Index Laws for Monoids

Theorem

These results are an extension of the results in Index Laws for Semigroups in which the domain of the indices is extended to include all integers.

Let $\left ({S, \odot}\right)$ be a monoid whose identity is $e_S$.

Let $a \in S$ be invertible for $\odot$.

Let $n \in \N$.

Let $a^n = \odot^n \left({a}\right)$ extend the definition in Power of an Element to include the identity as an index:

$a^n = \begin{cases} e_S : & n = 0 \\ a^x \odot a : & n = x + 1 \end{cases}$

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

Also, for each $n \in \N$ we can define:

$a^{-n} = \left({a^{-1}}\right)^n$

(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 we have the following results:

Negative Index

$\forall n \in \Z: \left({a^n}\right)^{-1} = a^{-n} = \left({a^{-1}}\right)^n$

Sum of Indices

$\forall m, n \in \Z: a^{n+m} = a^n \odot a^m$

Product of Indices

$\forall m, n \in \Z: a^{n m} = \left({a^m}\right)^n = \left({a^n}\right)^m$

Also see

• Integral Multiple

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 20$