Embedding distributive lattices in the Σ02 enumeration degrees

The local structure of the enumeration degrees Ge is the partially ordered set of the enumeration degrees below the first jump 0e of the least enumeration degree 0e. Cooper [3] shows that Ge consists exactly of the Σ2 enumeration degrees, degrees which contain Σ2 sets, or equivalently consist entirely of Σ 0 2 sets. In investigating structural complexity of Ge, the natural question of what other structures are embeddable in Ge arises. For example, if we view Ge as a countable partial ordering, we might ask what other partial orderings are embedded in Ge. The complete answer to this question is provided by Bianchini [2], who proves that every countable partial ordering can be embedded densely in Ge, i.e. in any nonempty interval of Σ2 enumeration degrees; see also Sorbi [11] for a published proof of Bianchini’s result. As Ge is an interval of enumeration degrees, Ge is a countable upper semi-lattice with least and greatest elements. In this article we investigate the further question of characterizing special types of partially ordered structures, lattices, that are embeddable in Ge. We start by outlining preliminary results on this topic. McEvoy and Cooper [8] prove that the standard embedding ι of the Turing degrees in the enumeration degrees preserves greatest lower bounds for low c.e. degrees, i.e., if a,b, c ∈ R and a′ = b′ = c′ = 0T ′, then