Index sets of quotient objects of the post numeration

In theory of algorithms we actively study equivalence relations on r.e. sets. Each such relation generates a factorization of the Post numeration. The present article is devoted to a study of these factorizations from the point of view of general theory of numerations [i]. It turns out that the thoery of complete numerations, developed by A. I. Mal'tsev and Yu. L. Ershov, and also the notions and results from [2-4] play an important role in this study. We also establish a close connection of the theory of numerations with the results of Arslanov [5] and of Jockusch and Solovay [6]. In particular, Proposition 1 is a successfull generalization of Arslanov's fixed point theorem. In Sec. 1 we study certain classes of precomplete numerations and in Secs. 2-4 we apply the obtained results to the study of important quotient objects of the Post numeration. The index sets of these numerations are investigated in detail.