Index Laws/Product of Indices/Notation

Notation for Index Laws: Product 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 product of indices:


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

This result can be expressed:
 * $a^{n m} = \left({a^n}\right)^m = \left({a^m}\right)^n$

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


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

or:


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