@misc{10498/35406,
year = {2024},
url = {http://hdl.handle.net/10498/35406},
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).},
publisher = {Institute of Mathematics of the Czech Academy of Sciences},
keywords = {Perfect numerical semigroup},
keywords = {saturated numerical semigroup},
keywords = {Arf numerical semigroup},
keywords = {covariety},
keywords = {Frobenius number},
keywords = {genus},
keywords = {algorithm},
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, José Carlos},
}