Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
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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, da 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.