# Commutativity theorems for groups and semigroups

### Francisco Araújo

Colégio Planalto, Lisboa, Portugal### Michael Kinyon

University of Denver, USA and Universidade de Lisboa, Portugal

## 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.

## Cite this article

Francisco Araújo, Michael Kinyon, Commutativity theorems for groups and semigroups. Port. Math. 74 (2017), no. 3, pp. 243–255

DOI 10.4171/PM/2005