Almost universal cupping and diamond embeddings

Abstract

In this paper, we study the cupping property in the d.r.e. degrees, and prove the existence of a d.r.e. degrees d with almost universal cupping property and nonzero r.e. degrees b<d and a such that b isolates d in the d.r.e. degrees, and b and a form a minimal pair. As a corollary, we have that for any nonzero r.e. degree c≤a and incomplete d.r.e. degree e≥d, 0,c,e and 0′ form a diamond embedding in the d.r.e. degrees.