Generators of a fraction of a numerical semigroup

Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{ x \in \mathbb{N} \ | \ dx \in S\}$ 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 $\frac{S}{d}$.