generators and monoid homomorphisms

Question

Is it true that any homomorphism $f: M \to N$ between two monoids $M$ and $N$ maps generators of $M$ to generators of $N$? I am having trouble proving it to myself.

Answer 1

It has no reason to be true :

If you consider the monoid on one generator, $M = <a>$. Note that $M$ is isomorphic to $\mathbb{N}$, associating to each word $w = aa\ldots a$ the number $n$ of $a$ in that word. We consider morphisms from $M$ to itself, then we can send $a$ to however many $a$'s we want

For instance, if you send $a$ on $a$, you get the identity morphism, it works. But if you chose to send $a$ on $aa$, then you also get a monoid morphism, which associates to each word $w=aa\ldots a$ of length $n$, the word $w'=aa\ldots a$ of length $2n$, containing twice as many $a$'s. Through the previous isomorphism with $\mathbb{N}$. This corresponds to the monoid endomorphism of $\mathbb{N}$ defined by $f(n) = 2n$.

Similarly, if you chose to send $a$ on the word containing $k$ times the letter $a$, then you define the monoid endomorphism of $\mathbb{N}$ $f(n) = kn$.

These are all perfectly valid morphism, and only the identity sends the generators on the generators. And it stays false if you add more generators

Answer 2

Another good reason this isn't true is that a monoid homomorphism is defined with no reference to generators, but monoids can be generated by many different choices of generating sets. For instance, $\mathbb Z$ may be generated either by $\{1\}$ or by $\{-1\}$. Thus every homomorphism $\mathbb Z\to\mathbb Z$ would have to send both $1$ and $-1$ to both $1$ and $-1$!