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The Reconstruction of Trees from Their Automorphism Groups
Matatyahu Rubin
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Contemporary Mathematics
1993; 274 pp; softcover
Volume: 151
ISBN-10: 0-8218-5187-X
ISBN-13: 978-0-8218-5187-6
List Price: US$68
Member Price: US$54.40
Order Code: CONM/151
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Trees, sometimes called semilinear orders, are partially ordered sets in which every initial segment determined by an element is linearly ordered. This book focuses on automorphism groups of trees, providing a nearly complete analysis of when two trees have isomorphic automorphism groups. Special attention is paid to the class of \(\aleph _0\)-categorical trees, and for this class the analysis is complete. Various open problems, mostly in permutation group theory and in model theory, are discussed, and a number of research directions are indicated. Aimed at graduate students and researchers in model theory and permutation group theory, this self-contained book will bring readers to the forefront of research on this topic.

Readership

Graduate students and researchers in model theory and permutation group theory.

Reviews

"The author has made it easy for the interested reader to penetrate this material to any desired depth by providing both a two-page summary and a twenty-seven page introduction. Both of these, as well as the main body of the book, are well motivated, intuitive, and clearly written. The author is obviously sensitive to the readers."

-- Mathematical Reviews

Table of Contents

  • An extended introduction
  • Some preliminaries concerning interpretations, groups and \(\aleph _0\)-categoricity
  • A new reconstruction theorem for Boolean algebras
  • The completion and the Boolean algebra of a U-tree
  • The statement of the canonization and reconstruction theorems
  • The canonization of trees
  • The reconstruction of the Boolean algebra of a U-tree
  • The reconstruction of \(PT({\mathrm Exp}(M))\)
  • Final reconstruction results
  • Observations, examples and discussion
  • Augmented trees
  • The reconstruction of \(\aleph _0\)-categorical trees
  • Nonisomorphic 1-homogeneous chains which have isomorphic automorphism groups
  • Bibliography
  • A list of notations and definitions
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