Compatibility Of Continuous Fuzzy Negation And Two Partial Orders On Bounded Lattices

The concept of fuzzy set was proposed by Zadeh in 1965.Fuzzy logic,as a part of it,has achieved rapid development in both theory and applications fields.Fuzzy implications,fuzzy negations and triangular norms(conorms)are important operators in fuzzy logic,and their properties have attracted the attention of re?searchers.Yager has generalized triangular norms(conorms)to be uninorms has applied to many fields.In recent years,scholars have generalized the uninorms to be 2-uninorms,and proposed the partial order on bounded lattice induced by 2-uninorm.In this paper,we discuss the compatibility of continuous fuzzy negations in the unit interval and characterize the law of importation using the compatibility,and then induce two kinds of partial order on the bounded lattice by 2-uninorms and implications satisfying the law of importation.The structure of this thesis is organized as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and rele-vant conclusions about fuzzy implications,triangular norms,implications on bound-ed lattice and 2-uninorms on bounded lattice used in this paper are given.Chapter Two:The compatibility of continuous fuzzy negations.On the one hand,the concept of compatibility of fuzzy negations with respect to 2-uninorm is introduced,and the situations that continuous fuzzy negations satisfying compat-ibility with respect to two types of conjunctive 2-uninorms are characterized.On the other hand,we give some applications of the compatibility of continuous fuzzy negations.Firstly,we define the law of importation with respect to conjunctive 2-uninorms,and introduce related properties of it.By using the compatibility of continuous fuzzy negations,the equivalent characterization that law of importation with respect to conjunction 2-uninorms on fuzzy implications satisfying continuousα-natural negation is discussed.On this basis,we provide indirectly the charac-terization of the first class of generalized distributive equation.Chapter Three:Two kinds of partial orders on bounded lattices.Firstly,we define the concept of the partial order ≤U2 induced by the 2-uninorm and the partial order ≤I induced by the implications satisfying the law of importation,and describe respectively the partial orders induced by four types of 2-uninorms.Secondly,by the order-preserving bijection,we give the relationship between ≤Iφ and ≤I,and prove that(L,≤I)is a lattice if and only if(L,≤Iφ)is a lattice.Finally,we show by an example that the partial order ≤U2 and the partial order ≤I are irrelevant.