The Truth Value Algebra of Type 2 Fuzzy Sets: Order Convolutions of Functions on the Unit Interval
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) Appendix: Properties of the Operations on M Bibliography Index Summary and Exercises appear at the end of each chapter.