Point Symmetric 2-Structures |
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Authors: | Helmut Karzel Jaros?aw Kosiorek Andrzej Matra? |
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Institution: | 1. Zentrum Mathematik, T.U. M??nchen, 80290, M??nchen, Germany 2. Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ?o?nierska 14, 10-561, Olsztyn, Poland
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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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