Medial groupoid varieties defined by infinite cyclic groups
type: Item
instance of: scholarly article
title: Medial groupoid varieties defined by infinite cyclic groups
zbMATH Open document ID: 1181.08003
author: J. Blot
published in: Algebra Universalis
publication date: 2009-12-14T00:00:00Z
review text: Let \(G\) be an abelian group with two generators \(\alpha\) and \(\beta\). The group ring \(\mathbb{Z}[G]\) can be considered as a groupoid under the operation \(x\cdot y=\alpha x+\beta y\). The set of all identities satisfied by this groupoid is denoted by \(\Sigma(G;\alpha,\beta)\). If the constant \(e\), representing the zero of \(\mathbb{Z}[G]\), is added to the similarity type, the set of all identities satisfied by \(\langle\mathbb{Z}[G];\cdot,e\rangle\) is denoted by \(\Sigma^e(G;\alpha,\beta)\). The author shows that when \(H\) is the infinite cyclic group generated by \(\delta\) and \(a\) and \(b\) are relatively prime integers, then the set \(\Sigma(H;\delta^a,\delta^b)\) has a finite basis if and only if \((a,b)\) is \((\pm1,\pm1)\), \((\pm1,\mp1)\), \((1,0)\) or \((0,1)\). (Earlier, a finite basis for the case \((a,b)=(\pm1,\pm1)\) was found by \textit{J. Ježek} and \textit{T. Kepka} [``Equational theories of medial groupoids'', Algebra Univers. 17, 174--190 (1983; Zbl 0538.08008)].) Also, finitely based sets of the form \(\Sigma^e(H;\delta^a,\delta^b)\) are characterized. Besides that, the author proves that the set \(\Sigma^e(G;\alpha,\beta)\) is always finitely based whenever the group \(G\) is finite.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 20N02
zbMATH DE Number: 5646233
zbMATH Keywords: medial groupoid
zbMATH Keywords: interchange law
zbMATH Keywords: finite basis
zbMATH Keywords: independent basis
MaRDI profile type: Publication
full work available at URL: s00012-009-0016-0
OpenAlex ID: W4234127691
cites work: Equational theories of medial groupoids
cites work: Identities common to four Abelian group operations with zero
cites work: Covering in the lattice of equational theories and some properties of term finite theories
cites work: Q3909096
DOI: 10.1007/S00012-009-0016-0
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description: scientific article; zbMATH DE number 5646233