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A Variant of the Notion of Semicreative Set

This paper introduces the notion of cW10-creative set, which strengthens that of semicreative set in a similar way as complete creativity strengthens creativity. Two results are proven, both of which imply that not all semicreative sets are cW10-creative. First, it is shown that semicreative Dedekind cuts cannot be cW10-creative; the existence of semicreative Dedekind cuts was shown by Soare. Secondly, it is shown that (i) if A ⊕ B, the join of A and B, is cW10-creative, then either A or B is cW10-creative, and (ii) the same is not true with ‘cW10-creative’ replaced by ‘semicreative’. Moreover, sets A, B which provide a counterexample for (ii) can be constructed within any given nonrecursive r.e. T-degrees a, b. MSC: 03D30.