Patch extensions and trajectory colorings of slim rectangular lattices

详细信息    查看全文

• 作者：Gábor Czédli
• 关键词：06C10 ; rectangular lattice ; patch lattice ; slim semimodular lattice ; congruence lattice ; lattice coloring ; quasi ; coloring ; quasiordering ; fork extension ; multi ; fork extension ; patch extension
• 刊名：Algebra Universalis
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：72
• 期：2
• 页码：125-154
• 全文大小：509 KB
• 参考文献：1. Czédli G.: The matrix of a slim semimodular lattice. Order 29, 85-03 (2012) CrossRef
  2. Czédli G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Universalis 67, 313-45 (2012) CrossRef
  3. Czédli G.: Coordinatization of join-distributive lattices. Algebra Universalis 71, 385-04 (2014) CrossRef
  4. Czédli, G.: The asymptotic number of planar, slim, semimodular lattice diagrams. Order (submitted); arXiv:1206.3679
  5. Czédli G.: Finite convex geometries of circles. Discrete Math. 330, 61-5 (2014) CrossRef
  6. Czédli, G.: Quasiplanar diagrams and slim semimodular lattices. Order (submitted); arXiv:1212.6904
  7. Czédli, G., Dékány, T., Ozsvárt, L., Szakács, N., Udvari, B.: On the number of slim, semimodular lattices. Math. Slovaca (submitted); arXiv:1208.6173
  8. Czédli G., Gr?tzer G.: Notes on planar semimodular lattices. VII. Resections of planar semimodular lattices. Order 30, 847-58 (2013)
  9. Czédli, G., Gr?tzer, G.: Planar semimodular lattices and their diagrams. In: Gr?tzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications. Birkh?user Verlag, Basel (2014, in press)
  10. Czédli G., Ozsvárt L., Udvari B.: How many ways can two composition series intersect?. Discrete Math. 312, 3523-536 (2012) CrossRef
  11. Czédli G., Schmidt E.T.: The Jordan-H?lder theorem with uniqueness for groups and semimodular lattices. Algebra Universalis 66, 69-9 (2011) CrossRef
  12. Czédli G., Schmidt E.T.: Slim semimodular lattices. I. A visual approach. Order 29, 481-97 (2012) CrossRef
  13. Czédli G., Schmidt E.T.: Composition series in groups and the structure of slim semimodular lattices. Acta Sci. Math. (Szeged) 79, 369-90 (2013)
  14. Czédli G., Schmidt E.T.: Slim semimodular lattices. II. A description by patchwork systems. Order 30, 689-21 (2013) CrossRef
  15. Gr?tzer, G.: General Lattice Theory, 2nd edn. Birkh?user, Basel (1998)
  16. Gr?tzer G.: The Congruences of a Finite Lattice. A Proof-by-picture Approach. Birkh?user, Boston (2006)
  17. Gr?tzer G.: Lattice Theory: Foundation. Birkh?user, Basel (2011)
  18. Gr?tzer G.: Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices. Algebra Universalis 69, 301-04 (2013) CrossRef
  19. Gr?tzer G.: A technical lemma for congruences of finite lattices. Algebra Universalis 72, 53-6 (2014) CrossRef
  20. Gr?tzer, G.: Congruences of fork extensions of lattices. arXiv:1307.8404
  21. Gr?tzer G., Knapp E.: Notes on planar semimodular lattices. I. Construction. Acta Sci. Math. (Szeged), 73, 445-62 (2007)
  22. Gr?tzer G., Knapp E.: Notes on planar semimodular lattices. II. Congruences. Acta Sci. Math. (Szeged), 74, 23-6 (2008)
  23. Gr?tzer G., Knapp E.: Notes on planar semimodular lattices. III. Congruences of rectangular lattices. Acta Sci. Math. (Szeged), 75, 29-8 (2009)
  24. Gr?tzer 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, 3-6 (2010)
  25. Gr?tzer G., Lakser H., Schmidt E.T.: Congruence lattices of finite semimodular lattices. Canad. Math. Bull. 41, 290-97 (1998) CrossRef
  26. Gr?tzer G., Schmidt E.T.: A short proof of the congruence representation theorem for semimodular lattices. Algebra Universalis 71, 65-8 (2014) CrossRef
  27. Jakubík, J.: Congruence relations and weak projectivity in lattices. ?asopis Pěst. Mat. 80, 206-16 (1955) (Slovak)
  28. Kelly D., Rival I.: Planar lattices. Canad. J. Math. 27, 636-65 (1975) CrossRef
  29. Schmidt E.T.: Congruence lattices and cover preserving embeddings of finite length semimodular lattices. Acta Sci. Math. Szeged 77, 47-2 (2011)
  
• 作者单位：Gábor Czédli (1)
  
  1. University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, 6720, Szeged, Hungary
  
• ISSN：1420-8911

文摘

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.