## Extension of embeddings in the computably enumerable degrees.(English)Zbl 0988.03063

The paper presents a solution to an important, long-standing problem in the Turing degrees, the problem of extensions of given embeddings. More precisely, let $$P$$, $$Q$$ be partial orderings with $$P$$ a sub-order of $$Q$$ and such that there is an embedding $$\nu$$ of $$P$$ into the computably enumerable Turing degrees. The main result of the paper gives a necessary and sufficient condition when the embedding $$\nu$$ can be extended to an embedding of $$Q$$ into the c.e. degrees. This problem has already been solved for many related structures, such as c.e. tt-degrees and c.e. wtt-degrees, but nowhere it was so hard and complicated. It was preceded by a number of partial results and ends a long period of development in Computability Theory. It can be viewed also as a significant step in the theory of the priority method and a great advancement into its practice.

### MSC:

 03D25 Recursively (computably) enumerable sets and degrees
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