Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties/1

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 $x \circ y = y$ and $y \circ x = x$.

Then for all $z \in S$:
 * $\paren {z \circ x} \mathrel \RR \paren {z \circ y}$

and:
 * $\paren {x \circ z} \mathrel \RR \paren {y \circ z}$

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

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

We have:

Similarly:

Hence by definition of $\RR$:


 * $\forall z \in S: \paren {z \circ x} \mathrel \RR \paren {z \circ y}$

Then:

and:

Hence by definition of $\RR$:


 * $\forall z \in S: \paren {x \circ z} \mathrel \RR \paren {y \circ z}$