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Table of contents (16 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.
Editors and Affiliations
Bibliographic Information
Book Title: Mathematics of Fuzzy Sets
Book Subtitle: Logic, Topology, and Measure Theory
Editors: Ulrich Höhle, Stephen Ernest Rodabaugh
Series Title: The Handbooks of Fuzzy Sets
DOI: https://doi.org/10.1007/978-1-4615-5079-2
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media New York 1999
Hardcover ISBN: 978-0-7923-8388-8Published: 31 December 1998
Softcover ISBN: 978-1-4613-7310-0Published: 05 November 2012
eBook ISBN: 978-1-4615-5079-2Published: 06 December 2012
Series ISSN: 1388-4352
Edition Number: 1
Number of Pages: XII, 716
Topics: Mathematical Logic and Foundations, Calculus of Variations and Optimal Control; Optimization, Operations Research/Decision Theory