Steffen Lempp, Manuel Lerman, and Reed Solomon Embedding finite lattices into the computably enumerable degrees — a status survey

We note here that the lattice embeddings problem is currently the primary remaining obstacle toward showing the decidability of the V3-theory of the c.e. degrees (in the language of partial ordering) as the former obviously forms

a subproblem of the latter. If the lattice embeddings problem can be shown to have a decidable (and “reasonable”) solution, then one would hope to show the remainder of the V3-theory also to be decidable using the techniques of Slaman and Soare [14] in their solution of the extension of embeddings problem (i.e., given two finite partial orders embedding of V into the c.e. degrees can be extended to an emoedding oi

and of Ambos-Spies, Jockusch, Shore, and Soare [1] in their work on the promptly simple degrees (i.e., those c.e. degrees not forming half of a minimal pair).