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Volume 13
Issue 11
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Article

Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps

by
María A. Moreno-Frías
1,*,† and
José Carlos Rosales
2,†
1
Department of Mathematics, Faculty of Sciences, University of Cádiz, E-11510 Cádiz, Spain
2
Department of Algebra, Faculty of Sciences, University of Granada, E-18071 Granada, Spain
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2025, 13(11), 1744; https://doi.org/10.3390/math13111744 (registering DOI)
Submission received: 25 March 2025 / Revised: 13 May 2025 / Accepted: 23 May 2025 / Published: 24 May 2025
Versions Notes

Abstract

If P is a nonempty finite subset of positive integers, then A(P)={S∣Sisanumericalsemigroup,S∩P=∅andmax(P)istheFrobeniusnumberofS}. In this work, we prove that A(P) is a covariety; therefore, we can arrange the elements of A(P) in the form of a tree. This fact allows us to present several algorithms, including one that calculates all the elements of A(P), another that obtains its maximal elements (with respect to the set inclusion order) and one more that computes the elements of A(P) that cannot be expressed as an intersection of two elements of A(P), that properly contain it.
Keywords: Frobenius number; gap; multiplicity; algorithm; covariety; irreducible element; R variety Frobenius number; gap; multiplicity; algorithm; covariety; irreducible element; R variety

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MDPI and ACS Style

Moreno-Frías, M.A.; Rosales, J.C. Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps. Mathematics 2025, 13, 1744. https://doi.org/10.3390/math13111744

AMA Style

Moreno-Frías MA, Rosales JC. Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps. Mathematics. 2025; 13(11):1744. https://doi.org/10.3390/math13111744

Chicago/Turabian Style

Moreno-Frías, María A., and José Carlos Rosales. 2025. "Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps" Mathematics 13, no. 11: 1744. https://doi.org/10.3390/math13111744

APA Style

Moreno-Frías, M. A., & Rosales, J. C. (2025). Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps. Mathematics, 13(11), 1744. https://doi.org/10.3390/math13111744

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