Definition:Power of Element/Notation

Notation for Power of Element
Let $\left({S, \circ}\right)$ be an algebraic structure.

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:
 * $\forall n \in \N_{>0}: a^{n + 1} = \circ^{n + 1} a = \left({\circ^n a}\right) \circ a= \left({a^n}\right) \circ a$