Skip to main content
SpringerLink
Log in

On the Largest Numbers of Congruences of Finite Lattices

  • Published:
Order Aims and scope Submit manuscript

Abstract

We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czédli’s recent work, we prove that the third, fourth and fifth largest numbers of congruences of an n–element lattice are: 5 ⋅ 2n− 5 if n ≥ 5, 2n− 3 and 7 ⋅ 2n− 6 if n ≥ 6, respectively. We also determine the structures of the n–element lattices having 5 ⋅ 2n− 5, 2n− 3, respectively 7 ⋅ 2n− 6 congruences, along with the structures of their congruence lattices.

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. Czédli, G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Universalis 67, 313–345 (2012)

    Article  MathSciNet  Google Scholar 

  2. Czédli, G.: Representing a monotone map by principal lattice congruences. Acta Mathematica Hungarica 147, 12–18 (2015)

    Article  MathSciNet  Google Scholar 

  3. Czédli, G.: The ordered set of principal congruences of a countable lattice. Algebra Universalis 75, 351–380 (2016)

    Article  MathSciNet  Google Scholar 

  4. Czédli, G.: An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices. Acta Sci. Math. (Szeged) 82, 3–18 (2016)

    Article  MathSciNet  Google Scholar 

  5. Czédli, G.: Representing some families of monotone maps by principal lattice congruences. Algebra Universalis 77, 51–77 (2017)

    Article  MathSciNet  Google Scholar 

  6. Czédli, G.: Complete congruence lattices of two related modular lattices. Algebra Universalis 78, 251–289 (2017)

    Article  MathSciNet  Google Scholar 

  7. Czédli, G.: Cometic functors and representing order–preserving maps by principal lattice congruences. Algebra Universalis 79, 59 (2018). https://doi.org/10.1007/s00012--018--0545--5

    Article  MathSciNet  MATH  Google Scholar 

  8. Czédli, G.: A note on finite lattices with many congruences. Acta Universitatis Matthiae Belii, Series Mathematics Online, 22–28 (2018)

  9. Czédli, G.: Finite semilattices with many congruences. Order 36(2), 233–247 (2019)

    Article  MathSciNet  Google Scholar 

  10. Czédli, G., Mureşan, C.: On Principal congruences and the number of congruences of a lattice with more ideals than filters. Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, arXiv:abs/06394 [math.RA] (2019)

  11. Freese, R.: Computing congruence lattices of finite lattices. Proc. Amer. Math. Soc. 125, 3457–3463 (1997)

    Article  MathSciNet  Google Scholar 

  12. Giuntini, R., Mureşan, C., Paoli, F.: PBZ∗–lattices: Ordinal and horizontal sums. arXiv:abs/1811.01869 [math.RA]

  13. Grätzer, G: General Lattice Theory. Birkhäuser Akademie, Basel (1978)

    Book  Google Scholar 

  14. Grätzer, G: The Congruences of Finite Lattices. A “proof–by–picture”Approach, 2nd edn. Birkhäuser–Springer, Cham (2016)

    Book  Google Scholar 

  15. Grätzer, G.: The order of principal congruences of a bounded lattice. Algebra Universalis 70, 95–105 (2013)

    Article  MathSciNet  Google Scholar 

  16. Grätzer, G.: Homomorphisms and principal congruences of bounded lattices. I. Isotone maps of principal congruences. Acta Sci. Math. (Szeged) 82, 353–360 (2016)

    Article  MathSciNet  Google Scholar 

  17. Grätzer, G.: Homomorphisms and principal congruences of bounded lattices. II. Sketching the proof for sublattices. Algebra Universalis 78, 291–295 (2017)

    Article  MathSciNet  Google Scholar 

  18. Grätzer, G.: Homomorphisms and principal congruences of bounded lattices. III. The independence theorem. Algebra Universalis 78, 297–301 (2017)

    Article  MathSciNet  Google Scholar 

  19. Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. III. Rectangular lattices. Acta Sci. Math. (Szeged) 75, 29–48 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Grätzer, G: Universal Algebra, 2nd edn. Springer Science+Business Media, LLC, New York (2008)

    Book  Google Scholar 

  21. Grätzer, G., Lakser, H.: Homomorphisms of distributive lattices as restrictions of congruences. Canad. J. Math. 38, 1122–1134 (1986)

    Article  MathSciNet  Google Scholar 

  22. Grätzer, G., Lakser, H.: Notes on the Set of Principal Congruences of a Finite Lattice. I. Some Preliminary Results, Algebra Universalis, arXiv:abs/1705.05319 [math.RA]

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Mureşan, C.: On the Cardinalities of the Sets of Congruences, Ideals and Filters of a Lattice, Analele Universităţii Bucureşti. Seria Informatică. Proceedings of the Workshop Days of Computer Science (DACS) 2015 LXII, affiliated workshop of the 11th edition of the conference Computability in Europe, pp. 55–68. University of Bucharest, Bucharest (2015)

  26. Mureşan, C.: Cancelling Congruences of Lattices, While Keeping Their Filters and Ideals, arXiv:abs/1710.10183v2 [math.RA]

  27. Mureşan, C., Kulin, J.: Some Extremal Values of the Number of Congruences of a Finite Lattice, arXiv:abs/1801.05282v2 [math.RA]

  28. Ploščica, M.: Uncountable critical points for congruence lattices. Algebra Universalis 76, 415–429 (2016)

    Article  MathSciNet  Google Scholar 

  29. Růžička, P.: Free trees and the optimal bound in Wehrung’s theorem. Fund. Math. 198, 217–228 (2008)

    Article  MathSciNet  Google Scholar 

  30. Schmidt, E.T.: A Survey on Congruence Lattice Representations. Teubner–Texte zur Mathematik, Leipzig (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the research grant Proprietà d’Ordine Nella Semantica Algebrica delle Logiche Non–classiche of Università degli Studi di Cagliari, Regione Autonoma della Sardegna, L. R. 7/2007, n. 7, 2015, CUP: F72F16002920002, as well as the NFSR of Hungary (OTKA), grant number K 115518.

We thank the anonymous referee for devoting his or her time to carefully reading our paper, for correcting our terminology and providing useful suggestions on how to improve the form of this article, and for undergoing this refereing work with responsibility and professionalism.

Author information

Authors and Affiliations

  1. University of Bucharest, Bucharest, Romania

    Claudia Mureşan

  2. University of Szeged, Szeged, Hungary

    Júlia Kulin

Authors
  1. Claudia Mureşan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Júlia Kulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudia Mureşan.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mureşan, C., Kulin, J. On the Largest Numbers of Congruences of Finite Lattices. Order 37, 445–460 (2020). https://doi.org/10.1007/s11083-019-09514-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-019-09514-2

Keywords

Search

Navigation