A version of Harrington's DELTA3-automorphism technique for the lattic
e of recursively enumerable sets is introduced and developed by reprov
ing Soare's Extension Theorem. Then this automorphism technique is use
d to show two technical theorems: the High Extension Theorem I and the
High Extension Theorem II. These theorems and other technical theorem
s 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 wa
s shown independently by Harrington and Soare). These techniques are a
lso 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 fi
lter formed by A is isomorphic to the lattice of r.e. sets.