Partially Ordered Rings and Semi-Algebraic Geometry (London Mathematical Society Lecture Note Series, Series Number 37)

The Purpose Of This Unique Book Is To Establish Purely Algebraic Foundations For The Development Of Certain Parts Of Topology. Some Topologists Seek To Understand Geometric Properties Of Solutions To Finite Systems Of Equations Or Inequalities And Configurations Which In Some Sense Actually Occur In The Real World. Others Study Spaces Constructed More Abstractly Using Infinite Limit Processes. Their Goal Is To Determine Just How Similar Or Different These Abstract Spaces Are From Those Which Are Finitely Described. However, As Topology Is Usually Taught, Even The First, More Concrete Type Of Problem Is Approached Using The Language And Methods Of The Second Type. Professor Brumfiel's Thesis Is That This Is Unnecessary And, In Fact, Misleading Philosophically. He Develops A Type Of Algebra, Partially Ordered Rings, In Which It Makes Sense To Talk About Solutions Of Equations And Inequalities And To Compare Geometrically The Resulting Spaces. The Importance Of This Approach Is Primarily That It Clarifies The Sort Of Geometrical Questions One Wants To Ask And Answer About Those Spaces Which Might Have Physical Significance. Gregory W. Brumfiel. Includes Index. Bibliography: P. 273-277. Cover London Mathematical Society Lecture Note Series 37 Partially Ordered Rings and Semi-Algebraic Geometry Copyright 9780521228459 Contents Preface Introduction CHAPTER I - PARTIALLY ORDERED RINGS 1.2. Existence of Orders 1.3. Extension and Contraction of Orders 1.4. Simple refinements of orders 1.5. Remarks on the Categories (PORNN) and (PORCK) 1.6. Remarks on Integral Domains 1.7. Some Examples CHAPTER II - HOMOMORPHISMS AND CONVEX IDEALS 2.2. Convex Hulls 2.3. Maximal Convex Ideals and Prime Convex Ideals 2.6. Semi-Noetherian Rings 2.7. Convex Ideals and Intersections of Orders 2.8. Some Examples 3.2. Sufficiency of Positive Multiplicative Sets 3.3. Refinements of an Order Induced by Certain Localizations 3.5. Concave Multiplicative Sets 3.6. The Shadow of 1 3.7. Localization at a Prime Convex Ideal 3.8. Localization in (PORCK) 3.9. Applications of Localization, I - Some Properties of Convex Prime Ideals 3.10. Applications of Localization, II - Zero Divisors 3.11. Applications of Localization, III - Minimal Primes, Isolated Sets of Primes, and Associated Invariants 3.12. Operators on the Set of Orders on a Ring CHAPTER IV - SOME CATEGORICAL NOTIONS 4.2. Fibre Sums 4.3. Direct and Inverse Limits 4.4. Some Examples CHAPTER V - THE PRIME CONVEX IDEAL SPECTRUM 5.2. Some Topological Properties 5.6. The Structure Sheaf, I - A First Approximation on Basic Open Sets 5.7. The Structure Sheaf, II - The Sheaf Axioms for Basic Open Sets CHAPTER VI - POLYNOMIALS 6.2. Adjoining Roots 6.3. A Universal Bound on the Roots of Polynomials 6.4. A "Going-Up" Theorem for Semi-Integral Extensions CHAPTER VII - ORDERED FIELDS 7.2. Function Theoretic Properties of Polynomials 7.3. Sturm's Theorem 7.4. Dedekind Cuts; Archimedean and Non-Archimedean Extensions 7.5. Orders on Simple Field Extensions 7.6. Total Orders and Signed Places 7.7. Existence of Signed Places 8.1. Introduction and Notation 8.2. Some Properties of RHJ-Algebras 8.3. Real Curves 8.4. Signed Places on Function Fields 8.5. Characterization of Non-Negative Functions 8.6. Derived Orders 8.7. A Preliminary Inverse Function Theorem 8.8. Algebraic Simple Points, Dimension, Codimension and Rank 8.9. Stratification of Semi-Algebraic Sets 8.10. Krull Dimension 8.11. Orders on Function Fields 8.12. Discussion of Total Orders on R(x,y) 8.13. Brief Discussion of Structure Sheaves I - The rational structure sheaf II - The semi-algebraic structure sheaf APPENDIX - The Tarski-Seidenberg Theorem BIBLIOGRAPHY LIST OF NOTATION INDEX