Index Laws/Sum of Indices/Notation

Notation for Index Laws: Sum 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:
 * $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \map {\circ^n} a$

Recall the index law for sum of indices:


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

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

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


 * $\left({n + m}\right) a = n a + m a$

or:


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