Homomorphism of Powers/Natural Numbers

Theorem

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

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

Let $n \in \N$.

Let $\odot^n$ and $\oplus^n$ be the $n$th powers of $\odot$ and $\oplus$, respectively.

Then:

$\forall a \in T_1: \forall n \in \N: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$

Proof

Consider the natural numbers $\N$ defined as a naturally ordered semigroup.

Then the result follows from Homomorphism of Powers: Naturally Ordered Semigroup.

$\blacksquare$