Because a monoid $(X,\ast)$ ensures by definition the existence of a neutral element $e\in X$ and the associativity of its binary operation $”\ast”,$ we can define a new kind of operation in it, called the *exponentiation* of its elements.

Let \((X,\ast)\) be a monoid, \(x\in X\), and \(n\) a natural number. We define the **exponentiation to the \(n\)-th power** as the binary operation $”\ast”$ applied \(n\) times to the element \(x\). For \(n=0\), we set \(x^0:=e\). Formally:

\[x^n :=

\begin{cases}

e & \text{ if } n=0 \\

x\ast x^{n-1} & \text{ if } n > 0.

\end{cases}\]

In the above definition, $e\in X$ denotes the unique neutral element of $X.$

| | | | | created: 2014-06-08 23:03:49 | modified: 2019-02-10 01:53:57 | by: *bookofproofs* | references: [581]

[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983