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Numerical semigroups closed under addition of their divisors

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Abstract

A numerical semigroup S is closed under addition of its divisors (\({\mathcal {C}}\)-semigroup) if the following condition holds: if \(s\in S\setminus \{0\}\) and d is a non trivial divisor of s then \(s+d\in S\). In this paper we prove that the set of all \({\mathcal {C}}\)-semigroups is a Frobenius variety. As a consequence, we give algorithms to compute the set \({\mathcal {C}}\)-semigroups with fixed multiplicity, genus and Frobenius number.

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References

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Acknowledgements

We thank the anonymous referees for their detailed suggestions and comments, which have greatly improved this manuscript.

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Authors and Affiliations

  1. Departamento de Álgebra, Universidad de Granada, 18071, Granada, Spain

    J. C. Rosales

  2. Departamento de Matemática, Universidade de Évora, 7000, Évora, Portugal

    M. B. Branco

Authors
  1. J. C. Rosales
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  2. M. B. Branco
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Correspondence to M. B. Branco.

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The fist author was partially supported by the research group FQM-343 FQM-5849 (Junta de Andalucia/Feder) and by the Project MTM2017-84890-P (MINECO/FEDER, UE). The second author is supported by the Project FCT PTDC/MAT/73544/2006). 2010 Mathematics Subject Classification: 20M14, 11D07.

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Rosales, J.C., Branco, M.B. Numerical semigroups closed under addition of their divisors. AAECC 32, 665–680 (2021). https://doi.org/10.1007/s00200-020-00420-4

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  • DOI: https://doi.org/10.1007/s00200-020-00420-4

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