Acta Mathematica Hungarica

, Volume 147, Issue 1, pp 12–18 | Cite as

Representing a monotone map by principal lattice congruences

  • G. Czédli


For a lattice L, let Princ (L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer proved that bounded ordered sets (in other words, posets with 0 and 1) are, up to isomorphism, exactly the Princ (L) of bounded lattices L. Here we prove that for each 0-separating boundpreserving monotone map ψ between two bounded ordered sets, there are a lattice L and a sublattice K of L such that, in essence, ψ is the map from Princ (K) to Princ (L) that sends a principal congruence to the congruence it generates in the larger lattice.

Key words and phrases

principal congruence lattice congruence ordered set order poset quasi-colored lattice preordering quasiordering monotone map 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Czédli, The ordered set of principal congruences of a countable lattice, algebra universalis, to appear; see also at
  2. 2.
    Czédli G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Universalis 67, 313–345 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Czédli, Representing some families of monotone maps by principal lattice congruences, Algebra Universalis, submitted in April, 2015.Google Scholar
  4. 4.
    K. P. Bogart, R. Freese and J. P. S. Kung (editors), The Dilworth Theorems. Selected papers of Robert P. Dilworth, Birkhäuser Boston, Inc. (Boston, MA, 1990), xxvi+465 pp. ISBN: 0-8179-3434-7Google Scholar
  5. 5.
    G. Grätzer, The Congruences of a Finite Lattice. A Proof-by-picture Approach, Birkhäuser (Boston, 2006).Google Scholar
  6. 6.
    Grätzer G.: The order of principal congruences of a bounded lattice. Algebra Universalis 70, 95–105 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grätzer G., Lakser H., Schmidt E. T.: Congruence lattices of finite semimodular lattices. Canad. Math. Bull. 41, 290–297 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huhn A. P.: On the representation of distributive algebraic lattices. III. Acta Sci. Math. (Szeged) 53, 11–18 (1989)MathSciNetGoogle Scholar
  9. 9.
    Růžička P.: Free trees and the optimal bound in Wehrung’s theorem. Fund. Math. 198, 217–228 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Schmidt E. T.: The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice. Acta Sci. Math. (Szeged) 43, 153–168 (1981)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

Personalised recommendations