Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order defined by "$x\leq y$ if and only if there exists $z\in E$ with $y=x+z$", so that it is an ordered monoid. If we consider a subset $F\subseteq E$, we can ask whether it is noetherian as an ordered set, i.e., whether every nonempty subset of $F$ has a maximal element.
It is not hard to see that if $E$ is finitely generated, then the noetherian subsets of $E$ are precisely the finite ones. So, we could ask for the converse: If the noetherian subsets of $E$ are precisely the finite ones, is then $E$ finitely generated?
I think this is not true, but was not able to come up with a counterexample. So, here is my question:
What is an example of a monoid as above, but not finitely generated, whose noetherian subsets are precisely the finite ones?