# Powers of Commutative Elements in Monoids

## Theorem

These results are an extension of the results in Powers of Commutative Elements in Semigroups in which the domain of the indices is extended to include all integers.

Let $\left ({S, \circ}\right)$ be a monoid whose identity is $e_S$.

Let $a, b \in S$ be invertible elements for $\circ$ that also commute.

Then the following results hold.

### Commutativity of Powers

$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$

### Product of Commutative Elements

$\forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$