Regular semigroup

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 $$cac = c$$.

Stronger properties

 * Weaker than::Group
 * Weaker than::Clifford semigroup
 * Weaker than::Completely regular semigroup
 * Weaker than::Inverse semigroup