# Completely regular semigroup

From Groupprops

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This article defines a semigroup property: a property that can be evaluated to true/false for any given semigroup

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## Contents

## Definition

A **completely regular semigroup** is a semigroup (i.e., a set with associative binary operation) where the subsemigroup generated by any element is a group under the induced multiplication.

Note that the identity elements for these groups need not, in general, coincide. If they *do* coincide, then the completely regular semigroup is a group.