Definition:Power of Element/Semigroup

Definition
Let $\left({T, \oplus}\right)$ be a semigroup which has no identity element.

Let $a \in T$.

For $n \in \N_{>0}$, the $n$th power of $a$ (under $\oplus$) is defined as:


 * $\oplus^n a = \begin{cases}

a & : n = 1 \\ \left({\oplus^m a}\right) \oplus a & : n = m + 1 \end{cases}$

That is:
 * $a^n = \underbrace{a \oplus a \oplus \cdots \oplus a}_{n \text{ copies of } a} = \oplus^n \left({a}\right)$

which from the General Associativity Theorem is unambiguous.

Also see

 * Definition:Power of Element of Magma with Identity