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

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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 $a$.


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 Semigroup Axiom $\text S 0$: Closure we take it for granted that $\struct {S, \circ}$ is closed under $\circ$.

From Semigroup Axiom $\text S 1$: Associativity we take it for granted that $\circ$ is associative.

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


We have:

\(\ds \forall z \in S: \, \) \(\ds z \circ x\) \(=\) \(\ds \paren {z \circ y} \circ \paren {z \circ x}\) Properties of Idempotent Semigroup: $1$
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ x} \circ \paren {z \circ x}\) \(=\) \(\ds \paren {z \circ x} \circ \paren {z \circ y} \circ \paren {z \circ x}\)
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ x}\) \(=\) \(\ds \paren {z \circ x} \circ \paren {z \circ y} \circ \paren {z \circ x}\) Definition of Idempotent Operation

Similarly:

\(\ds \forall z \in S: \, \) \(\ds z \circ y\) \(=\) \(\ds \paren {z \circ x} \circ \paren {z \circ y}\) Properties of Idempotent Semigroup: $1$
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ y} \circ \paren {z \circ y}\) \(=\) \(\ds \paren {z \circ y} \circ \paren {z \circ x} \circ \paren {z \circ y}\)
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ y}\) \(=\) \(\ds \paren {z \circ y} \circ \paren {z \circ x} \circ \paren {z \circ y}\) Definition of Idempotent Operation

Hence by definition of $\RR$:

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

$\Box$


Then:

\(\ds \forall z \in S: \, \) \(\ds x \circ z\) \(=\) \(\ds x \circ \paren {z \circ x} \circ z\) Definition of Idempotent Operation
\(\ds \) \(=\) \(\ds x \circ \paren {z \circ y} \circ \paren {z \circ x} \circ z\) Properties of Idempotent Semigroup: $1$
\(\ds \) \(=\) \(\ds \paren {x \circ z} \circ \paren {y \circ z} \circ \paren {x \circ z}\) Semigroup Axiom $\text S 1$: Associativity

and:

\(\ds \forall z \in S: \, \) \(\ds y \circ z\) \(=\) \(\ds y \circ \paren {z \circ y} \circ z\) Definition of Idempotent Operation
\(\ds \) \(=\) \(\ds y \circ \paren {z \circ x} \circ \paren {z \circ y} \circ z\) Properties of Idempotent Semigroup: $1$
\(\ds \) \(=\) \(\ds \paren {y \circ z} \circ \paren {x \circ z} \circ \paren {y \circ z}\) Semigroup Axiom $\text S 1$: Associativity

Hence by definition of $\RR$:

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

$\blacksquare$


Sources

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