To branch or not to branch: branching and non-branching in the medvedev lattice of pi(0,1) classes
Steffen Lempp, Christopher P. Alfeld · 2007
A P01 classes can be represented as the set of infinite paths through a computable tree. For classes P and Q we say that P is Medvedev above Q, P ≥M Q, if there exists a computable functional which maps P into Q. In this thesis I classify branching and non-branching degrees and present several related results. In particular, I define P to be inseparable if, for every clopen class C such that P ∩ C ≠ O ≠ P ∩ Cc, P ∩ C and P ∩ Cc are Medvedev comparable. Say P is separable if it not inseparable. I prove that inseparable (separability) is an invariant of a Medvedev degree and equivalent to non-branching (branching). With this as a starting line I then present two additional, more specialized, categories of non-branching and similar categories for branching and prove a variety of separation and structural results.