Commutativity theorems for groups and semigroups

Abstract

In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup S we have x^p y^p = y^p x^p and x^q y^q = y^q x^q for all x,y\in S where p and q are relatively prime, then S is commutative. In a separative or inverse semigroup S , if there exist three consecutive integers i such that (xy)^i = x^i y^i for all x,y\in S , then S is commutative. Finally, if S is a separative or inverse semigroup satisfying (xy)^3=x^3y^3 for all x,y\in S , and if the cubing map x\mapsto x^3 is injective, then S is commutative.