On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings

This paper is a continuation of ideas presented by Davvaz [Roughness in rings, Inform. Sci., 164 (2004) 147-163; Roughness based on fuzzy ideals, Inform. Sci., 176 (2006) 2417-2437]. We introduce the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring, and give some properties of such ideals. Also, we discuss the relations between the upper and lower rough prime (primary) ideals and the upper and lower approximations of their homomorphism images.     

Roughness in MV-algebras

In this paper, by considering the notion of an MV-algebra, we consider a relationship between rough sets and MV-algebra theory. We introduce the notion of rough ideal with respect to an ideal of an MV-algebra, which is an extended notion of ideal in an MV-algebra, and we give some properties of the lower and the upper approximations in an MV-algebra.     

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 rough sets based on relations

Rough set theory has been proposed by Pawlak as a tool for dealing with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations. This paper studies arbitrary binary relation based generalized rough sets. In this setting, a binary relation can generate a lower approximation operation and an upper approximation operation, but some of common properties of classical lower and upper approximation operations are no longer satisfied. We investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations.This paper also explores the relationships between the lower or the upper approximation operation generated by the intersection of two binary relations and those generated by these two binary relations, respectively. Through these relationships, we prove that two different binary relations will certainly generate two different lower approximation operations and two different upper approximation operations.     

粗糙集的粗糙度

设U是全集,R是U上的等价关系,(U,R)是相应的近似空间,则粗相等关系≈是幂集P(U)上的等价关系,其商集为P(U)／≈,而商集P(U)／≈是一个分配格,本文考虑两种特殊情况,使得在这两种特殊情况下粗糙度有类似于集合论的包容排斥原理,同时我们还把此结论推广到粗糙模糊集上。     

模糊关系下的粗集粗糙度

本文研究粗糙集的粗糙度问题。首先仔细分析了Pawlak粗糙集的粗糙度,得到了粗糙集粗糙度包容相斥原理;然后将结果推广到模糊集理论领域,对研究模糊关系下模糊粗糙集理论有一定的作用。     

On rough set and fuzzy sublattice

Let L be a lattice with the least element 0 and the greatest element 1 and let θ be a full congruence relation on L. In this paper, the notion of θ-upper and θ-lower approximations of a fuzzy subset of L is introduced and some important properties will be studied.     

一种改进的图像增强算法及其应用

为改进图像增强算法,使之更适合医学领域图片的处理,采用了粗糙集的上逼近和下逼近思想,将图像分为物体区和背景区,使用不同的函数进行增强,进而提出了一种改进的基于粗糙集的增强算法,并首次应用于医学图像处理领域.实验结果显示改进的基于粗糙集的增强效果优于直方图均衡化方法.     

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.     

Roughness in modules

The theory of rough sets was proposed by Pawlak in 1982. The relations between rough sets and algebraic systems have been already considered by many mathematicians. Important algebraic structures are groups, rings and modules. Rough groups and rough rings have been investigated by Biswas and Nanda, Kuroki and Wang, and Davvaz. In this paper, we consider an R-module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an R-module, which is an extended notion of a submodule in an R-module. We also give some properties of the lower and the upper approximations in an R-module.     

Textural approach to generalized rough sets based on relations

This paper presents a discussion on rough set theory from the textural point of view. A texturing is a family of subsets of a given universal set U satisfying certain conditions which are generally basic properties of the power set. The suitable morphisms between texture spaces are given by direlations defined as pairs (r,R) where r is a relation and R is a corelation. It is observed that the presections are natural generalizations for rough sets; more precisely, if (r,R) is a complemented direlation, then the inverse of the relation r (the corelation R) is actually a lower approximation operator (an upper approximation operator).     

合成信息系统与子信息系统

本文给出了对象合成信息系统、属性合成信息系统、对象子信息系统及属性子信息系统的定义,分别讨论了它们的上下近似算子与原信息系统的上下近似算子之间的关系．并给出了它们的一些实际应用。     

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．     

On Characteristics of Information System Homomorphisms

The notion of information system homomorphism as a powerful tool to study the relation between two information systems was introduced by J.W. Grzymala-Busse. In this work, we will present some characteristics of information system homomorphism, which reveal the interdependence of the three mappings, namely, object mapping, attribute mapping and value domain mapping. Besides, given a partition on universe, we can derive a new information system homomorphism defining a partition on universe identical with the partition given. In the mean time, some invariant characteristics of upper approximation and lower approximation under information system homomorphism are investigated. At last, we establish a surjection between rough sets of information systems under an information system homomorphism.     

粗糙模糊集的格论性质

Let U denote a finite and nonempty set called the universe, and P(U) a power set. Suppose R is an equiva-lence relation on U. Consider the equivalence relation ≈ (X≈Y←→^-RX=^-R and RX=RY, X,Y, ∈ F(U)) on F(U),the quotient set denoted by F(U)/≈. In this paper we show that F(U)/≈ is a distributive lattice.     

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings

In this paper, we present the notions of equiprime fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of a nearring. We characterize these fuzzy ideals using level subsets and fuzzy points. If f: N → M is an onto nearring homomorphism, we show that the map defines a one-to-one correspondence between the set of all f-invariant (alternatively with sup property) equiprime (3-prime and c-prime, respectively) fuzzy ideals of N and the set of all equiprime (3-prime and c-prime, respectively) fuzzy ideals of M. Finally, we define fuzzy cosets determined by generalized fuzzy ideals; obtain fundamental results and isomorphism theorems.     

Roughness in Cayley graphs

In this paper, rough approximations of Cayley graphs are studied, and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition for pseudo-Cayley graphs containing Cayley graphs is proposed, and a rough approximation is expanded to pseudo-Cayley graphs. In addition, rough vertex pseudo-Cayley graphs and rough pseudo-Cayley graphs are introduced. Some theorems are provided from which properties such as connectivity and optimal connectivity are derived. This approach opens new research fields, such as data networks.     

信息系统同态的性质及上下近似的不变性

作为研究信息系统之间关系的一种有力工具,信息系统同态的概念首先由J.W.Grzyrnala-Busse引入。本文研究了信息系统同态的一些性质,揭示了构成信息系统同态的对象映射、属性映射与值域映射之间的相互依赖性。由给定的论域上的划分,导出了一个新的信息系统同态,该同态对论域的划分与给定划分相同。同时,讨论了信息系统在同态的意义下,上近似与下近似的不变性,为粗糙集理论的进一步研究奠定了理论基础。     

Minimization of axiom sets on fuzzy approximation operators

Axiomatic characterization of approximation operators is an important aspect in the study of rough set theory. In this paper, we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators.