Algebra universalis

, Volume 70, Issue 1, pp 95–105 | Cite as

The order of principal congruences of a bounded lattice

  • G. Grätzer


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.

2010 Mathematics Subject Classification

Primary: 06B10 Secondary: 06A06 

Key words and phrases

principal congruence order 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birkhoff G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. vol. 25, rev. ed. Amer. Math. Soc., New York (1948)Google Scholar
  2. 2.
    Grätzer, G.: The Congruences of a Finite Lattice. A Proof-by-Picture Approach. Birkhäuser Boston (2006)Google Scholar
  3. 3.
    Grätzer G.: Lattice Theory: Foundation. Birkhäuser Verlag, Basel (2011)zbMATHCrossRefGoogle Scholar
  4. 4.
    Grätzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad Math. Bull. 41, 290–297 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Grätzer G., Schmidt E.T.: On inaccessible and minimal congruence relations. I. Acta Sci Math. (Szeged) 21, 337–342 (1960)zbMATHGoogle Scholar
  6. 6.
    Grätzer G., Schmidt E.T.: On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13, 179–185 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Grätzer G., Schmidt E.T.: A lattice construction and congruence-preserving extensions. Acta Math. Hungar. 66, 275–288 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grätzer G., Schmidt E.T.: Congruence-preserving extensions of finite lattices to semimodular lattices. Houston J. Math. 27, 1–9 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grätzer G., Schmidt E.T.: Regular congruence-preserving extensions of lattices. Algebra Universalis 46, 119–130 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Grätzer G., Wehrung F.: Proper congruence-preserving extensions of lattices. Acta Math. Hungar. 85, 175–185 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Grätzer G. and Wehrung, F., editors: Lattice Theory: Empire. Special Topics and Applications. Birkhäuser, Basel, forthcoming.Google Scholar
  13. 13.
    Ore O.: Theory of equivalence relations. Duke Math. J. 9, 573–627 (1942)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Schmidt E.T.: Über die Kongruenzverbänder der Verbände. Publ. Math. Debrecen 9, 243–256 (1962)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

Personalised recommendations