Definition:Power of Element/Notation/Semigroup

Notation for Power of Element of Semigroup

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 $n a$ or $n \cdot a$, that is, $n$ times $a$.

Thus:

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

and in general:

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

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 $n a$.