Regular semigroup

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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 $cac = c$ .