Symmetric 2-Structures,a Classification

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.