We investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given n ≥ 1, there exists an r.e. degree d such that the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ admits an embedding of the n-atom Boolean algebra B n preserving (least and) greatest element, but also such that there is no (n + 1)-tuple of pairwise incomparable r.e. degrees above d which pairwise join to 0' (and (...) hence, the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ does not admit a greatest-element-preserving embedding of any lattice L which has n + 1 co-atoms, including B n + 1 ). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of R has infinitely many one-types. (shrink)

We give an algorithm for deciding whether an embedding of a finite partial order [Formula: see text] into the enumeration degrees of the [Formula: see text]-sets can always be extended to an embedding of a finite partial order [Formula: see text].

We show that every finite lattice is embeddable into the Σ 0 2 enumeration degrees via a lattice-theoretic embedding which preserves 0 and 1.

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

We present a necessary and sufficient condition for the embeddability of a finite principally decomposable lattice into the computably enumerable degrees preserving greatest element.

Let b and c be r.e. Turing degrees such that b>c. We show that there is an r.e. degree a such that b>a>c and all lattices containing a critical triple, including the lattice M5, cannot be embedded into the interval [c, a].