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Relative enumerability and 1-genericity. (English) Zbl 1260.03079

Summary: A set of natural numbers \(B\) is computably enumerable in and strictly above (or c.e.a. for short) another set \(C\) if \(C <_{T} B\) and \(B\) is computably enumerable in \(C\). A Turing degree \(\mathbf b\) is c.e.a. \(\mathbf c\) if \(\mathbf b\) and \(\mathbf c\) respectively contain \(B\) and \(C\) as above. In this paper, it is shown that if \(\mathbf b\) is c.e.a. \(\mathbf c\) then \(\mathbf b\) is c.e.a. some 1-generic \(\mathbf g\).

MSC:

03D28 Other Turing degree structures
03D25 Recursively (computably) enumerable sets and degrees
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