An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $X$ by $Y$, i.e., the set of all products $xy$ with $x \in X$ and $y \in Y$). In particular, an ideal $I$ of $S$ is finitely generated if there exist finite sets $X, Y \subseteq S$ such that $I = X \cup XS = Y \cup SY$.
The non-empty ideals of $S$ form a semigroup under setwise multiplication that I'll denote by $\mathcal I(S)$ and call the ideal semigroup of $S$. What is known about the following?
Problem. Given a class $\mathcal C$ of semigroups, prove or disprove that $\mathcal I(H)$ is (semigroup-)isomorphic to $\mathcal I(K)$, for some $H, K \in \mathcal C$, if and only if $H$ is isomorphic to $K$.
Isomorphic semigroups have isomorphic ideal semigroups, so the only non-trivial aspect of the problem lies in the ''only if'' direction. I'm particularly interested in the case when $\mathcal C$ is a class (if not the class) of cancellative, commutative monoids with trivial group of units.
The ideal semigroup of a commutative monoid $H$ is isomorphic to the ideal semigroup of the factor monoid $H/H^\times$, where $H^\times$ is the group of units of $H$ and two elements of $H$ are identified in the quotient if and only if they differ by a unit. So, the condition on the group of units is kind of necessary.
