Definition:Power of Element/Semigroup

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.

Also see

 * Definition:Power of Element of Magma with Identity