Skip to main content
SpringerLink
Log in

Slim Semimodular Lattices. I. A Visual Approach

  • Published:
Order Aims and scope Submit manuscript

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.

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., Schmidt, E.T.: How to derive finite semimodular lattices from distributive lattices? Acta Math. Hung. 121, 277–282 (2008)

    Article  MATH  Google Scholar 

  2. 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)

    Google Scholar 

  3. Czédli, G., Schmidt, E.T.: The Jordan–Hölder theorem with uniqueness for groups and semimodular lattices. Algebra Univers. (2011, to appear)

  4. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1951)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Grätzer, G., Nation, J.B.: A new look at the Jordan–Hölder theorem for semimodular lattices. Algebra Univers. 64, 309–311 (2010)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Stern, M.: Semimodular Lattices. Theory and Applications, Encyclopedia of Mathematics and its Applications, vol. 73. Cambridge University Press (1999)

Download references

Author information

Authors and Affiliations

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

    Gábor Czédli

  2. Mathematical Institute, Budapest University of Technology and Economics, Műegyetem rkp. 3, 1521, Budapest, Hungary

    E. Tamás Schmidt

Authors
  1. Gábor Czédli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Tamás Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gábor Czédli.

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

Reprints 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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-011-9215-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation