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

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

Also see

• Powers of Commutative Elements in Semigroups
• Powers of Commutative Elements in Groups