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Second-Level Numerical Semigroups

Author

Listed:
  • David Llena

    (Departamento de Matemáticas, Universidad de Almería, E-04120 Almería, Spain
    These authors contributed equally to this work.)

  • José Carlos Rosales

    (Departamento de Álgebra, E-18071 Granada, Spain
    These authors contributed equally to this work.)

Abstract

Let S be a numerical semigroup with multiplicity m ( S ) . Then, S is called a second-level numerical semigroup if x + y + z − m ( S ) ∈ S for every { x , y , z } ⊆ S ∖ { 0 } . In this paper, we present some algorithms to compute all the second-level numerical semigroups with multiplicity, genus, and a Frobenius fixed number. For m and r , which are positive integers, such that m < r and gcd ( m , r ) = 1 , we show that there exists the minimal second-level numerical semigroup with multiplicity m containing r . We solve the Frobenius problem for these semigroups and show that they satisfy Wilf’s conjecture.

Suggested Citation

  • David Llena & José Carlos Rosales, 2025. "Second-Level Numerical Semigroups," Mathematics, MDPI, vol. 13(4), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:593-:d:1588668
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