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The Handbooks of Fuzzy Sets

Mathematics of Fuzzy Sets

Logic, Topology, and Measure Theory

Authors: Höhle, Ulrich, Rodabaugh, S.E.

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  • ISBN 978-1-4615-5079-2
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About this book

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14).
Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications.
Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact or complete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval.
Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton&endash;Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

Table of contents (16 chapters)

Table of contents (16 chapters)

Buy this book

eBook $189.00
price for USA in USD
  • ISBN 978-1-4615-5079-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $349.99
price for USA in USD
Softcover $249.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Mathematics of Fuzzy Sets
Book Subtitle
Logic, Topology, and Measure Theory
Authors
Series Title
The Handbooks of Fuzzy Sets
Series Volume
3
Copyright
1999
Publisher
Springer US
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4615-5079-2
DOI
10.1007/978-1-4615-5079-2
Hardcover ISBN
978-0-7923-8388-8
Softcover ISBN
978-1-4613-7310-0
Series ISSN
1388-4352
Edition Number
1
Number of Pages
XII, 716
Topics

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