Iterative hyperidentities for theA n,m varieties of semigroups

Iterative hyperidentities are hyperidentities of the special formF ^a (x ₁,...,x _k =F ^a+b (x ₁,...,x _k ). This type of hyperidentity has been considered by Denecke and Pöschel, and by Schweigert. Here we consider iterative hyperidentities for the variety A_n,m of commutative semigroups satisfyingx ⁿ =x ^n+m ,n,m 1. We introduce two parameters(m, n) and(m) associated withn andm, and show thatA _nn,m satisfies the iterative hyperidentitiesF (x ₁,...,x _k =F ^+b (x ₁,...,x _k ) for every arityk. Moreover, the numbers and are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for A_n,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.Presented by W. Taylor.Research supported by NSERC of Canada