Research On Modular Equations Between Several Types Of Aggregate Functions

The functional equations involving aggregation functions have always been one of the core topics in the field of information aggregation.Among them,the modularity equation has been a hot research topic for scholars at home and abroad in recent years.This thesis is dedicated to studying the modularity equation of several types of aggregation functions and seeking the analytic structure of the solutions to satisfy the modularity equation.The main content of this thesis is as follows:Chapter One:Preliminaries.This chapter briefly recalls some preliminaries that are necessary for this thesis,including basic notions and related algebraic properties of t-norms,t-conorms,uninorms,nullnorms,T-uninorms,S-uninorms,bi-uninorms,2-uninorms,overlap and grouping functions,as well as some fundamental conclusions concerning modularity equations involving binary aggregation functions.Chapter Two:The modularity equation between a class of aggregation functions with annihilator and well-studied types of aggregation functions.Based on the three subclasses of associative k-CAOA,this chapter is divided into three parts.In the first part,the modularity equation between T-uninorms and well-studied types of aggregation functions including t-norms,t-conorms,proper uninorms,T-uninorms,proper nullnorms,S-uninorms,bi-uninorms is systematically discussed.It is proved that the modularity equation between T-uninorms and partial aggregation functions has no solutions,and in the case of existence of solutions,the necessary and sufficient conditions from them to satisfy the modularity equation are given,and then the structural characterizations of the solutions are obtained.In the second part,we discuss the modularity equation between S-uninorms and well-studied types of aggregation functions dually.In the third part,we systematically study the modularity equation between bi-uninorms and well-studied types of aggregation functions,and prove that the modularity equations between t-norms,t-conorms,uninorms,nullnorms and bi-uninorms are unsolvable.At the same time,we give the corresponding structural characterizations of a special class of bi-uninorms satisfying the modularity equation.Since we study the modularity equation between these types of aggregation functions with annihilator and well-studied types of aggregation functions in a more general case,the results of this part are a further extension and improvement of the previous research results.Chapter Three:The modularity equation between 2-uninorms and overlap(grouping)functions.Based on the five subcategories and corresponding structural characteristics of 2-uninorms,this chapter systematically discusses the modularity equation between 2-uninorms and overlap functions and the modularity equation between 2-uninorms and grouping functions.It is proved that the modularity equation between certain some pairs of 2-uninorms and overlap(grouping)functions has no solutions,and in the case of existence of solutions,the corresponding structural characterizations of the 2-uninorms and overlap(grouping)functions satisfying the modularity equation are given.Theoretically,the content of this part is a comprehensive study of the modularity equation between 2-uninorms and overlap(grouping)functions,and is also a theoretical generalization of the modularity equation between uninorms and overlap(grouping)functions and the modularity equation between nullnorms and overlap(grouping)functions.In application,the research results of this part provide an important theoretical basis and a new mathematical model for practical problems such as information aggregation and decision analysis.