Correlations between e-degrees and T-degrees (Q1909815)

Let \(d_e (A)\) and \(d_T(A)\) be the equivalence classes of \(A\) under the enumeration and Turing reducibilities, respectively. Several inclusion relations between them are proved: 1. \(d_e (C \oplus \overline C) \nsubseteq d_T (C \oplus \overline C)\) for any set \(C\); 2. There is a quasi-minimal set \(A\) such that \(d_e (A) \nsubseteq d_T (A)\), where \(A\) is quasi-minimal if \(f\) is recursive whenever the graph of \(f\) is \(e\)-reducible to \(A\); 3. \(d_T (C) \nsubseteq d_e (C)\) for any set \(C\). An open question is mentioned at last: whether there exists a non- r.e. set \(A\) such that \(\exists C(d_T (C) \subseteq d_e (A))\).