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 $\odot$ that also commute.

Then the following results hold.