Definition:Power of Element/Monoid

Definition
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.