Mathematics of Fuzzy Sets : Logic, Topology, and Measure Theory / edited by Ulrich Höhle, Stephen Ernest Rodabaugh
(The Handbooks of Fuzzy Sets Series ; 3)
| データ種別 | 電子ブック |
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| 出版者 | Boston, MA : Springer US : Imprint: Springer |
| 出版年 | 1999 |
| 本文言語 | 英語 |
| 大きさ | XII, 716 p : online resource |
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| 内容注記 | 1. Many-valued logic and fuzzy set theory 2. Powerset operator foundations for poslat fuzzy set theories and topologies Introductory notes to Chapter 3 3. Axiomatic foundations of fixed-basis fuzzy topology 4. Categorical foundations of variable-basis fuzzy topology 5. Characterization of L-topologies by L-valued neighborhoods 6. Separation axioms: Extension of mappings and embedding of spaces 7. Separation axioms: Representation theorems, compactness, and compactifications 8. Uniform spaces 9. Extensions of uniform space notions 10. Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity 11. Fundamentals of generalized measure theory 12. On conditioning operators 13. Applications of decomposable measures 14. Fuzzy random variables revisited |
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| 一般注記 | 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 conditioni ng 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 |
| 著者標目 | Höhle, Ulrich editor Rodabaugh, Stephen Ernest editor SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Operations research LCSH:Decision making LCSH:Mathematical logic LCSH:Calculus of variations FREE:Mathematics FREE:Mathematical Logic and Foundations FREE:Calculus of Variations and Optimal Control; Optimization FREE:Operation Research/Decision Theory |
| 分 類 | DC23:511.3 |
| 巻冊次 | ISBN:9781461550792 
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| ISBN | 9781461550792 |
| URL | http://dx.doi.org/10.1007/978-1-4615-5079-2 |
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