Homomorphism of Powers/Integers

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Theorem

Let $\struct {T_1, \odot}$ and $\struct {T_2, \oplus}$ be monoids.

Let $\phi: \struct {T_1, \odot} \to \struct {T_2, \oplus}$ be a (semigroup) homomorphism.

Let $a$ be an invertible element of $T_1$.

Let $n \in \Z$.

Let $\odot^n$ and $\oplus^n$ be as defined as in Index Laws for Monoids.


Then:

$\forall n \in \Z: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$


Proof

By Homomorphism of Powers: Natural Numbers, we need show this only for negative $n$, that is:

$\forall n \in \N^*: \map \phi {\map {\odot^{-n} } a} = \map {\oplus^{-n} } {\map \phi a}$

But by Homomorphism with Identity Preserves Inverses:

$\map \phi {a^{-1} } = \paren {\map \phi a}^{-1}$

Hence by Homomorphism of Powers: Natural Numbers:

$\map {\oplus^{-n} } {\map \phi a} = \map {\oplus^n} {\map \phi {a^{-1} } } = \map \phi {\map {\odot^n} {a^{-1} } } = \map \phi {\map {\odot^{-n} } a}$

$\blacksquare$


Sources