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Finite final segments of the d.c.e. Turing degrees

Steffen Lempp, Yiqun Liu, Yong Liu, Keng Meng Ng, Cheng Peng, Guohua Wu

Abstract

We prove that every finite distributive lattice is isomorphic to a final segment of the d.c.e. Turing degrees (i.e., the degrees of differences of computably enumerable sets). As a corollary, we are able to infer the undecidability of the EAE-theory of the d.c.e. degrees in the language of partial ordering.

Finite final segments of the d.c.e. Turing degrees

Abstract

We prove that every finite distributive lattice is isomorphic to a final segment of the d.c.e. Turing degrees (i.e., the degrees of differences of computably enumerable sets). As a corollary, we are able to infer the undecidability of the EAE-theory of the d.c.e. degrees in the language of partial ordering.
Paper Structure (40 sections, 53 theorems, 74 equations, 4 figures, 1 table)

This paper contains 40 sections, 53 theorems, 74 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basics on Lattice Theory
  3. The Requirements and Conflicts
  4. The setup
  5. Initial version of our requirements
  6. The S-requirements rephrased
  7. The R-requirements rephrased
  8. Three examples
  9. A first discussion of the potential conflicts
  10. A Single U-set
  11. Introduction
  12. The priority tree
  13. Functionals manipulated at S-node and R-node
  14. beta star and beta sharp
  15. Preliminaries
  16. ...and 25 more sections

Key Result

Theorem 1.2

There is a maximal incomplete d.c.e. Turing degree $\bm{d}$; in particular, the d.c.e. Turing degree are not densely ordered.

Figures (4)

  • Figure 1: Pictures of lattices
  • Figure 2: S-requirements for each lattice
  • Figure 3: The priority tree for 3-element chain
  • Figure 4: The priority tree for the 3-element chain

Theorems & Definitions (146)

  • Definition 1.1
  • Theorem 1.2: D.C.E. Nondensity Theorem (Cooper, Harrington, Lachlan, Lempp, Soare CHLLS91)
  • Theorem 1.3
  • Theorem 1.4
  • proof
  • Theorem 2.1: see Gr11
  • Theorem 2.2: see Gr11
  • Lemma 2.3
  • proof
  • Definition 3.1
  • ...and 136 more