Hyperhypersimple sets and Q 1 ‐reducibility

We prove that the c.e. Q 1 ‐degrees are not dense, and there exists a c.e. Q 1 ‐degree with no minimal c.e. predecessors. It is proved that if M 1 and M 2 are maximal sets such thatM 1 ≡ Q 1M 2thenM 1 ≤ 1 M 2orM 2 ≤ 1 M 1 . We also show that there exist infinite collections of Q 1 ‐degrees{ a i } i ∈ ωand{ b i } i ∈ ωsuch that the following hold: (i) for every i , j ,a i < Q 1a i + 1,b j + 1< Q 1b j , anda i < Q 1b j , (ii) each a i consists entirely of maximal sets; and (iii) each b j consists entirely of non‐maximal hyperhypersimple sets.