Idempotent Semigroup/Properties/2

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Property of Idempotent Semigroup

Let $\struct {S, \circ}$ be an idempotent semigroup.


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

Then for all $z \in S$:

$x \circ z \circ y \circ z = x \circ z$

and:

$y \circ z \circ x \circ z = y \circ z$


Proof

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


\(\ds \forall z \in S: \, \) \(\ds x \circ z \circ y \circ z\) \(=\) \(\ds x \circ y \circ z \circ y \circ z\) as $x \circ y = x$ by hypothesis
\(\ds \) \(=\) \(\ds x \circ y \circ z\) Definition of Idempotent Operation: $\paren {y \circ z} \circ \paren {y \circ z} = y \circ z$
\(\ds \) \(=\) \(\ds x \circ z\) as $x \circ y = x$ by hypothesis


\(\ds \forall z \in S: \, \) \(\ds y \circ z \circ x \circ z\) \(=\) \(\ds y \circ x \circ z \circ x \circ z\) as $y \circ x = y$ by hypothesis
\(\ds \) \(=\) \(\ds y \circ x \circ z\) Definition of Idempotent Operation: $\paren {x \circ z} \circ \paren {x \circ z} = x \circ z$
\(\ds \) \(=\) \(\ds y \circ z\) as $y \circ x = y$ by hypothesis

$\blacksquare$


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