Research On The Cross-Migrativity Between Overlap(Grouping) Functions And Common Aggregation Functions

Aggregation functions have important applications in image processing,data analysis,classification problems and other fields.To meet different requirements in different domains,there exist different types of aggregation functions.Choosing the right and appropriate aggregation function is an important task.Therefore,it is necessary to understand the characteristics of these aggregation functions.As one of the important properties,the cross-migrativity is very interesting in theory and applications.Although many interesting results have been obtained about the cross-migrativity of various aggregation functions,the study of the cross-migrativity between different aggregation functions is far from complete.To fill this gap,the cross-migrativity between overlap（grouping）functions and common aggregation functions is studied.The main research of this paper are as follows:First of all,the main research background and status quo of common aggregation functions（including triangular norms,triangular conorms,uninorms,nullnorms,overlap functions and grouping functions）and their cross-migrativity properties are briefly reviewed.In this background,the motivation and originality of this research are analyzed,and the innovation points are pointed out.Secondly,the definitions of triangular norm,triangular conorm,uninorm,nullnorm,overlap function and grouping function,as well as some basic properties that will be used in this research are introduced.Thirdly,the cross-migrativity of triangular norms（triangular conorms）and overlap（grouping）functions is discussed.Whenαtakes the special values 0,1 and the neutral element e of uninorms,we get that（T,O）is 0-cross-migrative and（T,O）is 1-cross-migrative iff O=T.When triangular norm is continuous,we obtain a necessary condition forα-cross-migrativity by using the order sum of triangular norms.In any case,（S,O）does not satisfyα-cross-migrativity.The same is true of grouping functions.Fourthly,we discuss the cross-migrativity of uninorm and overlap（grouping）function.On the one hand,we show the equivalent conditions for the solution of theα-cross-migrativity equation whenαtakes some special values.Afterwards,when the uninorm U belongs to five general classes,(i.e.,U_min,U_max,U_ide,U_rep and U_cos),we confirm that except(U_min,O)isα-cross-migrative,whereα∈（0,e）and(U_cos,min,O)isα-cross-migrative,whereα∈(0,λ],the other cases are notα-cross-migrative.The same is true of grouping functions.Fifthly,when uninorm U∈COU,（U,O）isα-cross-migrative only when U is conjunction andαsatisfies certain conditions.In addition,we also obtain some equivalent characterizations when（U,O）satisfiesα-cross-migrativity.The same is true of grouping functions.Finally,theα-cross-migrativity of nullnorms and overlap（grouping）functions is discussed.It is concluded that nullnorms and overlap（grouping）functions can not possibly satisfiesα-cross-migrativity.