Index Laws for Monoids/Sum of Indices

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Theorem

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

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

Let $n \in \N$.

Let $a^n = \circ^n \left({a}\right)$ be defined as the power of an element of a monoid:

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

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


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

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

Then:

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


Proof

For each $c \in S$ which is invertible for $\circ$, we define the mapping $g_c: \Z \to S$ as:

$\forall n \in \Z: g_c \left({n}\right) = \circ^n \left({c}\right)$

By the Index Law for Monoids: Negative Index, $g_a \left({n}\right)$ is invertible for all $n \in \Z$.

By definition of Power of Element of Monoid, the restriction of $g_a$ to $\N$ is a homomorphism from $\left({\N, +}\right)$ to $\left({S, \circ}\right)$.

From the definition of Power of Element of Monoid, $g_a \left({0}\right)$ is the identity for $\circ$ by definition.

Hence, by the Extension Theorem for Homomorphisms, there is a unique homomorphism $h_a: \left({\N, +}\right) \to \left({S, \circ}\right)$ which coincides in $\N$ with $g_c$.

But by Homomorphism with Identity Preserves Inverses:

\(\, \displaystyle \forall n > 0: \, \) \(\displaystyle h_a \left({-n}\right)\) \(=\) \(\displaystyle \left({h_a \left({n}\right)}\right)^{-1}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({a^n}\right)^{-1}\)
\(\displaystyle \) \(=\) \(\displaystyle a^{-n}\)
\(\displaystyle \) \(=\) \(\displaystyle g_a \left({-n}\right)\)


Hence $h_a = g_a$ and so $g_a$ is a homomorphism and so the result follows.

$\blacksquare$


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