基于对象的广义粗糙近似算子的拓扑性质

粗糙集理论是一种处理不确定性问题的数学工具。近似算子是粗糙集理论中的核心概念，基于等价关系的Pawlak近似算子可以推广为基于一般二元关系的广义粗糙近似算子。近似算子的拓扑结构是粗糙集理论的重点研究方向。文中主要研究基于对象的广义粗糙近似算子诱导拓扑的性质，证明了广义近似空间中所有可定义集形成拓扑的充分条件也是其必要条件，研究了该拓扑的正则、正规性等拓扑性质；给出了串行二元关系与其传递闭包可以生成相同拓扑的等价条件；讨论了该拓扑与任意二元关系下基于对象的广义粗糙近似算子所诱导拓扑之间的相互关系。     

基于强对称关系的广义粗糙集模型

粗糙集的公理化是该理论研究的重要课题之一。文中在分析对称关系下粗糙近似算子的特征公理基础上,提出强对称二元关系。对比等价关系的性质,讨论该二元关系的一些重要特征,给出对称关系成为强对称关系的充要条件,并研究其下广义粗糙集的性质,得到强对称关系下粗糙近似算子的公理化特征。利用相应的特征公理与精确集之间的联系,探讨一般二元关系下论域中精确集的一些重要特征,对拓广粗糙集理论及其应用提供一定帮助。     

严格局部自反关系下的广义粗糙集

经典粗糙集是在等价关系基础上建立的一类不确定性理论方法。研究一般二元关系下的广义粗糙集,不仅可以拓宽粗糙集理论的应用范围,而且也能从一定的角度进一步阐释经典粗糙集的基本性质。在考虑自反粗糙近似算子的基础上,提出了严格局部自反关系的定义,讨论了严格局部自反关系下广义粗糙集的性质,给出了其公理化特征。结合自反粗糙近似算子,研究了一般二元关系下广义粗糙集中的精确集,得出了一些重要的结论。     

类传递关系下的广义粗糙集及其公理化

提出了正向类传递与反向类传递二元关系,分别考虑了基于这两种二元关系的广义粗糙集,探讨了它们各自的性质,给出了相应粗糙近似算子的公理化特征。分析了这两类广义粗糙集与其它相关二元关系下广义粗糙集之间的联系,得到了一些重要的结果。     

无限论域上粗糙集的拓扑结构

讨论了当论域不限制是有限集时满足自反、传递关系的广义近似空间中的近似算子的拓扑结构;证明了论域 上满足自反、传递关系的集合与其上所有的拓扑的集合是一一对应的;指出了该拓扑空间的拓扑基。     

无限论域上粗糙集的拓扑结构

讨论了当论域不限制是有限集时满足自反、传递关系的广义近似空间中的近似算子的拓扑结构;证明了论域上满足自反、传递关系的集合与其上所有的拓扑的集合是一一对应的;指出了该拓扑空间的拓扑基.     

二元关系性质判定算法的研究与实现

为了对二元关系的性质进行快速准确的判定，通过分析二元关系性质的定义和相关定理，给出了二元关系的五种性质判定方法的算法描述。该算法的重点是判定自反＼反自反性质、对称、反对称性质和传递性质。在计算机上进行了编程实现．并对二元关系性质的判定算法进行了测试。实验结果证明，该算法具有很强的可操作性，可以快速准确地判定二元关系的性质。     

类对称关系下的广义粗糙集模型

针对等价关系下的经典粗糙集,定义弱对称与局部强对称二元关系,构造相应的广义粗糙集模型。给出这2种模型的公理化特征,并将两者结合,得到强对称二元关系下的广义粗糙集模型。理论分析证明,论域上任何集合均为广义精确集的充要条件是其二元关系为强对称关系,即可以利用该模型刻画经典粗糙集中的广义精确集。     

有限集上二元关系性质的判定

离散数学中,二元关系自反、反自反、对称、反对称和传递性的判定是学习等价关系、相容关系和偏序关系的重要基础知识,为使读者灵活掌握二元关系五种性质的判定,分别从关系性质的定义、关系矩阵、关系图和关系运算四方面对二元关系五种性质的判定方法进行了探讨。     

基于R-蕴含算子的近似空间的拓扑性质

1引言 粗糙集理论[1]是由Pawlak首先提出的一种处理不完备知识和数据的新的数学工具,经过20余年的研究和发展,粗糙集已经在理论和应用上取得了长足的发展.近年来,许多文献讨论了粗糙集与模糊集之间的关系,提出了多种粗糙集模型[7～12].特别地,把三角模、蕴含算子应用于粗糙近似算子的定义[13～15],丰富了粗糙集研究内容和应用领域.最近,文[16]研究了Pawlak近似空间与拓扑空间之间的关系,指出Pawlak上、下近似算子分别是一个拓扑空间的闭包算子和内部算子.本文进一步研究基于R-蕴含算子的近似空间与模糊拓扑之间的关系.证明了自反和-传递近似空间可生成一个模糊拓扑空间,且在一定条件下,模糊拓扑空间也可以生成一个基于-蕴含算子的广义模糊近似空间.这些结果为利用拓扑学方法研究粗糙集理论提供了基础.     

Note on “Generalized rough sets based on reflexive and transitive relations”

In this note, we show that, in the paper [Generalized rough sets based on reflexive and transitive relations, Information Sciences 178 (2008) 4138-4141], the topology induced by a preorder (i.e., a reflexive and transitive binary relation) is nothing but the well-known Alexandrov topology of the preorder. In addition, we point out that a topology satisfying certain kind of compactness condition proposed in the above mentioned paper is exactly an Alexandrov topology.     

On intuitionistic fuzzy rough sets and their topological structures

In this paper, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of intuitionistic fuzzy approximation operators are examined. Relationships between intuitionistic fuzzy rough set approximations and intuitionistic fuzzy topologies are then discussed. It is proved that the set of all lower approximation sets based on an intuitionistic fuzzy reflexive and transitive approximation space forms an intuitionistic fuzzy topology; and conversely, for an intuitionistic fuzzy rough topological space, there exists an intuitionistic fuzzy reflexive and transitive approximation space such that the topology in the intuitionistic fuzzy rough topological space is just the set of all lower approximation sets in the intuitionistic fuzzy reflexive and transitive approximation space. That is to say, there exists an one-to-one correspondence between the set of all intuitionistic fuzzy reflexive and transitive approximation spaces and the set of all intuitionistic fuzzy rough topological spaces. Finally, intuitionistic fuzzy pseudo-closure operators in the framework of intuitionistic fuzzy rough approximations are investigated.     

Generalized rough sets based on reflexive and transitive relations

In this paper, we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite. It is proved that there exists a one-to-one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition.     

Algorithm and axiomatization of rough fuzzy sets based finite dimensional fuzzy vectors

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.     

Similarity of binary relations based on rough set theory and topology: an application for topological structures of matroids

In this paper, we propose an integration of rough sets, matroids and topology to exploit the advantages of three theories. First, we consider topologies induced by binary relations and illustrate that binary relations can be researched by means of topology. Next, similarity of binary relations based on rough set theory and topology is introduced and the fact that every binary relation is solely similar to some preorder relation is proved. Finally, as an application of the similarity, topological structures of matroids are given.     

模糊粗糙集拓扑性质的进一步研究

首先指出文献《基于Lukasiewicz三角模及其剩余蕴涵的模糊粗糙集》中定理7的结论不成立,并给出了反例。其次,从两个方面对上述文献进行了修正：（1）当R是自反模糊关系时,T′={A∈F（U）|R（A）=A}是一模糊拓扑;（2）当R是自反、传递的模糊关系时,上述文献中的结论成立。最后,给出了模糊集A为模糊拓扑T的开集的充分必要条件,从而得到了模糊拓扑T的另外几个性质。     

Algorithm and axiomatization of rough fuzzy sets based finite dimensional fuzzy vectors

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.