Definition:Power of Element/Monoid/Invertible Element

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Definition

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.

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 $b$:

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