**Fields Institute Monographs**

Volume: 12;
2000;
166 pp;
Hardcover

MSC: Primary 03; 12;
Secondary 26

Print ISBN: 978-0-8218-0943-3

Product Code: FIM/12

List Price: $62.00

Individual Member Price: $49.60

**Electronic ISBN: 978-1-4704-3139-6
Product Code: FIM/12.E**

List Price: $62.00

Individual Member Price: $49.60

# Ordered Exponential Fields

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*Salma Kuhlmann*

A co-publication of the AMS and Fields Institute

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).

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## Ordered Exponential Fields

- Cover Cover11
- Title page v6
- Dedication vii8
- Quote ix10
- Contents xi12
- Introduction xiii14
- Preliminaries on valued and ordered modules 120
- Non-archimedean exponential fields 1534
- Valuation theoretic interpretation of the growth and Taylor axioms 3352
- The exponential rank 4968
- Construction of exponential fields 6584
- Models for the elementary theory of the reals with restricted analytic functions and exponentiation 7796
- Exponential Hardy fields 89108
- The model theory of contraction groups 117136
- Bibliography 155174
- Index 159178
- List of notation 163182
- Back Cover Back Cover1186