AUTOMORPHISMS OF THE LATTICE OF RECURSIVELY-ENUMERABLE SETS

A version of Harrington's DELTA3-automorphism technique for the lattice of recursively enumerable sets is introduced and developed by reproving Soare's Extension Theorem. Then this automorphism technique is used to show two technical theorems: the High Extension Theorem I and the High Extension Theorem II. These theorems and other technical theorems are used to show: for all high r.e. degrees h and for all r.e. sets A there is an r.e. set B in h such that these two sets have isomorphic principal filters of r.e. sets. In addition it is shown that for any nonrecursive r.e. set A, there is a high r.e. set B such that A and B are automorphic in the lattice of recursively enumerable sets (this was shown independently by Harrington and Soare). These techniques are also used to show that if A is a coinfinite r.e. set such that ABAR is semi-low2 and A has the outer splitting property then the principal filter formed by A is isomorphic to the lattice of r.e. sets.