r‐Maximal sets and Q1,N‐reducibility

Abstract

We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and M∖A≡Q1,NB, then M∖A≤mB. We prove that the c.e. Q1,N‐degrees are not dense. We also show that there exist infinite collections of Q1,N‐degrees false{aifalse}i∈ω and false{bjfalse}j∈ω such that the following hold: (i) for every i, j, boldai<Q1,Nboldai+1, boldbj+1<Q1,Nboldbj and boldai<Q1,Nboldbj, (ii) each ai consists entirely of r‐maximal sets, and (iii) each bj consists entirely of non‐r‐maximal hyperhypersimple sets.