Abstract
We classify symmetric 2-structures \({(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}\), i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ \({(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}\) ”. To every chain \({K \in \mathfrak{K}}\) one can associate a reflection \({\widetilde{K}}\) in K. Then (*) is equivalent to “ \({(**) \, \, \forall K \in \mathfrak{K} : \widetilde{K}(\mathfrak{K}) = \mathfrak{K}}\) ” and one can define an orthogonality “ \({\perp}\) ” for chains \({K, L \in \mathfrak{K}}\) by “ \({K \perp L \Leftrightarrow K \neq L \wedge \widetilde{K}(L) = L}\) ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K. For one of these classes (called point symmetric 2-structures) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.
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Dedicated to Momme Thomsen on the occasion of his 70th birthday
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Karzel, H., Kosiorek, J. & Matraś, A. Symmetric 2-Structures, a Classification. Results. Math. 65, 347–359 (2014). https://doi.org/10.1007/s00025-013-0349-z
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DOI: https://doi.org/10.1007/s00025-013-0349-z