Anna Romanowska – Mal'tsev products of varieties, Part 1

Warsaw University of Technology, Faculty of Mathematics and Information Science
For two varieties V and W of the same type, the Mal'tsev product VoW consists of all algebras A of the same type, with a congruence such that the quotient belongs to W, and every congruence class that is a subalgebra of A belongs to V. The Mal'tsev product of two varieties is usually a quasivariety, but need not itself be a variety.

We describe an equational basis for the variety generated by the Mal'tsev product of any two varieties (of a type without nullary operations) in terms of the identities satisfied in each of the factors. Then, we consider Mal'tsev product VoS, where S is the variety of the same type as V, equivalent to the variety of semilattices. We provide a sufficient condition for such a product to be a variety, and discuss some situations when it is not.