Inverse semigroup

This is a variation of group|Find other variations of group | Read a survey article on varying group

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 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 $cac=c$.

Relation with other properties

Stronger properties

• Group
• Clifford semigroup

Weaker properties

• Regular semigroup

External links

• Planetmath page