On linearly ordered sets and permutation groups of countable degree
The following problem is partially solved: Does there exist a permutation group G on \(\Omega\) that has precisely m orbits in the power set \({\mathcal P}(\Omega)?\) And can this question be settled within ZFC? More exactly it is proved that there are precisely \(\aleph_ 1\) equivalence classes of countable linearly ordered sets under piecewise order-preserving bijections, and the group of such permutations of \({\mathbb{Q}}\) has precisely \(\aleph_ 1\) orbits in \({\mathcal P}({\mathbb{Q}})\).
scientific article; zbMATH DE number 4114614