We show that there is a generalized high degree which is a minimal cover of a minimal degree. This is the highest jump class one can reach by finite iterations of minimality. This result also answers an old question by Lerman.

We present some natural examples of countable Borel equivalence relations E, F with E ≤ B F such that there does not exist a continuous reduction from E to F.

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

We develop a new method for constructing hyperimmune minimal degrees and construct an ANR degree which is a minimal cover of a minimal degree.

The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors for guidance and advice and my fellow officers and our publisher for their work and support. To all of the members who answered my calls to chair or serve on this or that committee, I offer my thanks as well. Your (...) work was both needed and appreciated.A major component of the efforts of the Association is devoted to our publications. My first important task as President was to deal with the need to reorganize the reviews section of the JSL and eventually to move it to the BSL. Appropriately enough, my first administrative job for the Association, some thirty years ago, was to serve on a committee to plan a reorganization of the reviews. I thank all those who helped with this transition and who took over the task of running the new reviews section. I hope that it will be another thirty years before further major changes are needed in this area and that someone else will be making them. (shrink)

We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

We prove that for any computably enumerable degree c, if it is cappable in the computably enumerable degrees, then there is a d.c.e. degree d such that c d = 0′ and c ∩ d = 0. Consequently, a computably enumerable degree is cappable if and only if it can be complemented by a nonzero d.c.e. degree. This gives a new characterization of the cappable degrees.

Say a set Gω is 1-generic if for any eω, there is a string σG such that {e}σ↓ or τσ↑). It is known that can be split into two 1-generic degrees. In this paper, we generalize this and prove that any nonzero computably enumerable degree can be split into two 1-generic degrees. As a corollary, no two computably enumerable degrees bound the same class of 1-generic degrees.