Index Laws for Monoids

Theorem
These results are an extension of the results in Index Laws for Semigroup in which the domain of the indices is extended to include all integers.

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

Let $a \in S$ be invertible for $\circ$.

Let $n \in \N$.

Let $a^n$ be the $n$th power of $a$:


 * $a^n = \begin{cases}

e : & n = 0 \\ a^{n-1} \circ a : & n > 0 \\ \left({a^{-n}}\right)^{-1} : & n < 0 \end{cases}$

Then we have the following results:

Also see

 * Definition:Integral Multiple