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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 order ..."
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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 ′
Jumps of minimal degrees below 0
 J. London Math. Soc
, 1996
"... Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has be ..."
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Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has been the relationship between quantifier complexity of the definitions of sets in arithmetic as expressed by the jump operator and the basic notion of relative computability as expressed by the ordering of the (Turing) degrees. In this paper we
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.
A single minimal complement for the c.e. degrees
"... Abstract. We show that there exists a minimal (Turing) degree b<0 ′ such that for all nonzero c.e. degrees a, 0 ′ = a ∨ b. Since b is minimal this means that b complements all c.e. degrees other than 0 and 0 ′. Since every nc.e. degree bounds a nonzero c.e. degree, b complements every nc.e. deg ..."
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Abstract. We show that there exists a minimal (Turing) degree b<0 ′ such that for all nonzero c.e. degrees a, 0 ′ = a ∨ b. Since b is minimal this means that b complements all c.e. degrees other than 0 and 0 ′. Since every nc.e. degree bounds a nonzero c.e. degree, b complements every nc.e. degree other than 0 and 0 ′. 1.
Arch. Math. Logic (2011) 50:33–44 DOI 10.1007/s0015301001983 Mathematical Logic A superhigh diamond in the c.e. ttdegrees
"... Abstract The notion of superhigh computably enumerable (c.e.) degrees was first ..."
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Abstract The notion of superhigh computably enumerable (c.e.) degrees was first
Embedding the Diamond Lattice in the c.e. ttDegrees with Superhigh Atoms
"... Abstract. The notion of superhigh computably enumerable (c.e.) degrees was first introduced by Mohrherr in [7], where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to ..."
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Abstract. The notion of superhigh computably enumerable (c.e.) degrees was first introduced by Mohrherr in [7], where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in [4] that the diamond lattice can be embedded into the c.e. ttdegrees preserving 0 and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh. 1