SpringerLink
Log in

Minimal representations of a finite distributive lattice by principal congruences of a lattice

  • Algebra
  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

Let the finite distributive lattice D be isomorphic to the congruence lattice of a finite lattice L. Let Q denote those elements of D that correspond to principal congruences under this isomorphism. Then Q contains 0, 1 ∈ D and all the join-irreducible elements of D. If Q contains exactly these elements, we say that L is a minimal representation of D by principal congruences of the lattice L.

We characterize finite distributive lattices D with a minimal representation by principal congruences with the property that D has at most two dual

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. G. Czédli, Representing a monotone map by principal lattice congruences, Acta Math. Hungar., 147 (2015), 12–18.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. G. Czédli, Cometic functors and representing order-preserving maps by principal lattice congruences, Algebra Universalis (2017), submitted.

    Google Scholar 

  6. G. Czédli, Characterizing fully principal congruence representable distributive lattices, Algebra Universalis (2017), in press.

    Google Scholar 

  7. G. Czédli, On the set of principal congruences in a distributive congruence lattice of an algebra, Acta Sci. Math (Szeged), submitted.

  8. G. Czédli, G. Grätzer, and H. Lakser, Congruence structure of planar semimodular lattices: The General Swing Lemma, Algebra Universalis (2017), in press.

    Google Scholar 

  9. G. Grätzer, Lattice Theory: Foundation, Birkhäuser Verlag, Basel, 2011.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. G. Grätzer, A technical lemma for congruences of finite lattices, Algebra Universalis, 72 (2014), 53.

    Article  MathSciNet  Google Scholar 

  12. G. Grätzer, The Congruences of a Finite Lattice. A “Proof-by-Picture” Approach. Second edition, Birkhäuser Verlag, Basel, 2016.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. G. Grätzer, Homomorphisms and principal congruences of bounded lattices. II. Sketching the proof for sublattices, Algebra Universalis (2017), in press.

    Google Scholar 

  15. G. Grätzer, Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem, Algebra Universalis (2017), in press.

    Google Scholar 

  16. G. Grätzer, Characterizing representability by principal congruences for finite distributive lattices with a join-irreducible unit element, Acta Math Szeged (2017), in press.

    Google Scholar 

  17. G. Grätzer and H. Lakser, Some preliminary results on the set of principal congruences of a finite lattice, Algebra Universalis (2017), in press.

    Google Scholar 

  18. G. Grätzer and F. Wehrung (eds.), Lattice Theory: Special Topics and Applications. Volume 1, Birkhäuser Verlag, Basel, 2014.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Manitoba Winnipeg, Manitoba, Canada, R3T 2N2

    George Grätzer

  2. Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

    Harry Lakser

Authors
  1. George Grätzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harry Lakser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Grätzer.

Additional information

Communicated by G. Czédli

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grätzer, G., Lakser, H. Minimal representations of a finite distributive lattice by principal congruences of a lattice. ActaSci.Math. 85, 69–96 (2019). https://doi.org/10.14232/actasm-017-060-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-017-060-9

AMS Subject Classification (2010)

Key words and phrases

Search

Navigation