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1999 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Mathematics of Fuzzy Sets

Logic, Topology, and Measure Theory

herausgegeben von: Ulrich Höhle, Stephen Ernest Rodabaugh

Verlag: Springer US

Buchreihe : The Handbooks of Fuzzy Sets Series

Enthalten in: Professional Book Archive

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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.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
The mathematics of fuzzy sets is the mathematics of lattice-valued maps. Lattice-valued maps play different roles and have different connotations in different areas of mathematics.
Ulrich Höhle, Stephen Ernest Rodabaugh
Chapter 1. Many-Valued Logic And Fuzzy Set Theory
Abstract
Rather early in the (short) history of fuzzy sets it became clear that there is an intimate relationship between fuzzy set theory and many-valued logic. In the early days of fuzzy sets the main connection was given by fuzzy logic — in the understanding of this notion in those days: and this was as switching logic within a multiple-valued setting.
S. Gottwald
Chapter 2. Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies
Abstract
This chapter summarizes those powerset operator foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and preimage of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure theory and analysis, and topology. We first outline such foundations for the fixed-basis case—where the lattice of membership values or basis is fixed for objects in a particular category, and then extend these foundations to the variable-basis case—where the basis is allowed to vary from object to object within a particular category. Such foundations underlie almost all chapters of this volume. Additional applications include justifications for the Zadeh Extension Principle [19] and characterizations of fuzzy associative memories in the sense of Kosko [9]. Full proofs of all results, along with additional material, are found in Rodabaugh [16]—no proofs are repeated even when a result below extends its counterpart of [16]; some results are also found in Manes [11] and Rodabaugh [14, 15].
S. E. Rodabaugh
Introductory Notes To Chapter 3
Abstract
The purpose of this section is to understand the conception of space 1 and the role of lattice-valued maps in topology. In particular, we answer the question to what extent topology is based on classical set theory.
U. Höhle
Chapter 3. Axiomatic Foundations Of Fixed-Basis Fuzzy Topology
Abstract
This paper gives the first comprehensive account on various systems of axioms of fixed-basis, L-fuzzy topological spaces and their corresponding convergence theory. In general we do not pursue the historical development, but it is our primary aim to present the state of the art of this field. We focus on the following problems:
U. Höhle, A. P. Šostak
Chapter 4. Categorical Foundations of Variable-Basis Fuzzy Topology
Abstract
This chapter lays categorical foundations for topology and fuzzy topology in which the basis of a space—the lattice of membership values—is allowed to change from one object to another within the same category (the basis of a space being distinguished from the basis of the topology of a space). It is the goal of this chapter to create foundations which answer all the following questions in the affirmative:
S. E. Rodabaugh
Chapter 5. Characterization Of L-Topologies By L-Valued Neighborhoods
Abstract
It is well known that L-topologies can be characterized by L-neighborhood systems (cf. Subsection 6.1 in [14]). The aim of this paper is to give a characterization of a subclass of L-topologies by crisp systems of L-valued neighborhoods. This subclass consists of stratified and transitive L-topologies and covers simultaneously probabilistic L-topologies and [0, l]-topologies determined by fuzzy neighborhood spaces. We present this characterization depending on the structure of the underlying lattice L. In the case of probabilistic L-topologies it is remarkable to see that the structure of complete MV-algebras (cf. [2, 12]) is sufficient, while in all other case the complete distributivity of the underlying lattice L seems to be essential. Further, if L is given by the real unit interval [0, 1], then the Booleanization of [0, l]-topologies corresponding to fuzzy neighborhood spaces exists. Hence fuzzy neighborhood spaces can be chararcterized by two different types of many valued neighborhoods — namely by Booelan valued neighborhoods or by [0, l]-valued neighborhoods as the name “fuzzy neighborhood space” suggests (cf. Remark 3.17, Proposition 5.1). Moreover, [0, l]-fuzzifying topologies and [0, l]-topologies of fuzzy neighborhood spaces are equivalent concepts. Finally, we underline the interesting fact that a special class of stratified and transitive, [0, l]-topological spaces is induced by Menger spaces which form an important subclass of probabilistic metric spaces (cf. Example 5.6).
U. Höhle
Chapter 6. Separation Axioms: Extension of Mappings And Embedding of Spaces
Abstract
The present chapter is intended to give a self-contained development of some of the topics of fuzzy topology (54A40) that involve L-real-valued continuous functions on L-topological spaces. The four major themes include separation of L-sets by continuous L-real functions, insertion of continuous L-real functions between two comparable L-real functions, extension of continuous L-real functions from a subspace to the entire space and embedding L-topological spaces into products. (In actual fact, it is the embedding of L-Tychonoff spaces into L-cubes which falls into that category of results.)
T. Kubiak
Chapter 7. Separation Axioms: Representation Theorems, Compactness, and Compactifications
Abstract
In the historical development of general topology, the searches for appropriate compactness axioms and appropriate separation axioms are closely intertwined with each other. That such intertwining is important is proven by both the Alexandrov and Stone-Čech compactifications; that such intertwining is to be expected follows from the duality between compactness and separation—the former restricts, and the latter increases, the number of open sets; and that such intertwining is categorically necessary is proven by the categorical nature of the Stone-Čech compactification and its relationship to the Stone representation theorems. Furthermore, these compactifications and many other well-known results justify the compact Hausdorff spaces of traditional mathematics.
S. E. Rodabaugh
Chapter 8. Uniform Spaces
Abstract
Uniform spaces are the carriers of notions such as uniform convergence, uniform continuity, precompactness, etc.. In the case of metric spaces, these notions were easily defined. However, for general topological spaces such distance- or size-related concepts cannot be defined unless we have somewhat more structure than the topology itself provides. So uniform spaces lie between pseudometric spaces and topological spaces, in the sense that a pseudometric induces a uniformity and a uniformity induces a topology.
W. Kotzé
Chapter 9. Extensions Of Uniform Space Notions
Abstract
A large part of mathematics is based on the notion of a set and on binary logic. Statements are either true or false and an element either belongs to a set or not. In order to accommodate the idea of a sliding transition between the two states: true and false, and to generalise the concept of a subset of a given set, Zadeh introduced the notion of a fuzzy subset in a now-famous paper: [52]. For the record, let us recall that if X is a set and A is a subset of X then the characteristic function, denoted 1 A , is defined by Thus 1 A ∈ 2 X . In [52], an element μ ∈ I X , where I denotes the closed unit interval, was called a fuzzy set in X, with μ(x) being interpreted as the degree to which x belongs to the fuzzy set μ. Since the elements μ ∈ I X are generalisations of subsets of X, it is more accurate to refer to them as fuzzy subsets of X and we shall adopt this terminology here.
M. H. Burton, J. Gutiérrez García
Chapter 10. Fuzzy Real Lines And Dual Real Lines As Poslat Topological, Uniform, And Metric Ordered Semirings With Unity
Abstract
Nontrivial examples of objects and morphisms are fundamentally important to establishing the credibility of a new category or discipline such as lattice-dependent or fuzzy topology; and often the justifications of the importance of certain objects and the importance of certain morphisms are intertwined. In [33], we established classes of variable-basis morphisms between different fuzzy real lines and between different dual real lines, but left untouched the issue of the canonicity of these objects. In this chapter, we attempt to demonstrate the canonicity of these spaces stemming from the interplay between arithmetic operations and underlying topological structures. We shall summarize the definitions of fuzzy addition and fuzzy multiplication on the fuzzy real lines and indicate their joint-continuity—along with that of the addition and multiplication on the usual real line—with respect to the underlying poslat topologies, as well as the quasi-uniform and uniform continuity (in the case of fuzzy addition and addition) with respect to the underlying quasi-uniform, uniform, and metric structures. These results not only help establish fuzzy topology w.r.t. objects, but enrich our understanding of traditional arithmetic operations.
S. E. Rodabaugh
Chapter 11. Fundamentals of a Generalized Measure Theory
Abstract
In this chapter, we try to present a coherent survey on some recent attempts in building a theory of generalized measures. Our main goal is to emphasize a minimal set of axioms both for the measures and their domains, and still to be able to prove significant results. Therefore we start with fairly general structures and enrich them with additional properties only if necessary.
E. P. Klement, S. Weber
Chapter 12. On Conditioning Operators
Abstract
The construction of conditional events (so-called measure-free conditioning) has a long history and is one of the fundamental problems in non-deterministic system theory (cf. [6]). In particular, the iteration of measure-free conditioning is still an open problem. The present paper tries to make a contribution to this question. In particular, we give an axiomatic introduction of conditioning operators which act as binary operations on the universe of events. The corresponding axiom system of this type of operators focus special attention on the intuitive understanding that the event ‘α given β’ is somewhere in “between” ‘α and β’ and ‘β implies α’. A detailed motivation of these axioms can be found in Section 2 and Proposition 6.4.
U. Höhle, S. Weber
Chapter 13. Applications Of Decomposable Measures
Abstract
We shall give a brief overview of some applications of special non-additive measures-so-called decomposable measures including the corresponding integration theory, which form the basis for pseudo-analysis. We present applications to optimization problems, nonlinear partial differential equations, optimal control.
E. Pap
Chapter 14. Fuzzy Random Variables Revisited
Abstract
Fuzzy-valued functions, as an extension of set-valued functions, were considered in connection with a categorical theory of fuzzy sets, since the mid 1970’s (see Negoita and Ralescu [16]. Indeed, even earlier, fuzzy-valued functions with some special properties were studied: fuzzy relations (Zadeh [28]).
D. A. Ralescu
Backmatter
Metadaten
Titel
Mathematics of Fuzzy Sets
herausgegeben von
Ulrich Höhle
Stephen Ernest Rodabaugh
Copyright-Jahr
1999
Verlag
Springer US
Electronic ISBN
978-1-4615-5079-2
Print ISBN
978-1-4613-7310-0
DOI
https://doi.org/10.1007/978-1-4615-5079-2