# Index Laws for Monoid

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## Theorem

### Sum of Indices

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

For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$:

$a^n = \begin{cases} e & : n = 0 \\ a^x \circ a & : n = x + 1 \end{cases}$

That is:

$a^n = \underbrace{a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \left({a}\right)$

while:

$a^0 = e$

Then:

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

### Product of Indices

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.

For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$.

Then:

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