On The Structure And Migrativity Of Associative Aggregation Functions

It is well known that most problems in fuzzy set theory ultimately lead to the selection of appropriate fuzzy connectives.At the fundamental level,this provides a sound basis for fuzzy set theory.One part of my research con-cerns the structure of fuzzy connectives:structure of associative aggregation functions.The process of merging a given number of data into a represen-tative value is called aggregation and the function performing this process is called an aggregation function,which can be used in the field of economics and finance,pattern recognition and image processing,data fusion,and so on.There are many different classes of aggregation functions and their se-lections depend on the context where they are going to be applied.Besides,associative aggregation functions have the computational advantage of being readily extendable for any number of input arguments,which implies that any associative extended function is completely determined by its binary function.From a mathematical point of view,this investigation can build bridges between many-valued logics and the theory of functional equations.More-over,appropriate fuzzy connectives usually need to satisfy certain additional properties.The study of these properties typically results in solving func-tional equations.Another part of my research aims to solve such functional equations,in particular equations related to migrativity.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 and hence can be used in image processing and decision-making.In image processing,the property is needed to darken and lighten a certain part of an image.In decision-making,the ordering of inputs maybe relevant,even though the result of modifying one or another evaluation by a given ratio should be the same.The main results are as follows:Chapter 2 studies the structure of 2-uninorms.Both the lower and up-per underlying functions of 2-uninorms are uninorms.We divide the class of 2-uninorms into five mutually exclusive subclasses based on their bound-ary behaviour,as determined by the absorbing element and the conjunc-tive/disjunctive nature of underlying uninorms,which every 2-uninorm is known to have.For each of these subclasses,we fully characterize the struc-ture of its members,and for two subclasses,we fully characterize the structure of them under an additional continuity assumption.Chapter 3 focuses on the construction of uninorms(including t-norms and t-conorms)by paving.Inspired by paving construction method,some new operations defined on unit interval,including t-norms,t-conorms and uninorms,are constructed from a given t-norm defined on the unit interval and a discrete one defined on an index-set:a discrete t-norm,a discrete t-superconorm,a discrete t-conorm or a discrete uninorm.Dually,we also define some uninorms(including t-norms and t-conorms)from a t-conorm and a discrete t-norm,a discrete t-subnorm,a discrete t-conorm or a discrete uninorm.Chapter 4 studies the migrativity of 2-uninorms and t-norms,the mi-grativity of 2-uninorms and t-conorms,the migrativity of semi-t-operators.On the one hand,Based on the result of 2-uninorms in Chapter 2,We divide the class of 2-uninorms into five mutually exclusive subclasses to discuss the migrativity of 2-uninorms over t-norms,the migrativity of t-norms over 2-uninorms,the migrativity of 2-uninorms over t-conorms,the migrativity of t-conorms over 2-uninorms.On the other hand,the migrative property for semi-t-operators is studied.All solutions of the migrativity equation for all possible combinations of semi-t-operators are analyzed and characterized.