Semilattice sums of algebras and Mal’tsev products of varieties

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 V is a strongly irregular variety with no nullary operations and at least one non-unary operation, and S is the variety...

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Bibliographic Details
Published in:Algebra universalis 2020-08, Vol.81 (3), Article 33
Main Authors: Bergman, C., Penza, T., Romanowska, A. B.
Format: Article
Language:English
Subjects:
Algebra
Mathematics
Mathematics and Statistics
Sums
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Summary: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 V is a strongly irregular variety with no nullary operations and at least one non-unary operation, and S is the 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 base for the product from an equational base for V . However, if 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 V ∘ S .
ISSN:0002-5240
1420-8911
DOI:10.1007/s00012-020-00656-8