Abstract
A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. After exploring some elementary properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Czédli, G., Schmidt, E.T.: How to derive finite semimodular lattices from distributive lattices? Acta Math. Hung. 121, 277–282 (2008)
Czédli, G., Schmidt, E.T.: Some results on semimodular lattices. In: Contributions to General Algebra, vol. 19 (Proc. Olomouc Conf. 2010), pp. 45–56. Johannes Hein, Klagenfurt (2010)
Czédli, G., Schmidt, E.T.: The Jordan–Hölder theorem with uniqueness for groups and semimodular lattices. Algebra Univers. (2011, to appear)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1951)
Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)
Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. I. Construction. Acta Sci. Math. (Szeged) 73, 445–462 (2007)
Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. III. Rectangular lattices. Acta Sci. Math. (Szeged) 75, 29–48 (2009)
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)
Grätzer, G., Nation, J.B.: A new look at the Jordan–Hölder theorem for semimodular lattices. Algebra Univers. 64, 309–311 (2010)
Kelly, D., Rival, I.: Planar lattices. Can. J. Math. 27, 636–665 (1975)
Stern, M.: Semimodular Lattices. Theory and Applications, Encyclopedia of Mathematics and its Applications, vol. 73. Cambridge University Press (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the NFSR of Hungary (OTKA), grant no. K77432, and by TÁMOP-4.2.1/B-09/1/KONV-2010-0005.
Rights and permissions
About this article
Cite this article
Czédli, G., Schmidt, E.T. Slim Semimodular Lattices. I. A Visual Approach. Order 29, 481–497 (2012). https://doi.org/10.1007/s11083-011-9215-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-011-9215-3