We define $\mathbf{N} = \{0,1,2,...\}$ and $ \mathbf{N}^* = \{1,2,...\}$, each a monoid with addition and multiplication respectively. I am looking for monoid morphisms between these two monoids.
For example, $f :\mathbf{N}^* \rightarrow \mathbf{N} $, where $f({p_1}^{r_1}...{p_n}^{r_n}) = r_1 + ... + r_n$ and $f(1)=0$ is a monoid morphism.
I am having trouble finding a morphism on the opposite direction, but any examples of such morphisms, or sources on the subject would be appreciated.
Ok so it turns out it is a simple problem to find all such monoid morphisms. For morphisms from $\mathbf{N}$, they are uniquely determined by their image at $1$, and from $\mathbf{N}^*$ they are uniquely determined by any function $P \rightarrow\mathbf{N}$ where $P$ are primes.