Semilattice sums of algebras and Mal'tsev products of varieties

TL;DR

This paper characterizes when the Mal'tsev product of certain algebraic varieties forms a variety, focusing on semilattice sums of algebras in strongly irregular varieties.

Contribution

It establishes conditions under which the Mal'tsev product of a strongly irregular variety and a semilattice variety is itself a variety, and provides an equational basis.

Findings

Mal'tsev product of strongly irregular variety and semilattice variety is a variety.

The product consists of semilattice sums of algebras in the original variety.

Regular varieties may not produce a variety under the Mal'tsev product.

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 $\mathcal{V} \circ \mathcal{S}$.