Friedberg splittings of recursively enumerable sets
The authors investigate Friedberg splittings and in particular their orbits and degrees for various classes of r.e. sets. The starting point was the following conjecture: All Friedberg splittings of (simple) sets are automorphic. The authors show that a true Friedberg splitting (\(t\)- split) is not enough to guarantee automorphism. But the authors introduce a new notion of splitting (an \(e\)-splitting) and show that \(e\)-splittings of an r.e. set are automorphic. The authors also discuss another conjecture: Do \(f\)-creative sets form an orbit? Although they have not answered this question, they do classify the degrees of \(f\)-creative sets as exactly the promptly simple ones. Furthermore, since there are \(e\)- creative sets of all promptly simple degrees, there is an orbit in \(\Aut({\mathcal E})\) realizing exactly the promptly simple degrees. At last, the authors examine some other hemiproperties. In particular, they give proofs of results that there are nonhalfhemisimple sets yet there are completely halfhemisimple degrees.
scientific article; zbMATH DE number 416873
scientific article