Analytical Characterization Of (N, U)-Fuzzy Implications For Consistent Modulo Satisfaction Of The Input Law With Continuous Basis Operators

The law of importation plays an important role at both sides of theory and application in fuzzy logic.As we all know,the law of importation between fuzzy implications and some conjunctions,such as t-norms and several common types of uninorms,has been widely concerned.Although a lot of research has been carried out and many results have been achieved,the problem of solving the equation about the law of importation has not been completely solved.Therefore,this thesis will study the characterization of fuzzy implication solutions with a continuous natural negation to the law of importation with respect to a fixed uninorm having continuous underlying operators.According to the types of underlying operators T and S of a fixed uninorm,we will divide them into six cases to discuss the fuzzy implication solutions separately.The structure of this thesis is organized as follows:Chapter One:Preliminaries.This chapter recalls some basic definitions and results about fuzzy implication,α-natural negation,uninorm,t-norm and t-conorm which will be used in the rest of this thesis.At the same time,it recalls the definitions of the law of importation and U-compatibility and the equivalence characterization theorem between them.Chapter Two:The law of importation between fuzzy implications and four types of uninorms with either idempotent or continuous Archimedean underlying operators are studied.It is divided into the following situations:1.T and S are both idempotent;2.T is Archimedean and S is idempotent;3.T is idempotent and S is Archimedean;4.T and S are both Archimedean.In this chapter,by studying the U-compatibility between these types of uninorms and continuous fuzzy negations,we obtain the fuzzy implication solutions with a continuous α-natural negation that satisfy the law of importation with respect to these types of uninorms.Chapter Three:On the foundation of Chapter Two,we study the law of importation between fuzzy implications and two types of uninorms of which at least one of the underlying operators are given by ordinal sum.It is divided into the following situations:1.T and S are both given by proper ordinal sums;2.One of T and S is given by proper ordinal sum,and the other is idempotent or continuous Archimedean.This chapter discusses the necessary and sufficient conditions for the U-compatibility between these two types of uninorms and continuous fuzzy negations.Then,according to the characterization theorem of the law of importation and U-compatibility,we obtain the fuzzy implication solutions that satisfy the law of importation.