Special involutions

Summary

This article takes the form of a survey of results concerning a particular kind of involution — called, for want of a better term, ‘special’ — on a semigroup or ring. Most of the material is based on recent joint work by David Easdown (Sydney) and the author ([4], [5]); the remainder, comprising Section 3, is taken from [7].

Introduction

By an involution on a semigroup S we mean a mapping * : S → S, x ↦ x*, such that

Note that the second requirement ensures that * is a permutation of S further, if S is commutative then the identity permutation of S is an involution. An involution on a ring R is a mapping * : R → R that is an automorphism of the group (R, +) and an involution on the semigroup (R, ·).

We shall be concerned here with the following type of involution.

Definition. An involution * on a semigroup [ring] S is special if and only if, for every nonempty finite subset T of S,

Although our condition might, at first sight, appear somewhat complicated — and even artificial — it is satisfied frequently and in a variety of natural contexts. First, let us consider some examples.

Example 1. On a group G, inversion (x ↦ x⁻¹) is a special involution.

This is a trivial matter. Let T be a nonempty subset (not necessarily finite) of G, let t be an arbitrary member of T and let u, v ∈ G be such that tt⁻¹ = uv⁻¹. Then u = v.

Example 2. Let S denote the free semigroup on a nonempty set X.