Mathematics of Fuzzy Sets
Logic, Topology, and Measure Theory
Published on 5. November 2012
Book
Paperback/Softback
XII, 716 pages
978-1-4613-7310-0 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1999
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
XII, 716 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 39 mm
Weight
1083 gr
ISBN-13
978-1-4613-7310-0 (9781461373100)
DOI
10.1007/978-1-4615-5079-2
Schweitzer Classification
Other editions
Additional editions
Book
12/1998
Kluwer Academic Publishers
€320.99
Shipment within 15-20 days
Persons
Patrik Eklund develops applications based on many-valued representation of information. Information typically resides in the form of expressions and terms as integrated in knowledge structures, so that term functors, extendable to monads, become important instrumentations in applications. Categorical term constructions with applications to Goguen's category have been recently achieved (cf. Fuzzy Sets and Syst. 298, 128-157 (2016)). Information representation supported by such monads, and as constructed over monoidal closed categories, inherits many-valuedness in suitable ways also in implementations.
Javier Gutie¿rrez Garci¿a has been interested in many-valued structures since the late 1990s. Over recent years these investigations have led him to a deeper understanding of the theory of quantales as the basis for a coherent development of many-valued structures (cf. Fuzzy Sets and Syst. 313 43-60 (2017)).
Since the late 1980s the research work of Ulrich Höhle has been motivated by a non-idempotent extension of topos theory. A result of these activities is a non-commutative and non-idempotent theory of quantale sets which can be expressed as enriched category theory in a specific quantaloid (cf. Fuzzy Sets and Syst. 166, 1-43 (2011), Theory Appl. Categ. 25(13), 342-367 (2011)). These investigations have also led to a deeper understanding of the theory of quantales. Based on a new concept of prime elements, a characterization of semi-unital and spatial quantales by six-valued topological spaces has been achieved (cf. Order 32(3), 329-346 (2015)). This result has non-trivial applications to the general theory of C*-algebras.
Since the beginning of the 1990s the research work of Jari Kortelainen has been directed towards preorders and topologies as mathematical bases of imprecise information representation. This approach leads to the use of category theory as a suitable metalanguage. Especially, in cooperation with Patrik Eklund, his studies focus on categorical term constructions over specific categories (cf. Fuzzy Sets and Syst. 256, 211-235 (2014)) leading to term constructions over cocomplete monoidal biclosed categories (cf. Fuzzy Sets and Syst. 298, 128-157 (2016)).
Content
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.