Research On The Properties Of Several Classes Of Aggregation Operators Such As Overlap,Grouping Functions

The process of combining a large number of given data into representative values is often accomplished by so-called aggregation functions(also called aggregation operators).Such a process is needed in many fields,from mathematics and computer science to economics and social science,so aggregation functions have become an important tool in many application fields.In recent decades,aggregation functions have been proposed and developed rapidly in various applications,such as t-norms,t-conorms,uninorms,uni-nullnorms and n-uninorms.As a typical generalization of t-norms and t-conorms,the uninorms are proved to be useful in decision making,expert system,fuzzy system simulation,neural network,optimization and control,and many other fields.Moreover,all kinds of special classes of uninorms,and some general aggregation operators whose underlying operators are uninorms,such as uni-nullnorms and 2-uninorms are proposed and studied one after another.In addition,overlap and grouping functions as two important aggregation operators are proposed around 2010 and have rapid development because of their wide application in image processing,classification and decision-making.Among them,the distributivity,modularity and migrativity of aggregation operators which are related to the solution of functional equations have become the research hot-spot due to their applications in measure theory,information theory and system theory.For general aggregation operators,such as uni-nullnorms、2-uninorms,especially the research between them and overlap and grouping functions which are unnecessarily associative and have neutral element is not much on the related problems,and there are still many problems to be solved.Based on the above reasons,we studies the distributivity and modularity between some general aggregation operators and overlap and grouping aggregation functions in this dissertation.It can provide theoretical support for enriching and expanding the application of aggregation operators in related fields.There are the following aspects:(i)In this thesis we study the distributive equations between uni-nullnorms with continuous Archimedean underlying operators and overlap and grouping functions.The sufficient and necessary conditions and structures of the distributivity of eight classes of uni-nullnorms over overlap functions are discussed,and the corresponding conditions and structures for grouping functions are also characterized.In addition,the distributive results of overlap and grouping functions over conjunction and disjunction uni-nullnorms are studied,and the characterizations of the complete structures of idempotent uni-nullnorms are also given.(ii)The modularity between uni-nullnorms or 2-uninorms and overlap and grouping functions are studied.First of all,the results about the modularity conditions of overlap and grouping functions for uninorms and nullnorms under some additional conditions are generalized to the corresponding conclusions in any case.On this basis,the necessary and sufficient conditions and structures of the modularity equations on uni-nullnorms and 2-uninorms as the generalized aggregation operators of uninorms and nullnorms over overlap and grouping functions are described.(iii)The distributivity between idempotent uninorms,idempotent uni-nullnorms and idempotent null-uninorms and overlap(grouping)functions are studied.First,by introducing the weakly overlap and weakly grouping functions which generalize the overlap and grouping functions respectively,and comparing the values of the associated functions of idempotent uninorms at the end points with the weak coefficients,the distributive properties of any idempotent uninorms over overlap and grouping functions are described,which greatly extend the relevant existing conclusions.In addition,the equivalent conditions and structures on satisfying the distributive equations of idempotent uni-nullnorms and idempotent null-uninorms as two kinds of aggregation operators based on idempotent uninorms over overlap(grouping)functions are studied.