Special involutions
W. D. Munn · Cambridge University Press eBooks · 1996
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 .