Power of an Element

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Theorem

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

Then:

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

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 $n x$.

$\blacksquare$