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 phrase|[[quick phrase::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 $cac=c$.

Relation with other properties

Stronger properties

• Group
• Clifford semigroup
• Completely regular semigroup
• Inverse semigroup

External links

• Planetmath page