# Regular element in a semigroup

From Groupprops

## Definition

Let be a semigroup, i.e., a set with an associative binary operation. An element is termed a **regular element** of if it satisfies the following equivalent conditions:

- There exists such that
- There exists such that and . Such a is termed an inverse element (in the semigroup sense) for

A semigroup where every element is regular is termed a regular semigroup, and a semigroup where every element has a unique inverse element in the above sense is termed an inverse semigroup.