Description
Springer Mathematics of Fuzzy Sets Logic Topology and Measure Theory by Ulrich Höhle, S. E. Rodabaugh
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). _x000D_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. _x000D_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. _x000D_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._x000D_ Table of contents : - _x000D_
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. Hoehle, A.P. Sostak. 4. Categorical Foundations of Variable-Basis Fuzzy Topology; S.E. Rodabaugh. 5. Characterization of L-Topologies by L-Valued Neighborhoods; U. Hoehle. 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. Kotze. 9. Extensions of Uniform Space Notions; M.H. Burton, J. Gutierrez Garcia. 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. Hoehle, S. Weber. 13. Applications of Decomposable Measures; E. Pap. 14. Fuzzy Random Variables Revisited; D.A. Ralescu._x000D_