Definition:Power of Element/Magma

Definition
Let $\struct {S, \circ}$ be a magma which has no identity element.

Let $a \in S$.

Let the mapping $\circ^n a: \N_{>0} \to S$ be recursively defined as:


 * $\forall n \in \N_{>0}: \circ^n a = \begin{cases}

a & : n = 1 \\ \paren {\circ^r a} \circ a & : n = r + 1 \end{cases}$

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

Also see

 * Definition:Power of Element of Magma with Identity