Skip to main content
SpringerLink
Log in

The order of principal congruences of a bounded lattice

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.

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. Birkhoff G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. vol. 25, rev. ed. Amer. Math. Soc., New York (1948)

  2. Grätzer, G.: The Congruences of a Finite Lattice. A Proof-by-Picture Approach. Birkhäuser Boston (2006)

  3. Grätzer G.: Lattice Theory: Foundation. Birkhäuser Verlag, Basel (2011)

    Book  MATH  Google Scholar 

  4. Grätzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad Math. Bull. 41, 290–297 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grätzer G., Schmidt E.T.: On inaccessible and minimal congruence relations. I. Acta Sci Math. (Szeged) 21, 337–342 (1960)

    MATH  Google Scholar 

  6. Grätzer G., Schmidt E.T.: On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13, 179–185 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grätzer G., Schmidt E.T.: A lattice construction and congruence-preserving extensions. Acta Math. Hungar. 66, 275–288 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grätzer G., Schmidt E.T.: Congruence-preserving extensions of finite lattices to semimodular lattices. Houston J. Math. 27, 1–9 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Grätzer G., Schmidt E.T.: Regular congruence-preserving extensions of lattices. Algebra Universalis 46, 119–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grätzer G., Schmidt E.T.: On the Independence Theorem of related structures for modular (arguesian) lattices. Studia Sci. Math. Hungar. 40, 1–12 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Grätzer G., Wehrung F.: Proper congruence-preserving extensions of lattices. Acta Math. Hungar. 85, 175–185 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grätzer G. and Wehrung, F., editors: Lattice Theory: Empire. Special Topics and Applications. Birkhäuser, Basel, forthcoming.

  13. Ore O.: Theory of equivalence relations. Duke Math. J. 9, 573–627 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schmidt E.T.: Über die Kongruenzverbänder der Verbände. Publ. Math. Debrecen 9, 243–256 (1962)

    MathSciNet  MATH  Google Scholar 

  15. Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    G. Grätzer

Authors
  1. G. Grätzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Grätzer.

Additional information

Presented by G. Czédli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grätzer, G. The order of principal congruences of a bounded lattice. Algebra Univers. 70, 95–105 (2013). https://doi.org/10.1007/s00012-013-0242-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-013-0242-3

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation