Abstract
In this paper we study those submonoids of \(\mathbb {N}^d\) with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of \(\mathbb {N}^d\) and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.
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Acknowledgements
This paper was originally motivated by a question formulated by Antonio Campillo and Félix Delgado about \(\mathcal {C}\)-semigroups during a talk of the fourth author at the GAS seminar of the SINGACOM research group. The question is answered in a wider context by Corollary 7. Part of this paper was written during a visit of the second author to the Universidad de Cádiz (Spain) and to the IEMath-GR (Universidad de Granada, Spain), he thanks these institutions for their warm hospitality. The authors would like to thank Antonio Campillo, Félix Delgado and Pedro A. García-Sánchez for useful suggestions and comments. The authors also thank the referee for many helpful observations.
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The first author were partially supported by Junta de Andalucía research group FQM-366 and by the project MTM2017-84890-P. The second author was partially supported by the research groups FQM-024 (Junta de Extremadura/FEDER funds) and by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and by the project MTM2017-84890-P. The third author were partially supported by Junta de Andalucía research group FQM-343 and by the project MTM2017-84890-P. The fourth author was partially supported by Junta de Andalucía research group FQM-366, by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and by the project MTM2017-84890-P.
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García-García, J.I., Ojeda, I., Rosales, J.C. et al. On pseudo-Frobenius elements of submonoids of \(\mathbb {N}^d\). Collect. Math. 71, 189–204 (2020). https://doi.org/10.1007/s13348-019-00267-0
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DOI: https://doi.org/10.1007/s13348-019-00267-0
Keywords
- Affine semigroups
- Numerical semigroups
- Frobenius elements
- Pseudo-Frobenius elements
- Apéry sets
- Gluings
- Free resolution
- Irreducible semigroups