# Are simple commutative monoids monogeneous?

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.

Let $$L=\{0,1\}$$, considered as a commutative monoid under multiplication. If $$M$$ is any commutative monoid, there is a homomorphism $$f:M\to L$$ which sends all invertible elements to $$1$$ and all non-invertible elements to $$0$$. If $$M$$ is simple, then either $$f$$ must be an isomorphism or $$f$$ must fail to be surjective. If $$f$$ is not surjective, that means every element of $$M$$ is invertible so $$M$$ is a group. Then $$M$$ must also be simple as a abelian group, so it is cyclic of prime order.
So, the only simple commutative monoids (up to isomorphism) are $$L$$ and cyclic groups of prime order. In particular, they are all generated by a single element.