Unary iterative hyperidentities for semigroups and inverse semigroups

Iterative hyperidentities are hyperidentities that have the special form F-a((x) over bar) = Fa+b((x) over bar), (x) over bar = (x(1),...,x(k)), for some k greater than or equal to 1. Such hyperidentities have been studied by Deneche and Poschel, Schweigert, and Wismath. In [9], minimal iterative hyperidentities of arities k greater than or equal to 2 were found for certain varieties of semigroups. Here we consider unary iterative hyperidentities and the semigroup varieties V-n,V-m defined by identities x(n) = x(n+m). For each such variety V-n,V-m, we produce the minimal unary iterative hyperidentity it satisfies; and for each unary iterative hyperidentity F-a(x) = Fa+b(x), we find the largest semigroup variety to satisfy it. We also solve the same two problems for varieties of inverse semigroups, in the first consideration of hyperidentities for varieties of inverse semigroups. In both the ordinary and inverse cases, we also examine the interaction of the variety-to-hyperidentity and hyperidentity-to-variety processes.