Given a integer Set, if the distances between integers in the set are still in the set, what mathematical term should be used to describe that nature? Or what natures the set have?

For example, $x,y \in \{1,2,3,4\}$, $m=|x-y|$, m still $\in \{1,2,3,4\}$.
I want to know if there is a discontinuous integet set that has the nature?