Idempotent Semigroup/Properties/1

Property of Idempotent Semigroup

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

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

Then for all $z \in S$:

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

and:

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

Proof

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

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

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

$\blacksquare$