Numerical semigroups that are not intersections of d-squashed semigroups

Abstract

We say that a numerical semigroup is d-squashed if it can be written in the form S =1/N(a1, . . . , ad) ∩ Z for N, a1, . . . , ad positive integers with gcd(a1, . . . , ad) = 1. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of d-squashed semigroups for any fixed d, and we prove some partial results towards this conjecture.