Regular element in a semigroup

Definition

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

• There exists $b\in S$ such that $aba=a$
• There exists $c\in S$ such that $aca=a$ and $cac=c$. Such a $c$ is termed an inverse element (in the semigroup sense) for $a$

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.