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Patch extensions and trajectory colorings of slim rectangular lattices

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Abstract

With the help of our new tools in the title, we give an efficient representation of the congruence lattice of a slim rectangular lattice by an easy-to-visualize quasiordering on the set of its meet-irreducible elements or, equivalently, on the set of its trajectories.

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References

  1. Czédli G.: The matrix of a slim semimodular lattice. Order 29, 85–103 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Czédli G.: Coordinatization of join-distributive lattices. Algebra Universalis 71, 385–404 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Czédli, G.: The asymptotic number of planar, slim, semimodular lattice diagrams. Order (submitted); ar**v:1206.3679

  5. Czédli G.: Finite convex geometries of circles. Discrete Math. 330, 61–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czédli, G.: Quasiplanar diagrams and slim semimodular lattices. Order (submitted); ar**v:1212.6904

  7. Czédli, G., Dékány, T., Ozsvárt, L., Szakács, N., Udvari, B.: On the number of slim, semimodular lattices. Math. Slovaca (submitted); ar**v:1208.6173

  8. Czédli G., Grätzer G.: Notes on planar semimodular lattices. VII. Resections of planar semimodular lattices. Order 30, 847–858 (2013)

    MATH  Google Scholar 

  9. Czédli, G., Grätzer, G.: Planar semimodular lattices and their diagrams. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications. Birkhäuser Verlag, Basel (2014, in press)

  10. Czédli G., Ozsvárt L., Udvari B.: How many ways can two composition series intersect?. Discrete Math. 312, 3523–3536 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Czédli G., Schmidt E.T.: The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices. Algebra Universalis 66, 69–79 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Czédli G., Schmidt E.T.: Slim semimodular lattices. I. A visual approach. Order 29, 481–497 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Czédli G., Schmidt E.T.: Composition series in groups and the structure of slim semimodular lattices. Acta Sci. Math. (Szeged) 79, 369–390 (2013)

    MathSciNet  Google Scholar 

  14. Czédli G., Schmidt E.T.: Slim semimodular lattices. II. A description by patchwork systems. Order 30, 689–721 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)

  16. Grätzer G.: The Congruences of a Finite Lattice. A Proof-by-picture Approach. Birkhäuser, Boston (2006)

    MATH  Google Scholar 

  17. Grätzer G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)

    MATH  Google Scholar 

  18. Grätzer G.: Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices. Algebra Universalis 69, 301–304 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grätzer G.: A technical lemma for congruences of finite lattices. Algebra Universalis 72, 53–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grätzer, G.: Congruences of fork extensions of lattices. ar**v:1307.8404

  21. Grätzer G., Knapp E.: Notes on planar semimodular lattices. I. Construction. Acta Sci. Math. (Szeged), 73, 445–462 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Grätzer G., Knapp E.: Notes on planar semimodular lattices. II. Congruences. Acta Sci. Math. (Szeged), 74, 23–36 (2008)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  24. Grätzer G., Knapp E.: Notes on planar semimodular lattices. IV. The size of a minimal congruence lattice representation with rectangular lattices. Acta Sci. Math. (Szeged), 76, 3–26 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Grätzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad. Math. Bull. 41, 290–297 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grätzer G., Schmidt E.T.: A short proof of the congruence representation theorem for semimodular lattices. Algebra Universalis 71, 65–68 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jakubík, J.: Congruence relations and weak projectivity in lattices. Časopis Pěst. Mat. 80, 206–216 (1955) (Slovak)

  28. Kelly D., Rival I.: Planar lattices. Canad. J. Math. 27, 636–665 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schmidt E.T.: Congruence lattices and cover preserving embeddings of finite length semimodular lattices. Acta Sci. Math. Szeged 77, 47–52 (2011)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Presented by G. Grätzer

This research was supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences” of project number “TÁMOP-4.2.2.A-11/1/KONV-2012-0073”, and by NFSR of Hungary (OTKA), grant number K83219.

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Czédli, G. Patch extensions and trajectory colorings of slim rectangular lattices. Algebra Univers. 72, 125–154 (2014). https://doi.org/10.1007/s00012-014-0294-z

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  • DOI: https://doi.org/10.1007/s00012-014-0294-z

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