Definition:Power of Element/Semigroup

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Definition

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.


Notation

Let $\paren {S, \circ}$ be a semigroup.

Let $a \in S$.

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 $n a$, that is, $n$ times $a$.


Thus:

$a^1 = \circ^1 a = a$

and in general:

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


When the operation is addition of numbers or another commutative operation derived from addition, the following symbology is often used:

$n a = \begin{cases} a & : n = 1 \\ \paren {n - 1} a + a & : n > 1 \end{cases}$

Sometimes, for clarity, $n \cdot a$ is preferred to $n a$.


Also see


Sources