Illinois Journal of Mathematics

Modular numerical semigroups with embedding dimension equal to three

Aureliano M. Robles-Pérez and José Carlos Rosales

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In this paper, we give explicit descriptions of all numerical semigroups, generated by three positive integer numbers, that are the set of solutions of a Diophantine inequality of the form $ax \operatorname{mod} b\leq x$.

Article information

Illinois J. Math., Volume 55, Number 1 (2011), 77-88.

First available in Project Euclid: 19 December 2012

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Zentralblatt MATH identifier

Primary: 11D75: Diophantine inequalities [See also 11J25] 20M14: Commutative semigroups
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]


Robles-Pérez, Aureliano M.; Rosales, José Carlos. Modular numerical semigroups with embedding dimension equal to three. Illinois J. Math. 55 (2011), no. 1, 77--88. doi:10.1215/ijm/1355927028.

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  • R. Apéry, Sur les branches superlinéaires des courbes algébriques, C. R. Math. Acad. Sci. Paris 222 (1946), 1198–1200.
  • V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc. 598 (1997).
  • M. Bullejos and J. C. Rosales, Proportionally modular Diophantine inequalities and the Stern–Brocot tree, Math. Comp. 78 (2009), 1211–1226.
  • E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970), 748–751.
  • J. L. Ramírez Alfonsín, The Diophantine Frobenius problem, Oxford Univ. Press, Oxford, 2005.
  • J. C. Rosales, Symmetric modular Diophantine inequalities, Proc. Amer. Math. Soc. 134 (2006), 3417–3421.
  • J. C. Rosales and P. A. García-Sánchez, Finitely generated commutative monoids, Nova Science Publishers, New York, 1999.
  • J. C. Rosales and P. A. García-Sánchez, Numerical semigroups, Developments in Mathematics, vol. 20, Springer, New York, 2009.
  • J. C. Rosales, P. A. García-Sánchez, J. I. García-García and J. M. Urbano-Blanco, Proportionally modular Diophantine inequalities, J. Number Theory 103 (2003), 281–294.
  • J. C. Rosales, P. A. García-Sánchez and J. M. Urbano-Blanco, Modular Diophantine inequalities and numerical semigroups, Pacific J. Math. 218 (2005), 379–398.
  • J. C. Rosales, P. A. García-Sánchez and J. M. Urbano-Blanco, The set of solutions of a proportionally modular Diophantine inequality, J. Number Theory 128 (2008), 453–467.
  • J. C. Rosales and J. M. Urbano-Blanco, Opened modular numerical semigroups, J. Algebra 306 (2006), 368–377.
  • J. J. Sylvester, Problem 7382, Mathematical questions, with their solutions, from The Educational Times, vol. 41 (W. J. C. Miller, ed.), Francis Hodgson, London, 1884, p. 21.