Generators of a fraction of a numerical semigroup

Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{x\in \mathbb{N}\mathrm{\mid }dx\in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $\frac{S}{d}$ in terms of the minimal generators of $S$ and provide sharp upper bounds for the embedding dimension of $\frac{S}{d}$.