Let $H$ be a monoid (written multiplicatively). We call $H$ Archimedean if, for all $a, b \in H$ such that $b$ is a non-unit, there is an integer $k \ge 1$ with $b^k \in HaH$; strongly Archimedean if, for each $a \in H$, there is an integer $k \ge 1$ such that $HaH$ contains any product of any $k$ non-units of $H$; and duo if $aH = Ha$ for all $a \in H$. For instance, commutative monoids are duo and numerical monoids are strongly Archimedean.

    We will outline a proof of the following results: (1) The sets of (non-empty) ideals of two strongly Archimedean, cancellative, duo monoids are isomorphic as semigroups under the induced operation of setwise multiplication if and only if the monoids themselves are isomorphic up to units. (2) The same holds upon restriction to finitely generated ideals in Archimedean, cancellative, duo monoids, where a finitely generated ideal of the monoid $H$ is an ideal $I$ of $H$ such that $I = XH = HY$ for some (non-empty) finite sets $X, Y \subseteq H$.

    If time permits, we will also discuss how these results can be applied to answer some special cases of a problem of Kobayashi from the early 1980s. The talk is based on joint work with Pedro A.~Garc\'ia-S\'anchez (University of Granada, Spain).