Special Involutions: A Survey

W. D. Munn
Department of Mathematics
University of Glasgow
Glasgow G12 8QW, U.K.
e-mail: wdm@maths.gla.ac.uk

Abstract:

An involution on a semigroup $S$ is a mapping $^*:S\rightarrow S$, $x\mapsto x^*$ such that

$$(\forall \; x,y \in S) \quad (xy)^*=y^*x^* \mbox{ and } x^{**}=x.$$

As in [1], we say that an involution $^*$ on $S$ is special if and only if, for every nonempty finite subset $T$ of $S$,

$$(\exists \;t\in T)(\forall \; u,v\in T) \quad tt^*=uv^*\Rightarrow u=v.$$

These concepts extend readily from semigroups to algebras over certain subfields of the complex field.

The study of special involutions will be traced from its origin in a result on semigroup algebras to some recent developments. It turns out that many naturally occurring involutions are of this type.

Reference

1. D. Easdown and W. D. Munn. On semigroups with involution. Bull. Austral. Math. Soc. 48 (1993), 93-100.