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 (??, ≤) of c. e. degrees under Turing reducibility. We demonstrate that the latticeM₅ 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 M₅ can be embedded below every nonlow₂ 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 nonlow₂ degrees. We note also that there are embeddings of M₅ in which all five degrees are array noncomputable, and in which the bottom degree is the computable degree 0 but the other four are array noncomputable. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)