Non-isolated quasi-degrees

A set A ⊆ ω is called 2-computably enumerable (2-c.e.), if there are computably enumerable sets A1 and A2 such that A1 −A2 = A. A set A ⊆ ω is quasi-reducible to a set B ⊆ ω (A ≤Q B), if there is a computable function g such that for all x ∈ ω we have x ∈ A if and only if Wg(x) ⊆ B. This reducibility was introduced by Tennenbaum (see [1], p.207) as an example of a reducibility which differs from T -reducibility on the class of computably enumerable sets. A set A is quasi-equivalent to a set B (A ≡Q B), if A ≤Q B and B ≤Q A. It is not hard to see that the relation ≡Q is an equivalence relation. The class of all sets, which are quasi-equivalent to a set A, is called the quasi-degree (Q-degree) of A and usually is denoted by a small italic letter a. The Q-degree of a 2-c.e. set is called a properly 2-c.e. degree, if it doesn’t contain any c.e. sets. The interest to the study of the algebraic structure of Q-degrees arose after results of Dobritsa and Belegradek (see [2]). It follows from these results that every obtained property of the structure gives a property of the classes of finitely generated subgroups of algebraically closed groups. The further study of the algebraic structure of Q-degrees was conducted in [3], [4] and [5]. In this paper we study the isolation property of the algebraic structure of Q-degrees. The results of the paper supplement the results of [6]. In [6] we showed that isolated from below and from above 2-c.e. Q-degrees are dense in the structure of c.e. Q-degrees; in this paper we show that non-isolated from below 2-c.e. Q-degrees are also dense in the structure of c.e. Q-degrees and that non-isolated from above 2-c.e. Q-degrees are downward dense in the structure of c.e. Q-degrees. We adopt the standard notational conventions, found, for instance, in [6]. In particular, we write [s] after functionals and formulas to indicate that every functional or parameter therein is evaluated at stage s. We also use the following definition: Definition 1.1 (i) A degree d is called isolated from below, if there is a c.e. degree b Q d such that for all c.e. degree a, if a ≥Q d, then b ≤Q a. A degree d is called non-isolated from above, if there is no such c.e. degree b >Q d. We also assume in the statements of our theorems that none of the sets below is ω, since ω has a Q-degree strictly below that of any other set.