Numerical semigroups from open intervals

Abstract

Consider an interval in the positive rationals, I ⊆ Q . Set S(I) = {m ∈ N : ∃ n ∈ N , mn ∈ I}. This turns out to be a numerical semigroup, and has been the subject of considerable recent investigation (see Chapter 4 of [2] for an introduction). Special cases include modular numerical semigroups (see [4]) where I = [mn , m n−1 ] (m,n ∈ N ), proportionally modular numerical semigroups (see [3]) where I = [mn , m n−s ] (m,n, s ∈ N), and opened modular numerical semigroups (see [5]) where I = (mn , m n−1 ) (m,n ∈ N ). We consider instead arbitrary open intervals I = (a, b). We show that this set of semigroups coincides with the set of semigroups generated by closed and half-open intervals. Consequently, this class of semigroups contains modular numerical semigroups, proportionally modular numerical semigroups, as well as opened modular numerical semigroups. We also compute two important invariants of these numerical semigroups: the Frobenius number g(S(I)) and multiplicity m(S(I)).