In this paper, we prove that there exist computably enumerable degrees a and b such that $\mathbf{a} > \mathbf{b}$ and for any degree x, if x ≤ a and x is a minimal degree, then $\mathbf{x}.

We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable. We also show that every finite poset labeled with elements of (where is the partition of induced by the generalized high/low hierarchy) can be embedded in preserving the labels. Note that no condition is imposed on the labels.

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.

In this paper, we improve a result of Seetapun and prove that above any nonzero, incomplete recursively enumerable degree a, there is a high2 r.e. degree c>ac>a witnessing that a is locally noncappable . Theorem 1.1 provides a scheme of obtaining high2 nonboundings , as all known high2 nonboundings, such as high2 degrees bounding no minimal pairs, high2 plus-cuppings, etc.

We employ techniques related to Lempp and Lerman's “iterated trees of strategies” to directly measure a Σ5-predicate and use this in showing the index set of the cuppable r.e. sets to be Σ5-complete. We also show how certain technical devices arise naturally out of the iterated-trees context, in particular, links arise as manifestations of a generalized notion of “stage”.