Abstract
In a loop (L, +) each a ∈ L defines a permutation a+: L → L; x → a + x. Here (L, +) is called of exponent 2 if a+ o a+ = id for all a ∈ L. Then L+ ≔ {a+¦a ∈ L} is a reflection structure in the sense of [3] satisfying additional conditions and the complete graph Γ with set of vertices L can be endowed with a parallelism ∥ between edges.
The relations between the structures (L, +), L+ and the complete graph with parallelism Γ are investigated if further conditions are assumed for one of these structures. So for “+” is assumed that (L, +) is a K-loop or a group.
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Research supported by M.U.R.S.T. and by C.N.R. (G.N.S.A.G.A.).
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Zizioli, E. Connections between loops of exponent 2, reflection structures and complete graphs with parallelism. Results. Math. 38, 187–194 (2000). https://doi.org/10.1007/BF03322442
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DOI: https://doi.org/10.1007/BF03322442