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Reflection Spaces, Partial K-Loops and K-Loops

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Abstract

Starting from a paper by Sperner (Archiv der Mathematik 5:458–468, 1954) we introduce the notion reflection space and inside this concept, transitive elliptic and non-elliptic reflection spaces. On a transitive and non-elliptic reflection space we can apply the so called K-loop derivation and obtain a K-loop with fibration. In order to achieve similar results for elliptic reflection spaces we have to claim further conditions (the existence of a so called midpoint domain) and to extend the notion K-loop to partial K-loop with fibration.

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References

  1. Ellers E., Sperner E.: Einbettung eines desarguesschen Ebenenkeims in eine projektive Ebene. Abh. Math. Sem. Univ. Hamburg 25, 206–230 (1962)

    Article  Google Scholar 

  2. Karzel, H.: Gruppentheoretische Begründung metrischer Geometrien. Vorlesungsausarbeitung von G. Graumann, Hamburg (1963)

  3. Karzel, H., Kroll, H.-J.: Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt (1988)

  4. Karzel, H., Sörensen, K., Windelberg, D.: Einführung in die Geometrie. UTB 184. Göttingen (1973)

  5. Karzel, H., Marchi, M., Taherian, S.-Gh.: Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality. Results Math. (2010). doi:10.1007/s00025-010-0070-0

  6. Sörensen K.: Elliptische Ebenen. Mitt. Math. Ges. Hamburg 10, 277–296 (1976)

    MathSciNet  MATH  Google Scholar 

  7. Sörensen K.: Elliptische Räume. Mitt. Math. Ges. Hamburg 18, 159–167 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Sperner, E.: Ein gruppentheoretischer Beweis des Satzes von Desargues in der absoluten Axiomatik. Archiv der Mathematik 5, 458–468 (1954); reprinted in: H. Karzel and K. Sörensen: Wandel von Begriffsbildungen in der Mathematik. Wissenschaftliche Buchgesellschaft Darmstadt 177–188 (1984)

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Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, 80290, Munich, Germany

    Helmut Karzel

  2. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

    Sayed-Ghahreman Taherian

Authors
  1. Helmut Karzel
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  2. Sayed-Ghahreman Taherian
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Correspondence to Helmut Karzel.

Additional information

Dedicated to Heinrich Wefelscheid, the main founder of the theory of K-loops on the occasion of his 70th birthday

The author S.-G. Taherian was financially supported by DAAD in summer 2010.

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Karzel, H., Taherian, SG. Reflection Spaces, Partial K-Loops and K-Loops. Results. Math. 59, 213–218 (2011). https://doi.org/10.1007/s00025-010-0085-6

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  • DOI: https://doi.org/10.1007/s00025-010-0085-6

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