

Abstract

The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $\mathcal {V}$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $\mathcal {S}$ is the variety, of the same type as $\mathcal {V}$, equivalent to the variety of semilattices, then the Mal’tsev product $\mathcal {V}\circ \mathcal {S}$ is a variety. It consists precisely of semilattice sums of algebras in $\mathcal {V}$. We derive an equational base for the product from an equational base for $\mathcal {V}$. However, if $\mathcal {V}$ is a regular variety, then the Mal’tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $\mathcal {V}\circ \mathcal {S}$.