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

Abstract

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