Generalized T-norms And Fuzzy Implication Operators

In the paper,we study the properties and construction of generalized t-norms ,and thesolutions of distributivity equations and functional equation I(x,n(x)) = n(x).Specifically speaking,two topics are discussed,one is generalized t-norms,the other is functional equation.In Chapter 1. Definition and properties of t-norms,t-conorms and fuzzy implication operators are reviewed. Variety and construction of t-norms and t-conorms are introduced. In particular,the relations of continuous t-norms,continuous t-conorms and related operators are given. Finally,several kinds of fuzzy implication operators and their basic properties are recalled.In Chapter 2. Firstly,the concepts of discrete t-subnorms are introduced,the general methods of structuring discrete t-subnorms and discrete t-norms are given. The construction of smoothly discrete t-subnorms are studied . Then,the concepts of nullnorms on lattices are introduced. The properties of nullnorms on lattices are discussed,and the general methods of structuring nullnorms on lattices are given. Nullnorms on product lattice,which is the special nullnorms on lattices,are studied. The properties of nullnorms on product lattices are discussed,and the general methods of structuring nullnorms on product lattice are given.In Chapter 3. Firstly,we investigate [r→(t∧s)]∧[(r→t)≡(r→s)],[r→(t∨s)]≡[(r→t)∨(r→s)],[(p∧q)→r]≡[(p→r)∨(q→r)],[(p∨q)→r]≡[(p→r)∧(q→r)] and give the solutions of the above equations,where I is any QL-implication or D-implication. Then,we investigate I(x,n{x))=n(x) and give the solutions of the above equations,where I is generated by a continuous t-norms,a continuous t-conorms and a strong negation operator.