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Numerical Semigroups That Are Not Intersections of d-Squashed Semigroups

Published online by Cambridge University Press:  20 November 2018

M. A. Moreno ,
J. Nicola ,
E. Pardo  and
H. Thomas
M. A. Moreno
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain e-mail: mariangeles.moreno@uca.esenrique.pardo@uca.es
J. Nicola
Affiliation:
Département de Mathématiques, Université de Genève, Switzerland e-mail: jacqnicola@hotmail.com
E. Pardo
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain e-mail: mariangeles.moreno@uca.esenrique.pardo@uca.es
H. Thomas
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3 e-mail: hugh@math.unb.ca
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Abstract

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We say that a numerical semigroup is $d$-squashed if it can be written in the form

$$S\,=\,\frac{1}{N}\langle {{a}_{1}},\,.\,.\,.\,,\,{{a}_{d}}\rangle \,\cap \,\mathbb{Z}$$

for $N$, ${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$ positive integers with $\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$. 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.

Keywords

20M1406F0546L80numerical semigroupsquashed semigroupproportionally modular semigroup
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 598 - 612
Copyright
Copyright © Canadian Mathematical Society 2009

References

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