Highness and Local Noncappability

Abstract

Let a be a nonzero incomplete c.e. degree. Say that a is locally noncappable if there is a c.e. degree c above a such that no nonzero c.e. degree below c can form a minimal pair with a, and c is a witness of such a property of a. Seetapun proved that every nonzero incomplete c.e. degree is locally noncappable, and Stephan and Wu proved recently that such witnesses can always be chosen as high₂ degrees. This latter result is optimal as certain classes of c.e. degrees, such as nonbounding degrees, plus-cupping degrees, etc., cannot have high witnesses. Here, a c.e. degree is nonbounding if it bounds no minimal pairs, and is plus-cupping if every nonzero c.e. degree below it is cuppable.

In this paper, we prove that for any nonzero incomplete c.e. degree a, there exist two incomparable c.e. degrees c, d > a witnessing that a is locally noncappable, and that c ∨ d, the joint of c and d, is high. This result implies that both classes of the plus-cuppping degrees and the nonbounding c.e. degrees do not form an ideal, which was proved by Li and Zhao by two separate constructions.