Abstract. We say that A ≤LR B if every B-random number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.

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 ′