Extensions of Embeddings below Computably Enumerable Degrees

Extensions of Embeddings in the Computably Enumerable Degrees

Embeddings into the Computably Enumerable Degrees

We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directions which have been pursued to try to solve the problem, and present some related

The Non-isolating Degrees Are Upwards Dense in the Computably Enumerable Degrees

The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degree d is isolated by a c.e. degree a if a < d is the greatest c.e. degree below d. A computably enumerable degree c is non-isolating if no d.c.e. degree above c is isolated by c. Obviously, 0 is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Arslan...

Toward the Definability of the Array Noncomputable Degrees

by Stephen M. Walk 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 array noncomputable degrees and definability in the partial ordering (R,≤) of c.e. degrees. First we demonstrate that the lattice M5 cannot be embedded into the c.e. degrees below every array noncomputabl...

Universal computably enumerable sets and initial segment prefix-free complexity

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that s...