Q-degrees of $n$-c.e. sets

Abstract

In this paper we study Q-degrees of $n$-computably enumerable ($n$-c.e.) sets. It is proved that $n$-c.e. sets form a true hierarchy in terms of Q-degrees, and that for any $n\ge 1$ 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. Studying weak density properties of $n$-c.e. Q-degrees, we prove that for any $n\ge 1$, 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 $A-B<_QA\equiv_Q\emptyset'$, and there are no c.e. sets for which the Q-degrees are strongly between $A-B$ and $A$.