  From any poset isomorphic to the poset of gaps of a numerical semigroup $S$
with the order induced by $S$, one can recover $S$. As an application, we prove
that two different numerical semigroups cannot have isomorphic posets (with
respect to set inclusion) of ideals whose minimum is zero. We also show that
given two numerical semigroups $S$ and $T$, if their ideal class monoids are
isomorphic, then $S$ must be equal to $T$.
