# 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:

$\forall n \in \N_{>0}: a^{n + 1} = \circ^{n + 1} a = \paren {\circ^n a} \circ a = \paren {a^n} \circ a$