Definition:Power of Element/Monoid

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

Let $a \in S$.

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


 * $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}$