\begin{definition}[Definition:Power of Element/Magma]
Let $\struct {S, \circ}$ be a magma which has no identity element.
Let $a \in S$.
Let the mapping $\circ^n a: \N_{>0} \to S$ be recursively defined as:
:$\forall n \in \N_{>0}: \circ^n a = \begin{cases}
a & : n = 1 \\
\paren {\circ^r a} \circ a & : n = r + 1
\end{cases}$
The mapping $\circ^n a$ is known as the '''$n$th power of $a$ (under $\circ$)'''.
\end{definition}