Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ N | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is g+ceil((d−1)f/2), where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups