We show that there is a non-GL2 degree which is a minimal cover of a minimal degree. This answers a question by Lerman.

We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.

The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X -recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p (...) Y , one also says that X is pmaj-reducible to Y . It is shown that between degrees not above the Halting problem the pmaj-reducibility coincides with the Turing reducibility and that therefore only recursive sets have a strongly hyperimmune-free Turing degree while those sets Turing above the Halting problem bound uncountably many sets with respect to the pmaj-relation. So pmaj-reduction is identical with Turing reduction on a class of sets of measure 1. Moreover, every pmaj-degree coincides with the corresponding Turing degree. The pmaj-degree of the Halting problem is definable with the pmaj-relation. (shrink)