Power of Element/Semigroup

Informal Definition
Let $\left({S, \circ}\right)$ be a semigroup.

Let $a \in S$.

Let $n \in \N_{>0}$.

Then $a^n$ is defined as:
 * $\underbrace{a \circ \ldots \circ a}_{\hbox{$n$ copies of $a$}}$

Definition
Let $\left({S, \circ}\right)$ be a semigroup.

Let $a \in S$.

Let $n \in \N_{>0}$.

Let $\left({a_1, a_2, \ldots, a_n}\right)$ be the ordered $n$-tuple defined by $a_k = a$ for each $k \in \N_n$.

Then:


 * $\displaystyle \prod_{k \mathop = 1}^n a_k =: \circ^n a$

In a general semigroup, we usually write $\circ^n a$ as $a^n$.

The construct $a^n$ is called the $n$th power of $a$.

In a semigroup in which $\circ$ is addition, or derived from addition, this can be written $n a$, that is, $n$ times $a$.

It can be defined inductively as:


 * $a^n = \begin{cases}

a & : n = 1 \\ a^{n-1} \circ a & : n > 1 \end{cases}$

or


 * $n a = \begin{cases}

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

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