1st Edition

# 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 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 2^{3} and 2^{5} 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 m^{n }

The automorphism group of m^{n}

Convex normal functions

The De Morgan algebras H(m^{n})

Appendix: Properties of the Operations on M

Bibliography

Index

Summary and Exercises appear at the end of each chapter.

### Biography

**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.

"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 topics 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 Setsis 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-Universität Jena, and Affiliated Researcher, European Centre for Soft Computing