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

  Published:2009-12-01
 Printed: Dec 2009
  • M. A. Moreno
  • J. Nicola
  • E. Pardo
  • H. Thomas
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Abstract

We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ 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 is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{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.
Keywords: numerical semigroup, squashed semigroup, proportionally modular semigroup numerical semigroup, squashed semigroup, proportionally modular semigroup
MSC Classifications: 20M14, 06F05, 46L80 show english descriptions Commutative semigroups
Ordered semigroups and monoids [See also 20Mxx]
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
20M14 - Commutative semigroups
06F05 - Ordered semigroups and monoids [See also 20Mxx]
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
 

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