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A generalization of Wilf’s conjecture for generalized numerical semigroups

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Abstract

A numerical semigroup is a submonoid of \({\mathbb {N}}\) with finite complement in \({\mathbb {N}}\). A generalized numerical semigroup is a submonoid of \({\mathbb {N}}^{d}\) with finite complement in \({\mathbb {N}}^{d}\). In the context of numerical semigroups, Wilf’s conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf’s conjecture, to the class of \({\mathcal {C}}\)-semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf’s conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by García-García, Marín-Aragón, and Vigneron-Tenorio.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, Viale Ferdinando Stagno d’Alcontres 31, 98166, Messina, Italy

    Carmelo Cisto & Rosanna Utano

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Michael DiPasquale, Zachary Flores & Chris Peterson

  3. Università Mediterranea di Reggio Calabria, DIIES, Via Graziella, Feo di Vito, Reggio Calabria, Italy

    Gioia Failla

Authors
  1. Carmelo Cisto
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  2. Michael DiPasquale
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  3. Gioia Failla
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  4. Zachary Flores
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  5. Chris Peterson
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  6. Rosanna Utano
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Corresponding author

Correspondence to Michael DiPasquale.

Additional information

Communicated by Fernando Torres.

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The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM. DiPasquale, Peterson, and Flores would like to thank the University of Messina for its financial support. The research of DiPasquale, Peterson, and Flores was also supported in part by NSF Grant DMS-1830676.

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Cisto, C., DiPasquale, M., Failla, G. et al. A generalization of Wilf’s conjecture for generalized numerical semigroups. Semigroup Forum 101, 303–325 (2020). https://doi.org/10.1007/s00233-020-10085-7

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  • DOI: https://doi.org/10.1007/s00233-020-10085-7

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