Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties/6

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 $a$.

The quotient structure $\struct {S / \RR, \circ_\RR}$ is a commutative idempotent semigroup.

The equivalence classes under $\RR$ are anticommutative subsemigroups of $\struct {S, \circ}$.

Proof
From Idempotent Semigroup: Relation induced by Inverse Element: $5$:
 * $\RR$ is a congruence relation on $\struct {S, \circ}$.

Hence $\struct {S / \RR, \circ_\RR}$ is indeed a quotient structure, and Quotient Structure is Well-Defined applies.

From Quotient Structure of Semigroup is Semigroup, we have that $\struct {S / \RR, \circ_\RR}$ is a semigroup.

Then we have:

Thus $\struct {S / \RR, \circ_\RR}$ is an idempotent semigroup.