@misc{10481/93872,
year = {2024},
month = {7},
url = {https://hdl.handle.net/10481/93872},
abstract = {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).},
organization = {Proyecto de Excelencia de la Junta de Andalucía
ProyExcel 00868 and Proyecto de investigación del Plan Propio–UCA 2022-2023
(PR2022-011)},
organization = {Junta de Andalucía group
FQM-298 and Proyecto de investigación del Plan Propio–UCA 2022-2023 (PR2022-004)},
organization = {Junta de Andalucía group FQM-343},
publisher = {Springer},
keywords = {perfect numerical semigroup},
keywords = {saturated numerical semigroup},
keywords = {Arf numerical semigroup},
title = {The covariety of perfect numerical semigroups with fixed Frobenius number},
doi = {10.21136/CMJ.2024.0379-23},
author = {Moreno Frías, María Ángeles and Rosales González, José Carlos},
}