The purpose of this short paper is to clarify and correct the proof of the main result contained in [1]: namely, that there exists no low maximal d. c. e. degree. There we gave a simple proof obtained as an immediate corollary of the following posited extension of the Robinson Splitting Theorem ([1, Theorem 1.7]: For any c. e. set A,Denis Hirschfeldt (private communication) was the first to notice a problem with the particular application of the Recursion Theorem in the proof of this result, one which does not occur in the original Robinson proof [5].We present below the following reformulation of the use of the Recursion Theorem sufficient to correct the proof of our main result (the non-existence of a low maximal d. c. e. 1) ), via a degree-theoretic extension of the Robinson Splitting Theorem:Theorem For any d. c. e. degree l, any c. e. degree a, if l is low and l < a, then there are d. c. e. degrees a 0 , a 1 such that l < a 0 , a 1 < a and a 0 ∪ a 1 = a. P r o o f. Let L be a d. c. e. set of low degree. Let L = L 0 − L 1 for some c. e. sets L 0 , L 1 such that L 0 ⊃ L 1 . Let f be a 1 − 1 computable function such that L 0 = {f (x) : x ∈ ω}, and M = f −1 (L 1 ). Then M is c. e. and M ≤ T L. (M is called Lachlan's set for L.)Given a c. e. set A, and a d. c. e. set L, assume that L < T A and L has low degree. First we construct ω-c. e. sets A 0 , A 1 to satisfy the following requirements:where i = 0, 1, e ∈ ω, and {Φ e : e ∈ ω} is a standard list of all partial computable (p. c.) functionals Φ. Let a = deg T (A), l = deg T (L) and a i = deg T (A i ⊕ L) for i = 0, 1. Then a i is an ω-c. e. degree, for each i = 0, 1. By the R-requirement, a 0 ∪ a 1 = a, and by the S-requirements, l ≤ a i < a, i ∈ {0, 1}. Then by the S-and R-requirements we will have l < a i .We will construct ω-c. e. sets so that they are c. e. in Lachlan's set M for L. The theorem follows from the following result from [2].Proposition 1 (Arslanov, LaForte and Slaman [2]) . Let A and C be sets such that C is c. e., A is c. e. in C, C ≤ T A, and A is ω-c. e. Then deg(A) is d.

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We show that the intersection of the class of 2-REA degrees with that of the ω-r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy.

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We show that for any computably enumerable (c. e.) set A and any ∆ 0 2 set L, if L is low and L

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a b s t r a c tThis paper continues the project, initiated in (Arslanov, Cooper and Kalimullin 2003) [3], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting.The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′′ -priority construction of a Π 0 1 e-degree unsplittable over a 3-c.e. e-degree below it.

In this chapter we investigate set-theoretic properties and the Turing degree structure of the hierarchy of ∆ 0 2 -sets, which is well known in the literature as the Ershov hierarchy.