# Automorphisms of the Lattice of Recursively Enumerable Sets

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*Peter Cholak*

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.

#### Reviews & Endorsements

Significant work … clearly a must for workers in the area and for those looking towards studying amorphism groups of other related areas.

-- Journal of Symbolic Logic

#### Table of Contents

# Table of Contents

## Automorphisms of the Lattice of Recursively Enumerable Sets

- Table of Contents v6 free
- Chapter I: Introduction 110 free
- Chapter II: The Extension Theorem Revisited 514 free
- Chapter III: The High Extension Theorems 4554
- Chapter IV: The Proof of the High Extension Theorem I 5463
- Chapter V: The Proof of the High Extension Theorem II 8392
- Chapter VI: Lowness Notions in the Lattice of R.E. Sets 115124
- Bibliography 149158