Index Laws for Semigroup

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

Let $a \in S$.

Let $n \in \N^*$.

Let $\odot^n \left({a}\right) = a^n$ be defined as in Power of an Element:


 * $a^n = \begin{cases}

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

... that is:
 * $a^n = a \odot a \odot \cdots \left({n}\right) \cdots \odot a = \odot^n \left({a}\right)$

(Note that the notation $a^n$ and $\odot^n \left({a}\right)$ mean the same thing. The former is more compact and readable, but the latter is more explicit and can be more useful when more than one structure is under consideration.)

Then the following results hold: