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 when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal'tsev product V • S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal'tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V • S.

Mal’tsev products of varieties, I

“…Recall that a strongly irregular variety is a variety V t defined by a set of regular identities and a strongly irregular identity t(x, y) = x, where t(x, y) is a binary term containing both variables x and y. The variety S of semilattices may be considered as the variety S τ of any plural type τ (definitionally) equivalent to S. (See [4] for details.) Theorem 1.6 [4].…”

“…The variety S of semilattices may be considered as the variety S τ of any plural type τ (definitionally) equivalent to S. (See [4] for details.) Theorem 1.6 [4]. If V t is a strongly irregular variety of a plural type τ and S τ is the variety of the same type τ equivalent to the variety of semilattices, then V t • S τ is a variety.…”

“…Also in [4], it was shown how to derive an equational base for the product V t • S τ from equational bases for V t and S τ , and even more generally, how to derive all identities true in the product V • S τ , for any variety V of Ω-algebras of a plural type τ .…”

“…However, it is possible that V • U W is a variety but V • W is not. It was shown in [4] that the Mal'tsev product S • S, where S is the variety of semilattices, is not a variety. But since S ⊆ S • S, it follows that S • S S is the variety S.…”

“…We use notation and conventions similar to those of [4], [19,20]. For details and further information concerning quasivarieties and Mal'tsev product of quasivarieties, we refer the reader to [14,15], and then also to [2] and [20,Ch.…”