Index Laws for Semigroup

Theorem
Let $\left ({S, \circ}\right)$ be a semigroup.

Let $a \in S$.

Let $n \in \N_{>0}$.

Let $\circ^n \left({a}\right) = a^n$ be the $n$th power of $a$:


 * $a^n = \begin{cases}

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

Then the following results hold: