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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 and Michael Kinyon

(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