Quasi-minimal enumeration degrees and minimal Turing degrees (Q1568759)

The principal result in this paper is the following: Theorem. There exist an e-degree \({\mathbf a}\) and a family of e-degrees \(\{{\mathbf b}_i: i\in I\}\) of cardinality \(2^{\aleph_0}\) such that for all \(i\in I\), (i) \({\mathbf b}_i\) is minimal total, (ii) \({\mathbf a}\) is quasi-minimal, (iii) \({\mathbf a}<{\mathbf b}_i\). Answering a related question raised by Solon, it is shown that there exists a nontotal enumeration degree which is not e-hyperimmune.