# Element Commutes with Square in Semigroup

## Theorem

Let $\left({S, \circ}\right)$ be a semigroup.

Let $x \in S$.

Then $x$ commutes with $x \circ x$.

## Proof

 $\displaystyle x \circ \left({x \circ x}\right)$ $=$ $\displaystyle \left({x \circ x}\right) \circ x$ by definition of semigroup: $\circ$ is associative

$\blacksquare$