Definition:Power of Element/Magma

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

Let $a \in T$.

Let the mapping $\oplus^n: \N_{>0} \to T$ be defined as:


 * $\forall n \in \N_{>0}: \oplus^n a = f_a \left({n}\right)$

where $f_a: \N_{>0} \to T$ is the recursively defined mapping:


 * $\forall n \in S: f_a \left({n}\right) = \begin{cases}

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

The mapping $\oplus^n a$ is known as the $n$th power of $a$ (under $\oplus$).

Also see

 * Definition:Power of Element of Magma with Identity