Skip to main content
SpringerLink
Log in

Computation of the ω-primality and asymptotic ω-primality with applications to numerical semigroups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give an algorithm to compute the ω-primality of finitely generated atomic monoids. Asymptotic ω-primality is also studied and a formula to obtain it in finitely generated quasi-Archimedean monoids is proven. The formulation is applied to numerical semigroups, obtaining an expression of this invariant in terms of its system of generators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. F. Anderson and S. T. Chapman, How far is an element from being prime, Journal of Algebra and its Applications 9 (2010), 779–789.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. F. Anderson, S. T. Chapman, N. Kaplan and D. Torkornoo, An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), 96–108.

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Blanco, P. A. García-Sánchez and A. Geroldinger, Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids, Illinois Journal of Mathematics 55 (2011), 1385–1414.

    MATH  MathSciNet  Google Scholar 

  4. M. Delgado, P. A. García-Sánchez and J. Morais, “NumericalSgps”: a GAP package for numerical semigroups, http://www.gap-system.org/Packages/numericalsgps.html

  5. L. Diracca, On a generalization of the exchange property to modules with semilocal endomorphism rings, Journal of Algebra 313 (2007), 972–987.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten, Mathematische Zeitschrift 17 (1923), 228–249.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. I. García-García and A. Vigneron-Tenorio, OmegaPrimality, a package for computing the omega primality of finitely generated atomic monoids, Handle: http://hdl.handle.net/10498/15961 (2014).

  8. P. A. García Sánchez, I. Ojeda and A. Sánchez-R-Navarro, Factorization invariants in half-factorial affine semigroups, International Journal of Algebra and Computation 23 (2013), 111–122.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Geroldinger, Chains of factorizations in weakly Krull domains, Colloquium Mathematicum 72 (1997), 53–81.

    MATH  MathSciNet  Google Scholar 

  10. A. Geroldinger and F. Halter-Koch, Non-unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Vol. 278, Chapman & Hall/CRC, Boca Raton, FL, 2006.

    MATH  Google Scholar 

  11. A. Geroldinger and W. Hassler, Local tameness or v-Noetherian monoids, Journal of Pure and Applied Algebra 212 (2008), 1509–1524.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. G. Levin, On commutative, nonpotent, archimedean semigroups, Pacific Journal of Mathematics 27 (1968), 365–371.

    Article  MATH  MathSciNet  Google Scholar 

  13. L. Redéi, The Theory of Finitely Generated Commutative Monoids, Pergamon Press, Oxford-Edinburgh-New York, 1965.

    Google Scholar 

  14. J. C. Rosales and J. I. García-García, Hereditary archimedean commutative semigroups, International Mathematical Journal 5 (2002), 467–472.

    Google Scholar 

  15. J. C. Rosales and P. A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Science Publishers, Inc., Commack, NY, 1999.

    MATH  Google Scholar 

  16. J. C. Rosales, P. A. García-Sánchez and J. I. García-García, Irreducible ideals of finitely generated commutative monoids, Journal of Algebra 238 (2001), 328–344.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. C. Rosales, P. A. García-Sánchez and J. I. García-García, Atomic commutative monoids and their elasticity, Semigroup Forum 68 (2004), 64–86.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Cádiz, E-11510, Puerto Real, Cádiz, Spain

    J. I. García-García & M. A. Moreno-Frías

  2. Departamento de Matemáticas, Universidad de Cádiz, E-11405, Jerez de la Frontera, Cádiz, Spain

    A. Vigneron-Tenorio

Authors
  1. J. I. García-García
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Moreno-Frías
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Vigneron-Tenorio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. I. García-García.

Additional information

Partially supported by the grant MTM2010-15595 and Junta de Andalucía group FQM-366.

Partially supported by MTM2008-06201-C02-02 and Junta de Andalucía group FQM-298.

Partially supported by the grant MTM2007-64704 (with the help of FEDER Program), MTM2012-36917-C03-01 and Junta de Andalucía group FQM-366.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García-García, J.I., Moreno-Frías, M.A. & Vigneron-Tenorio, A. Computation of the ω-primality and asymptotic ω-primality with applications to numerical semigroups. Isr. J. Math. 206, 395–411 (2015). https://doi.org/10.1007/s11856-014-1144-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-014-1144-6

Keywords

Search

Navigation