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.