Opis: The Truth Value Algebra of Type-2 Fuzzy Sets - Elbert Walker, Carol Walker, John Harding

This mongraph is on the topic of type-2 fuzzy sets from a mathematical as opposed to an engineering perspective. It will be of interest to mathematicians who work in fields related to fuzzy sets as well as those who work with universal algebra. The book will make connections to logic, but will focus on truth value algebras which is fundamental to the subject. The underlying principles of this book will lead to connections with many other fields of mathematics like lattice theory, topological algebras, complex algebras and category theory. A major feature of this book will be the emphasis on applications as the subalgebra of normal convex functions will be covered. "This book, written by experts in algebra, is a comprehensive and original treatment of operations on fuzzy sets of type-2, i.e., fuzzy sets whose membership degrees are fuzzy sets of the unit interval. The theoretical material in the book is useful for practitioners of fuzzy modeling and inference in computer science, engineering, and social sciences. The book could also be used as a text for a special topic course in algebra." -Hung T. Nguyen, Emeritus Professor of Mathematical Sciences, New Mexico State University "At the beginning of fuzzy set theory, some easy results were published and this field was not fully respected by mathematicians. This opinion has changed thanks to deep mathematical results that cannot be ignored as 'cheap' or 'easy.' ... However, a deep algebraic approach to fuzzy sets was limited to special cases, in particular MV-algebras (Roberto Cignoli, Itala D'Ottaviano, and Daniele Mundici). Experienced algebraists have now contributed to filling in this gap. The authors of this book show how much general algebra can say about fuzzy truth values. On the other hand, fuzzy concepts contribute by interesting examples of algebras, automorphisms, varieties, categories, etc. The book is based on decades of investigation performed by leading experts in the field. They present the latest results in a form readable by all mathematicians, starting from upper undergraduate level. The presentation of ideas is surprisingly easy to follow, although it proceeds to advanced theoretical results. The organization of the material is very carefully elaborated. Type-2 fuzzy sets give rise to rather complex algebras, while the mainstream of research deals with the easier case of type-1 fuzzy sets; nevertheless, the book contributes also to this simpler discipline. Readers will be surprised how rich this topic is and how advanced knowledge has been collected recently. Open questions and suggestions for further investigation are presented and they inspire future research. The book will certainly influence the following decade of research of fuzzy logical operations ... ." -Professor Mirko Navara, Department of Cybernetics, Czech Technical University "The Truth Value Algebra of Type-2 Fuzzy Sets is an intensive study of the algebraic structure of Map(I; I). It is based on results of the authors and also other researchers in the field ... . This book presents a self-contained course for readers who have successfully completed undergraduate math." -Dr. Rudolf Seising, Professor for the History of Science, Friedrich-Schiller-Universitat Jena, and Affiliated Researcher, European Centre for Soft ComputingThe Algebra of Truth Values Preliminaries Classical and fuzzy subsets The truth value algebra of type-2 fuzzy sets Simplifying the operations Examples Properties of the Type-2 Truth Value Algebra Preliminaries Basic observations Properties of the operations Two partial orderings Fragments of distributivity Subalgebras of the Type-2 Truth Value Algebra Preliminaries Type-1 fuzzy sets Interval-valued fuzzy sets Normal functions Convex functions Convex normal functions Endmaximal functions The algebra of sets Functions with finite support Automorphisms Preliminaries Sets, fuzzy sets, and interval-valued fuzzy sets Automorphisms of M with the pointwise operations Characterizing certain elements of M Automorphisms of M with the convolution operations Characteristic subalgebras of M T-Norms and T-Conorms Preliminaries Triangular norms and conorms T-norms and t-conorms on intervals Convolutions of t-norms and t-conorms Convolutions of continuous t-norms and t-conorms T-norms and t-conorms on L Convex Normal Functions Preliminaries Straightening the order Realizing L as an algebra of decreasing functions Completeness of L Convex normal upper semicontinuous functions Agreement convexly almost everywhere Metric and topological properties T-norms and t-conorms Varieties Related to M Preliminaries The variety V(M) Local finiteness A syntactic decision procedure The algebra E of sets in M Complex algebras of chains Varieties and complex algebras of chains The algebras 23 and 25 revisited Type-2 Fuzzy Sets and Bichains Preliminaries Birkhoff systems and bichains Varieties of Birkhoff systems Splitting bichains Toward an equational basis Categories of Fuzzy Relations Preliminaries Fuzzy relations Rule bases and fuzzy control Additional variables The type-2 setting Symmetric monoidal categories The Finite Case Preliminaries Finite type-2 algebras Subalgebras The partial orders determined by convolutions The double order Varieties related to mn The automorphism group of mn Convex normal functions The De Morgan algebras H(mn) Appendix: Properties of the Operations on M Bibliography Index Summary and Exercises appear at the end of each chapter.

Szczegóły: The Truth Value Algebra of Type-2 Fuzzy Sets - Elbert Walker, Carol Walker, John Harding

Tytuł: The Truth Value Algebra of Type-2 Fuzzy Sets
Autor: Elbert Walker, Carol Walker, John Harding
Wydawnictwo: Productivity Press Inc
ISBN: 9781498735278
Rok wydania: 2016
Ilość stron: 254
Oprawa: Twarda
Waga: 0.5 kg

Recenzje: The Truth Value Algebra of Type-2 Fuzzy Sets - Elbert Walker, Carol Walker, John Harding