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 ′

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.

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 long-standing 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

Abstract. We show that there exists a minimal (Turing) degree b<0 ′ such that for all non-zero 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 n-c.e. degree bounds a non-zero c.e. degree, b complements every n-c.e. degree other than 0 and 0 ′. 1.