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Results in Mathematics

, Volume 63, Issue 3–4, pp 1409–1420 | Cite as

Chain Geometry Determined by the Affine Group

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Abstract

Chain geometry associated with an affine group and with a linear group is studied. In particular, closely related to the respective chain geometries affine partial linear spaces and generalizations of sliced spaces are defined. The automorphisms of thus obtained structures are determined.

Mathematics Subject Classification (2010)

Primary 51B99 51A45 Secondary 51B20 

Keywords

Chain geometry affine transformation linear group affine partial linear space sliced space 
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Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Benz W.: Vorlesungen über Geometrie der Algebren. Springer, Berlin/Heidelberg/New York (1973)MATHCrossRefGoogle Scholar
  2. 2.
    Gorodowienko J., Prażmowska M., Prażmowski K.: Elementary characterizations of some classes of reducts of affine spaces. J. Geom 89, 17–33 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Herzer A., Meuren S.: Ein Axiomensystem für partielle affine Räume. J. Geom. 50, 124–142 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Karzel H.: Loops related to geometric structures. Quasigroups Related Syst 15, 47–76 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Karzel H., Meissner H.: Geschlitzte Inzidenzgruppen und normale Fastmoduln. Abh. Math. Sem. Univ. Hamburg 31, 69–88 (1967)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Karzel H., Kosiorek J., Matraś A.: Properties of auto- and antiautomorphisms of maximal chain structures and their relations to i-perspectivities. Results Math. 50(1–2), 81–92 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Karzel, H., Kroll, H.-J.: Perspectivities in circle geometries. In: Plaumann, P. et al. (eds.) Geometry—von Staudt’s point of view. Proc. NATO Adv. Study Inst. Bad Windsheim 1980, D. Reidel, Dordrecht, pp. 51–99 (1981)Google Scholar
  8. 8.
    Karzel H., Pieper I.: Bericht über geschlitzte Inzidenzgruppen. Jber. Deutsch. Math.-Verein. 70, 70–114 (1970)Google Scholar
  9. 9.
    Meuren S.: Partial affine spaces of dimension ≥ 3. J. Geom. 56, 113–125 (1996)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Wefelscheid, H.: Über die Automorphsmengruppen von Hyperbelstrukturen. Beitr. Geom. Algebra, Proc. Symp. Duisburg, pp. 337–343 (1976), (1977)Google Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BiałystokBiałystokPoland

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