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.

Definition: Exponentiation in a Monoid

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 $X.$

Definitions: 1 2 3
Proofs: 4 5 6
Propositions: 7

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983