Ulrich Hohle;S E Hc6hle UlrichRodabaugh
Release Date: 31 December 1998
Format: Hardback
Pages: 716
Category:
Set TheoryISBN: 9780792383888
ISBN10: 0792383885
Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide muchneeded 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 12), general topology (Chapters 310), and measure and probability theory (Chapters 1114). Chapter 1 deals with nonclassical logics and their syntactic and semantic foundations. Chapter 2 details the latticetheoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using latticevalued mappings as a fundamental tool. Chapter 3 focuses on the fixedbasis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variablebasis case, providing a categorical unification of locales, fixedbasis topological spaces, and variablebasis compactifications. Chapter 5 relates latticevalued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in latticevalued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of StoneCechcompactification and Stonerepresentation 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 orcomplete 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 measurefree conditioning. Chapter 13 presents elements of pseudoanalysis 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.
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