Lattice embeddings and array noncomputable degrees

We focus on a particular class of computably enumerable (c. e.) degrees, the array noncomputable degrees defined by Downey, Jockusch, and Stob, to answer questions related to lattice embeddings and definability in the partial ordering (? less than or equal to) of c. e. degrees under Turing reducibility. We demonstrate that the lattice M-5 cannot be embedded into the c. e. degrees below every array noncomputable degree, or even below every nonlow array noncomputable degree. As Downey and Shore have proved that M-5 can be embedded below every nonlow(2) degree, our result is the best possible in terms of array noncomputable degrees and jump classes. Further, this result shows that the array noncomputable degrees are definably different from the nonlow(2) degrees. We note also that there are embeddings of M-5, in which all live degrees are array noncomputable, and in which the bottom degree is the computable degree 0 but the other four are array noncomputable. (C) 2004 WILEY-VCH Verlag GmbH & CO. KGaA. Weinheim.