Abstract

We consider the strongest forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities which are counterparts to 1- and i-reducibility, respectively, in the same way that nm- and npm-reducibility are counterparts to m- and pm-reducibility, respectively, we bring out the structure of the strong reducibilities. By further restricting n1- and nm-reducibility we are able to define infinite families of reducibilities which isomorphically embed the r. e. Turing degrees. Thus the many well-known results in the theory of the r. e. Turing degrees have counterparts in the theory of strong reducibilities. We are also able to positively answer the question of whether there exist distinct reducibilities ≤y and ≤a between ≤e and ≤m such that there exists a non-trivial ≤y-contiguous ≤z degree