A Study On Some Problems Of Fuzzy Implication

In Chapter 1. Definition and properties of t-norms, t-conorms and fuzzy implication op-erators are reviewed. variety and construction of t-norms and t-conorms are introduced. Inparticular, the relations of continuous t-norms, continuous t-conorms and related operators aregiven. Finally, several kinds of fuzzy implication operators and their basic properties are recalled.In Chapter 2.we prove that convergence of fuzzy implications is equivalent with the uniformconvergence of them, and then character any continuous implications by means of all strongnegations. Specially, we yet find that any S-implication is fully determined by its value on thediagonal line in domain of definition. Moreover, a new method to construct fuzzy implicationsis suggested.In Chapter 3. we present two methods which allow to construction the multiplicative andthe additive generator of a representable uninorm from its partial derivatives.First,the methodscan be used for conjunctive uninorm whose multiplicative generator has a non-zero derivativeat 0 and for disjunctive uninorm whose multiplicative generator has a non-zero derivative at1.Second,the methods can be used for representable uninorm whose additive generator has aderivative continuous at [0,1].In Chapter 4. we study the functional equation S(S1(x,y),T(x,y)) = S(x,y), where S,S1are two continuous t-conorms and T is a continuous t-norm. Some interesting methods forsolving this type of equations are introduced.