Index_Laws_for_Monoids

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 $\struct {S, \circ}$ 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 $a$ $a^n = \begin{cases} e : & n = 0 \\ a^{n - 1} \circ a : & n > 0 \\ \paren {a^{-n}}^{-1} : & n < 0 \end{cases}$ Then we have the following results