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# Portugaliae Mathematica

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**Volume 74, Issue 3, 2017, pp. 243–255**

**DOI: 10.4171/PM/2005**

Published online: 2018-02-08

Commutativity theorems for groups and semigroups

Francisco Araújo^{[1]}and Michael Kinyon

^{[2]}(1) Colégio Planalto, Lisboa, Portugal

(2) University of Denver, USA and Universidade de Lisboa, Portugal

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.

*Keywords: *Separative semigroup, inverse semigroup, completely regular, commutativity theorems

Araújo Francisco, Kinyon Michael: Commutativity theorems for groups and semigroups. *Port. Math.* 74 (2017), 243-255. doi: 10.4171/PM/2005