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 $n \geq 1$, there exists an r.e. degree $\mathbf{d}$ such that the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ admits an embedding of the $n$-atom Boolean algebra $\mathscr{B}_n$ preserving (least and) greatest element, but also such that there is no $(n + 1)$-tuple of pairwise incomparable r.e. degrees above $\mathbf{d}$ which pairwise join to $\mathbf{0'}$ (and hence, the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ does not admit a greatest-element-preserving embedding of any lattice $\mathscr{L}$ which has $n + 1$ co-atoms, including $\mathscr{B}_{n + 1}$). 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 $\mathbf{R}$ has infinitely many one-types.