Index Laws for Monoids

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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 $\struct {S, \circ}$ 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:

Negative Index

$\forall n \in \Z: \paren {a^n}^{-1} = a^{-n} = \paren {a^{-1} }^n$

Sum of Indices

$\forall m, n \in \Z: a^{n + m} = a^n \circ a^m$

Product of Indices

$\forall m, n \in \Z: a^{n m} = \paren {a^m}^n = \paren {a^n}^m$