# Inverse 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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QUICK PHRASES: semigroup where every element has a uniqueinversein the semigroup sense (i.e., without reference to neutral elements)

## Contents

## Definition

### Symbol-free definition

An **inverse semigroup** is a semigroup (i.e., a set with an associative binary operation) where every element has a unique inverse in the semigroup sense.

### Definition with symbols

A semigroup is termed an **inverse semigroup** if for every , there is a unique satisfying the conditions and .