On groups generated by involutions of a semigroup

TL;DR

This paper explores the structure of groups generated by involutions on semigroups, analyzing their properties for specific classes and characterizing which groups can arise from such involutions.

Contribution

It provides a detailed investigation of the groups generated by involutions on semigroups and characterizes the groups that can be realized as such involution-generated groups.

Findings

Characterization of groups generated by involutions on certain semigroups

Identification of classes of semigroups with specific involution groups

Conditions under which a group is isomorphic to C(S) for some semigroup S

Abstract

An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function f:S->S that satisfies f(xy)=f(y)f(x) and f(f(x))=x for all x,y in S. The set I(S) of all such involutions on S generates a subgroup C(S)=<I(S)> of the symmetric group Sym(S) on the set S. We investigate the groups C(S) for certain classes of semigroups S, and also consider the question of which groups are isomorphic to C(S) for a suitable semigroup S.