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

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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:

\(\text {(1)}: \quad\) \(\ds \paren {a \circ b} \circ a\) \(=\) \(\ds a\) Definition of $\RR$
\(\text {(2)}: \quad\) \(\ds b \circ \paren {a \circ b}\) \(=\) \(\ds b\) Definition of $\RR$


Then we have:

\(\text {(3)}: \quad\) \(\ds a \circ \paren {a \circ b}\) \(=\) \(\ds a \circ b\) Definition of Idempotent Operation
\(\text {(4)}: \quad\) \(\ds \paren {a \circ b} \circ b\) \(=\) \(\ds a \circ b\) Definition of Idempotent Operation


Then:

\(\ds \forall z \in S: \, \) \(\ds a \circ z\) \(\RR\) \(\ds \paren {a \circ b} \circ z\) Idempotent Semigroup: Relation induced by Inverse Element: $1$, $(1)$ and $(3)$
\(\, \ds \land \, \) \(\ds z \circ a\) \(\RR\) \(\ds z \circ \paren {a \circ b}\)
\(\ds \forall z \in S: \, \) \(\ds \paren {a \circ b} \circ z\) \(\RR\) \(\ds b \circ z\) Idempotent Semigroup: Relation induced by Inverse Element: $2$, $(2)$ and $(4)$
\(\, \ds \land \, \) \(\ds z \circ \paren {a \circ b}\) \(\RR\) \(\ds z \circ b\)
\(\ds \leadsto \ \ \) \(\ds \forall z \in S: \, \) \(\ds z \circ a\) \(\RR\) \(\ds z \circ b\) Definition of Equivalence Relation: $\RR$ is transitive
\(\ds \land \ \ \) \(\ds a \circ z\) \(\RR\) \(\ds b \circ z\)


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

$\blacksquare$


Sources

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