Embeddings in the strong reducibilities between 1 and npm

We consider the strongest (most restricted) forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities (which we call n1- and ni-reducibility) 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 (under the natural relation on reducibilities `strong with respect to') 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 z between e and m such that there exists a non-trivial y-contiguous z degree.