Properties And Construction Of Some New Kinds Of Aggregation Operators

The process of combining or merging a collection of data into a representable value is called aggregation,and the mathematical model of this process is called aggregation operator.There are a lot of literatures on aggregation.Especially in recent decades,with the development of science and technology,information aggregation has become more and more important.According to statistics,aggre-gation operators are mainly used in the fields that need comprehensive judgment or fusion of expert opinions,such as decision-making,statistics,artificial intelli-gence,pattern recognition,image processing,data mining and fusion,multi cri-teria decision making,game theory and so on.In practical application,different aggregation operators should be selected according to different application back-ground.And the extensive practical application promotes the theoretical study of aggregation operators.Therefore,the study on the structures and properties of aggregation operators has become a hot topic in recent years.Among them,the study on the properties of aggregation operators boils down to solving cor-responding functional equations.This paper mainly discusses the distributivity and migrativity of some aggregation operators.The distributivity equations of aggregation operators play an important role in the theory of fuzzy set and fuzzy logic.In fact,the problem of distributivity is related to the so-called pseudo-analysis,where the structure of R as a vector space is replaced by the structure of semi-ring on any interval[a,b],denoting the corresponding operations as pseudo-addition and pseudo-multiplication.In this context,the known t-norms,t-conorms,uninorms and nullnorms,have been used to model the mentioned pseudo-operations.Therefore,this leads to a new research direction,that is,the distributivity between aggregation operators.Migrativity of aggregation operators is a very important property in the s-tudy of information aggregation model.In recent years,there has been a growing interest in studying the notion of α-migrativity and generalizations.The interest of this property comes from its applications,such as image processing,the nature of the image itself does not change when a part of the image shrunk in proportion;decision making,it has nothing to do with the sequence of information selection when repeated and partial information is aggregated into a whole conclusion.As is pointed out by Mesiar et al.,it is important to ensure in some applications that variations in the value of some functions caused by considering just a giv-en fraction of one of the input variables is independent of the actual choice of variable.At present,the researches on aggregation operators are mostly focused on u-nit interval.The total order of real number field does bring a lot of convenience to the research work,and the conclusions are also beautiful.However,the complex practical problems make the research on partial order more meaningful.There-fore,the research on aggregation operators on bounded lattice attracts more and more attention.This paper mainly obtains the following research results:(1)In Chapter 3,we study the distributivity of uni-nullnorms and the dis-tributivity for 2-uninorms over semi-t-operators.The distributivity equations between uni-nullnorms and continuous t-(co)norms are solved first,and the ob-tained solutions extend the ones of these equations between uninorms and con-tinuous t-(co)norms.Then,distributivity between uni-nullnorms and uninorms is characterized.It is found that when uni-nullnorms and uninorms satisfy the distributivity equations,the structures of uni-nullnorms are basically unchanged,but the structures of uninorms are not only idempotent but also different on the anti diagonal region.Finally,the distributivity equations between uni-nullnorms and nullnorms are discussed.According to whether the zero elements of uni-nullnorms and nullnorms are equal,these equations under different conditions are fully described.The results show that nullnorms have different structures un-der different conditions.Distributivity equations for each type of 2-uninorms over semi-t-operators are discussed respectively.The results are complete and extend the ones of distributivity between uninorms(nullnorms)and semi-t-operators.(2)In Chapter 4,we study the migrativity of Mayor’s aggregation opera-tors and the migrativity between 2-uninorms and nullnorms.The migrativity equations between Mayor’s aggrega.tion operators and semi-t-operators(semi-uninorms)are fully characterized.And because of the abstract structure of May-ors aggregation operators,the results of this part are different from the ones of migrativity between other aggregation operators and semi-t-operators(semi-uninorms).According to whether the zero elements of nullnorms and 2-uninorms are equal,migrativity between nullnorms and 2-uninorms is fully characterized.These solutions generalize the ones of migrativity between nullnorms and uni-norms.(3)In Chapter 5,we generalize the definition of uni-nullnorms on unit in-terval to bounded lattice.If there is a uni-nullnorm on any bounded lattice,then some properties of the uni-nullnorm on bounded lattices can be obtained.Based on these properties,a construction method of uni-nullnorms on bounded lattices is given.Then,it is shown that there is not necessarily idempotent uni-nullnorm on any bounded lattice,and two construction methods of idempotent uni-nullnorms on some special bounded lattices are provided.