Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties/5

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 $a$.

$\RR$ is a congruence relation for $\circ$.

Proof
From Idempotent Semigroup: Relation induced by Inverse Element: $4$:


 * $\RR$ is an equivalence relation.

Let $a, b \in S$ be arbitrary such that $a \mathrel \RR b$.

Then:

Then we have:

Then:

As $a$ and $b$ are arbitrary, the result follows by Congruence Relation iff Compatible with Operation.