The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets

This paper provides a continuation of ideas presented by Davvaz and Mahdavipour [B. Davvaz, M. Mahdavipour, Roughness in modules, Inform. Sci. 176 (2006) 3658-3674]. The notion of hypermodule is a generalization of the notion of module. In this paper, we consider the quotient hypermodule M/A and interpret the lower and upper approximations as subsets of the quotient hypermodule M/A. Then, we introduce the concept of quotient rough sub-hypermodule. Also, using the concept of fuzzy sets, we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub-hypermodules and level rough sets. This relation is characterized as a necessary and sufficient condition.     

Generalized lower and upper approximations in a ring

In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism of a ring are introduced, and related properties are investigated. The notions of generalized lower and upper approximation operators, constructed by means of a set-valued mapping, which is a generalization of the notion of lower and upper approximation of a ring, are provided. We also propose the notion of generalized lower and upper approximations with respect to an ideal of a ring which is an extended notation of rough ideal introduced lately by Davvaz [B. Davvaz, Roughness in rings, Information Science 164 (2004) 147-163] in a ring and discuss some significant properties of them.     

Invertible approximation operators of generalized rough sets and fuzzy rough sets

This paper studies the classes of rough sets and fuzzy rough sets. We discuss the invertible lower and upper approximations and present the necessary and sufficient conditions for the lower approximation to coincide with the upper approximation in both rough sets and fuzzy rough sets. We also study the mathematical properties of a fuzzy rough set induced by a cyclic fuzzy relation.     

The lower and upper approximations in a fuzzy group

In this paper, we shall introduce the notion of a rough subgroup with respect to a normal subgroup of a group, and give some properties of the lower and the upper approximations in a group. Also, we will discuss a rough subgroup with respect to a t-level subset of a fuzzy normal subgroup.     

对模糊集、粗糙集和Vague集的比较研究

模糊集、粗糙集和Vague集三种理论都是对经典集合理论的扩展,使得集合论的应用扩展到了含糊的、不确定性的问题领域。介绍了三种集合的基本思想,重点分析三种理论的区别和内在联系,同时对三种理论的发展及应用作了一些探讨性研究。     

Reference points and roughness

In this paper, we introduce the notion of a reference point and provide local approximations for a subset of the universe. The notion of a reference point naturally gives rise to a rough approximations framework, wherein several approximations are possible on the same set. Also, we present an extension to the decision theoretic rough set model by using reference points.     

基于直觉模糊粗糙集的属性约简

针对Jensen下近似定义的局限性,提出一种新的等价类形式的近似算子表示,并将其推广到直觉模糊环境.在此基础上,将相对正域、相对约简、相对核等粗糙集的知识约简概念推广到直觉模糊环境,提出一种直觉模糊信息系统的启发式属性约筒算法.实例计算表明.该方法比Jensen的属性约简方法更为合理有效.     

基于模糊粗糙集的因素权重分配方法

基于模糊集与粗糙集的融合,提出一种对多因素分配权重的简单方法．该方法与传统的权重分配方法有明显的区别．首先利用模糊聚类分析对数据进行聚类,并提取最佳聚类;然后应用粗糙集中属性的重要程度对各个特征因素进行客观地分配权重．实例分析验证了所提出方法的有效性．     

模糊集和粗糙集

介绍和比较了模糊集和粗糙集各自的理论和特点,同时讨论了两种理论相结合来研究问题的方法。     

Rough approximations on a complete completely distributive lattice with applications to generalized rough sets

The generalizations of rough sets considered with respect to similarity relation, covers and fuzzy relations, are main research topics of rough set theory. However, these generalizations have shown less connection among each other and have not been brought into a unified framework, which has limited the in-depth research and application of rough set theory. In this paper the complete completely distributive (CCD) lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed. These definitions result from the concept of cover introduced on a CCD lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers. When T-similarity relation is considered, the existing fuzzy rough sets are the special cases of our proposed approximations on a CCD lattice. Thus these generalizations of rough sets are brought into a unified framework, and a wider mathematical foundation for rough set theory is established.     

Topological approaches to covering rough sets

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the topological properties of this type of rough sets, study the interdependency between the lower and the upper approximation operations, and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. Lastly, axiomatic systems for the lower approximation operation and the upper approximation operation are constructed.     

基于布尔矩阵的模糊粗糙集代数运算与表示定理

主要研究模糊粗糙集理论基本概念与基本运算的矩阵表示,用布尔矩阵对模糊粗糙集理论中的基本概念进行描述,并通过布尔矩阵运算性质研究、揭示和刻画模糊粗糙集知识空间的基本代数性质.文中定义了布尔矩阵\"与积\"和\"或积\"两种逻辑运算,分别对模糊粗糙集理论中的模糊可能(fuzzy diamond)算子和模糊必然(fuzzy box)算子计算过程进行描述,对模糊粗糙集理论的基本概念和基本代数性质给出了基于布尔矩阵的表示定理,为基于模糊粗糙集理论的知识表示与知识获取提供了一种能行与可计算的思路与方法.     

一种粗糙模糊控制器的设计与仿真

研究了应用粗糙集理论上、下近似定义对模糊控制规则进行拓展设计粗糙模糊控制器的方法,完成了水箱水位粗糙模糊控制器的设计,并分别进行了模糊控制和粗糙模糊控制仿真研究,研究结果表明该粗糙模糊控制能有效改善模糊控制的性能。     

Atanassov’s intuitionistic fuzzy grade of hypergroups

This paper deals with connections between hypergroupoids and Atanassov’s intuitionistic fuzzy sets. First a sequence of join spaces is associated with a hypergroupoid H; the length of the sequence is called Atanassov’s intuitionistic fuzzy grade of H. Second, a theorem about the existence of a hypergroup with Atanassov’s intuitionistic fuzzy grade equal to n is proved. Furthermore, some properties of the complete hypergroups in connection with this argument are presented and discussed.     

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.     

一种广义直觉模糊粗糙集模型

Atanassov直觉模糊集是对Zadeh模糊集最有影响的一种扩充和发展。为进一步拓展Pawlak粗糙集对多重不确定性信息的处理能力,将直觉模糊集引入粗糙集,采用构造性方法提出了一种广义直觉模糊粗糙集模型。首先,介绍了直觉模糊集在一个特殊格上的等价定义,对直觉模糊近似空间的两个基本要素(直觉模糊逻辑算子和直觉模糊关系)进行了研究,证明了一些重要的性质定理;在此基础上,建立了等价关系下的直觉模糊粗糙集模型;最后,对所提模型的性质进行了分类验证与讨论。     

基于程度不可区分关系的粗糙集模型

不可区分关系是粗糙集理论的基础。针对信息系统,提出了程度不可区分关系的概念来刻画信息系统中对象的可区分性程度的差异。提出了基于程度不可区分关系的粗糙集模型,并讨论了模型的基本性质,最后研究了基于程度不可区分关系的粗糙近似算子与Pawlak近似算子的关系。     

基于相似Rough集的模糊检索策略

Ｒｏｕｇｈ集理论作为一种具有模糊边界的集合理论,被广泛运用于不确定环境下的信息处理。文章探讨了一种基于相似关系Ｒｏｕｇｈ集的模糊查询技术,它是对普通Ｒｏｕｇｈ集在数据库中应用的推广,能有效地提高查询的灵活度及效率。     

粗集理论在调查数据可信度评判中的应用

在教育领域中常常会进行教学评价、学习评价等调查活动,而学生填写的数据有很大的随意性。为评判调查数据的可信度,引入粗集理论,先对属性集合进行初步约简,再利用条件属性的重要度确定约简中各属性权重,最后由模糊评判方法来衡量调查数据的可信度。其中,模糊关系矩阵由选择可信度来确定。实例分析结果表明,该方法是可行的、有效的。