【作者】 张廷海；

【导师】 覃锋；

【作者基本信息】 江西师范大学 ， 应用数学， 2023， 博士

【摘要】 将大量给定数据合并成具有代表性数值的过程常常由所谓的聚合函数(又称聚合算子)来完成.从数学和计算机科学到经济和社会科学的诸多领域都需要这样的处理过程,因此,聚合函数成为许多应用领域的一种重要工具.近几十年,各类不同应用环境下的聚合函数不断被提出并得到较快发展,比如,-模、-余模、一致模、一致零模以及9)-一致模等.其中,作为-模和-余模的一种典型概括形式的一致模被证明在决策、专家系统、模糊系统模拟、神经网络、优化与控制等许多领域有着广泛应用,并且各类特殊一致模类,以及以它们为基础算子的一致零模、2-一致模等一些一般性聚合算子也相继被提出和研究.另外,重叠和群组函数是两类因其在图像处理、分类和决策问题上的广泛应用而于2010年前后被提出的重要聚合算子,它们在理论和应用方面也得到迅速的发展.其中,聚合算子的分配性、模态性、迁移性等与函数方程求解有关的性质因其在测度论、信息论以及系统论等方面的应用而成为研究的热点.但大多相关问题都集中于某些特殊聚合算子之间的讨论,对于一般性聚合算子,比如一致零模以及2-一致模,特别是它们与非结合、不必含单位元的重叠和群组函数之间的相关问题的研究不多且尚有许多需要解决的问题.基于以上原因,本学位论文就几类一般性聚合算子与重叠和群组信息聚合函数关于分配性、模态性等性质进行了研究,从而为丰富和拓展更多聚合算子在相关领域的应用提供理论支撑.具体有以下几方面:(i)研究了具有连续阿基米德基础算子的一致零模与重叠函数(群组函数)的分配方程.给出了八类一致零模关于重叠函数(群组函数)满足分配方程的充要条件及其算子的结构刻画.此外,还研究了重叠函数(群组函数)关于合取(析取)一致零模的分配性,获得了幂等一致零模结构的另一种完全刻画.(ii)研究了一致零模和2-一致模与重叠和群组函数的模态性.首先,将附加一定条件下的一致模和零模关于重叠与群组函数模条件的有关结果推广到任意情形下的相应结论,并在此基础上继续刻画了一致模与零模的概括性聚合算子一致零模、以及一般性2-一致模关于重叠与群组函数满足模方程的充要条件和结构.(iii)研究了幂等一致模,以及幂等一致零模和幂等零一致模与重叠(群组)函数的分配性.首先,通过引入分别概括重叠和群组函数的弱重叠和弱群组函数,并通过对其弱系数与幂等一致模的相关函数在端点处的值进行比较刻画了任意幂等一致模与重叠和群组函数的分配性,从而极大地推广了已有的有关结论.另外,在此基础上研究了以幂等一致模为基础算子的两类聚合算子幂等一致零模和幂等零一致模与重叠(群组)函数满足分配方程的等价条件和结构.更多还原

【Abstract】 The process of combining a large number of given data into representative values is often accomplished by so-called aggregation functions(also called aggregation operators).Such a process is needed in many fields,from mathematics and computer science to economics and social science,so aggregation functions have become an important tool in many application fields.In recent decades,aggregation functions have been proposed and developed rapidly in various applications,such as t-norms,t-conorms,uninorms,uni-nullnorms and n-uninorms.As a typical generalization of t-norms and t-conorms,the uninorms are proved to be useful in decision making,expert system,fuzzy system simulation,neural network,optimization and control,and many other fields.Moreover,all kinds of special classes of uninorms,and some general aggregation operators whose underlying operators are uninorms,such as uni-nullnorms and 2-uninorms are proposed and studied one after another.In addition,overlap and grouping functions as two important aggregation operators are proposed around 2010 and have rapid development because of their wide application in image processing,classification and decision-making.Among them,the distributivity,modularity and migrativity of aggregation operators which are related to the solution of functional equations have become the research hot-spot due to their applications in measure theory,information theory and system theory.For general aggregation operators,such as uni-nullnorms、2-uninorms,especially the research between them and overlap and grouping functions which are unnecessarily associative and have neutral element is not much on the related problems,and there are still many problems to be solved.Based on the above reasons,we studies the distributivity and modularity between some general aggregation operators and overlap and grouping aggregation functions in this dissertation.It can provide theoretical support for enriching and expanding the application of aggregation operators in related fields.There are the following aspects:(i)In this thesis we study the distributive equations between uni-nullnorms with continuous Archimedean underlying operators and overlap and grouping functions.The sufficient and necessary conditions and structures of the distributivity of eight classes of uni-nullnorms over overlap functions are discussed,and the corresponding conditions and structures for grouping functions are also characterized.In addition,the distributive results of overlap and grouping functions over conjunction and disjunction uni-nullnorms are studied,and the characterizations of the complete structures of idempotent uni-nullnorms are also given.(ii)The modularity between uni-nullnorms or 2-uninorms and overlap and grouping functions are studied.First of all,the results about the modularity conditions of overlap and grouping functions for uninorms and nullnorms under some additional conditions are generalized to the corresponding conclusions in any case.On this basis,the necessary and sufficient conditions and structures of the modularity equations on uni-nullnorms and 2-uninorms as the generalized aggregation operators of uninorms and nullnorms over overlap and grouping functions are described.(iii)The distributivity between idempotent uninorms,idempotent uni-nullnorms and idempotent null-uninorms and overlap(grouping)functions are studied.First,by introducing the weakly overlap and weakly grouping functions which generalize the overlap and grouping functions respectively,and comparing the values of the associated functions of idempotent uninorms at the end points with the weak coefficients,the distributive properties of any idempotent uninorms over overlap and grouping functions are described,which greatly extend the relevant existing conclusions.In addition,the equivalent conditions and structures on satisfying the distributive equations of idempotent uni-nullnorms and idempotent null-uninorms as two kinds of aggregation operators based on idempotent uninorms over overlap(grouping)functions are studied.更多还原