PreprintCC BY 4.0arXiv
The covariety of numerical semigroups with some prescribed pseudo-Frobenius numbers
Abstract
If S is a numerical semigroup, then we denote by m(S), F(S) and PF(S) its multiplicity, its Frobenius number and its set of pseudo-Frobenius numbers, respectively. Let π« be a finite nonempty set of positive integers. In this paper, we prove that π(π«)={S| F(S)=max(π«), π«βPF(S)} is a covariety. The associated tree provides algorithms to compute π(π«), its elements with a prescribed type or genus, and the set of numerical semigroups S satisfying PF(S)=π«. We show that every element of π(π«) has a unique minimal system of generators relative to π(π«), and we call its cardinality the π(π«)-rank. Finally, we characterize the elements of π(π«) with π(π«)-rank one.
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BibTeX
@misc{hernandezyanes2026covariety,
title = {{The covariety of numerical semigroups with some prescribed pseudo-Frobenius numbers}},
author = {William Giovanni HernΓ‘ndez-Yanes and JosΓ© Carlos Rosales},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09425},
eprint = {2611.09425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The covariety of numerical semigroups with some prescribed pseudo-Frobenius numbers
AU - William Giovanni HernΓ‘ndez-Yanes
AU - JosΓ© Carlos Rosales
PY - 2026
DA - 2026/11/06
PB - arXiv
DO - 10.48550/arxiv.2611.09425
UR - https://arxiv.org/abs/2611.09425
AB - If S is a numerical semigroup, then we denote by m(S), F(S) and PF(S) its multiplicity, its Frobenius number and its set of pseudo-Frobenius numbers, respectively. Let π« be a finite nonempty set of positive integers. In this paper, we prove that π(π«)={S| F(S)=max(π«), π«βPF(S)} is a covariety. The associated tree provides algorithms to compute π(π«), its elements with a prescribed type or genus, and the set of numerical semigroups S satisfying PF(S)=π«. We show that every element of π(π«) has a unique minimal system of generators relative to π(π«), and we call its cardinality the π(π«)-rank. Finally, we characterize the elements of π(π«) with π(π«)-rank one.
ER -