The covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías; José Carlos Rosales

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 3, page 697-714
  • ISSN: 0011-4642

Abstract

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Let S be a numerical semigroup. We say that h ∈ ℕ ∖ S is an isolated gap of S if { h - 1 , h + 1 } ⊆ S . A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by m ( S ) the multiplicity of a numerical semigroup S . A covariety is a nonempty family 𝒞 of numerical semigroups that fulfills the following conditions: there exists the minimum of 𝒞 , the intersection of two elements of 𝒞 is again an element of 𝒞 , and S ∖ { m ( S ) } ∈ 𝒞 for all S ∈ 𝒞 such that S ≠ min ( 𝒞 ) . We prove that the set 𝒫 ( F ) = { S : S is a perfect numerical semigroup with Frobenius number F } is a covariety. Also, we describe three algorithms which compute: the set 𝒫 ( F ) , the maximal elements of 𝒫 ( F ) , and the elements of 𝒫 ( F ) with a given genus. A Parf -semigroup (or Psat -semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (or saturated numerical semigroup), respectively. We prove that the sets Parf ( F ) = { S : S is a Parf -numerical semigroup with Frobenius number F } and Psat ( F ) = { S : S is a Psat -numerical semigroup with Frobenius number F } are covarieties. As a consequence we present some algorithms to compute Parf ( F ) and Psat ( F ) .

How to cite

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Moreno-Frías, María Ángeles, and Rosales, José Carlos. "The covariety of perfect numerical semigroups with fixed Frobenius number." Czechoslovak Mathematical Journal 74.3 (2024): 697-714. <http://eudml.org/doc/299298>.

@article{Moreno2024,
abstract = {Let $S$ be a numerical semigroup. We say that $h\in \mathbb \{N\} \backslash S$ is an isolated gap of $S$ if $\lbrace h-1,h+1\rbrace \subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $\{\rm m\} (S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathcal \{C\}$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $\mathcal \{C\},$ the intersection of two elements of $\mathcal \{C\}$ is again an element of $\mathcal \{C\}$, and $S\backslash \lbrace \{\rm m\}(S)\rbrace \in \mathcal \{C\}$ for all $S\in \mathcal \{C\}$ such that $S\ne \min (\mathcal \{C\}).$ We prove that the set $\mathcal \{P\}(F)=\lbrace S\colon S$ is a perfect numerical semigroup with Frobenius number $F\rbrace $ is a covariety. Also, we describe three algorithms which compute: the set $\mathcal \{P\}(F),$ the maximal elements of $\mathcal \{P\}(F)$, and the elements of $\mathcal \{P\}(F)$ with a given genus. A $\{\rm Parf\}$-semigroup (or $\{\rm Psat\}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (or saturated numerical semigroup), respectively. We prove that the sets $\{\rm Parf\}(F)=\lbrace S\colon S$ is a $\{\rm Parf\}$-numerical semigroup with Frobenius number $F\rbrace $ and $\{\rm Psat\}(F)=\lbrace S\colon S$ is a $\{\rm Psat\}$-numerical semigroup with Frobenius number $F\rbrace $ are covarieties. As a consequence we present some algorithms to compute $\{\rm Parf\}(F)$ and $\{\rm Psat\}(F).$},
author = {Moreno-Frías, María Ángeles, Rosales, José Carlos},
journal = {Czechoslovak Mathematical Journal},
keywords = {perfect numerical semigroup; saturated numerical semigroup; Arf numerical semigroup; covariety; Frobenius number; genus; algorithm},
language = {eng},
number = {3},
pages = {697-714},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The covariety of perfect numerical semigroups with fixed Frobenius number},
url = {http://eudml.org/doc/299298},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Moreno-Frías, María Ángeles
AU - Rosales, José Carlos
TI - The covariety of perfect numerical semigroups with fixed Frobenius number
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 697
EP - 714
AB - Let $S$ be a numerical semigroup. We say that $h\in \mathbb {N} \backslash S$ is an isolated gap of $S$ if $\lbrace h-1,h+1\rbrace \subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by ${\rm m} (S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathcal {C}$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $\mathcal {C},$ the intersection of two elements of $\mathcal {C}$ is again an element of $\mathcal {C}$, and $S\backslash \lbrace {\rm m}(S)\rbrace \in \mathcal {C}$ for all $S\in \mathcal {C}$ such that $S\ne \min (\mathcal {C}).$ We prove that the set $\mathcal {P}(F)=\lbrace S\colon S$ is a perfect numerical semigroup with Frobenius number $F\rbrace $ is a covariety. Also, we describe three algorithms which compute: the set $\mathcal {P}(F),$ the maximal elements of $\mathcal {P}(F)$, and the elements of $\mathcal {P}(F)$ with a given genus. A ${\rm Parf}$-semigroup (or ${\rm Psat}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (or saturated numerical semigroup), respectively. We prove that the sets ${\rm Parf}(F)=\lbrace S\colon S$ is a ${\rm Parf}$-numerical semigroup with Frobenius number $F\rbrace $ and ${\rm Psat}(F)=\lbrace S\colon S$ is a ${\rm Psat}$-numerical semigroup with Frobenius number $F\rbrace $ are covarieties. As a consequence we present some algorithms to compute ${\rm Parf}(F)$ and ${\rm Psat}(F).$
LA - eng
KW - perfect numerical semigroup; saturated numerical semigroup; Arf numerical semigroup; covariety; Frobenius number; genus; algorithm
UR - http://eudml.org/doc/299298
ER -

References

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