The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated

Maria Angeles Moreno-Frías; José Carlos Rosales

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 3, page 439-454
  • ISSN: 0862-7959

Abstract

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Let Δ be a numerical semigroup. In this work we show that 𝒥 ( Δ ) = { I ∪ { 0 } : I is an ideal of Δ } is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set 𝒥 a ( Δ ) = { S ∈ 𝒥 ( Δ ) : max ( Δ ∖ S ) = a } for a given a ∈ Δ . As a consequence, we obtain another algorithm that computes all the elements of 𝒥 ( Δ ) with a fixed genus.

How to cite

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Moreno-Frías, Maria Angeles, and Rosales, José Carlos. "The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated." Mathematica Bohemica 149.3 (2024): 439-454. <http://eudml.org/doc/299503>.

@article{Moreno2024,
abstract = {Let $\Delta $ be a numerical semigroup. In this work we show that $\mathcal \{J\}(\Delta ) =\lbrace I\cup \lbrace 0\rbrace \colon I \mbox\{ is an ideal of \} \Delta \rbrace $ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set $\mathcal \{J\}_a(\Delta )=\lbrace S\in \mathcal \{J\}(\Delta )\colon \max (\Delta \backslash S)=a\rbrace $ for a given $a\in \Delta .$ As a consequence, we obtain another algorithm that computes all the elements of $\mathcal \{J\}(\Delta )$ with a fixed genus.},
author = {Moreno-Frías, Maria Angeles, Rosales, José Carlos},
journal = {Mathematica Bohemica},
keywords = {numerical semigroup; ideal; Frobenius restricted variety; embedding dimension; Frobenius number; restricted Frobenius number; genus; multiplicity; Arf numerical semigroup; saturated semigroup},
language = {eng},
number = {3},
pages = {439-454},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated},
url = {http://eudml.org/doc/299503},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Moreno-Frías, Maria Angeles
AU - Rosales, José Carlos
TI - The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 3
SP - 439
EP - 454
AB - Let $\Delta $ be a numerical semigroup. In this work we show that $\mathcal {J}(\Delta ) =\lbrace I\cup \lbrace 0\rbrace \colon I \mbox{ is an ideal of } \Delta \rbrace $ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set $\mathcal {J}_a(\Delta )=\lbrace S\in \mathcal {J}(\Delta )\colon \max (\Delta \backslash S)=a\rbrace $ for a given $a\in \Delta .$ As a consequence, we obtain another algorithm that computes all the elements of $\mathcal {J}(\Delta )$ with a fixed genus.
LA - eng
KW - numerical semigroup; ideal; Frobenius restricted variety; embedding dimension; Frobenius number; restricted Frobenius number; genus; multiplicity; Arf numerical semigroup; saturated semigroup
UR - http://eudml.org/doc/299503
ER -

References

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