Embedding countable partial orderings in the enumeration degrees and the -enumeration degrees

Abstract

In this article, we examine the complexity of three degree structures in terms of the quantity of partial orderings embeddable in them. We prove that every countable partial ordering can be embedded in any interval of enumeration degrees with upper bound, a degree which contains a member with a good approximation. Next we prove that a similar result is true of the ω-enumeration degrees. Finally we consider the structure of the ∑02ω-enumeration degree modulo iterated jump and show that there as well one can embed densely every countable partial ordering.