On pseudo-Frobenius elements of submonoids of \mathbb {N}^d
- J. I. Garcia-Garcia
- I. Ojeda
- José Carlos Rosales González
- Alberto Vigneron-Tenorio
ISSN: 0010-0757
Year of publication: 2020
Volume: 71
Fascicle: 1
Pages: 189-204
Type: Article
More publications in: Collectanea mathematica
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.
Funding information
This paper was originally motivated by a question formulated by Antonio Campillo and F?lix Delgado about C \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} -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 C?diz (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, F?lix Delgado and Pedro A. Garc?a-S?nchez for useful suggestions and comments. The authors also thank the referee for many helpful observations.Funders
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