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


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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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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