Homomorphism of Powers

Theorem
Let $$\left({T_1, \odot_1}\right)$$ and $$\left({T_2, \odot_2}\right)$$ be semigroups.

Let $$\phi: \left({T_1, \odot_1}\right) \to \left({T_2, \odot_2}\right)$$ be a homomorphism.

Naturally Ordered Semigroup
Let $$\left({S, \circ; \preceq}\right)$$ be a naturally ordered semigroup.

Let $$\odot_1^n$$ and $$\odot_2^n$$ be as defined as in Recursive Mapping to Semigroup.

Then $$\forall a \in T_1: \forall n \in \left({S^*, \circ; \preceq}\right): \phi \left({\odot_1^n \left({a}\right)}\right) = \odot_2^n \left({\phi \left({a}\right)}\right)$$.

Natural Numbers
Let $$n \in \mathbb{N}$$.

Let $$\odot_1^n$$ and $$\odot_2^n$$ be as defined as in Index Laws for Semigroups.

Then $$\forall a \in T_1: \forall n \in \mathbb{N}: \phi \left({\odot_1^n \left({a}\right)}\right) = \odot_2^n \left({\phi \left({a}\right)}\right)$$.

Integers
Let $$\left({T_1, \odot_1}\right)$$ and $$\left({T_2, \odot_2}\right)$$ be monoids.

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

Let $$n \in \mathbb{Z}$$.

Let $$\odot_1^n$$ and $$\odot_2^n$$ be as defined as in Index Laws for Monoids.

Then $$\forall n \in \mathbb{Z}: \phi \left({\odot_1^n \left({a}\right)}\right) = \odot_2^n \left({\phi \left({a}\right)}\right)$$.

Naturally Ordered Semigroup
Can be proved by the Principle of Finite Induction.

Natural Numbers
Follows directly from Homomorphism of Powers: Naturally Ordered Semigroup as the Natural Numbers are a Naturally Ordered Semigroup.

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

$$\forall n \in \mathbb{N}^*: \phi \left({\odot_1^{-n} \left({a}\right)}\right) = \odot_2^{-n} \left({\phi \left({a}\right)}\right)$$

But $$\phi \left({a^{-1}}\right) = \left({\phi \left({a}\right)}\right)^{-1}$$ by Homomorphism with Identity Preserves Inverses.

Hence:

$$\odot_2^{-n} \left({\phi \left({a}\right)}\right) = \odot_2^n \left({\phi \left({a^{-1}}\right)}\right) = \phi \left({\odot_1^{-n} \left({a^{-1}}\right)}\right) = \phi \left({\odot_1^{-n} \left({a}\right)}\right)$$

by Homomorphism of Powers: Natural Numbers.