Computably enumerable sets and related issues

A topical direction in the theory of algorithms is studying reducibilities of arithmetic sets. Post defined concepts of $m$-, $tt$-, and $T$-reducibilities of arithmetic sets; subsequently, other kinds of reducibilities were introduced. A $T$-reducibility is being studied quite intensively today. Here a number of remarkable results were obtained. However, many questions concerning $T$-reducibility still await a solution. Progress in $tt$-reducibility is less noticeable. For an $m$-reducibility, exhaustive solutions were derived in several directions – especially if consideration was limited to computably enumerable sets. In the present review, we consider different aspects associated with computably enumerable sets and m-reducibility. Among these are an algebraic description of the structure of these formations both in upper and lower parts, definability, decidability problems, and so on. Many of the results cited in the paper are widely scattered over the literature. This makes it impossible to conceptualize the overall picture and spectrum of available investigations. It is also worth mentioning that a series of books and papers are not easily accessible to domestic specialists.