Automorphisms of the lattice of recursively enumerable sets
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 1995: Volume 113, Number 541
ISBNs: 978-0-8218-2601-0 (print); 978-1-4704-0120-7 (online)
MathSciNet review: 1227497
MSC: Primary 03D25
This work explores the connection between the lattice of recursively enumerable (r.e.) sets and the r.e. Turing degrees. Cholak presents a degree-theoretic technique for constructing both automorphisms of the lattice of r.e. sets and isomorphisms between various substructures of the lattice. In addition to providing another proof of Soare's Extension Theorem, this technique is used to prove a collection of new results, including: every nonrecursive r.e. set is automorphic to a high r.e. set; and for every nonrecursive r.e. set $A$ and for every high r.e. degree h there is an r.e. set $B$ in h such that $A$ and $B$ form isomorphic principal filters in the lattice of r.e. sets.
Mathematicians interested in recursion theory, mainly logicians and theoretical computer scientists.
Table of Contents
- I. Introduction
- II. The extension theorem revisited
- III. The high extension theorems
- IV. The proof of the high extension theorem I
- V. The proof of the high extension theorem II
- VI. Lowness notions in the lattice of r.e. sets