Hyperhypersimple sets and Q₁ -reducibility

Abstract

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<Q1boldai+1, boldbj+1<Q1boldbj, and boldai<Q1boldbj, (ii) each ai consists entirely of maximal sets; and (iii) each bj consists entirely of non‐maximal hyperhypersimple sets.

r‐Maximal sets and Q1,N‐reducibility

“…Analogous result for c.e. ‐degrees will be proved in this section with slight modification of the proof of [3, Theorem 3.2]. Theorem The c.e.…”

“…sequence (cf. [3]) such that, for all i , j , (i), (ii)if , then , (iii) and . So we have two cases: …”

Agreement reducibility

“…There is a lot of work on understanding how s‐ and s 1 ‐reducibility behave in terms of immunity properties, particularily on sets (e.g., [5, 6, 22–25]). E.g., the following theorem implies that the s‐degree of any hh‐immune set contains infinitely many s 1 ‐degrees.…”

“…We collect some facts about agreement reducibility that follow from ones for s 1 ‐reducibility. The following argument is modified from [5, Theorem 2.2]. Lemma If set A is not immune and set B is c.e., then , so .…”

“…In § 3.2, we look at what s 1 ‐reducibility can tell us about agreement reducibility. Much of the work on s 1 ‐reducibility, e.g., that in [5, 22, 24], was done in terms of Q 1 ‐reducibility, where if there is a computable function such that and if , then and are disjoint. If , then , so results about one reducibility can be converted into results about the other.…”