Definition:Power of Element/Monoid

Definition
Let $\left({S, \circ}\right)$ be a monoid whose identity element is $e$.

Let $a \in S$.

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

The definition $a^n = \circ ^n \left({a}\right)$ as the $n$th power of $a$ in a semigroup can be extended to allow an exponent of $0$:


 * $a^n = \begin{cases}

e & : n = 0 \\ a^{n-1} \circ a & : n > 0 \end{cases}$

or


 * $n \cdot a = \begin{cases}

a & : n = 0 \\ \left({n - 1}\right) a \circ a & : n > 0 \end{cases}$

The validity of this definition follows from the fact that a monoid has an identity element.