The Truth Value Algebra of Type-2 Fuzzy Sets Order Convolutions of Functions on the Unit Interval

Type-2 fuzzy sets extend both ordinary and interval-valued fuzzy sets to allow distributions, rather than single values, as degrees of membership. Computations with these truth values are governed by the truth value algebra of type-2 fuzzy sets. The Truth Value Algebra of Type-2 Fuzzy Sets: Order Convolutions of Functions on the Unit Interval explores the fundamental properties of this algebra and the role of these properties in applications. Accessible to anyone with a standard undergraduate mathematics background, this self-contained book offers several options for a one- or two-semester course. It covers topics increasingly used in fuzzy set theory, such as lattice theory, analysis, category theory, and universal algebra. The book discusses the basics of the type-2 truth value algebra, its subalgebra of convex normal functions, and their applications. It also examines the truth value algebra from a more algebraic and axiomatic view.

The 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 operationsCharacterizing certain elements of M Automorphisms of M with the convolution operationsCharacteristic 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 convolutionsThe 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.

John Harding is a professor in the Department of Mathematical Sciences at New Mexico State University. He is the author/coauthor of roughly 70 papers, president of the International Quantum Structures Association, and member of the editorial board of Order and the advisory board of Mathematica Slovaca. His research focuses on order theory and its applications, particularly applications to topology and logic, the foundations of quantum mechanics, completions, and fuzzy sets. He earned his Ph.D. in mathematics from McMaster University. Carol Walker was a professor in the Department of Mathematical Sciences at New Mexico State University before retiring in 1996. She was department head for 14 years and associate dean of arts and sciences and director of the Arts and Sciences Research Center for three years. She is the author/coauthor of more than 35 papers as well as several textbooks and technical manuals. Her research focuses on algebra, including abelian group theory, applications of category theory to abelian groups and modules, and algebraic aspects of the mathematics of fuzzy sets. She earned her Ph.D. in mathematics from New Mexico State University. Elbert Walker was a professor in the Department of Mathematical Sciences at New Mexico State University before retiring in 1987. He then worked at the U.S. National Science Foundation for two years. He is the author/coauthor of about 95 research papers and several books. His research interests include abelian group theory, statistics, and the mathematics of fuzzy sets and fuzzy logic. He earned his Ph.D. in mathematics from the University of Kansas.