Skip to main content
SpringerLink
Log in

The Monotone Catenary Degree of Krull Monoids

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c mon (H) of H is the smallest integer m with the following property: for each \({a \in H}\) and each two factorizations z, z′ of a with length |z| ≤  |z′|, there exist factorizations z = z 0, ... ,z k  = z′ of a with increasing lengths—that is, |z 0| ≤  ... ≤  |z k |—such that, for each \({i \in [1,k]}\) , z i arises from z i-1 by replacing at most m atoms from z i-1 by at most m new atoms. Up to now there was only an abstract finiteness result for c mon (H), but the present paper offers the first explicit upper and lower bounds for c mon (H) in terms of the group invariants of G.

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. Blanco, V., García-Sánchez, P.A., Geroldinger, A.: Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids. Ill. J. Math. (in press)

  2. Chang G.-W.: Every divisor class of Krull monoid domains contains a prime ideal. J. Algebra 336, 370–377 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chapman S.T., García-Sánchez P.A., Llena D.: The catenary and tame degree of numerical monoids. Forum Math. 21, 117–129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chapman S.T., García-Sánchez P.A., Llena D., Ponomarenko V., Rosales J.C.: The catenary and tame degree in finitely generated commutative cancellative monoids. Manuscr. Math. 120, 253–264 (2006)

    Article  MATH  Google Scholar 

  5. Coykendall J., Smith W.W.: On unique factorization domains. J. Algebra 332, 62–70 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delgado, M., García-Sánchez, P.A., Morais, J.: “Numericalsgps”: a gap package on numerical semigroups. http://www.gap-system.org/Packages/numericalsgps.html

  7. Diaconis P., Graham R.L., Sturmfels B.: Primitive partition identities, combinatorics, Paul Erdös is eighty. J. Bolyai Math. Soc. 1, 173–192 (1993)

    Google Scholar 

  8. Foroutan A.: Monotone chains of factorizations. In: Badawi, A. (ed.) Focus on commutative rings research, pp. 107–130. Nova Science Publishers, New York (2006)

    Google Scholar 

  9. Foroutan, A., Geroldinger, A.: Monotone chains of factorizations in C-monoids. In: Arithmetical Properties of Commutative Rings and Monoids, Lect. Notes Pure Appl. Math., vol. 241, pp. 99–113. Chapman & Hall/CRC, London (2005)

  10. Foroutan A., Hassler W.: Chains of factorizations and factorizations with successive lengths. Commun. Algebra 34, 939–972 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Freeze M., Schmid W.A.: Remarks on a generalization of the Davenport constant. Discrete Math. 310, 3373–3389 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Geroldinger A.: Additive group theory and non-unique factorizations. In: Geroldinger, A., Ruzsa, I. (eds) Combinatorial Number Theory and Additive Group Theory. Advanced Courses in Mathematics CRM Barcelona, pp. 1–86. Birkhäuser, Basel (2009)

    Chapter  Google Scholar 

  13. Geroldinger A., Grynkiewicz D.J., Schaeffer G.J., Schmid W.A.: On the arithmetic of Krull monoids with infinite cyclic class group. J. Pure Appl. Algebra 214, 2219–2250 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geroldinger A., Grynkiewicz D.J., Schmid W.A.: The catenary degree of Krull monoids. I. J. Théor. Nombres Bordx. 23, 137–169 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Geroldinger, A., Halter-Koch, F.: Non-Unique Factorizations. In: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, London (2006)

  16. Geroldinger A., Hassler W.: Arithmetic of Mori domains and monoids. J. Algebra 319, 3419–3463 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Halter-Koch F.: Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  18. Hassler W.: Properties of factorizations with successive lengths in one-dimensional local domains. J. Commut. Algebra 1, 237–268 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kainrath, F.: Arithmetic of Mori domains and monoids: the global case (manuscript)

  20. Kim H.: The distribution of prime divisors in Krull monoid domains. J. Pure Appl. Algebra 155, 203–210 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim H., Park Y.S.: Krull domains of generalized power series. J. Algebra 237, 292–301 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lev V.F.: The rectifiability threshold in abelian groups. Combinatorica 28, 491–497 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Omidali, M.: The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math. (in press)

  24. Philipp A.: A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings. Semigroup Forum 81, 424–434 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Philipp, A.: A precise result on the arithmetic of non-principal orders in algebraic number fields. J. Algebra Appl. (in press)

  26. Philipp A.: A characterization of arithmetical invariants by the monoid of relations. Semigroup Forum 81, 424–434 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Reinhart, A.: On integral domains that are C -monoids. Houston J. Math. (in press)

  28. Schmid, W.A.: The inverse problem associated to the Davenport constant for C 2 ⊕ C 2 ⊕ C 2n , and applications to the arithmetical characterization of class groups. Electron. J. Comb. 18(1) (2011) (Research Paper 33)

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Alfred Geroldinger

  2. School of Mathematics, South China Normal University, 510631, Guangzhou, People’s Republic of China

    Pingzhi Yuan

Authors
  1. Alfred Geroldinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pingzhi Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pingzhi Yuan.

Additional information

This work was supported by the Austrian Science Fund FWF (Projects W1230-N13 and P21576-N18), and by the National Science Fund NSF of China (Project Number 10971072).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Geroldinger, A., Yuan, P. The Monotone Catenary Degree of Krull Monoids. Results. Math. 63, 999–1031 (2013). https://doi.org/10.1007/s00025-012-0250-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-012-0250-1

Mathematics Subject Classification

Keywords

Search

Navigation