# Inverse semigroup

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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 phrase|[[quick phrase::semigroup where every element has a unique inverse in the semigroup sense (i.e., without reference to neutral elements)}}

## 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 $(S,*)$ is termed an inverse semigroup if for every $a \in S$, there is a unique $c \in S$ satisfying the conditions $aca = a$ and $cac = c$.