Numerical Semigroups That Are Not Intersections of d-Squashed Semigroups

Moreno, M. A. ; Nicola, J. ; Pardo, E. ; Thomas, H.

Canadian mathematical bulletin, Tome 52 (2009) no. 4, pp. 598-612

Résumé

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.

Détail

DOI : 10.4153/CMB-2009-059-4

Mots-clés : 20M14, 06F05, 46L80, numerical semigroup, squashed semigroup, proportionally modular semigroup