Definition:Power of Element
Definition
Magma
Let $\struct {S, \circ}$ be a magma which has no identity element.
Let $a \in S$.
Let the mapping $\circ^n a: \N_{>0} \to S$ be recursively defined as:
- $\forall n \in \N_{>0}: \circ^n a = \begin{cases}
a & : n = 1 \\ \paren {\circ^r a} \circ a & : n = r + 1 \end{cases}$
The mapping $\circ^n a$ is known as the $n$th power of $a$ (under $\circ$).
Magma with Identity
Let $\struct {S, \circ}$ be a magma with an identity element $e$.
Let $a \in S$.
Let the mapping $\circ^n a: \N \to S$ be recursively defined as:
- $\forall n \in S: \circ^n a = \begin{cases}
e & : n = 0 \\ \paren {\circ^r a} \circ a & : n = r + 1 \end{cases}$
The mapping $\circ^n a$ is known as the $n$th power of $a$ (under $\circ$).
Semigroup
Let $\struct {S, \circ}$ be a semigroup which has no identity element.
Let $a \in S$.
For $n \in \N_{>0}$, the $n$th power of $a$ (under $\circ$) is defined as:
- $\circ^n a = \begin{cases} a & : n = 1 \\ \paren {\circ^m a} \circ a & : n = m + 1 \end{cases}$
That is:
- $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a}$
which from the General Associativity Theorem is unambiguous.
Monoid
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 $b$:
- $b^{-n} = \paren {b^{-1} }^n$
Group
Let $\struct {G, \circ}$ be a group whose identity element is $e$.
Let $g \in G$.
Let $n \in \Z$.
The definition $g^n = \map {\circ^n} g$ as the $n$th power of $g$ in a monoid can be extended to allow negative values of $n$:
- $g^n = \begin{cases}
e & : n = 0 \\ g^{n - 1} \circ g & : n > 0 \\ \paren {g^{-n} }^{-1} & : n < 0 \end{cases}$
or
- $n \cdot g = \begin{cases}
e & : n = 0 \\ \paren {\paren {n - 1} \cdot g} \circ g & : n > 0 \\ -\paren {-n \cdot g} & : n < 0 \end{cases}$
The validity of this definition follows from the group axioms: $g$ has an inverse element.
Ring
Let $\struct {R, +, \circ}$ be a ring.
Let $r \in R$.
Let $n \in \Z_{>0}$ be the set of strictly positive integers.
The $n$th power of $r$ in $R$ is defined as the $n$th power of $r$ with respect to the semigroup $\struct {R, \circ}$:
- $\forall n \in \Z_{>0}: r^n = \begin {cases}
r & : n = 1 \\ r^{n - 1} \circ r & : n > 1 \end {cases}$
If $R$ is a ring with unity where $1_R$ is that unity, the definition extends to $n \in \Z_{\ge 0}$:
- $\forall n \in \Z_{\ge 0}: r^n = \begin {cases}
1_R & : n = 0 \\ r^{n - 1} \circ r & : n > 0 \end {cases}$
Field
Let $\struct {F, +, \circ}$ be a field with zero $0_F$ and unity $1_F$.
Let $a \in F^*$ where $F^*$ denotes the set of elements of $F$ without the zero $0_F$.
Let $n \in \Z$ be an integer.
The $n$th power of $a$ in $F$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$:
- $\forall n \in \Z: a^n = \begin {cases}
1_F & : n = 0 \\ a^{n - 1} \circ a & : n > 0 \\ \paren{a^{-1}}^{-n} & : n < 0 \end {cases}$
The definition of $n$th power of $a$ in $F$ as the the $n$th power of $a$ with respect to the monoid $\struct {F, \circ}$ can be extended to $0_F$ for positive values of $n$.
For all $n \in \Z_{\ge 0}$ the $n$th power of $0_F$ in $F$ is defined:
- $\paren{0_F}^n = \begin {cases}
1_F & : n = 0 \\ 0_F & : n > 0 \end {cases}$
It should be noted that for all $n < 0$ the $n$th power of $0_F$ is not defined.
Notation
Let $\circ^n a$ be the $n$th power of $a$ under $\circ$.
The usual notation for $\circ^n a$ in a general algebraic structure is $a^n$, where the operation is implicit and its symbol omitted.
In an algebraic structure in which $\circ$ is addition, or derived from addition, this can be written $n a$ or $n \cdot a$, that is, $n$ times $a$.
Thus:
- $a^1 = \circ^1 a = a$
and in general:
- $\forall n \in \N_{>0}: a^{n + 1} = \circ^{n + 1} a = \paren {\circ^n a} \circ a = \paren {a^n} \circ a$