Q-degrees of \(n\)-c.e. sets (Q938805)

The paper is devoted to the study of Q-degrees of \(n\)-computably enumerable (\(n\)-c.e.) sets. The following results are proved: (1) \(n\)-c.e. sets form a true hierarchy in terms of Q-degrees; (2) for any \(n \geq 1\) there exists a \(2n\)-c.e. Q-degree which bounds no noncomputable c.e. Q-degree, but any \((2n+1)\)-c.e. non \(2n\)-c.e. Q-degree bounds a c.e. noncomputable Q-degree; (3) for any \(n \geq 1\), properly \(n\)-c.e. Q-degrees are dense in the ordering of c.e. Q-degrees but there exist c.e. sets \(A\) and \(B\) such that \(A-B <_{Q} A \equiv_{Q} \emptyset '\) and there are no c.e. sets for which the Q-degrees are strongly between \(A-B\) and \(A\).