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.

Recall:

From $x \circ y = y$ and $y \circ x = x$
 * $(1): \quad \forall z \in S: z \circ x \circ z \circ y = z \circ y$

using Idempotent Semigroup Properties: $1$

From $y \circ z = y$ and $z \circ y = z$
 * $(2): \quad \forall x \in S: y \circ x \circ z \circ x = y \circ x$

using Idempotent Semigroup Properties: $2$, exchanging $z$ and $x$

We have:

and:

Hence the result, by definition of $\RR$.