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?
Meanwhile this was answered on MO. There, David Lampert showed that there is no such example.