Homomorphism of Powers/Natural Numbers

Theorem

Let $\left({T_1, \odot}\right)$ and $\left({T_2, \oplus}\right)$ be semigroups.

Let $\phi: \left({T_1, \odot}\right) \to \left({T_2, \oplus}\right)$ 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:

$\forall a \in T_1: \forall n \in \N: \phi \left({\odot^n \left({a}\right)}\right) = \oplus^n \left({\phi \left({a}\right)}\right)$

Proof

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

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

$\blacksquare$