Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T04:59:39.230Z Has data issue: false hasContentIssue false
  • English
  • Français

MODULAR DIOPHANTINE INEQUALITIES AND ROTATIONS OF NUMERICAL SEMIGROUPS

Part of: Diophantine equations Semigroups

Published online by Cambridge University Press:  01 June 2008

M. DELGADO  and
J. C. ROSALES
M. DELGADO
Affiliation:
Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687 4169-007 Porto, Portugal (email: mdelgado@fc.up.pt)
J. C. ROSALES*
Affiliation:
Departamento de Álgebra, Universidad de Granada, E-18071 Granada, Spain (email: jrosales@ugr.es)
*
✉For correspondence; e-mail: jrosales@ugr.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a numerical semigroup S, a positive integer a and a nonzero element m of S, we define a new numerical semigroup R(S,a,m) and call it the (a,m)-rotation of S. In this paper we study the Frobenius number and the singularity degree of R(S,a,m). Moreover, we observe that the rotations of ℕ are precisely the modular numerical semigroups.

Keywords

numerical semigroupsDiophantine inequalityFrobenius numbersingularity degree

MSC classification

Secondary: 20M14: Commutative semigroups 11D75: Diophantine inequalities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 3 , June 2008 , pp. 315 - 328
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The first author gratefully acknowledges support of the FCT through the CMUP. The second author was supported by the project MTM2004-01446 and FEDER founds.

References

[1]Apéry, R., ‘Sur les branches superlinéaires des courbes algébriques’, C. R. Acad. Sci. Paris 222 (1946), 1198–1200.Google Scholar
[2]Barucci, V., Dobbs, D. E. and Fontana, M., ‘Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains’, Mem. Amer. Math. Soc. 125 (1997), no. 598.Google Scholar
[3]Ramírez Alfonsín, J. L., The Diophantine Frobenius Problem, Oxford Lectures Series in Mathematics and its Applications, 30 (Oxford University Press, Oxford, 2005).CrossRefGoogle Scholar
[4]Rosales, J. C. and García-Sánchez, P. A., Finitely Generated Commutative Monoids (Nova Science Publishers, New York, 1999).Google Scholar
[5]Rosales, J. C., García-Sánchez, P. A., García-García, J. I. and Urbano-Blanco, J. M., ‘Proportionally modular Diophantine inequalities’, J. Number Theory 103 (2003), 281–294.CrossRefGoogle Scholar
[6]Rosales, J. C., García-Sánchez, P. A. and Urbano-Blanco, J. M., ‘Modular Diophantine inequalities and numerical semigroups’, Pacific J. Math. 218 (2005), 379–398.CrossRefGoogle Scholar
[7]Selmer, E. S., ‘On a linear diophantine problem of Frobenius’, J. Reine Angew. Math. 293–294 (1977), 1–17.Google Scholar