Idempotent Semigroup/Properties/2

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

$\blacksquare$