Abstract
This chapter summarizes some of the results of the algebraic structure of the computably enumerable (c.e.) sets since 1987, when the subject was covered in Soare. In addition to defining computable functions, there was in interest in defining computable generated sets. Church and Kleene defined a set of positive integers to be “recursively enumerable,” if it is the range of a recursive function. A little more was done with these sets until Post proposed a formal system for generating sets rather than computing their characteristic functions. Post showed that the normal sets are exactly the recursively enumerable (r.e.) sets, providing the empty set is added as an r.e. set. Post, however, thought not so much in formal systems as in informal terms and described the corresponding informal concept of effectively enumerable set or generated set. The chapter considers the various properties of Є: namely, definable properties, automorphisms, invariant properties, decidability and undecidability results, and miscellaneous results.
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