Idempotent Semigroup/Examples/Relation induced by Inverse Element/Properties/4

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 an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
Thus $\RR$ is seen to be reflexive.

Symmetry
Thus $\RR$ is seen to be symmetric.

Transitivity
Let $a \mathrel \RR b$ and $b \mathrel \RR c$.

Thus we have:

and:

Let:

We have:

and:

Hence using Idempotent Semigroup Properties: $1$:

Then we have:

and:

Hence using Idempotent Semigroup Properties: $2$:

Thus we have:

Hence by Idempotent Semigroup: Relation induced by Inverse Element: $3$:


 * $x \mathrel \RR z$

That is:

Thus $\RR$ is seen to be transitive.

$\RR$ has been shown to be reflexive, symmetric and transitive.

Hence by definition $\RR$ is an equivalence relation.