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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.
1. Many-Valued Logic and Fuzzy Set Theory; S. Gottwald. 2. Powerset Operator Foundations for Poslat Fuzzy Set Theories and Topologies; S.E. Rodabaugh. 3. Axiomatic Foundations of Fixed-Based Fuzzy Topology; U. Höhle, A.P. Sostak. 4. Categorical Foundations of Variable-Basis Fuzzy Topology; S.E. Rodabaugh. 5. Characterization of L-Topologies by L-Valued Neighborhoods; U. Höhle. 6. Separation Axioms: Extension of Mappings and Embedding of Spaces; T. Kubiak. 7. Separation Axioms: Representation Theorems, Compactness, and Compactifications; S.E. Rodabaugh. 8. Uniform Spaces; W. Kotzé. 9. Extensions of Uniform Space Notions; M.H. Burton, J. Gutiérrez García. 10. Fuzzy Real Lines and Dual Real Lines as Poslat Topological, Uniform, and Metric Ordered Semirings with Unity; S.E. Rodabaugh. 11. Fundamentals of Generalized Measure Theory; E.P. Klement, S. Weber. 12. On Conditioning Operators; U. Höhle, S. Weber. 13. Applications of Decomposable Measures; E. Pap. 14. Fuzzy Random Variables Revisited; D.A. Ralescu.