A Geometrical Characterization of Proportionally Modular Affine Semigroups

Abstract

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality \(f_1x_1+\cdots +f_nx_n \bmod b \le g_1x_1+\cdots +g_nx_n\), where \(g_1,\dots ,g_n\), \(f_1,\ldots ,f_n\in \mathbb {Z}\), and \(b\in \mathbb {N}\). In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms are provided to check if a semigroup S in \(\mathbb {N}^n\), with \(\mathbb {N}^n{\setminus } S\) a finite set, is a proportionally modular affine semigroup.

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Acknowledgements

This publication and research have been partially Granted by INDESS (Research Universitary Institute for Sustainable Social Development), Universidad de Cádiz, Spain. The authors were partially supported by the Junta de Andalucía Research Group FQM-366. The first author was supported by the Programa Operativo de Empleo Juvenil 2014–2020, which is financed by the European Social Fund within the Youth Guarantee initiative. The second, third, and fourth authors were partially supported by the Project MTM2017-84890-P (MINECO/FEDER, UE), and the fourth author was partially supported by the Project MTM2015-65764-C3-1-P (MINECO/FEDER, UE).

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Correspondence to A. Vigneron-Tenorio.

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Díaz-Ramírez, J.D., García-García, J.I., Sánchez-R.-Navarro, A. et al. A Geometrical Characterization of Proportionally Modular Affine Semigroups. Results Math 75, 99 (2020). https://doi.org/10.1007/s00025-020-01230-3

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Keywords

  • Affine semigroup
  • modular Diophantine inequalities
  • numerical semigroup
  • proportionally modular numerical semigroup

Mathematics Subject Classification

  • Primary 20M14
  • Secondary 68U05