Curve singularities with one Puiseux pair and value sets of modules over their local rings

In this paper we characterize the value set Delta of the R-modules of the form R+zR for the local ring R associated to a germ xi of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of Kahler differentials attached to xi, we recover some results of Delorme. From our characterization of Delta we introduce a proper subset of semimodules over the value semigroup of the ring R. Moreover, we provide a combinatorial algorithm to construct all possible semimodules in this subset for a given value semigroup.