Structural properties of the local Turing degrees

In this thesis we look at some properties of the local Turing Degrees, as a partial order. We first give discussion of the Turing Degrees and certain historical results, some translated into a form resembling the constructions we look at later. Chapter 1 gives a introduction to the Turing Degrees, Chapter 2 introduces the Local Degrees. In Chapter 3 we look at minimal Turing Degrees, modifying some historical results to use a priority tree, which we use in chapter 4 to prove the new result that every c.e. degree has the (minimal) meet property. Chapter 5 uses similar methods to establish existence of a high 2 degree that does not have the meet property.