Idempotent Semigroup/Examples/Relation induced by Inverse Element

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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 $a$.


Then the following properties can be deduced:

Properties

Property $1$

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:

$\paren {x \circ z} \mathrel \RR \paren {y \circ z}$


Property $2$

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

Then for all $z \in S$:

$\paren {x \circ z} \mathrel \RR \paren {y \circ z}$

and:

$\paren {z \circ x} \mathrel \RR \paren {z \circ y}$


Property $3$

Let:

\(\ds x \circ y\) \(=\) \(\ds y\)
\(\ds y \circ x\) \(=\) \(\ds x\)
\(\ds y \circ z\) \(=\) \(\ds y\)
\(\ds z \circ y\) \(=\) \(\ds z\)


Then:

$x \mathrel \RR z$


Property $4$

$\RR$ is an equivalence relation.


Property $5$

$\RR$ is a congruence relation for $\circ$.


Property $6$

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

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


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Sources

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