Research On Distributive Inequality And Modular Inequality Of Aggregation Function

In many practical problems,the fusion method based on aggregation functions is to combine multiple input information into one representative information.Aggregation functions are widely used in decision analysis,pattern recognition,fuzzy logic and expert systems,and play the important role in mathematics,computer science,statistics,economics and other disciplines.There are four main types of aggregation functions: conjunctive aggregation,disjunctive aggregation,averaging aggregation and mixed aggregation.Triangular norms and conorms,as the typical representatives of conjunctive and disjunctive aggregation functions respectively,play a significant role in fuzzy set theory.In addition,De Morgan triplet composed of triangular norms,triangular conorms and strong negation satisfying De Morgan’s law is also an important research content in fuzzy logic.The selection of aggregation functions in practical problems usually is translated into the study of corresponding function equations and inequalities.This thesis focuses on the generalized forms of distributive equation and modular equation of aggregate functions: distributive inequality and modular inequality.The main achievements of this thesis are as follows:Firstly,a summary is given of the relevant conclusions about the "almost distributive" De Morgan triplet and supplement proof of an important lemma.Next,this thesis considers the ordinal sum structure of triangular norms or triangular conorms in subdistributive De Morgan triplets.Three kinds of characterizations of semi De Morgan triplets that satisfy the subdistributive property are given: functional equation,derivative of one monotonic function,and strict triangular conorm.Secondly,research on submodular inequality of aggregation functions.The related properties of two aggregation functions satisfying submodular inequalities are analyzed in the sense of duality and isomorphism.In addition,in the cases where Archimedean triangular norm is submodular over Archimedean triangular conorm,we offer a characterization based on the convexity of the compound function of their additive generators.The submodular inequality between Archimedean triangular norms(or triangular conorms)are transformed into the new inequality of corresponding additiive generators.Moreover,it is also shown that triangular conorm is not submodular over triangular norm.Finally,the submodular inequality between the ordinal sum of conjunctors are considered.