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.