Opened modular numerical semigroups

We introduce the class of the opened modular numerical semigroups (for brevity, OM-semigroups) and obtain formulas for the Frobenius number, the singularity degree and the type of these semigroups. We prove that a numerical semigroup S has a Toms decomposition if and only if S is an intersection of OM-semigroups. We also characterize irreducible proportionally modular numerical semigroups in terms of OM-semigroups.