Index Laws/Product of Indices/Notation

Notation for Index Laws: Product of Indices
Let $\struct {S, \circ}$ be a semigroup.

Let $a \in S$.

Let $a^n$ be defined as the power of an element of a magma:


 * $a^n = \begin{cases}

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

... that is:
 * $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$

Recall the index law for product of indices:


 * $\circ^{n m} a = \circ^m \paren {\circ^n a} = \circ^n \paren {\circ^m a}$

This result can be expressed:
 * $a^{n m} = \paren {a^n}^m = \paren {a^m}^n$

When additive notation $\struct {S, +}$ is used, the following is a common convention:


 * $\paren {n m} a = m \paren {n a} = n \paren {m a}$

or:


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