Definition:Power of Element/Monoid

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

Let $a \in S$.

Let $n \in \N$.

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 $0$:

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


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

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

Invertible Element

Let $b \in S$ be invertible for $\circ$.

Let $n \in \Z$.

The definition $b^n = \map {\circ^n} b$ as the $n$th power of $b$ in $\left({S, \circ}\right)$ can be extended to include the inverse of $b$:

$b^{-n} = \paren {b^{-1} }^n$