Idempotent Semigroup/Examples

Examples of Idempotent Semigroups

Idempotent Semigroup with Relation induced by Inverse

Let $\struct {S, \circ}$ be an idempotent semigroup.

Let $\RR$ be the relation on $S$ defined as:

$\forall a, b \in S: a \mathrel \RR b \iff \paren {a \circ b \circ a = a \land b \circ a \circ b = b}$

That is, such that $a$ is the inverse of $b$ and $b$ is the inverse of $a$.

Then the following properties can be deduced: