Index Laws/Sum of Indices/Notation

Notation for Index Laws: Sum of Indices
Let $\left({T, \oplus}\right)$ be a semigroup.

Let $a \in T$.

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


 * $a^n = \begin{cases}

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

... that is:
 * $a^n = \underbrace{a \oplus a \oplus \cdots \oplus a}_{n \text{ copies of } a} = \oplus^n \left({a}\right)$

Recall the index law for sum of indices:


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

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

When additive notation $\left({T, +}\right)$ is used, the following is a common convention:


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

or:


 * $\forall m, n \in \N_{>0}: \left({n + m}\right) \cdot a = n \cdot a + m \cdot a$