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

TL;DR

This paper proves that all automorphisms of the truth-table degrees are fixed on some cone, using properties of 2-generic reals and the relationship between degrees and jumps.

Contribution

It establishes that every automorphism of the truth-table degrees is fixed on some cone, advancing understanding of their structural rigidity.

Findings

Automorphisms are fixed on some cone for all truth-table degrees.

Properties of 2-generic reals are used to analyze automorphisms.

Relationships between degrees and jumps are characterized.

Abstract

Let Dtt denote the set of truth-table degrees. A bijection p from Dtt to Dtt is an automorphism if for all truth-table degrees x and y we have x <=tt y if and only if p(x) <=tt p(y). We say an automorphism p is fixed on some cone if there is a degree b such that for all x >=tt b we have p(x) = x. We first prove that for every 2-generic real X we have X' is not tt below 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 some cone.