Iterative hyperidentities for theA n,m varieties of semigroups

Abstract

Iterative hyperidentities are hyperidentities of the special formFa(x1,...,xk=Fa+b(x1,...,xk). This type of hyperidentity has been considered by Denecke and Poschel, and by Schweigert. Here we consider iterative hyperidentities for the variety An,m of commutative semigroups satisfyingxn=xn+m,n,m ≥ 1. We introduce two parametersγ(m, n) andβ(m) associated withn andm, and show thatAnn,m satisfies the iterative hyperidentitiesFγ(x1,...,xk =Fγ+b(x1,...,xk) for every arityk. Moreover, the numbersγ andβ are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for An,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.