Mathematics of Fuzzy Sets

by Hohle, Ulrich; Rodabaugh, Stephen Ernest

• ISBN13: 9780792383888
• ISBN10: 0792383885
• Format: Hardcover
• Copyright: 1998-12-01
• Publisher: Kluwer Academic Pub

Summary

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 HamiltoneJacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

Table of Contents

Authors and Editors	vii
Foreword	ix
Introduction	1	(4)
Many-valued logic and fuzzy set theory	5	(86)
S. Gottwald
Powerset operator foundations for poslat fuzzy set theories and topologies	91	(26)
S.E. Rodabaugh
Introductory notes to Chapter 3	117	(6)
U. Hohle
Axiomatic foundations of fixed-basis fuzzy topology	123	(150)
U. Hohle
A.P. Sostak
Categorical foundations of variable-basis fuzzy topology	273	(116)
S.E. Rodabaugh
Characterization of L-topologies by L-valued neighborhoods	389	(44)
U. Hohle
Separation axioms: Extension of mappings and embedding of spaces	433	(48)
T. Kubiak
Separation axioms: Representation theorems, compactness, and compactifications	481	(72)
S.E. Rodabaugh
Uniform spaces	553	(28)
W. Kotze
Extensions of uniform space notions	581	(26)
M.H. Burton
J. Gutierrez Garcia
Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity	607	(26)
S.E. Rodabaugh
Fundamentals of generalized measure theory	633	(20)
E.P. Klement
S. Weber
On conditioning operators	653	(22)
U. Hohle
S. Weber
Applications of decomposable measures	675	(26)
E. Pap
Fuzzy random variable revisited	701	(10)
D. A. Ralescu
Index	711