Abstract
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.
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Acknowledgements
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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Communicated by Fernando Torrs.
D. Marín-Aragón—Member of the group FQM-366 Junta de Andalucía.
J. I. García-García was partially supported by MTM2014-55367-P and Junta de Andalucía group FQM-366. D. Marín-Aragón was partially supported by Junta de Andalucía group FQM-366. A. Vigneron-Tenorio was partially supported by MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and Junta de Andalucía group FQM-366.
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García-García, J.I., Marín-Aragón, D. & Vigneron-Tenorio, A. An extension of Wilf’s conjecture to affine semigroups. Semigroup Forum 96, 396–408 (2018). https://doi.org/10.1007/s00233-017-9906-1
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DOI: https://doi.org/10.1007/s00233-017-9906-1