Numerical Semigroups That Are Not Intersections ofd-Squashed Semigroups

AbstractWe say that a numerical semigroup isd-squashedif it can be written in the formforN,a1, … ,adpositive 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 Rosaleset 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 ofd-squashed semigroups for any fixedd, and we prove some partial results towards this conjecture.