The covariety of perfect numerical semigroups with fixed Frobenius number

• Published in: Czechoslovak mathematical journal Vol. 74; no. 3; pp. 697 - 714
• Main Authors: Moreno-Frías, María Ángeles, Rosales, José Carlos
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Algorithms; Analysis; Convex and Discrete Geometry; Group theory; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Semigroups; perfect numerical semigroup; Arf numerical semigroup; 11D07; genus; 20M14; Frobenius number; covariety; saturated numerical semigroup; algorithm; 13H10
• DOI: 10.21136/CMJ.2024.0379-23
• ISSN: 0011-4642

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 ≠ 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 ).