The Resource Partially ordered rings and semi-algebraic geometry, Gregory W. Brumfiel, (electronic resource)

Partially ordered rings and semi-algebraic geometry, Gregory W. Brumfiel, (electronic resource)

Label
Partially ordered rings and semi-algebraic geometry
Title
Partially ordered rings and semi-algebraic geometry
Statement of responsibility
Gregory W. Brumfiel
Creator
Subject
Genre
Language
  • eng
  • eng
Summary
The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology
Member of
Cataloging source
MiAaPQ
http://library.link/vocab/creatorName
Brumfiel, Gregory W
Dewey number
512.4
Illustrations
illustrations
Index
index present
Language note
English
LC call number
QA251.3
LC item number
.B78 1979
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
Series statement
  • London Mathematical Society lecture note series
  • London Mathematical Society Lecture Note Series
Series volume
  • 37
  • v.37
http://library.link/vocab/subjectName
  • Categories (Mathematics)
  • Commutative rings
Label
Partially ordered rings and semi-algebraic geometry, Gregory W. Brumfiel, (electronic resource)
Instantiates
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Content category
text
Content type code
  • txt
Contents
  • Cover; Title; Copyright; Contents; INTRODUCTION; CHAPTER I - PARTIALLY ORDERED RINGS; 1.1. Definitions; 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 - HOMOMORPIIISMS AND CONVEX IDEALS; 2.1. Convex Ideals and Quotient Rings; 2.2. Convex Hulls; 2.3. Maximal Convex Ideals and Prime Convex Ideals; 2.4. Relation between Convex Ideals in (A,β) and (A/I, β/I); 2.5. Absolutely Convex Ideals; 2.6. Semi-Noetherian Rings
  • 2.7. Convex Ideals and Intersections of Orders2.8. Some Examples; CHAPTER III- LOCALIZATION; 3.1. Partial Orders on Localized Rings; 3.2. Sufficiency of Positive Multiplicative Sets; 3.3. Refinements of an Order Induced by Certain Localizations; 3.4. Convex Ideals in (A,β) and (AT,βT); 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 Invariants3.12. Operators on the Set of Orders on a Ring; CHAPTER IV - SOME CATEGORICAL NOTIONS; 4.1. Fibre Products; 4.2. Fibre Sums; 4.3. Direct and Inverse Limits; 4.4. Some Examples; CHAPTER V - THE PRIME CONVEX IDEAL SPECTRUM; 5.1. The Zariski Topology Defined; 5.2. Some Topological Properties; 5.3. Irreducible Closed Sets in Spec(A,β); 5.4. Spec(A,β) as a Functor; 5.5. Disconnectedness of Spec(A,β); 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 Sets5.8. The Structure Sheaf, III - Definition; CHAPTER VI - POLYNOMIALS; 6.1. Polynomials as Functions; 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; 7.1. Basic Results; 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
  • CHAPTER VIII - AFFINE SEMT-ALGEBRAIC SETS8.1. 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
Dimensions
unknown
Extent
1 online resource (290 p.)
Form of item
online
Isbn
9781107360921
Media category
computer
Media type code
  • c
Specific material designation
remote
System control number
  • (CKB)2610000000004155
  • (EBL)1182389
  • (SSID)ssj0000461882
  • (PQKBManifestationID)11310690
  • (PQKBTitleCode)TC0000461882
  • (PQKBWorkID)10487048
  • (PQKB)10512952
  • (UkCbUP)CR9780511721533
  • (MiAaPQ)EBC1182389
  • (EXLCZ)992610000000004155
Label
Partially ordered rings and semi-algebraic geometry, Gregory W. Brumfiel, (electronic resource)
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Content category
text
Content type code
  • txt
Contents
  • Cover; Title; Copyright; Contents; INTRODUCTION; CHAPTER I - PARTIALLY ORDERED RINGS; 1.1. Definitions; 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 - HOMOMORPIIISMS AND CONVEX IDEALS; 2.1. Convex Ideals and Quotient Rings; 2.2. Convex Hulls; 2.3. Maximal Convex Ideals and Prime Convex Ideals; 2.4. Relation between Convex Ideals in (A,β) and (A/I, β/I); 2.5. Absolutely Convex Ideals; 2.6. Semi-Noetherian Rings
  • 2.7. Convex Ideals and Intersections of Orders2.8. Some Examples; CHAPTER III- LOCALIZATION; 3.1. Partial Orders on Localized Rings; 3.2. Sufficiency of Positive Multiplicative Sets; 3.3. Refinements of an Order Induced by Certain Localizations; 3.4. Convex Ideals in (A,β) and (AT,βT); 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 Invariants3.12. Operators on the Set of Orders on a Ring; CHAPTER IV - SOME CATEGORICAL NOTIONS; 4.1. Fibre Products; 4.2. Fibre Sums; 4.3. Direct and Inverse Limits; 4.4. Some Examples; CHAPTER V - THE PRIME CONVEX IDEAL SPECTRUM; 5.1. The Zariski Topology Defined; 5.2. Some Topological Properties; 5.3. Irreducible Closed Sets in Spec(A,β); 5.4. Spec(A,β) as a Functor; 5.5. Disconnectedness of Spec(A,β); 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 Sets5.8. The Structure Sheaf, III - Definition; CHAPTER VI - POLYNOMIALS; 6.1. Polynomials as Functions; 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; 7.1. Basic Results; 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
  • CHAPTER VIII - AFFINE SEMT-ALGEBRAIC SETS8.1. 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
Dimensions
unknown
Extent
1 online resource (290 p.)
Form of item
online
Isbn
9781107360921
Media category
computer
Media type code
  • c
Specific material designation
remote
System control number
  • (CKB)2610000000004155
  • (EBL)1182389
  • (SSID)ssj0000461882
  • (PQKBManifestationID)11310690
  • (PQKBTitleCode)TC0000461882
  • (PQKBWorkID)10487048
  • (PQKB)10512952
  • (UkCbUP)CR9780511721533
  • (MiAaPQ)EBC1182389
  • (EXLCZ)992610000000004155

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