The multiples of a numerical semigroup

Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d\in \mathbb{N}\setminus \{0\}$ such that $S=\{x\in \mathbb{N}\mid dx\in T\}$. In this paper we study the family of multiples of a (fixed) numerical semigroup. We also address the open problem of finding numerical semigroups of embedding dimension $e$ without any quotient of embedding dimension less than $e$, and provide new families with this property.