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Mathematics of Fuzzy Sets : Logic, Topology, and Measure Theory - Ulrich Hohle

Mathematics of Fuzzy Sets

Logic, Topology, and Measure Theory


Published: 31st December 1998
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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.

Authors and Editors
Introductionp. 1
Many-valued logic and fuzzy set theoryp. 5
Powerset operator foundations for poslat fuzzy set theories and topologiesp. 91
Introductory notes to Chapter 3p. 117
Axiomatic foundations of fixed-basis fuzzy topologyp. 123
Categorical foundations of variable-basis fuzzy topologyp. 273
Characterization of L-topologies by L-valued neighborhoodsp. 389
Separation axioms: Extension of mappings and embedding of spacesp. 433
Separation axioms: Representation theorems, compactness, and compactificationsp. 481
Uniform spacesp. 553
Extensions of uniform space notionsp. 581
Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unityp. 607
Fundamentals of generalized measure theoryp. 633
On conditioning operatorsp. 653
Applications of decomposable measuresp. 675
Fuzzy random variables revisitedp. 701
Indexp. 711
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792383888
ISBN-10: 0792383885
Series: The Handbooks of Fuzzy Sets
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 716
Published: 31st December 1998
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 24.77 x 17.78  x 4.45
Weight (kg): 1.16