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

One of the most basic measures of the complexity of a given partially ordered structure is the quantity of partial orderings embeddable in this structure. In the structure of the Turing degrees, DT , this problem is investigated in a series of results: Mostowski [15] proves that there is a computable partial ordering in which every countable partial ordering can be embedded. Kleene and Post [10] introduce the notion of a computably independent sequence of sets and prove the existence of a countable computably independent sequence of sets {Ai}i<ω, so that the Turing degree of every member Ai of this class is uniformly below 0 ′. Muchnik [16] proves the existence of a computably independent sequence of computably enumerable sets. Sacks [20] shows that one can embed any computable partial ordering using a computably independent sequence of sets, and as a corollary of the previously mentioned results obtains the embeddability of any countable partial ordering in the structure of the computably enumerable degrees, R. Finally Robinson [19] generalizes Sacks’ Density Theorem [22] by showing that one can embed any countable partial ordering in the computably enumerable degrees between any two given c.e. degrees b < a. (See Odifreddi [17, 18] for an extensive survey of these results.) The structure of the enumeration degrees De, which can be seen as an extension of the structure of the Turing degrees DT , naturally inherits this complexity. Further results on this topic are obtained by Case [2], who shows that any countable partial ordering can be embedded in the enumeration degrees below the e-degree of any given generic function, and Copestake [5], who shows that one can embed any countable partial ordering in the e-degrees below any given 1-generic enumeration degree. Lagemann [12] proves that the embedding of any countable partial ordering can be obtained below any nonzero ∆2 enumeration degree. Finally the density of the structure of the Σ2 enumeration degrees, G, proved by Cooper [3] is strengthened by Bianchini [1] who shows that every countable partial ordering can be embedded in any non-empty interval of Σ2 enumeration degrees; see also Sorbi [24] for a published proof of Bianchini’s result. In this article we study the embeddability problem further in the context of three different structures. We start with a slight improvement on the above mentioned embeddability results for the structure of the enumeration degrees. Then build onto our first result to solve the embeddability of countable partial orderings problem for the structure of the ω-enumeration degrees. Finally we apply our second result