Semilattice sums of algebras and Mal’tsev products of varieties

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2020-05-20
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Penza, T.
Romanowska, A. B.
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Bergman, Clifford
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Mathematics
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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.

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This is a post-peer-review, pre-copyedit version of an article published in Algebra universalis. The final authenticated version is available online at DOI: 10.1007/s00012-020-00656-8. Posted with permission.

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Wed Jan 01 00:00:00 UTC 2020
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