Touching structures which are no chain spaces
Let \(\Sigma=(P,C)\) be an incidence structure with \(P\) as set of points and \(C\) as set of blocks. The points \(p,q\) are called distant if \(p\neq q\) and if there exists a block through \(p,q\). The structure \(\Sigma\) is called a weak chain space if through three pairwise distant points there is exactly one block, if every block contains at least three distinct points, and if through every point there is at least one block. In the case that \(\Sigma\) is a weak chain space, blocks are called chains. To every point \(p\) of a weak chain space \(\Sigma\) there is associated the residual space \(\Sigma_p= (\Delta(p), C_p')\) where \(\Delta(p)\) is the set of all \(q\in P\) distant to \(p\), and where \(C_p'\) is the set of all \(c\setminus\{p\}\) with \(p\in c\in C\). The structure \(\Sigma=(P,C, (\rho_p)_{p\in P})\) is a contact structure if \((P,C)\) is a weak chain space, if \(\rho_p\) is for every \(p\in P\) an equivalence relation on the set \(C_p\) of all \(c\in C\) passing through \(p\), and if the following two conditions hold true. \(c,c'\in C_p\) and \(c\rho_pc'\) imply \(c=c'\) or \(c\cap c'=\{p\}\). To \(c\in C_p\) and \(q\) distant to \(p\) there exists \(c'\in C_p\) satisfying \(q\in c'\) and \(c \rho_pc'\). -- If all residual spaces of a contact structure \(\Sigma= (P,C, (\rho_p)_{p\in P})\) are partial affine spaces, then \(\Sigma\) is called a chain space. The authors present contact structures which are not chain spaces, and they are even able to find contact structures which are not chain spaces despite the fact that already suitable subspaces of residual spaces can be uniquely imbedded in affine, non-desarguesian planes. This is a substantial result, certainly unexpected, since the plane structure usually determines the whole space. The construction is based on permutation sets satisfying properties of transitivity. The existence of those permutation sets is proved by means of automorphism groups of classical chain geometries and semilinear mappings.
scientific article; zbMATH DE number 1384290