Algebraic Model Theory
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| Author | : Elisabeth Bouscaren |
| Publisher | : Springer |
| Total Pages | : 223 |
| Release | : 2009-03-14 |
| Genre | : Mathematics |
| ISBN | : 3540685219 |
This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski's proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.
| Author | : David Marker |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 342 |
| Release | : 2006-04-06 |
| Genre | : Mathematics |
| ISBN | : 0387227342 |
Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures
| Author | : Alexander Prestel |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 198 |
| Release | : 2011-08-21 |
| Genre | : Mathematics |
| ISBN | : 1447121767 |
Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.
| Author | : Katrin Tent |
| Publisher | : Cambridge University Press |
| Total Pages | : 259 |
| Release | : 2012-03-08 |
| Genre | : Mathematics |
| ISBN | : 052176324X |
Concise introduction to current topics in model theory, including simple and stable theories.
| Author | : Matthias Aschenbrenner |
| Publisher | : Princeton University Press |
| Total Pages | : 873 |
| Release | : 2017-06-06 |
| Genre | : Mathematics |
| ISBN | : 0691175438 |
Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.
| Author | : Jörg Flum |
| Publisher | : Springer |
| Total Pages | : 161 |
| Release | : 2006-11-14 |
| Genre | : Mathematics |
| ISBN | : 3540385444 |
| Author | : Yves Félix |
| Publisher | : Oxford University Press |
| Total Pages | : 483 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0199206511 |
A text aimed at both geometers needing the tools of rational homotopy theory to understand and discover new results concerning various geometric subjects, and topologists who require greater breadth of knowledge about geometric applications of the algebra of homotopy theory.
| Author | : Wilfrid Hodges |
| Publisher | : Cambridge University Press |
| Total Pages | : 322 |
| Release | : 1997-04-10 |
| Genre | : Mathematics |
| ISBN | : 9780521587136 |
This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory.
| Author | : E.G. Manes |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 364 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461298601 |
In the past decade, category theory has widened its scope and now inter acts with many areas of mathematics. This book develops some of the interactions between universal algebra and category theory as well as some of the resulting applications. We begin with an exposition of equationally defineable classes from the point of view of "algebraic theories," but without the use of category theory. This serves to motivate the general treatment of algebraic theories in a category, which is the central concern of the book. (No category theory is presumed; rather, an independent treatment is provided by the second chap ter.) Applications abound throughout the text and exercises and in the final chapter in which we pursue problems originating in topological dynamics and in automata theory. This book is a natural outgrowth of the ideas of a small group of mathe maticians, many of whom were in residence at the Forschungsinstitut für Mathematik of the Eidgenössische Technische Hochschule in Zürich, Switzerland during the academic year 1966-67. It was in this stimulating atmosphere that the author wrote his doctoral dissertation. The "Zürich School," then, was Michael Barr, Jon Beck, John Gray, Bill Lawvere, Fred Linton, and Myles Tierney (who were there) and (at least) Harry Appelgate, Sammy Eilenberg, John Isbell, and Saunders Mac Lane (whose spiritual presence was tangible.) I am grateful to the National Science Foundation who provided support, under grants GJ 35759 and OCR 72-03733 A01, while I wrote this book.
| Author | : Anatolij Ivanovic Mal'cev |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 331 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 364265374X |
As far back as the 1920's, algebra had been accepted as the science studying the properties of sets on which there is defined a particular system of operations. However up until the forties the overwhelming majority of algebraists were investigating merely a few kinds of algebraic structures. These were primarily groups, rings and lattices. The first general theoretical work dealing with arbitrary sets with arbitrary operations is due to G. Birkhoff (1935). During these same years, A. Tarski published an important paper in which he formulated the basic prin ciples of a theory of sets equipped with a system of relations. Such sets are now called models. In contrast to algebra, model theory made abun dant use of the apparatus of mathematical logic. The possibility of making fruitful use of logic not only to study universal algebras but also the more classical parts of algebra such as group theory was dis covered by the author in 1936. During the next twenty-five years, it gradually became clear that the theory of universal algebras and model theory are very intimately related despite a certain difference in the nature of their problems. And it is therefore meaningful to speak of a single theory of algebraic systems dealing with sets on which there is defined a series of operations and relations (algebraic systems). The formal apparatus of the theory is the language of the so-called applied predicate calculus. Thus the theory can be considered to border on logic and algebra.
