Powers of Commutative Elements in Semigroups

Theorem

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

Let $a, b \in S$ both be cancellable elements of $S$.

Then the following results hold:

Commutativity of Powers

$\forall m, n \in \N^*: a^m \circ b^n = b^n \circ a^m \iff a \circ b = b \circ a$

Product of Commutative Elements

$\forall n \in \N, n > 1: \left({x \circ y}\right)^n = x^n \circ y^n \iff x \circ y = y \circ x$