Definition:Power of Element

Theorem
Let $$\left({S, \circ}\right)$$ be an algebraic structure. Let $$x \in S$$.

Let $$\left({x_1, x_2, \ldots, x_n}\right)$$ be the ordered $n$-tuple defined by $$x_k = x$$ for each $$k \in \mathbb{N}_n$$.

Then:

$$\prod_{k=1}^n x_k = \circ^n x$$

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

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

It can be defined inductively as:

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

or

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

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

Proof
Follows directly from Recursive Mapping to Semigroup.