The covariety of perfect numerical semigroups with fixed Frobenius number
Metadatos
Afficher la notice complèteEditorial
Springer
Materia
perfect numerical semigroup saturated numerical semigroup Arf numerical semigroup
Date
2024-07-15Referencia bibliográfica
Moreno Frías, M.A. & Rosales, J.C. Czech Math J (2024). [https://doi.org/10.21136/CMJ.2024.0379-23]
Patrocinador
Proyecto de Excelencia de la Junta de Andalucía ProyExcel 00868 and Proyecto de investigación del Plan Propio–UCA 2022-2023 (PR2022-011); Junta de Andalucía group FQM-298 and Proyecto de investigación del Plan Propio–UCA 2022-2023 (PR2022-004); Junta de Andalucía group FQM-343Résumé
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 6= 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).