Advertisement

Designs, Codes and Cryptography

, Volume 53, Issue 1, pp 45–57 | Cite as

Combinatorial generalizations of generalized quadrangles of order (2, 2)

  • Andrzej Owsiejczuk
  • Małgorzata Prażmowska
Article

Abstract

We study substructures of a projective space PG(n, 2) represented in terms of elementary combinatorics of finite sets, which generalize the Sylvester’s representation of the generalized quadrangle of order (2, 2). Their synthetic properties are established and automorphisms are characterized.

Keywords

Fano projective space Combinatorial Grassmannian Veblen configuration Net (configuration) Generalized quadrangle 

Mathematics Subject Classification (2000)

51A45 51E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chow W.-L.: On the geometry of algebraic homogeneous spaces. Ann. Math. 50, 32–67 (1949)CrossRefGoogle Scholar
  2. 2.
    Coxeter H.S.M.: Desargues configurations and their collineation groups. Math. Proc. Camb. Phil. Soc. 78, 227–246 (1975)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hall J.I.: Classifying copolar spaces and graphs. Quart. J. Math. Oxford 33(2), 421–449 (1982)MATHCrossRefGoogle Scholar
  4. 4.
    Hilbert D., Cohn-Vossen P.: Anschauliche Geometrie. Springer, Berlin (1932) (English translation: Geometry and the Imagination, AMS Chelsea Publishing)MATHGoogle Scholar
  5. 5.
    Klin M.Ch., Pöschel R., Rosenbaum K.: Angewandte Algebra für Mathematiker und Informatiker. VEB Deutcher Verlag der Wissenschaften, Berlin (1988).Google Scholar
  6. 6.
    Łapiński M., Prażmowski K.: On set-theoretic and cyclic representation of the structure of barycentres. Demonstratio Mathematica 37, 619–638 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Payne S.E., Thas J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics, vol. 110. Pitman, Boston (1984)Google Scholar
  8. 8.
    Polster B.: A Geometrical Picture Book. Springer, New York (1998)MATHGoogle Scholar
  9. 9.
    Prażmowska M.: Multiplied perspectives and generalizations of Desargues configuration. Demonstratio Mathematica 39(4), 887–906 (2006)MATHMathSciNetGoogle Scholar
  10. 10.
    Prażmowska M., Prażmowski K.: The convolution of a partial Steiner triple system and a group. J. Geom. 85, 90–109 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Reye T.: Das Problem der Konfigurationen. Acta Math. 1, 93–96 (1882)CrossRefMathSciNetGoogle Scholar
  12. 12.
    van Maldeghem H.: Slim and bislim geometries. In: Topics in Diagram Geometry. Quad. Mat., vol. 12, pp. 227–254. Department of Mathematics, Secondary University, Napoli, Caserta (2003).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Secondary SchoolMichałowoPoland
  2. 2.Institute of MathematicsUniversity of BiałystokBiałystokPoland

Personalised recommendations