# Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory

Ulrich Höhle, S.E. Rodabaugh
Springer Science & Business Media, 1999 - Business & Economics - 716 pages
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–Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

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### Contents

 ManyValued Logic And Fuzzy Set Theory 5 Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies 91 Introductory Notes To Chapter 3 117 Axiomatic Foundations Of FixedBasis Fuzzy Topology 123 Categorical Foundations Of VariableBasis Fuzzy Topology 273 Characterization Of LTopologies By LValued Neighborhoods 389 Separation Axioms Extension Of Mappings And Embedding Of Spaces 433 Separation Axioms Representation Theorems Compactness And Compactifications 481
 Extensions Of Uniform Space Notions 581 Fuzzy Real Lines And Dual Real Lines As Poslat Topological Uniform And Metric Ordered Semirings With Unity 607 Fundamentals of a Generalized Measure Theory 633 On Conditioning Operators 653 Applications Of Decomposable Measures 675 Fuzzy Random Variables Revisited 701 Index 711 Copyright

 Uniform Spaces 553

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