Medial groupoid varieties defined by finite cyclic groups

Abstract

Let G be the abelian group generated by α and β. We write Σ(G; α, β) for the set of groupoid identities that are satisfied in the group ring \({\mathbb{Z}[G]}\) when the binary operation is αx + βy. When the constant e, representing the zero of \({\mathbb{Z}}\) , is added to the type, we write Σe(G; α, β) for the corresponding set of identities. In this paper, we assume that G is an infinite cyclic group with generator δ. We write Σa,b for Σ(G; δa, δb) and \({\Sigma^{e}_{a,b}}\) for Σe(G; δa, δb). For each pair of relatively prime integers a and b, we determine whether Σa,b is finitely based and whether \({\Sigma^{e}_ {a,b}}\) is finitely based. One of our results implies that Σe(G; α, β) is finitely based whenever G is finite.