Definition:Power of Element/Magma with Identity

Definition

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 $\circ$).

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:

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

Furthermore:

$a^0 = \circ^0 a = e$

Also see

• Definition:Power of Element of Magma

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers