In this paper we study structural properties of n-c. e. Q-degrees. Two theorems contain results on the distribution of incomparable Q-degrees. In another theorem we prove that every incomplete Q-degree forms a minimal pair in the c. e. degrees with a Q-degree. In a further theorem it is proved that there exists a c. e. Q-degree that is not half of a minimal pair in the c. e. Q-degrees.

We show that non-isolated from below 2-c.e. Q -degrees are dense in the structure of c.e. Q -degrees. We construct a 2-c.e. Q -degree, which can't be isolated from below not only by c.e. Q -degrees, but by any Q -degree. We also prove that below any c.e. Q -degree there is a 2-c.e. Q -degree, which is non-isolated from below and from above.

We prove a completeness criterion for quasi-reducibility and generalize it to higher levels of the arithmetical hierarchy. As an application of the criterion we obtain Q-completeness of the set of all pairs such that the prefix-free Kolmogorov complexity of x is less than n.