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Results in Mathematics

, Volume 42, Issue 1–2, pp 74–80 | Cite as

Loops, Reflection Structures and Graphs with Parallelism

  • Helmut Karzel
  • Silvia Pianta
  • Elena Zizioli
Article

Abstract

The correspondence between right loops (P, +) with the property “(*) ∀a, bP : a − (ab) − b” and reflection structures described in [4] is extended to the class of graphs with parallelism (P, ε, ∥). In this connection K-loops correspond with trapezium graphs, i.e. complete graphs with parallelism satisfying two axioms (T1), (T2) (cf. §3 ). Moreover (P, ε, ∥ +) is a structure loop (i.e. for each aP the map a + : PP; xa + x is an automorphism of the graph with parallelism (P, ε, ∥)) if and only if (P, +) is a K-loop or equivalently if (P, ε, ∥) is a trapezium graph.

AMS subject classification

20N05 05C15 

key words

right loops graphs with parallelism 

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References

  1. [1]
    Ellers, E., Karzel, H.: Kennzeichnung elliptischer Gruppenräume. Abh. Math. Sem. Univ. Hamburg 26 (1963), 55–77MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Gabrieli, E., Im, B. and Karzel, H.: Webs related to K-loops and Reflection Structures. Abh. Math. Sem. Univ. Hamburg. 69 (1999), 89–102MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Karzel, H.: Bericht über projective Inzidenzgruppen. J. Ber. DMV 67 (1964), 58–92MathSciNetGoogle Scholar
  4. [4]
    Karzel, H.: Recent Developments on Absolute Geometries and Algebraization by K-Loops. Discr. Math. 208/209 (1999), 387–409MathSciNetCrossRefGoogle Scholar
  5. [5]
    Kreuzer, A., Wefelscheid, H.: On K-loops of Finite Order. Result. Math. 25 (1994), 79–102MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Wähling. H.: Protective Inzidenzgruppoide und Fastalgebren. J. of Geom. 9 (1977), 109–126MATHCrossRefGoogle Scholar
  7. [7]
    Zizioli, E.: Connections Between Loops of Exponent 2, Reflection Structures and Complete Graphs with Parallelism. Result. Math. 38 (2000), 187–194MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenMünchenGermany
  2. 2.Dip. di Matematica e FisicaUniversità CattolicaBresciaItaly
  3. 3.Dip. di Matematica - Facoltà di IngegneriaUniversità di BresciaBresciaItaly

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