Definition:Power of Element/Magma with Identity

Definition
Let $\left({T, \oplus}\right)$ be a magma with an identity element $e$.

Let $a \in T$.

Let the mapping $\oplus^n: \N \to T$ be defined as:


 * $\forall n \in \N: \oplus^n a = g_a \left({n}\right)$

where $g_a: \N \to T$ is the recursively defined mapping:


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

e & : n = 0 \\ g_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 $\oplus$).

Notation
Furthermore:
 * $a^0 = \oplus^0 a = e$

Also see

 * Definition:Power of Element of Magma