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 $X.$
Definitions: 1 2 3
Proofs: 4 5 6
Propositions: 7
