Quotients of perfect numerical semigroups

Abstract

A numerical semigroup is said to be perfect if it does not contain any isolated gaps. In this paper we show that if $S$ is any numerical semigroup, then for each integer $n\ge 2$ there exists a perfect numerical semigroup $T$ such that $S=\frac{T}{n}$ (that is, $S=\left\{x\in \mathbb{N}:nx\in T\right\}$)