On finite hyperidentity bases for varieties of semigroups
type: Item
instance of: scholarly article
title: On finite hyperidentity bases for varieties of semigroups
zbMATH Open document ID: 0820.20067
DOI: 10.1007/BF01190763
published in: Algebra Universalis
publication date: 1995-02-22T00:00:00Z
review text: An identity \(E\) is hypersatisfied by a variety \(V\) if whenever the operation symbols in \(E\) are replaced by term functions of \(V\) of the appropriate arity then the resulting identity is an identity in \(V\); \(E\) is then said to be a hyperidentity of \(V\). The set of all hyperidentities of \(V\) is denoted by \(H(V)\). Just as with ordinary identities it is natural to ask whether \(H(V)\) is finitely based. This paper considers this in the special case where \(V\) is a variety of semigroups. This question has already been answered in the negative for several varieties of semigroups (including the variety of all semigroups [\textit{G. M. Bergman}, Aequationes Math. 23, 50-65 (1981; Zbl 0485.08005)]). This paper shows how Bergman's proof can be extended to cover all semigroup varieties except those satisfying an identity of the form \(x^ 2 = x^{2+m}\) for some positive \(m\). It is also shown that the variety of bands and all its subvarieties do not have finite bases of hyperidentities. In all these cases, the proofs depend on finding hyperidentities of arbitrarily high arity. The paper concludes by showing that the medial variety of semigroups (which is defined by the identity \(xyzw = xzyw\)) has the property that even its set of hyperidentities involving only one operation symbol, and that binary, is not finitely based.
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 724100
zbMATH Keywords: term functions
zbMATH Keywords: hyperidentities
zbMATH Keywords: variety of semigroups
zbMATH Keywords: variety of bands
zbMATH Keywords: finite bases of hyperidentities
zbMATH Keywords: medial variety of semigroups
author: Shelly L. Wismath
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: The lattice of semigroup varieties
cites work: Hypervarieties of a given type
cites work: Q4126563
cites work: Q3322157
cites work: Hyperidentities and hypervarieties
cites work: Hyperidentities for some varieties of commutative semigroups
cites work: Q4285329
cites work: A basis for the type \(\langle n\rangle\) hyperidentities of the medial variety of semigroups
full work available at URL: https://doi.org/10.1007/bf01190763
OpenAlex ID: W2022206058
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description: scientific article; zbMATH DE number 724100