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 |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.19 \ \text {(c)}$