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Hyperhypersimple sets and Q 1 ‐reducibility
Author(s) -
Chitaia Irakli
Publication year - 2016
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600045
Subject(s) - mathematics , combinatorics , degree (music) , set (abstract data type) , discrete mathematics , computer science , physics , programming language , acoustics
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.