The covariety of perfect numerical semigroups with fixed Frobenius number

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).