The Truth Value Algebra of Type-2 Fuzzy Sets Order Convolutions of Functions on the Unit Interval Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series

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 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)