Advertisement

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

Chapter
  • 430 Downloads
Part of the Foundations for Undergraduate Research in Mathematics book series (FURM)

Abstract

We give a self-contained introduction to numerical semigroups and present several open problems centered on their factorization properties.

Notes

Acknowledgements

The authors would like to thank an unknown referee for detailed comments that greatly improved this manuscript. They would also thank Nathan Kaplan for his discussions and input. All plots created using Sage [21], and all computations involving numerical semigroups utilize the GAP package numericalsgps [12].

References

  1. 1.
    Amos, J., Chapman, S., Hine, N., Paixao, J.: Sets of lengths do not characterize numerical monoids. Integers, 7(1): Paper-A50 (2007).Google Scholar
  2. 2.
    Barron, T., O’Neill, C., Pelayo, R.: On the set of elasticities in numerical monoids. Semigroup Forum 94: 37–50 (2017). Available at arXiv:math.CO/1409.3425.Google Scholar
  3. 3.
    Bowles, C., Chapman, S., Kaplan, N., Reiser, D.: On delta sets of numerical monoids. J. Algebra Appl. 5: 695–718 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapman, S., Daigle, J., Hoyer, R., Kaplan, N.: Delta sets of numerical monoids using nonminimal sets of generators. Comm. Algebra. 38: 2622–2634 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chapman, S., García-Sánchez, P., Llena, D., Malyshev, A., Steinberg, D.: On the delta set and the Betti elements of a BF-monoid. Arabian Journal of Mathematics. 1: 53–61 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chapman, S., Holden, M., Moore, T.: Full elasticity in atomic monoids and integral domains. The Rocky Mountain Journal of Mathematics. 36:1437–1455 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chapman, S., Hoyer, R., Kaplan, N.: Delta sets of numerical monoids are eventually periodic. Aequationes mathematicae. 77: 273–279 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chapman, S., Kaplan, N., Lemburg, T., Niles, A., Zlogar, C.: Shifts of generators and delta sets of numerical monoids. International Journal of Algebra and Computation. 24: 655–669 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chapman, S., O’Neill, C.: Factorization in the Chicken McNugget Monoid. Math. Mag. 91(5): 323–336 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Loera, J., O’Neill, C., Wilbourne, D.: Random numerical semigroups and a simplicial complex of irreducible semigroups. Electronic Journal of Combinatorics. 25: #P4.37 (2018).Google Scholar
  11. 11.
    Delgado, M., García-Sánchez, P.: numericalsgps, a GAP package for numerical semigroups. ACM Commun. Comput. Algebra 50 (2016), no. 1, 12–24.Google Scholar
  12. 12.
    Delgado, M., García-Sánchez, P., Morais, J.: NumericalSgps, A package for numerical semigroups, Version 1.2.0 (2019), (Refereed GAP package), https://gap-packages.github.io/numericalsgps.
  13. 13.
    García, S., O’Neill, C., Yih, S.: Factorization length distribution for affine semigroups I: numerical semigroups with three generators. to appear, European Journal of Combinatorics. Available at arXiv:1804.05135.Google Scholar
  14. 14.
    García-García, J., Moreno-Frías, M., Vigneron-Tenorio, A.: Computation of delta sets of numerical monoids. Monatshefte für Mathematik. 178: 457–472 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    García-Sánchez, P., Llena, D., Moscariello, A.: Delta sets for symmetric numerical semigroups with embedding dimension three. Aequationes Math. 91(2017), 579–600.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    García-Sánchez, P., Llena, D., Moscariello, A.: Delta sets for nonsymmetric numerical semigroups with embedding dimension three. Forum Math. 30(2018), 15–30.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    García-Sánchez, P., O’Neill, C., Webb, G.: On the computation of factorization invariants for affine semigroups. Journal of Algebra and its Applications. 18: 1950019, 21 pp (2019).Google Scholar
  18. 18.
    Geroldinger, A.: On the arithmetic of certain no integrally closed noetherian integral domains. Comm. Algebra. 19: 685–698 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glen, J., O’Neill, C., Ponomarenko, V., Sepanski, B.: Augmented Hilbert series of numerical semigroups. arXiv:1806.11148.Google Scholar
  20. 20.
    Colton, S., Kaplan, N.: The realization problem for delta sets of numerical semigroups. Journal of Commutative Algebra. 9: 313–339 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    The Sage Developers: SageMath, the Sage Mathematics Software System (Version 7.2). (2016) http://www.sagemath.org.
  22. 22.
    Sylvester, J.: On subinvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order. American Journal of Mathematics. 5: 134 (1882).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Mathematics and Statistics DepartmentSam Houston State UniversityHuntsvilleUSA
  2. 2.Mathematics and Statistics DepartmentSan Diego State UniversitySan DiegoUSA

Personalised recommendations