Semilinear tower of Steiner systems
The author considers infinite Steiner systems; more precisely, a Steiner system consists of a set \(\Omega\) of points and countably infinite subsets of \(\Omega\), blocks, such that every pair of points is contained in a unique block. By building a suitable semilinear tower of Steiner systems (i.e. any pair of distinct Steiner systems with two elements in common must have one contained in a single block of the other), the following result is proved: There exists an infinite Jordan group \((G, \Omega)\) such that (i) \((G, \Omega)\) is 3-transitive but not 3-primitive; (ii) \(G\) preserves a limit of Steiner systems on \(\Omega\); (iii) \(G\) preserves none of the following relations on \(\Omega\): linear or circular order, separation relation, relations induced on maximal chains and leaves of a tree; (iv) \(G\) preserves no nontrivial Steiner systems; (v) \(G_\alpha\) preserves a relation on maximal chains on \(\Omega \backslash \{\alpha\}\), \(\alpha \in \Omega\).
scientific article; zbMATH DE number 823495