Modular Equations Between A Class Of S-uniform Modules (T-uniform Modules) And Several Aggregation Operators

Aggregation operators are mathematical models that describe the aggregation of information.It can be used to aggregate expert opinions to make comprehensive judgments and decisions.Aggregation operators have a wide range of types,including triangular norms,triangular conorms,uninorms,nullnorms,t-operators,2-uninorms,n-uninorms,overlap functions and grouping functions,etc.Functional equations involving aggregation operators play an important role in fuzzy sets and fuzzy logic theory.The modularity equations involving aggregation operators are one important class of functional equations,which can be viewed as both one generalization of modular law in modular lattices and restricted associative equations and are also closely related to the distributivity equations.The modularity equations for different kinds of aggregation operators such as t-norms,t-conorms,uninorms,t-operators,2-uninorms,semi-uninorms,overlap functions and grouping functions have been extensively studied in the literature and many profound results have been obtained.This thesis focuses on the modularity equations between two types of aggregation operators and commonly-used aggregation operators such as t-norms,t-conorms,uninorms,t-operators.The structure of this thesis is organized as follows:Firstly,the modularity equations between a class of S-uninorms and t-norms,t-conorms,uninorms,t-operators,S-uninorms are studied.This chapter proves several cases do not satisfy the modularity equations,and gives structural characterizations of these aggregation operators satisfying modularity equations,including t-norms over S-uninorms,uninorms over S-uninorms,S-uninorms over S-uninorms,t-operators over S-uninorms.In addition,we give some negative examples of Suninorms that does't satisfying modularity equations.Secondly,the modularity equations between a class of T-uninorms and t-norms,t-conorms,uninorms,t-operators,T-uninorms,S-uninorms are studied.This chapter proves several cases do not satisfy the modularity equations,and gives structural characterizations of these aggregation operators satisfying modularity equations,including T-uninorms over t-conorms,T-uninorms over uninorms,T-uninorms over T-uninorms,T-uninorms over t-operators,T-uninorms over S-uninorms.Besides,we give some negative examples of T-uninorms that does't satisfying modularity equations.Finally,the structural characterizations between T-uninorms and S-uninorms satisfying the modularity equations partially generalize the structural characterizations of 2-uninorms.According to the way to characterize the structure of T-uninorms and S-uninorms satisfying the modularity equations,gives structural characterizations of the class of 2-uninorms with F(0,1)=k.