Mal'tsev products of varieties. I (Q829764)

Let \(\mathcal{V}\) and \(\mathcal{W}\) be varieties of algebras of the same similarity type, with no nullary operations. The Mal'tsev product \(\mathcal{V}\circ\mathcal{W}\) is a \textit{quasivariety} that consists of algebras with a congruence \(\theta\) such that the quotient \(A^\theta\) is in \(\mathcal{W}\) and each \(\theta\)-class \(a^\theta\) is in \(\mathcal{V}\). This paper describes the identities true in \(\mathcal{V}\circ\mathcal{W}\) in terms of the identities satisfied in \(\mathcal{V}\) and \(\mathcal{W}\) (Section 2). The main result (Theorem 3.3.) provides a new sufficient condition for \(\mathcal{V}\circ\mathcal{W}\) to be a variety. Finally, in Section 4, applications of this result are presented for \(\mathcal{W}\) being the variety of different kinds of generalizations of idempotent semigroups.