\begin{definition}[Definition:Power of Element/Monoid]
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $a \in S$.
Let $n \in \N$.
The definition $a^n = \map {\circ^n} a$ 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:
:$n \cdot a = \begin {cases}
e & : n = 0 \\
\paren {\paren {n - 1} \cdot a} \circ a & : n > 0
\end{cases}$
The validity of this definition follows from the fact that a monoid has an identity element.
\end{definition}