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Ordered Exponential Fields
Salma Kuhlmann, University of Saskatchewan, Saskatoon, SK, Canada
A co-publication of the AMS and Fields Institute.

Fields Institute Monographs
2000; 166 pp; hardcover
Volume: 12
ISBN-10: 0-8218-0943-1
ISBN-13: 978-0-8218-0943-3
List Price: US$59
Member Price: US$47.20
Order Code: FIM/12
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Model theoretic algebra has witnessed remarkable progress in the last few years. It has found profound applications in other areas of mathematics, notably in algebraic geometry and in singularity theory.

Since Wilkie's results on the o-minimality of the expansion of the reals by the exponential function, and most recently even by all Pfaffian functions, the study of o-minimal expansions of the reals has become a fascinating topic. The quest for analogies between the semi-algebraic case and the o-minimal case has set a direction to this research.

Through the Artin-Schreier Theory of real closed fields, the structure of the non-archimedean models in the semi-algebraic case is well understood. For the o-minimal case, so far there has been no systematic study of the non-archimedean models. The goal of this monograph is to serve this purpose.

The author presents a detailed description of the non-archimedean models of the elementary theory of certain o-minimal expansions of the reals in which the exponential function is definable. The example of exponential Hardy fields is worked out with particular emphasis. The basic tool is valuation theory, and a sufficient amount of background material on orderings and valuations is presented for the convenience of the reader.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).


Graduate students and research mathematicians interested in algebra, analysis, and model theory.


"This book is clearly and carefully written, ... it would be a useful addition to the library of anyone interested in algebraic model theory, valuation theory, or general exponentiation."

-- Mathematical Reviews

"This book can easily be read by those with little or no background in ordered structures or valuation theory ... the author has taken great care to include all the necessary material. Throughout, the presentation is well-motivated, and the discussion and proofs are clear and thorough. For those unfamiliar with ordered fields, this book will serve as a pleasant introduction to the subject. And for those already familiar with the subject, it is gratifying to see that the author has successfully dealt with the intriguing challenge of using the structure theory to describe the implications of the presence of an exponential function."

-- CMS Notes

Table of Contents

  • Preliminaries on valued and ordered modules
  • Non-archimedean exponential fields
  • Valuation theoretic interpretation of the growth and Taylor axioms
  • The exponential rank
  • Construction of exponential fields
  • Models for the elementary theory of the reals with restricted analytic functions and exponentiation
  • Exponential Hardy fields
  • The model theory of contraction groups
  • Bibliography
  • Index
  • List of notation
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