On the regular part of varieties of algebras
type: Item
instance of: scholarly article
title: On the regular part of varieties of algebras
zbMATH Open document ID: 0605.08004
DOI: 10.1007/BF01190914
author: B. George
published in: Algebra Universalis
publication date: 1986-01-01T00:00:00Z
review text: An identity is said to be normal (or regular) if the same variables appear on each side of it. The variety N(V) defined by all normal identities satisfied in the variety V of algebras is called the normal closure of V. The reviewer has shown that the normal closure of a semigroup variety V consists of subalgebras of rigid commutative bands (or sums of direct systems) of semigroups from V [Izv. Vyssh. Uchebn. Zaved., Mat. 1969, No.5(84), 61-68 (1969; Zbl 0199.336), Teor. Polugrupp Prilozh. 2, 69-74 (1971; Zbl 0244.20077)]. It was stated by the reviewer [in Tr. Mol. Uchen. Mat. Mekh. 124-130 (1969; Zbl 0276.08001)] that the same is true for arbitrary algebras. The present authors refute this statement by an example. The second example shows that the join of infinitely many nonregular varieties can be regular thus disproving another erroneous claim from (op.cit.). \textit{I. I. Mel'nik} [Uporjad. Mnozestv. Resetki 1, 56-65 (1971; Zbl 0244.08003)] has proved that if N(V) has a finite basis then V is also finitely based. In the paper under review a finitely based variety V is exposed for which the normal closure N(V) has no finite basis.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 3977072
zbMATH Keywords: regular varieties
zbMATH Keywords: normal identities
zbMATH Keywords: normal closure
zbMATH Keywords: nonregular varieties
zbMATH Keywords: finitely based variety
reviewed by: V. N. Saliĭ
MaRDI profile type: Publication
cites work: Q3487647
cites work: On a method of construction of abstract algebras
cites work: On equational classes of abstract algebras defined by regular equations
cites work: Q5538979
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description: scientific article; zbMATH DE number 3977072