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The theory of the degrees below 0
 J. London Math. Soc
, 1981
"... Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppers ..."
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Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns
Generalized high degrees have the complementation property
 Journal of Symbolic Logic
"... Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the ..."
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Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′
Strong minimal covers and a question of Yates: the story so far
 the proceedings of the ASL meeting
, 2006
"... Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1. ..."
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Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1.
CLASSES, STRONG MINIMAL COVERS AND HYPERIMMUNEFREE DEGREES
"... Abstract. We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmunefree degrees. 1. ..."
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Abstract. We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmunefree degrees. 1.