Definition:Power of Element/Monoid
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Definition
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $a \in S$.
Let $n \in \N$.
The definition $a^n = \map {\circ^n} a$ as the $n$th power of $a$ in a semigroup can be extended to allow an exponent of $0$:
- $a^n = \begin {cases} e & : n = 0 \\ a^{n - 1} \circ a & : n > 0 \end{cases}$
or:
- $n \cdot a = \begin {cases} e & : n = 0 \\ \paren {\paren {n - 1} \cdot a} \circ a & : n > 0 \end{cases}$
The validity of this definition follows from the fact that a monoid has an identity element.
Invertible Element
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 $b$:
- $b^{-n} = \paren {b^{-1} }^n$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.11)$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids