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The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

Year 2025, Volume: 38 Issue: 38, 12 - 28, 14.07.2025

Maria Angeles Moreno Frias , José Carlos Rosales

https://doi.org/10.24330/ieja.1575996

Abstract

In this paper we will show that
MED$(F,m)=\{S\mid S \mbox{ is a numerical semigroup with maximal embedding dimension, Frobenius number} ~F~ \mbox{and multiplicity}~ m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes MED$(F,m)$ and another one that calculates the elements of MED$(F,m)$ with a given genus.
If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in $ MED$(F,m)$, then there exists the smallest element of MED$(F,m)$ containing $X$. This element will be denoted by MED$(F,m)[X]$ and we will say that $X$ one of its MED$(F,m)$-system of generators. We will prove that every element $S$ of MED$(F,m)$ has a unique minimal MED$(F,m)$-system of generators and it will be denoted by MED$(F,m)$msg$(S).$ The cardinality of MED$(F,m)$msg$(S)$, will be called MED$(F,m)$-rank of $S.$ We will also see in this work, how all the elements of MED$(F,m)$ with a fixed MED$(F,m)$-rank are.

Keywords

Numerical semigroup ratio-covariety Frobenius number genus multiplicity algorithm

References

  • S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math., 89 (1967), 1073-1077.
  • R. Apery, Sur les branches superlineaires des courbes algebriques, C. R. Acad. Sci. Paris, 222 (1946), 1198-2000.
  • V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc., 125 (1997), 598 (78 pp).
  • J. Bertin and P. Carbonne, Semi-groupes d'entiers et application aux branches, J. Algebra, 49(1) (1977), 81-95.
  • V. Blanco and J. C. Rosales, The tree of irreducible numerical semigroup with fixed Frobenius number, Forum Math., 25(6) (2013), 1249-1261.
  • W. C. Brown and J. Herzog, One dimensional local rings of maximal and almost maximal length, J. Algebra, 151 (1992), 332-347.
  • J. Castellanos, A relation between the sequence of multiplicities and the semigroups of values of an algebroid curve, J. Pure Appl. Algebra, 43(2) (1986), 119-127.
  • M. Delgado, P. A. Garcia-Sanchez and J. Morais, NumericalSgps, A package for numerical semigroups, Version 1.3.1 (2022), Refereed GAP package, \verb+ https://gap-packages.github.io/numericalsgps+.
  • C. Delorme, Sous-monoides d'intersection complete de N, Ann. Scient. École Norm. Sup., (4)9 (1976), 145-154.
  • The GAP group, GAP-Groups, Algorithms, and Programming, Version 4.12.2, 2022, \verb+https://www.gap-system.org+.
  • E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc., 25 (1970), 748-751.
  • M. A. Moreno-Frias and J. C. Rosales, Ratio-covariety of numerical semigroups, Axioms, 13(3) (2024), 193 (13 pp).
  • J. C. Rosales, Principal ideals of numerical semigroups, Bull. Belg. Math. Soc., 10 (2003), 329-343.
  • J. C. Rosales and M. B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups, J. Pure Appl. Algebra, 171 (2002), 303-314.
  • J. C. Rosales and M. B. Branco, Irreducible numerical semigroups, Pacific J. Math., 209 (2003), 131-143.
  • J. C. Rosales and P. A. Garcia-Sanchez, Numerical Semigroups, Developments in Mathematics, Vol. 20, Springer, New York, 2009.
  • J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and M. B. Branco, Numerical semigroups with maximal embedding dimension, Int. J. Commut. Rings, 2 (2003), 47-53.
  • J. D. Sally, Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra, 56 (1979), 168-183.
  • K. Watanabe, Some examples of one dimensional Gorenstein domains, Nagoya Math. J., 49 (1973), 101-109.

Year 2025, Volume: 38 Issue: 38, 12 - 28, 14.07.2025

Maria Angeles Moreno Frias , José Carlos Rosales

https://doi.org/10.24330/ieja.1575996

Abstract

References

  • S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math., 89 (1967), 1073-1077.
  • R. Apery, Sur les branches superlineaires des courbes algebriques, C. R. Acad. Sci. Paris, 222 (1946), 1198-2000.
  • V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc., 125 (1997), 598 (78 pp).
  • J. Bertin and P. Carbonne, Semi-groupes d'entiers et application aux branches, J. Algebra, 49(1) (1977), 81-95.
  • V. Blanco and J. C. Rosales, The tree of irreducible numerical semigroup with fixed Frobenius number, Forum Math., 25(6) (2013), 1249-1261.
  • W. C. Brown and J. Herzog, One dimensional local rings of maximal and almost maximal length, J. Algebra, 151 (1992), 332-347.
  • J. Castellanos, A relation between the sequence of multiplicities and the semigroups of values of an algebroid curve, J. Pure Appl. Algebra, 43(2) (1986), 119-127.
  • M. Delgado, P. A. Garcia-Sanchez and J. Morais, NumericalSgps, A package for numerical semigroups, Version 1.3.1 (2022), Refereed GAP package, \verb+ https://gap-packages.github.io/numericalsgps+.
  • C. Delorme, Sous-monoides d'intersection complete de N, Ann. Scient. École Norm. Sup., (4)9 (1976), 145-154.
  • The GAP group, GAP-Groups, Algorithms, and Programming, Version 4.12.2, 2022, \verb+https://www.gap-system.org+.
  • E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc., 25 (1970), 748-751.
  • M. A. Moreno-Frias and J. C. Rosales, Ratio-covariety of numerical semigroups, Axioms, 13(3) (2024), 193 (13 pp).
  • J. C. Rosales, Principal ideals of numerical semigroups, Bull. Belg. Math. Soc., 10 (2003), 329-343.
  • J. C. Rosales and M. B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups, J. Pure Appl. Algebra, 171 (2002), 303-314.
  • J. C. Rosales and M. B. Branco, Irreducible numerical semigroups, Pacific J. Math., 209 (2003), 131-143.
  • J. C. Rosales and P. A. Garcia-Sanchez, Numerical Semigroups, Developments in Mathematics, Vol. 20, Springer, New York, 2009.
  • J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and M. B. Branco, Numerical semigroups with maximal embedding dimension, Int. J. Commut. Rings, 2 (2003), 47-53.
  • J. D. Sally, Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra, 56 (1979), 168-183.
  • K. Watanabe, Some examples of one dimensional Gorenstein domains, Nagoya Math. J., 49 (1973), 101-109.
There are 19 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Maria Angeles Moreno Frias

José Carlos Rosales

Early Pub Date October 30, 2024
Publication Date July 14, 2025
Submission Date January 14, 2024
Acceptance Date September 18, 2024
Published in Issue Year 2025 Volume: 38 Issue: 38

Cite

APA Moreno Frias, M. A., & Rosales, J. C. (2025). The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. International Electronic Journal of Algebra, 38(38), 12-28. https://doi.org/10.24330/ieja.1575996
AMA Moreno Frias MA, Rosales JC. The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. IEJA. July 2025;38(38):12-28. doi:10.24330/ieja.1575996
Chicago Moreno Frias, Maria Angeles, and José Carlos Rosales. “The Ratio-Covariety of Numerical Semigroups Having Maximal Embedding Dimension With Fixed Multiplicity and Frobenius Number”. International Electronic Journal of Algebra 38, no. 38 (July 2025): 12-28. https://doi.org/10.24330/ieja.1575996.
EndNote Moreno Frias MA, Rosales JC (July 1, 2025) The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. International Electronic Journal of Algebra 38 38 12–28.
IEEE M. A. Moreno Frias and J. C. Rosales, “The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number”, IEJA, vol. 38, no. 38, pp. 12–28, 2025, doi: 10.24330/ieja.1575996.
ISNAD Moreno Frias, Maria Angeles - Rosales, José Carlos. “The Ratio-Covariety of Numerical Semigroups Having Maximal Embedding Dimension With Fixed Multiplicity and Frobenius Number”. International Electronic Journal of Algebra 38/38 (July2025), 12-28. https://doi.org/10.24330/ieja.1575996.
JAMA Moreno Frias MA, Rosales JC. The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. IEJA. 2025;38:12–28.
MLA Moreno Frias, Maria Angeles and José Carlos Rosales. “The Ratio-Covariety of Numerical Semigroups Having Maximal Embedding Dimension With Fixed Multiplicity and Frobenius Number”. International Electronic Journal of Algebra, vol. 38, no. 38, 2025, pp. 12-28, doi:10.24330/ieja.1575996.
Vancouver Moreno Frias MA, Rosales JC. The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. IEJA. 2025;38(38):12-28.