Semi-algebraic function rings and reflectors of partially ordered rings / Niels Schwarz, James J. Madden.
By: Schwartz, Niels
Contributor(s): Madden, James JMaterial type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 1712.Publisher: Berlin ; New York : Springer, ©1999Description: 1 online resource (x, 279 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783540482840; 3540482849Subject(s): Semialgebraic sets | Ordered algebraic structures | Rings (Algebra) | Ordered algebraic structures | Rings (Algebra) | Semialgebraic sets | Ringen (wiskunde) | Geometria algebrica | AlgebraGenre/Form: Electronic books. Additional physical formats: Print version:: Semi-algebraic function rings and reflectors of partially ordered rings.; Print version:: Semi-algebraic function rings and reflectors of partially ordered rings.DDC classification: 510 s | 512/.4 LOC classification: QA3 | .L28 no. 1712 | QA564Online resources: Click here to access online
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Includes bibliographical references (pages 259-265) and index.
Preface -- List of contents -- Introduction -- 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 -- Bibliography -- List of categories -- Index.
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.