AsetAof nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E =({We}e∈ω,⊆). Most previously known automorphisms Φ of the structure E of

The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) deønable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results. 1 A Brief History of C.E. Sets G#del 1934 introduced the deønition of a (general) recursive function, and Church 1936 p...

Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the inclusion ordering of c.e. subsets of!, namely E = (fWngn2! ; ae). In the 1950's and 1960's Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating E-definable properties of A, like maximal, hh-simple, atomless, to the information content (usually the

this paper is to introduce the \Delta

This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here