Let $M$ be a simple commutative monoid.
Is there a surjective monoid morphism $\mathbb N\to M$?
If non-monogeneous simple commutative monoids do exist, what's known about them?
Edit. The commutative monoid $(M,+)$ is simple if it satisfies the following equivalent conditions:
$\bullet$ $M$ admits exactly two congruences,
$\bullet$ up to isomorphism, there are exactly two surjective morphisms $M\to N$,
$\bullet$ $M\ne0$ and any nonzero surjective morphisms $M\to N$ is an isomorphism.