Embedding distributive lattices preserving 1 below a nonzero recursively enumerable turing degree - Ambos-Spies, Klaus; Decheng, Ding; Fejer, Peter A. (Logical methods, 1993)
One way to try to gain an understanding of the various degree-theoretic structures which recursion theorists study is to see what lattices can be embedded into them. Lattice embeddings have been used to show that such structures have an undecidable theory (via embeddings as initial segments) and to show that the theory of such structures is decidable up to a certain quantifier level. Many results in recursion theory can be stated as results about lattice embeddings even if they were not originally phrased that way.
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