Unary hyperidentities for varieties of inverse semigroups (Q1279671)

Hyperidentities of an algebra resemble identities, but contain variables representing operations as well as elements. There has been considerable interest in hyperidentities of varieties of semigroups, which have a rich enough structure to be interesting, but are relatively accessible. Because of the extra operation involved, hyperidentities of inverse semigroups seem somewhat more difficult to handle. Thus the authors of this paper are considering only the case where the operation symbols are unary. They made a start on this endeavour in a previous paper [Semigroup Forum 55, No. 2, 221-231 (1997; Zbl 0885.20034)] where they considered only unary iterative hyperidentities. In this paper they investigate all unary hyperidentities for inverse semigroups and show that they can be associated with varieties defined by the following sets of identities: \(x=y\), \(xx=x\), \(xx^{-1}=yy^{-1}\), \(xx^{-1}=x^{-1}x\), \(x^nx^{-1}=x^{-1}x^n\) (\(n\geq 1\)), \(x^n=x^{n+m}\) (\(n,m\geq 1\)).