%0 Journal Article 
%A Moreno Frías, María Ángeles
%A Rosales, José Carlos
%T The covariety of perfect numerical semigroups with fixed Frobenius number
%D 2024
%@ 1572-9141

%U http://hdl.handle.net/10498/35406
%X 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).
%K Perfect numerical semigroup
%K saturated numerical semigroup
%K Arf numerical semigroup
%K covariety
%K Frobenius number
%K genus
%K algorithm
%~ Universidad de Cádiz