Medial groupoid varieties defined by finite cyclic groups

Abstract

Let G be the group generated by δ of finite order n and let a and b be integers such that G is generated by δ ^a and δ ^b. We write ${\Sigma_{a,b}^n}$ for the set of groupoid identities that are satisfied in the group ring ${\mathbb{Z}[G]}$ when the binary operation is δ ^a x + δ ^b y. For every positive integer n, we show that ${\Sigma_{1,1}^n}$ and ${\Sigma_{0,1}^n}$ are finitely based. When n is not a multiple of 6, we give a finite basis for ${\Sigma_{n-1,1}^n}$.