Advertisement

Algebra universalis

, Volume 77, Issue 1, pp 51–77 | Cite as

Representing some families of monotone maps by principal lattice congruences

  • Gábor Czédli
Article

Abstract

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.

Key words and phrases

principal congruence lattice congruence ordered set order poset quasi-colored lattice preordering quasiordering monotone map categorified lattice functor lattice category lifting diagrams 

2010 Mathematics Subject Classification

06B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bogart, K.P., Freese, R., Kung, J.P.S. (eds): The Dilworth Theorems. Selected papers of Robert P. Dilworth. Birkhäuser, Boston (1990)Google Scholar
  2. 2.
    Czédli G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Universalis 67, 313–345 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Czédli G.: Representing a monotone map by principal lattice congruences. Acta Math. Hungar. 147, 12–18 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Czédli G.: An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices. Acta Sci. Math. 82, 3–18 (2016)MathSciNetGoogle Scholar
  5. 5.
    Czédli G.: The ordered set of principal congruences of a countable lattice. Algebra Universalis 75, 351–380 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Czédli, G.: Diagrams and rectangular extensions of planar semimodular lattices. Algebra Universalis, to appear. arXiv:1412.4453
  7. 7.
    Czédli G., Schmidt E.T.: Finite distributive lattices are congruence lattices of almost-geometric lattices. Algebra Universalis 65, 91–108 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Funayama N., Nakayama T.: On the distributivity of a lattice of lattice-congruences. Proc. Imp. Acad. Tokyo 18, 553–554 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gillibert, P.; Wehrung, F.: From objects to diagrams for ranges of functors. Lecture Notes in Mathematics 2029. Springer, Heidelberg (2011)Google Scholar
  10. 10.
    Grätzer, G.: The Congruences of a Finite Lattice. A Proof-by-picture Approach. Birkhäuser, Boston (2006)Google Scholar
  11. 11.
    Grätzer G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Grätzer G.: The order of principal congruences of a bounded lattice. Algebra Universalis 70, 95–105 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grätzer G.: A technical lemma for congruences of finite lattices. Algebra Universalis 72, 53–55 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grätzer G.: Congruences and prime-perspectivities in finite lattices. Algebra Universalis 74, 351–359 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grätzer G., Knapp E.: Notes on planar semimodular lattices. III. Congruences of rectangular lattices. Acta Sci. Math. (Szeged) 75, 29–48 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Grätzer G., Lakser H.: Homomorphisms of distributive lattices as restrictions of congruences. Canad. J Math. 38, 1122–1134 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grätzer G., Lakser H.: Homomorphisms of distributive lattices as restrictions of congruences. II. Planarity and automorphisms. Canad. J Math. 46, 3–54 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grätzer G., Lakser H.: Representing homomorphisms of congruence lattices as restrictions of congruences of isoform lattices. Acta Sci. Math. (Szeged) 75, 393–421 (2009)MathSciNetGoogle Scholar
  19. 19.
    Grätzer G., Lakser H., Schmidt E.T.: Isotone maps as maps of congruences. I. Abstract maps. Acta Math. Hungar. 75, 105–135 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grätzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad. Math. Bull. 41, 290–297 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grätzer G., Lakser H., Schmidt E.T.: Isotone maps as maps of congruences. II. Concrete maps. Acta Math. Hungar. 92, 233–238 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grätzer G., Schmidt E.T.: On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13, 179–185 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Grätzer G., Schmidt E.T.: An extension theorem for planar semimodular lattices. Period. Math. Hungar. 69, 32–40 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huhn A. P.: On the representation of distributive algebraic lattices. III. Acta Sci. Math. (Szeged) 53, 11–18 (1989)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kamara M.: Zur Konstruktion vollständiger Polaritätsverbände. (German) J. Reine Angew. Math. 299/300, 280–286 (1978)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kelly D., Rival I.: Planar lattices. Canad. J. Math. 27, 636–665 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nation, J. B.: Notes on Lattice Theory. http://www.math.hawaii.edu/~jb/books.html
  28. 28.
    Růžička P.: Free trees and the optimal bound in Wehrung’s theorem. Fund. Math. 198, 217–228 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wehrung, F.: Schmidt and Pudlák’s approaches to CLP. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications I, pp. 235–296. Birkhäuser, Basel (2014)Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.University of Szeged, Bolyai InstituteSzegedHungary

Personalised recommendations