Abstract view
Numerical Semigroups That Are Not Intersections of $d$Squashed Semigroups


Published:20091201
Printed: Dec 2009
M. A. Moreno
J. Nicola
E. Pardo
H. Thomas
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
2squashed 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.
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]
