There is no low maximal d. c. e. degree - Corrigendum

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d.c.e. degrees. The purpose of this short paper is to clarify and correct the main result and proof contained in [1]. There we gave a simple proof that there exists no low maximal d.c.e. degree. This was obtained as an immediate corollary of the following strengthening of the Robinson Splitting Theorem (Theorem 1.7 of [1]): For any c.e. set A, any ∆02 low set L, if L <T A, then there is a c.e. splitting A0 ⊕A1 = A such that Ai ⊕ L <T 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 [6]. We present below a 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. degree), and to give a modified degree-theoretic extension of the Robinson Splitting Theorem, which (necessarily, it turns out) replaces our old Theorem 1.7: 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 a0,a1 such that l < a0,a1 < a and a0 ∪ a1 = a. Proof. Let L be a d.c.e. set of low degree. Let L = L0 − L1 for some c.e. sets L0, L1 such that L0 ⊃ L1. Let f be a 1−1 computable function such that L0 = {f(x) | x ∈ ω}, and M = f (L1). Then M is c.e. and M ≤T L. (M is called Lachlan’s set for L.) Given c.e. set A, and d.c.e. set L, assume that L <T A and L has low degree. First we construct ω-c.e. sets A0, A1 to satisfy the following requirements: R : A0, A1 ≤T A ∧ A ≤T A0 ⊕A1 Se,i: A 6= Φe(Ai ⊕ L) where i = 0, 1, e ∈ ω, and {Φe : e ∈ ω} is a standard enumeration of all partial computable (p.c.) functionals Φ. The first two authors would like to acknowledge the support of a grant under the Royal Society Joint Projects with the Former Soviet Union scheme. Downey and Yu (see [4]) have independently proved this. This is because Downey has proved (private communication) that in the old result of Cooper and Seetapun, and independently Li, that there exists a properly ∆ 2 degree a which cups all the non-zero c.e. degrees, the degree a can be made low. In fact, Lewis has conjectured (private communication) that his single minimal complement below 0 for the intermediate c.e. degrees (see [5]) can be made to be low.