generators and monoid homomorphisms 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.

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 $-1$!
