Illinois Journal of Mathematics

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

M. M. Arslanov and R. Sh. Omanadze

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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 $A$.

Article information

Illinois J. Math., Volume 51, Number 4 (2007), 1189-1206.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D25: Recursively (computably) enumerable sets and degrees
Secondary: 03D30: Other degrees and reducibilities


Arslanov, M. M.; Omanadze, R. Sh. Q-degrees of $n$-c.e. sets. Illinois J. Math. 51 (2007), no. 4, 1189--1206. doi:10.1215/ijm/1258138538.

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