Semigroup Forum

, Volume 96, Issue 2, pp 396–408 | Cite as

An extension of Wilf’s conjecture to affine semigroups

  • J. I. García-García
  • D. Marín-Aragón
  • A. Vigneron-Tenorio
Research Article


Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.


Affine semigroup Embedding dimension Frobenius number Genus Semigroup tree Wilf’s conjecture 



The authors would like to thank Shalom Eliahou and the referees for their helpful comments and suggestions related to this work.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • J. I. García-García
    • 1
  • D. Marín-Aragón
    • 1
    • 2
  • A. Vigneron-Tenorio
    • 2
  1. 1.Departamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible)Universidad de CádizPuerto RealSpain
  2. 2.Departamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible)Universidad de CádizJerez de la FronteraSpain

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