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 \(\mathscr{C}\) of numerical semigroups that fulfills the following conditions: there exists the minimum of \(\mathscr{C}\), the intersection of two elements of \(\mathscr{C}\) is again an element of \(\mathscr{C}\), and \(S\backslash\{{\rm m}(S)\}\in\mathscr{C}\) for all \(S\in\mathscr{C}\) such that \(S\neq\min(\mathscr{C})\). We prove that the set \({\mathscr{P}}(F)=\{S\colon\ S\ \text{is}\ \text{a}\ \text{perfect}\ \text{numerical}\ \text{semigroup}\ \text{with}\ \text{Frobenius}\ \text{number}\ F\}\) is a covariety. Also, we describe three algorithms which compute: the set \({\mathscr{P}}(F)\), the maximal elements of \({\mathscr{P}}(F)\), and the elements of \({\mathscr{P}}(F)\) 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).