The isomorphism problem for ideal class monoids of numerical semigroups

Abstract 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 .