模糊粒度空间的性质研究

在文献[16]基础上,进一步将模糊粒度空间推广到更一般地模糊等价关系上,研究了模糊粒度空间的性质,主要获得了3个结论.首先,引入了有序的等价关系集的概念,给出了下列的四个命题是等价的:(1) 给定一个模糊等价关系;(2) 给定一个等腰归一化伪距离;(3) 给定一个有序的粒度空间;(4) 给定一个有序的等价关系集.第二,通过模糊等价关系诱导的等腰归一化伪距离的投影距离和扩展距离,建立了模糊粒度空间上的距离,即是等腰归一化距离,并且给出了模糊粒度空间上距离度量的动态性质研究.最后,给出了模糊粒度空间与模糊等价关系之间的序关系,即它们的序是一致的.这些研究工作进一步完善了模糊粒度空间的理论,为模糊粒度计算提供了更为直观的数学理论和工具.     

定向空间的way below基

文章将连续定向空间的<<-基推广到了交连续定向空间和拟连续定向空间.并给出了定向拓扑的基与(拟)<<-基的关系,以及连续定向空间<<-基与其拟<<-基、交<<-基的关系.     

强极大TAF代数的交不可约D-模

本文研究了强极大TAF代数的交不可约D-模的有关性质.利用矩阵单元链的方法,证明了全上三角矩阵代数中任一D-模等于包含它的所有交不可约D-模的交,建立了强极大TAF代数的交不可约D-模与矩阵单元链之间的对应关系,推广了强极大TAF代数中关于理想的若干性质.     

拟半连续格和交半连续格

作为半连续格的推广, 引入了拟半连续格的概念. 讨论了拟半连续格的基本性质. 在拟半连续格上得到了类似于拟连续偏序集的一些主要结果. 同时探讨了半连续格、拟半连续格、交半连续格、交连续格、强连续格几种不同结构之间的关系. 最后,讨论了半连续函数空间仍是半连续格的条件.     

实赋范数性空间内的L~P-正交

F.Sullivan提出了L~P-正交的概念,刘证研究了Banach空间中L~P-正交的性质与一致凸空间的关系.本文证明了:设P>1,E是实赋范线性空间,若其内定义的L~p-正交满足:i)齐性;ii)可力性;iii)x(?)_L~py推出X(?)_jy,x,y∈E;iv)x(?)_Jy推出x(?)_L~py,x,y∈E,中任何一项,则E是一个抽象欧氏空间,而且有P=2.另外,本文还为R.C.James的一个未经证实的结论补充了证明.     

交连续dcpo的遗传性和不变性

讨论了交连续dcpo的遗传性和不变性,证明了如下结论:(1)交连续dcpo对于开子空间和闭子空间都是可遗传的;(2)交连续dcpo在加最大元和去最小元运算下保持交连续性;(3)交连续dcpo的收缩核为交连续dcpo.另外,给出了交连续的主理想刻画的一个直接证明;构造了反例说明交连续dcpo对于主滤子是不可遗传的;也构造了反例说明所有主滤子都交连续的一个dcpo,自身不必是交连续的.     

商空间拓扑结构合成

研究了商空间拓扑结构合成方法。首先对商空间理论中的多粒度世界概念进行拓展,证明了基于拓扑结构的多粒度世界按照拓扑粗细关系构成一个完备半序格,并将它称为拓扑粒度格。其次在拓扑粒度格上得出了拓扑向下合成和拓扑向上合成的形式化定义和数学描述,并证明了其数学描述的正确性。最后讨论了这两种合成方法的关系,以实际应用举例得到了求解问题的两个满足一定条件的解集,并讨论了拓扑结构合成方法的意义。     

Banach空间上可约化算子的谱分解

本文从谱分解的角度讨论了Banach空间上可约化算子,谱算子和可分解算子间的关系,并证明了以下主要结果: 1.设T∈B(X)是完全谱可约化的可分解算子,则对每个F∈B,成立着 2.设T∈B(X),则T是谱算子当且仅当T是具有性质(B)的完全谱可约化的可分解算子。     

加权Bergman空间上的约化子空间

主要证明在加权Bergman空间上符号为有限Blaskchke乘积的解析Toeplitz算子总是可约的,即至少有一个约化子空间.并且把该算子限制在该子空间上酉等价于加权Bergman位移.     

关于交半格同态构成的函数空间与FS-交连续Domain

定义了一类序结构—FS-交连续domain,讨论其相关性质并证明:(1)FS-交连续domain关于由Scott连续且保持非空有限交运算的函数构成的函数空间封闭,以(代数)FS-交连续domain为对象、以Scott连续函数为态射的范畴是Cartesian闭范畴;(2)任意分配可乘的有界完备domain是FS-交连续domain,从而紧连续dcpo的Smyth幂domain是FS-交连续domain.这些结果表明,FS-交连续domain是关于保非空有限交的连续映射构成的函数空间封闭的最恰当序结构.     

基于泛系粗糙集等价算子上的粒计算模型

标准粗糙集使用等价类作为粒来描述概念.本文弱化对等价关系的要求, 将更广泛的粒计算模型建立到泛系粗糙集上去.本文通过对全域的分割和覆盖来诱导出泛系粗糙集上的粒计算模型.     

Topology vs generalized rough sets

This paper investigates the relationship between topology and generalized rough sets induced by binary relations. Some known results regarding the relation based rough sets are reviewed, and some new results are given. Particularly, the relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial relations, reflexive relations and pre-order relations, respectively. We point that inverse serial relations are weakest relations which can induce topological spaces, and that different relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to reflexive relation based rough set model given recently are different, and gave a condition under which the both are the same topology.     

Optimal scale selection for multi-scale decision tables

Human beings often observe objects or deal with data hierarchically structured at different levels of granulations. In this paper, we study optimal scale selection in multi-scale decision tables from the perspective of granular computation. A multi-scale information table is an attribute-value system in which each object under each attribute is represented by different scales at different levels of granulations having a granular information transformation from a finer to a coarser labelled value. The concept of multi-scale information tables in the context of rough sets is introduced. Lower and upper approximations with reference to different levels of granulations in multi-scale information tables are defined and their properties are examined. Optimal scale selection with various requirements in multi-scale decision tables with the standard rough set model and a dual probabilistic rough set model are discussed respectively. Relationships among different notions of optimal scales in multi-scale decision tables are further analyzed.     

覆盖空间及粗糙集与拓扑的统一

引入覆盖空间,定义了其邻域、内部、闭包、测度等概念,研究了它们的性质.得出了粗糙集近似空间和拓扑空间都是具体覆盖空间的重要结论,从而用覆盖空间统一了粗糙集和拓扑.利用覆盖空间,得到了粗糙集和拓扑中更深刻的性质,从算子论和集合论的角度丰富和深化了粗糙集与拓扑的内容.     

基于覆盖上近似算子的拟阵结构及其关系

Covering-based rough sets,as a technique of granular computing,can be a useful tool for dealing with inexact,uncertain or vague knowledge in information systems.Matroids generalize linear independence in vector spaces,graph theory and provide well established platforms for greedy algorithm design.In this paper,we construct three types of matroidal structures of covering-based rough sets.Moreover,through these three types of matroids,we study the relationships among these matroids induced by six types of covering-based upper approximation operators.First,we construct three families of sets by indiscernible neighborhoods,neighborhoods and close friends,respectively.Moreover,we prove that they satisfy independent set axioms of matroids.In this way,three types of matroidal structures of covering-based rough sets are constructed.Secondly,we study some characteristics of the three types of matroid,such as dependent sets,circuits,rank function and closure.Finally,by comparing independent sets,we study relationships among these matroids induced by six types of covering-based upper approximation operators.     

Topological approximations of multisets

Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concept of rough set theory are information systems and approximation operators of approximation spaces. In this paper, we define and investigate three types of lower and upper multiset approximations of any multiset. These types based on the multiset base of multiset topology induced by a multiset relation. Moreover, the relationships between generalized rough msets and mset topologies are given. In addition, an illustrative example is given to illustrate the relationships between different types of generalized definitions of rough multiset approximations.     

模糊粗糙线性空间

将粗糙集理论引入到线性空间与模糊线性空间中,分别给出了上粗与下粗线性空间及模糊上粗与模糊下粗线性空间的概念,并研究了它们的有关性质,获得了一系列有意义的结果。     

Definability and textures

This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369.     

Matroidal approaches to rough sets via closure operators

This paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems.     

Fuzzy Closure Spaces vs. Fuzzy Rough Sets

This paper investigates the relationship among fuzzy rough sets, fuzzy closure spaces and fuzzy topology. It is shown that there exists a bijective correspondence between the set of all fuzzy reflexive approximation spaces and the set of all quasi-discrete fuzzy closure spaces satisfying a certain extra condition. Similar correspondence is also obtained between the set of all fuzzy tolerance approximation spaces and the set of all symmetric quasi-discrete fuzzy closure spaces satisfying a certain extra condition.