Definition:Power of Element/Magma with Identity

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

Let $a \in S$.

Let the mapping $\circ^n a: \N \to S$ be recursively defined as:


 * $\forall n \in S: \circ^n a = \begin{cases}

e & : n = 0 \\ \left({\circ^r a}\right) \circ a & : n = r + 1 \end{cases}$

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

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

Also see

 * Definition:Power of Element of Magma