Abstract
The correspondence between right loops (P, +) with the property “(*) ∀a, b ∈ P : a − (a − b) − 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 a ∈ P the map a + : P → P; x → a + 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.
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Dedicated to Heinrich Wefelscheid on the occasion of his 60th birthday
Research supported by M.I.U.R.
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Karzel, H., Pianta, S. & Zizioli, E. Loops, Reflection Structures and Graphs with Parallelism. Results. Math. 42, 74–80 (2002). https://doi.org/10.1007/BF03323555
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DOI: https://doi.org/10.1007/BF03323555