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Automorphisms of the Lattice of Recursively Enumerable Sets

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Electronic ISBN: 978-1-4704-0120-7
Product Code: MEMO/113/541.E
List Price: $46.00 MAA Member Price:$41.40
AMS Member Price: $27.60 Click above image for expanded view Automorphisms of the Lattice of Recursively Enumerable Sets Available Formats:  Electronic ISBN: 978-1-4704-0120-7 Product Code: MEMO/113/541.E  List Price:$46.00 MAA Member Price: $41.40 AMS Member Price:$27.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1131995; 151 pp
MSC: Primary 03;

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.

• Chapters
• 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
• Reviews

• 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
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Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 1131995; 151 pp
MSC: Primary 03;

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.

• Chapters
• 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
• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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