Cross-migrativity And Modularity Of Binary Operations In The Context Of The Study Of Uninorms

Aggregation is the process of combining several numerical values into a s-ingle representative value,and an aggregation function performs this operation.Since the 1980s.the rapid growth of the field of computer sciences has made aggregation functions become a hot research field.Considering the huge variety of potential aggregation functions,defining or choosing the right class of aggre-gation functions for a specific problem is a difficult task.In order to meet the different needs of different fields.experts have constructed a large number of different classes of aggregation functions.As a typical representative of aggre-gation functions,uninorms are widely used in several fields,such as information fusion,expert systems,fuzzy logic,decision making,image proessing and fuzzy neural networks.Semi-uninorms are generalizations of uninorms.and they also have quite important applications in the above fields.The study of aggregation functions focuses on the structure of the functions,the characterization and the properties in the context of specific applications.Among them,the study of the properties of aggregate functions is often translated into solving the correspond-ing functional equations.Examples of this research are the study of the properties of idemipotency,modularity.migrativitv,cross-migrativity and distributivity.The migrativity property ensures that variations of the value of a variable by a ratio do not vary the result in terms of the affected variable.Migrativity plays a key role in decision making and image processing.In decision making,the orderng of inputs may be relevant,even though the result,of modifying one or another evaluation by a given ratio should be the same.In image processing,the property is needed to darken and lighten a certain part of an image.Cross-migrativity is a form of promotion of migrativity.In a multi-step information aggregation process,cross-migrativity can ensure that even if the order of aggre-gation is different,it does not affect the global output.To some extent,it is a special case of associativity.The modularity property,like cross-migrativity,plays ail important role in the field of information fusion.In the process of information aggregation,partial information from different times,places or circumstances often needs to be merged into a global summary.In this information summary process,people want to be able to reduce the failure rate.The modules of these procedures,however,should conform as well as possible to rational human thought processes.Researchers hope to provide the decision maker with complcex decision supporting systems producing reasonable and acceptable results.After getting the solutions of the modularity equations for aggregation unctions,we can provide a referen value for the operator selection and failure rate reduction in some aggregation process.From a mathematical point of view.the modularity equation not only can be viewed as a restricted general associative equation,but can also be seen as a special case of the distributivity equation.The former has an important role in fuzzy set theory,the latter is highly useful in fuzzy logic.In this thesis,uninorms and their generalized form semi-uninroms,are the main research objects.We focus on the cross-inigrativity of uninorms and the mocdularity of semi-uninorms.The main results are summarized next:The first chapter is the introduction.We present the research background and current research state of migrativity.cross-migrativity and modularity of aggregation functions.In Chapter 2,we introduce the concepts of aggregation functions,triangu-lar norms,triangular conorms,semi-copulas,pseudo triangular norms,uninorms,semi-uninorms and semi-t-operators,especially the four common classes of uni-norms.In Chapter 3,we investigate the cross-migrativity of uninorms.Firstly,we limit the research to those uninorms that have the same neutral element,and give the sufficient,and necessarsy conditions for cross-migrativitv for all possible combi-nations of uninorms belonging to four common classes of uninorms.Secondly,we focus on the cross-migrativity for uninorms with different neutral elements.'The study shows that there is no cross-migrativity between representable uninorms and other classes of uninorms,so is the relation between conjunctive uninorms and disjunctive uninorms.Solving the cross-migrativity for other combinations of uninorms is equivalent to solving the cross-migrativity equation between local functions of these uninorms.In Chapter 4.we investigate the modularity property of t-seniinorms,t-serniconorms,t-norms and t-conorms.New solutions to the modularity equations of a continuous t-norm over a continuous t-seiminorm,a t-seminorm over a strict t-norm,a t-seninorm over a t-semiconorm and a t-semiconorm over a t-seminorm are characterized.In Chapter 5.we concentrate on the modularity property of semi-uninorms and semi-t-operators.Firstly,we discuss the modularity property for semi-uninorms belonging to Nmin and Nmax.The necessary and suffieient conditions are given for the case of commutative semi-uninorms.Moreover.we alao give neces-sary and suffieient conditions for the case of uninorms with continuousus underlying operators with respect.to semi-uninorms with continuous underlying operators.Secondly,Necessary and sufficeent conditions are given for a semi-t-operator over a semi-tminorm,a pseudo-uninorm over a semi-t-operator to satisfy the modu-larity equation.Finally.we discuss the modularity btetween semi-t-openrators.In this thesis,we discuss the cross-migrativity property of aggregation func-tions with different neutral elements.As we know,most of the previous work focus on aggregation functions having the same neutral element.This condition is not necessary in order to find out solutions of the cross-migrativity equations.In practical applications,the two functions in the migrative/cross-migrative func-tional equations usually have different neutral elements.So,our work explore the more general case of the cross-migrativity equations.Moreover,our study shows that the cross-migrativity of uninorms is completely determined by the cross-migrativity of their local regions.This means that the complexity of solving the cross-niigrativitv equations can be greatly reduced.This thesis also broadens the scope of applications of modularity equations.Our results obtained in Chapter 4 and 5 generalize results about modularity of corresponding aggregation functions in the literature.New methods are used to solve the cross-migrativity equation and the modularity equation.In the study of cross-migrativity,we use a ze-ro divisor as a classification condition.This classification method has not.been previously used.In order to solve the modularity equations.we use the accunmu-lation of infinite local regions to obtain the solutions in the whole region.The ideas mentioned above provide references for the study of functional equations and function characterizations.