Abstract
We show that every symmetric 2-structure \({(P,\mathfrak G_1,\mathfrak G_2,\mathfrak K)}\) of the class (III) [cf. Karzel H et al. (Result. Math., submitted)] is point symmetric, i.e. any two orthogonal chains \({A,B \in \mathfrak K}\) intersect in exactly one point and that any two points \({a,b \in P}\) have exactly one midpoint m : = a * b (with \({\widetilde m(a) = b}\) where \({\widetilde m}\) is the unique symmetry in the point m). \({ \widetilde{P} := \{\widetilde p \ | \ p \in P \}}\) is invariant, i.e. \({\forall a,b \in P : \widetilde a\circ \widetilde b\circ \widetilde a \in \widetilde P}\) . Therefore the pair \({(P,\widetilde{P})}\) is an invariant regular involution set and the loop derivation in a point \({o \in P}\) gives a K-loop (P, +) uniquely 2-divisible.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Karzel, H., Kosiorek, J. & Matraś, A. Point Symmetric 2-Structures. Results. Math. 59, 229–237 (2011). https://doi.org/10.1007/s00025-010-0091-8
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DOI: https://doi.org/10.1007/s00025-010-0091-8