r ‐Maximal sets and Q 1 , N ‐reducibility

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 ≡ Q 1 , NB , then M ∖ A ≤ m B . We prove that the c.e. Q 1 , N ‐degrees are not dense. We also show that there exist infinite collections of Q 1 , N ‐degrees{ a i } i ∈ ωand{ b j } j ∈ ωsuch that the following hold: (i) for every i , j ,a i < Q 1 , Na i + 1,b j + 1< Q 1 , Nb janda i < Q 1 , Nb j , (ii) each a i consists entirely of r ‐maximal sets, and (iii) each b j consists entirely of non‐ r ‐maximal hyperhypersimple sets.