A decomposition of partitions and numerical sets

Abstract

The aim of this work is to exhibit a decomposition of partitions of natural numbers and numerical sets. In particular, we obtain a decomposition of a sparse numerical set into the so called hook semigroups which turn out to be primitive. Since each Arf semigroup is sparse, we thus obtain a decomposition of any Arf semigroup into primitive numerical semigroups.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Antokoletz, E., Miller, A.: Symmetry and factorization of numerical sets and monoids. J. Algebra 247, 636–671 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arf, C.: Une interprétation algébrique de la suite ordres de multiplicité d’une branche algébrique. Proc. Lond. Math. Soc. 20, 256–287 (1949)

    MATH  Google Scholar 

  3. 3.

    Barucci, V., Dobbs, D.E., Fontana, M.: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Mem. Am. Math. Soc. 125(598), 1–77 (1997)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Constantin, H., Houston-Edwards, B., Kaplan, N.: Numerical sets, core partitions, and integer points in polytopes. Combinatorial and Additive Number Theory II—CANT, New York, NY, USA, 2015 and 2016: Springer Proc Math Stat, Springer, vol. 220, pp. 99–127 (2017)

  5. 5.

    Fulton, W.: Young tableaux, with application to representation theory and geometry. Cambridge University Press, New York (1997)

    Google Scholar 

  6. 6.

    García-Sánchez, P.A., Karakaş, H.İ., Heredia, B.A., Rosales, J.C.: Parametrizing Arf numerical semigroups. J. Algebra Appl. 16, 11 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    İlhan, S., Karakaş, H.İ.: Arf numerical semigroups. Turk. J. Math. 41, 1448–1457 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Keith, W.J., Nath, R.: Partitions with prescribed hooksets. J. Comb. Num. Thy. 3(1), 39–50 (2011)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Marzuola, J., Miller, A.: Counting numerical sets with small atoms. J. Comb. Thy. 117 A, 650–667 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Munuera, C., Torres, F., Villanueva, J.: Sparse numerical semigroups. Lecture Notes in Computer Science, Applied algebra, algebraic algorithms and error-correcting codes. Springer, Heidelberg, vol. 5527, pp 23–31 (2009)

  11. 11.

    Rosales, J.C., Garcia-Sánchez, P.A., Garcia, J.I., Branco, M.: Arf numerical semigroups. J. Algebra 276, 3–12 (2004)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Rosales, J.C., Garcia-Sánchez, P.A.: Numerical Semigroups. Springer, New York (2009)

    Google Scholar 

  13. 13.

    Tutaş, N., Karakaş, H.İ., Gümüşbaş, N.: Young tableaux and Arf partitions. Turk. J. Math. 43, 448–459 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nesrin Tutaş.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Fernando Torres.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Karakaş, H.İ., Tutaş, N. A decomposition of partitions and numerical sets. Semigroup Forum 101, 704–715 (2020). https://doi.org/10.1007/s00233-019-10080-7

Download citation

Keywords

  • Partition
  • Arf partition
  • Numerical set
  • Young Tableau
  • Hook set
  • Numerical semigroup
  • Arf semigroup
  • Arf closure