Monatshefte für Mathematik

, Volume 178, Issue 3, pp 457–472 | Cite as

Computation of Delta sets of numerical monoids

  • J. I. García-García
  • M. A. Moreno-Frías
  • A. Vigneron-Tenorio


Let \(\{a_1,\ldots ,a_p\}\) be the minimal generating set of a numerical monoid S. For any \(s\in S\), its Delta set is defined by \(\Delta (s)=\{l_{i}-l_{i-1}\mid i=2,\ldots ,k\}\) where \(\{l_1<\dots <l_k\}\) is the set \(\{\sum _{i=1}^px_i\mid s=\sum _{i=1}^px_ia_i \text { and } x_i\in \mathbb {N}\text { for all }i\}.\) The Delta set of a numerical monoid S, denoted by \(\Delta (S)\), is the union of all the sets \(\Delta (s)\) with \(s\in S.\) As proved in Chapman et al. (Aequationes Math. 77(3):273–279, 2009), there exists a bound N such that \(\Delta (S)\) is the union of the sets \(\Delta (s)\) with \(s\in S\) and \(s<N\). In this work, we obtain a sharpened bound and we present an algorithm for the computation of \(\Delta (S)\) that requires only the factorizations of \(a_1\) elements.


Delta set Non-unique factorization Numerical monoid Numerical semigroup 

Mathematics Subject Classification

Primary 20M14 Secondary 20M05 


  1. 1.
    Baginski, P., Chapman, S.T., Schaeffer, G.J.: On the delta set of a singular arithmetical congruence monoid. J. Théor. Nombres Bordeaux 20(1), 45–59 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bowles, C., Chapman, S.T., Kaplan, N., Reiser, D.: On delta sets of numerical monoids. J. Algebra Appl. 5(5), 695–718 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chapman, S.T., Daigle, J., Hoyer, R., Kaplan, N.: Delta sets of numerical monoids using nonminimal sets of generators. Comm. Algebra 38(7), 2622–2634 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapman, S.T., Garcia, P.A., Llena, D., Malyshev, A., Steinberg, D.: On the delta set and the Betti elements of a BF-monoid. Arab. J. Math. (Springer) 1(1), 53–61 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chapman, S.T., Hoyer, R., Kaplan, N.: Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77(3), 273–279 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chapman S.T., Kaplan N., Lemburg T., Niles A., Zlogar C.: Shifts of generators and delta sets of numerical monoids. J. Comm. AlgebraGoogle Scholar
  7. 7.
    Chapman S.T., Malyshev A., Steinberg D.: On the delta set of a 3-generated numerical monoid, submitted, (online available at
  8. 8.
    Clausen, M., Fortenbacher, A.: Efficient solution of linear Diophantine equations. J. Symbolic Comput. 8(1–2), 201–216 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    García-García J.I., Vigneron-Tenorio A.: DeltaSetsNumericalMonoids, a Mathematica package for computing the Delta sets of numerical semigroups. Online available at, (2015)
  10. 10.
    Geroldinger, A.: On the arithmetic of certain not integrally closed Noetherian integral domains. Comm. Algebra 19(2), 685–698 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Geroldinger A., Halter-Koch F.: Non-unique factorizations. Algebraic, combinatorial and analytic theory, Pure and Applied Mathematics (Boca Raton), 278. Chapman and Hall/CRC (2006)Google Scholar
  12. 12.
    Haarmann J., Kalauli A., Moran A., O’Neill C., Pelayo R.: Factorization properties of Leamer monoids, arXiv:1309.7477v1
  13. 13.
    Halter-Koch, F.: Arithmetical semigroups defined by congruences. Semigroup Forum 42(1), 59–62 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rosales, J.C., García-Sánchez, P.A.: Finitely generated commutative monoids. Nova Science Publishers Inc, New York (1999)zbMATHGoogle Scholar
  15. 15.
    Rosales, J.C., García-Sánchez, P.A.: Numerical semigroups, Developments in Mathematics, 20. Springer, New York (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de CádizCadizSpain
  2. 2.Departamento de MatemáticasUniversidad de CádizCadizSpain

Personalised recommendations