Pseudojump operators and ∏10 classes

Abstract

For a pseudojump operator Vx and a ∏ 10 class P, we consider properties of the set {Vx : X ∈P}. We show that there always exists X ∈P with Vx ≤T 0′ and that if P is Medvedev complete, then there exists X ∈P with Vx le;T 0′. We examine the consequences when Vx is Turing incomparable with VY for X ne; Y in P and when W eX = WWeY for all X, Y ∈ P. Finally, we give a characterization of the jump in terms of ∏ 10 classes. © Springer-Verlag Berlin Heidelberg 2007.