Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties

Example of Idempotent Semigroup
Let $\struct {S, \circ}$ be an idempotent semigroup.

Let $\RR$ be the relation on $S$ defined as:
 * $\forall a, b \in S: a \mathrel \RR b \iff \paren {a \circ b \circ a = b \land b \circ a \circ b = a}$

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

Then the following properties can be deduced: