We suggest some new ways to effectivize the definitions of several classes of simple sets. On this basis, new completeness criterions for simple sets are obtained. In particular, we give descriptions of the class of complete maximal sets.

In this paper we consider the we known method by E. Post of solving the problem of construction of recursively enumerable sets that have a degree intermediate between the degrees of recursive and complete sets with respect to a given reducibility. Post considered reducibilities ≤m, ≤btt, ≤tt and ≤T and solved the problem for al of them except ≤T. Here we extend Post's original method of construction of incomplete sets onto two wide classes of sub-Turing reducibilities what were studying in (...) [1, 2]. (shrink)