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Keywords:
numerical semigroup; ideal; Frobenius restricted variety; embedding dimension; Frobenius number; restricted Frobenius number; genus; multiplicity; Arf numerical semigroup; saturated semigroup
Summary:
Let $\Delta $ be a numerical semigroup. In this work we show that $\mathcal {J}(\Delta ) =\{I\cup \nobreak \{0\}\colon I \mbox { is an ideal of } \Delta \}$ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set $\mathcal {J}_a(\Delta )=\{S\in \mathcal {J}(\Delta )\colon \max (\Delta \backslash S)=a\}$ for a given $a\in \Delta .$ As a consequence, we obtain another algorithm that computes all the elements of $\mathcal {J}(\Delta )$ with a fixed genus.
References:
[1] Arf, C.: Une interprétation algébrique de la suite des ordres de multiplicité d'une branche algébrique. Proc. Lond. Math. Soc., II. Ser. 50 (1948), 256-287 French. DOI 10.1112/plms/s2-50.4.256 | MR 0031785 | Zbl 0031.07002
[2] Barucci, V.: Decomposition of ideals into irreducible ideals in numerical semigroups. J. Commut. Algebra 2 (2010), 281-294. DOI 10.1216/JCA-2010-2-3-281 | MR 2728145 | Zbl 1237.20056
[3] Barucci, V., Dobbs, D. E., Fontana, M.: Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analitycally Irreducible Local Domains. Memoirs of the American Mathematical Society 598. AMS, Providence (1997),\99999DOI99999 10.1090/memo/0598 . MR 1357822 | Zbl 0868.13003
[4] Campillo, A.: On saturations of curve singularities (any characteristic). Singularities. Part 1 Proceedings of Symposia in Pure Mathematics 40. AMS, Providence (1983), 211-220. DOI 10.1090/pspum/040.1 | MR 0713060 | Zbl 0553.14013
[5] Mata, F. Delgado de la, Jiménez, C. A. Núñez: Monomial rings and saturated rings. Géométrie algébrique et applications I Travaux en Cours 22. Hermann, Paris (1987), 23-34. MR 0907904 | Zbl 0636.14009
[6] Lipman, J.: Stable ideals and Arf rings. Am. J. Math. 93 (1971), 649-685. DOI 10.2307/2373463 | MR 0282969 | Zbl 0228.13008
[7] Moreno-Frías, M. A., Rosales, J. C.: Counting the ideals with given genus of a numerical semigroup. J. Algebra Appl. 22 (2023), Article ID 2330002, 21 pages. DOI 10.1142/S0219498823300027 | MR 4598665 | Zbl 07709969
[8] Núñez, A.: Algebro-geometric properties of saturated rings. J. Pure Appl. Algebra 59 (1989), 201-214. DOI 10.1016/0022-4049(89)90135-7 | MR 1007922 | Zbl 0701.14026
[9] Pham, F.: Fractions lipschitziennes et saturations de Zariski des algèbres analytiques complexes. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2 Gautier-Villars, Paris (1971), 649-654 French. MR 0590058 | Zbl 0245.32003
[10] Robles-Pérez, A. M., Rosales, J. C.: Frobenius restricted varieties in numerical semigroups. Semigroup Forum 97 (2018), 478-492. DOI 10.1007/s00233-018-9949-y | MR 3881853 | Zbl 1448.20050
[11] Rosales, J. C.: Principal ideals of numerical semigroups. Bull. Belg. Math. Soc. - Simon Stevin 10 (2003), 329-343. DOI 10.36045/bbms/1063372340 | MR 2016807 | Zbl 1051.20026
[12] Rosales, J. C., García-Sánchez, P. A.: Numerical Semigroups. Developments in Mathematics 20. Springer, New York (2009). DOI 10.1007/978-1-4419-0160-6 | MR 2549780 | Zbl 1220.20047
[13] Zariski, O.: General theory of saturation and of saturated local rings I. Saturation of complete local domains of dimension one having arbitrary coefficient fields (of characteristic zero). Am. J. Math. 93 (1971), 573-684. DOI 10.2307/2373462 | MR 0282972 | Zbl 0226.13013
[14] Zariski, O.: General theory of saturation and of saturated local rings II. Saturated local rings of dimension 1. Am. J. Math. 93 (1971), 872-964 \99999DOI99999 10.2307/2373741 . MR 0299607 | Zbl 0228.13007
[15] Zariski, O.: General theory of saturation and of saturated local rings III. Saturation in arbitrary dimension and, in particular, saturation of algebroid hypersurfaces. Am. J. Math. 97 (1975), 415-502. DOI 10.2307/2373720 | MR 0389893 | Zbl 0306.13009
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