THE STRUCTURE OF D.R.E. DEGREES

Abstract

This thesis is highly motivated by d.r.e. Nondensity Theorem (Cooper, Harrington, Lachlan, Lempp, and Soare in 1991), which states that there exists a maximal d.r.e. degree. Very few results on maximal d.r.e. degrees were found as the construction is too complicated to work with. Therefore, it is very interesting and challenging to know what other properties can a maximal d.r.e. degree have. This thesis contains two parts. Firstly, we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called (m,n)-cupping degree and show that there exists an isolated (2,\omega)-cupping degree, and there exists a proper (2,1)-cupping degree. Secondly, by generalizing the strategy for maximality we prove that every finite Boolean algebra can be embedded into d.r.e. degrees as a final segment. It generalizes the d.r.e. Nondensity Theorem.