Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups

[摘要] We say that a numerical semigroup is emph{$d$-squashed} if it canbe written in the form$$ S=frac 1 N langle a_1,dots,a_d angle cap mathbb{Z}$$for $N,a_1,dots,a_d$ positive integers with$gcd(a_1,dots, a_d)=1$.Rosales and Urbano have shown that a numerical semigroup is2-squashed if and only if it is proportionally modular.Recent works by Rosales emph{et al.} give a concrete example of anumerical semigroup that cannot be written as an intersection of$2$-squashed semigroups. We will show the existence of infinitelymany numerical semigroups that cannot be written as anintersection of $2$-squashed semigroups. We also will prove thesame result for $3$-squashed semigroups. We conjecture that thereare numerical semigroups that cannot be written as theintersection of $d$-squashed semigroups for any fixed $d$, and weprove some partial results towards this conjecture.

[学科分类] 数学（综合）

[关键词] numerical semigroup;squashed semigroup;proportionally modular semigroup