Properties of Idempotent Semigroup

Properties of Idempotent Semigroup

Property $1$

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:

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

Property $2$

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:

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