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Bounding non-\(\mathrm{GL}_2\) and r.e.a. (English) Zbl 1201.03025
The main result is to show that any non-GL\(_2\) degree is computably enumerable in and above some 1-generic degree. The proof splits into two parts. First the authors deal with the case that a is non-low\(_2\) and \(\Delta_2^0\). This filters through a result on linear orderings by, respectively, Harizanov, and (a re-interpretation of a result of) Hirschfeldt and Shore. Then the authors show that if a is non-GL\(_2\) then there is a degree \({\mathbf c}<{\mathbf a}\) such that a is c.e. and non-low\(_2\) over a.

MSC:
03D28 Other Turing degree structures
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