We show that the Q-degree of a hyperhypersimple set includes an infinite collection of Q 1 -degrees linearly ordered under ≤ Q 1 with order type of the integers and consisting entirely of hyperhypersimple sets. Also, we prove that the c.e. Q 1 -degrees are not an upper semilattice. The main result of this paper is that the Q 1 -degree of a hemimaximal set contains only one c.e. 1-degree. Analogous results are valid for Π 0 1 s 1 -degrees.
We prove that the c.e. Q1‐degrees are not dense, and there exists a c.e. Q1‐degree with no minimal c.e. predecessors. It is proved that if M1 and M2 are maximal sets such that M1≡Q1M2 then M1≤1M2 or M2≤1M1. We also show that there exist infinite collections of Q1‐degrees {ai}i∈ω and {bi}i∈ω such that the following hold: (i) for every i,j, boldai
We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has $K \not\leq_{ss} B$ (respectively, $K \not\leq_{\overline{s}} B$): here $\leq_{\overline{s}}$ is the finite-branch version of s-reducibility, $\leq_{ss}$ is the computably bounded version of $\leq_{\overline{s}}$, and $\overline{K}$ is the complement of the halting set. Restriction to $\Sigma^0_2$ sets provides a similar characterization of the $\Sigma^0_2$ hyperhyperimmune sets in terms of s-reducibility. We also showthat no $A \geq_{\overline{s}} \overline{K}$ is hyperhyperimmune. As a consequence, $deg_s (\overline{K})$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed
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