Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings

1999
ISBN: 978-3-540-66460-4
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 1712, 279 Seiten, Format (B × H): 155 mm x 233 mm, Gewicht: 454 g

Reihe: Lecture Notes in Mathematics

The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions associated with the spectra. The many different possible choices for these rings of functions are studied via reflections of partially ordered rings. Readers should feel comfortable using basic algebraic and categorical concepts. As motivational background some familiarity with real geometry will be helpful. The book aims at researchers and graduate students with an interest in real algebra and geometry, ordered algebraic structures, topology and rings of continuous functions.

Zielgruppe

Research

Autoren/Hrsg.

Madden, James J.

Schwartz, Niels

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

Preordered and partially ordered rings.- Reflective subcategories.- Totally ordered and real closed fields.- Real spectra of preordered rings.- Epimorphisms of reduced porings.- Functions and representable porings.- Semi-algebraic functions.- Comparing reflectors.- Constructing reflectors.- H-closed epireflectors.- Quotient-closed reflectors.- The real closure reflector.- Arities of reflectors and approximations by H-closed reflectors.- Epimorphic extensions of reduced porings.- Essential monoreflectors.- Reflections of totally ordered fields.- von Neumann regular f-rings.- Totally ordered domains.- Reduced f-rings.- Rings of continuous piecewise polynomial functions.- Rings of continuous piecewise rational functions.- Discontinuous semi-algebraic functions.- The lattice of H-closed monoreflectors.