Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties/3

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

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

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

Let:

Then:
 * $x \mathrel \RR z$

Proof
From we take it for granted that $\circ$ is associative.

Hence parentheses will be employed whenever it makes groupings of operations more clear.