The meet property in local degree structures
Benedict Richard Fabian Durrant · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York) · 2014
In this thesis we look at whether two different classes of local Turing degrees (the c.e. degrees, and the 1-generic degrees below 0') satisfy the meet property - where a degree a satisfies the meet property if it is incomputable and for all b < a there exists a non-zero degree c such that a ∧ c = 0. We first give a general discussion of the Turing Degrees and certain known results, before giving a brief introduction to priority arguments. This is followed by some more technical considerations (full approximation and minimal degree constructions) before the proof of two new theorems - the first concerning c.e. degrees and the meet property and the second concerning 1 − generic degrees and the meet property. Chapter 1 contains a broad introduction to the Turing Degrees, and Chapter 2 to the Local Degrees. In Chapter 3 we consider minimal degree constructions, which we use in Chapter 4 to prove our first new theorem - Theorem 4.2.1 Given any non-zero c.e. degree a and any degree b < a, there is a minimal degree m < a such that m ≰ b. From which we get Corollary 4.2.2 Every c.e. degree satisfies the meet property - answering a question first asked by Cooper and Epstein in the 1980s. In Chapter 5 we prove the second new theorem - Theorem 5.2.2 There exists a 1 − generic degree which does not satisfy the meet property - showing that a result from Kumabe in the 1990s does not extend to the case n = 1.