# Index Laws for Monoids/Sum of Indices It has been suggested that this article or section be renamed: Should have invertible elements referenced in its name One may discuss this suggestion on the talk page. It has been suggested that this page or section be merged into Index Laws/Sum of Indices/Monoid. (Discuss)

## Theorem

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

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

Let $n \in \N$.

Let $a^n = \map {\circ^n} a$ 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:

$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{\text {$n$instances} } = \map {\circ^n} a$

For each $n \in \N$ we define:

$a^{-n} = \paren {a^{-1} }^n$

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:

$\forall n \in \Z: \map {g_c} n = \map {\circ^n} c$

By the Index Law for Monoids: Negative Index, $\map {g_a} n$ 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 $\struct {\N, +}$ to $\struct {S, \circ}$.

From the definition of Power of Element of Monoid, $\map {g_a} 0$ is the identity for $\circ$.

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

 $\, \ds \forall n > 0: \,$ $\ds \map {h_a} {-n}$ $=$ $\ds \paren {\map {h_a} n}^{-1}$ $\ds$ $=$ $\ds \paren {a^n}^{-1}$ $\ds$ $=$ $\ds a^{-n}$ $\ds$ $=$ $\ds \map {g_a} {-n}$

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

$\blacksquare$