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)