# 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

View a complete list of semigroup properties

QUICK PHRASES: semigroup with all elements regular, regular being a weak version of invertible

## Contents

## Definition

### Symbol-free definition

A **regular semigroup** is a semigroup (i.e., a set with associative binary operation) in which every element is regular.

### Definition with symbols

A semigroup is termed a **regular semigroup** if every is regular, i.e.:

- For every , there exists a such that
- Equivalently, for every , there exists a such that and .