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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 $X.$

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

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