# Index Laws for Semigroup

## Theorem

### Sum of Indices

Let $\struct {S, \circ}$ be a semigroup.

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

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

That is:

$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$

Then:

$\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$

### Product of Indices

Let $\struct {S, \circ}$ be a semigroup.

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

Then:

$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$

## Source of Name

The name index laws originates from the name index to describe the exponent $y$ in the power $x^y$.