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Israel Journal of Mathematics

, Volume 206, Issue 1, pp 395–411 | Cite as

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

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

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.

Keywords

Numerical Semigroup Commutative Monoids Cancellative Semigroup Cancellative Monoid Archimedean Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [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.zbMATHMathSciNetGoogle Scholar
  4. [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. [5]
    L. Diracca, On a generalization of the exchange property to modules with semilocal endomorphism rings, Journal of Algebra 313 (2007), 972–987.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten, Mathematische Zeitschrift 17 (1923), 228–249.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [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. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    A. Geroldinger, Chains of factorizations in weakly Krull domains, Colloquium Mathematicum 72 (1997), 53–81.zbMATHMathSciNetGoogle Scholar
  10. [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.zbMATHGoogle Scholar
  11. [11]
    A. Geroldinger and W. Hassler, Local tameness or v-Noetherian monoids, Journal of Pure and Applied Algebra 212 (2008), 1509–1524.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    R. G. Levin, On commutative, nonpotent, archimedean semigroups, Pacific Journal of Mathematics 27 (1968), 365–371.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    L. Redéi, The Theory of Finitely Generated Commutative Monoids, Pergamon Press, Oxford-Edinburgh-New York, 1965.Google Scholar
  14. [14]
    J. C. Rosales and J. I. García-García, Hereditary archimedean commutative semigroups, International Mathematical Journal 5 (2002), 467–472.Google Scholar
  15. [15]
    J. C. Rosales and P. A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Science Publishers, Inc., Commack, NY, 1999.zbMATHGoogle Scholar
  16. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [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.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  • J. I. García-García
    • 1
  • M. A. Moreno-Frías
    • 1
  • A. Vigneron-Tenorio
    • 2
  1. 1.Departamento de MatemáticasUniversidad de CádizPuerto Real, CádizSpain
  2. 2.Departamento de MatemáticasUniversidad de CádizJerez de la Frontera, CádizSpain

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