Title:

The meet property in local degree structures

In this thesis we look at whether two different classes of local Turing degrees (the c.e. degrees, and the 1generic 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 nonzero 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 nonzero 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.
