Definition:Power of Element/Notation/Semigroup

Notation for Power of Element of Semigroup
Let $\left({S, \oplus}\right)$ be a semigroup.

Let $a \in S$.

Thus the power of an element of a semigroup is defined recursively as:


 * $a^n = \begin{cases}

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

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

When the operation is addition of numbers or another commutative operation derived from addition, the following symbology is often used:


 * $n a = \begin{cases}

a & : n = 1 \\ \left({n - 1}\right) a + a & : n > 1 \end{cases}$

Sometimes, for clarity, $n \cdot a$ is preferred to $n a$.