Medial groupoid varieties defined by infinite cyclic groups
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.
scientific article; zbMATH DE number 5646233
scientific article