Automorphisms of the truth-table degrees are fixed on a cone

Abstract

Let D_tt denote the set of truth-table degrees. A bijection π : D_tt → D_tt is an automorphism if for all truth-table degrees x and y we have x ≤_tt y ⇔ π (x) ≤_tt π (y). We say an automorphism π is fixed on a cone if there is a degree b such that for all x ≥_tt b we have π (x) = x. We first prove that for every 2-generic real X we have X'≰_tt X ⊕ 0'. We next prove that for every real X ≥_tt 0' there is a real Y such that Y ⊕ 0' ≡_tt Y' ≡_tt X. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on a cone.