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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Birkhoff G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. vol. 25, rev. ed. Amer. Math. Soc., New York (1948)
Grätzer, G.: The Congruences of a Finite Lattice. A Proof-by-Picture Approach. Birkhäuser Boston (2006)
Grätzer G.: Lattice Theory: Foundation. Birkhäuser Verlag, Basel (2011)
Grätzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad Math. Bull. 41, 290–297 (1998)
Grätzer G., Schmidt E.T.: On inaccessible and minimal congruence relations. I. Acta Sci Math. (Szeged) 21, 337–342 (1960)
Grätzer G., Schmidt E.T.: On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13, 179–185 (1962)
Grätzer G., Schmidt E.T.: A lattice construction and congruence-preserving extensions. Acta Math. Hungar. 66, 275–288 (1995)
Grätzer G., Schmidt E.T.: Congruence-preserving extensions of finite lattices to semimodular lattices. Houston J. Math. 27, 1–9 (2001)
Grätzer G., Schmidt E.T.: Regular congruence-preserving extensions of lattices. Algebra Universalis 46, 119–130 (2001)
Grätzer G., Schmidt E.T.: On the Independence Theorem of related structures for modular (arguesian) lattices. Studia Sci. Math. Hungar. 40, 1–12 (2003)
Grätzer G., Wehrung F.: Proper congruence-preserving extensions of lattices. Acta Math. Hungar. 85, 175–185 (1999)
Grätzer G. and Wehrung, F., editors: Lattice Theory: Empire. Special Topics and Applications. Birkhäuser, Basel, forthcoming.
Ore O.: Theory of equivalence relations. Duke Math. J. 9, 573–627 (1942)
Schmidt E.T.: Über die Kongruenzverbänder der Verbände. Publ. Math. Debrecen 9, 243–256 (1962)
Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by G. Czédli.
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-013-0242-3