# Regular 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
View a complete list of semigroup properties
QUICK PHRASES: semigroup with all elements regular, regular being a weak version of invertible

## 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 $(S,*)$ is termed a regular semigroup if every $a \in S$ is regular, i.e.:

• For every $a \in S$, there exists a $b \in S$ such that $aba = a$
• Equivalently, for every $a \in S$, there exists a $c \in S$ such that $aca = a$ and $cac = c$.