Research On Some Problems About Uninorms With Continuous Underlying Operators

As the mathematical model corresponding to information fusion, Ag-gregation operators perform the combination of several inputs into a single output. In the practical applications, the construction and choice of aggrega-tion operators is an important and difficult issue. The choice of appropriate aggregation operator has a crucial impact to information fusion. As an im-portant one in the family of aggregation operators, uninorms have many good mathematical properties and have been successfully applied in different fields. But in the references many results about the structure, characteriza-tion, the related functional equations and so on, are often based on the usual classes of uninorms.This dissertation focuses on the research of uninorms with continuous underlying operators. This dissertation studies the characterization, the dis-tributive and conditionally distributive equations over the continuous tri-angular conorms. This dissertation includes three parts. In the first part, we characterize this class of uninorms and offer two construction methods of uninorms. In the second part, we research the distributivity and condi-tional distributivity of uninorms with continuous underlying operators over the continuous triangular conorms. In the third part, the characterization of discrete uninorms having smooth underlying operators on finite chains is discussed.The main results are as follows.The background and the main innovation points of this dissertation are offered in the section of introduction.Chapter 1 is devoted to the definitions, technical terms and the related main results used in this dissertation.Chapter 2 studies the characterization of the class of uninorms with continuous underlying operators. The full characterization of uninorms with continuous and Archimedean underlying operators is offered in this chapter. Our results can be viewed as complement to those proved by Fodor and De Baets. Moreover, the full characterization of uninorms with the idempotent underlying triangular norms or conorms is presented in this chapter. On the other hand, based on the above results, two construction methods of uni-norms are obtained and the corresponding sufficient and necessary conditions are offered here. These results are helpful to the construction and the choice of aggregation operators in practical applications.Chapter 3 studies the distributivity and conditional distributivity of uni-norms over continuous triangular conorms. One of the open problems pointed out in the Linz2000 recalled by Klement is the so called distributivity and conditional distributivity of a uninorm and a continuous triangular conor-m. This problem is related to the so-called pseudo-analysis and construc-tion of aggregation operators based on integrals. Ruiz and Torrens solved distributivity and conditional distributivity of a uninorm over a continuous triangular conorm for the usual four classes of uninorms and proved that distributivity and conditional distributivity are equivalent for these cases. This chapter breaks through the limit of the usual four classes of uninorms and is precisely devoted to distributivity and conditional distributivity of a uninorm over a continuous triangular conorm for the class of uninorms with continuous underlying operators. Partial results are obtained. If the uni-norm is conditionally distributive over a continuous triangular conorm, then the triangular conorm is maximum or has only one ordinal summand; if the uninorm is conditionally distributive over a strict triangular conorm, then the uninorm belongs to the class of representable uninorms; there exists no uninorm that is conditionally distributive over a nilpotent triangular conorm; some properties of uninorms is offered for the case of continuous triangular conorm having one ordinal summand. From these obtained results, it is d-educed that distributivity and conditional distributivity are equivalent for this case. The results in this chapter can be viewed as complement to those obtained by Ruiz and Torrens, and bring us a step closer to the complete solutions of the open problem recalled by Klement.Chapter 4 focuses on the discrete uninorms on finite chain. It is nec-essary to deal with the uninorm defined on finite chain in many real appli-cations, for example, fuzzy control. There are some papers which deal with the discrete counterparts of the usual classes of uninorms on finite chains. This chapter is precisely devoted to the discrete counterparts of the class of uninorms with continuous underlying operators, i.e., the class of discrete uninorms with smooth underlying operators. Some mathematical properties are discussed, the discrete uninorm take the minimum or maximum in the area of A(e). Based on three unary functions, a full characterization of all discrete uninorms having smooth underlying operators is obtained. These re-sults in this chapter supply the theoretical foundation for the real application of uninoms.