Roughness in n-ary hypergroups

This paper is aimed at studying the rough sets within the context of the commutative n-ary hypergroups. For this approach, the presentation of a detailed study of the various types of n-ary subhypergroups is of the greatest importance. This study was initiated in [V. Leoreanu-Fotea, Several types of n-ary subhypergroups, Italian Journal of Pure and Applied Mathematics, in press]. Subsequently, we shall use the findings in this study to define the upper and lower approximation of a subset with respect to an invertible n-ary subhypergroup of a commutative n-ary hypergroup. In the final part, the concept of n-ary rough subhypergroup is introduced.     

The lower and upper approximations in a hypergroup

This paper presents a relationship between rough sets and hypergroup theory. We analyze the lower and upper approximations of a subset, with respect to an invertible subhypergroup and we consider some particular situations. Moreover, the notion of a rough subhypergroup is introduced. Finally, fuzzy rough subhypergroups are introduced and characterized.     

Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice

This paper considers the relationships among L-fuzzy sets, rough sets and n-ary hypergroup theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy n-ary subhypergroup are introduced and studied. Then, a new algebraic structure called (invertible) L-fuzzy rough n-ary subhypergroups is defined, and the (strong) homomorphism of lower and upper L-fuzzy rough approximation operators is studied.     

Fuzzy soft hypergroups

We propose the concept of fuzzy soft hypergroups and explore two particular classes of them, namely fuzzy soft closed hypergroups and fuzzy soft ultraclosed hypergroups. Moreover, we study the image and inverse image of a fuzzy soft (closed, ultraclosed) hypergroup under a fuzzy soft map.     

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

Interval-valued fuzzy n-ary subhypergroups of n-ary hypergroups

This paper provides a continuation of ideas presented by Davvaz and Corsini (J Intell Fuzzy Syst 18(4):377–382, 2007). Our aim in this paper is to introduce the concept of quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set. This concept is a generalized concept of quasicoincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup of a n-ary hypergroup. This newly defined interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup is a generalization of the usual fuzzy n-ary subhypergroup. Finally, we consider the concept of implication-based interval-valued fuzzy n-ary subhypergroup in an n-ary hypergroup; in particular, the implication operators in ￡ukasiewicz system of continuous-valued logic are discussed.     

双论域上粗糙集的矩阵定义

提出了两个关系矩阵的序偶取小乘法的概念,并利用关系矩阵和布尔列向量重量上乘法和下乘法的有关结论,给出了计算双论域上任一集合上下近似的具体算法,从而使得双论域上粗糙集的计算程序化。     

On probabilistic n-ary hypergroups

The T-fuzzy n-ary subhypergroups of an n-ary hypergroup are defined by using triangular norms and some related properties are hence obtained. In particular, we consider the probabilistic version of n-ary hypergroups by using random sets and show that the fuzzy n-ary hypergroups defined by triangular norms are consequences of some probabilistic n-ary hypergroups under certain conditions. Some results on n-ary hypergroups recently given by Davvaz and Corsini are extended.     

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.     

Rough approximations based on intersection of indiscernibility,similarity and outranking relations

Rough sets theory has proved to be a useful mathematical tool for dealing with the vagueness and granularity in information tables. Classical definitions of lower and upper approximations were originally introduced with reference to an indiscernibility relation. However, indiscernibility relation is still restrictive for many applications. Many real-world problems deal with assignment of some objects to some preference-ordered decision classes. And, the objects are described by a finite set of qualitative attributes and quantitative attributes. In this paper, we construct the indiscernibility relation for the subset of nominal attributes, the outranking relation for the subset of ordinal attributes, and the similarity relation for the subset of quantitative attributes. Then the global binary relation is generated by the intersection of indiscernibility relation, outranking relation and similarity relation. New definitions of lower and upper approximations of the upward and downward unions of decision classes are proposed based on the global relation. We also prove that the lower and upper approximation operations satisfy the properties of rough inclusion, complementarity, identity of boundaries, and monotonicity.     

UPPER AND LOWER APPROXIMATIONS OF FUZZY SETS

The upper and lower approximations of a fuzzy subset with respect to an indistinguish-ability operator are studied. Their relations with fuzzy rough sets are also investigated.     

粗集理论的矩阵方法

粗糙集理论是近年来发展起来的一种有效的处理不精确、不确定信息的理论,在机器学习及数据挖掘等领域获得了成功的应用,该文用矩阵的方法来研究粗糙集,即从一个二元关系矩阵出发,给出粗糙集上下近似的矩阵描述,实际上是用矩阵的方法重新定义上下近似,矩阵的方法不仅提供了上下近似的简单的计算方法,也提供了一种新的推理的方法,我们还把矩阵方法用于信息系统的约简。     

Knowledge acquisition in incomplete fuzzy information systems via the rough set approach

Abstract: Machine learning can extract desired knowledge from training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete and incomplete data sets. If attribute values are known as possibility distributions on the domain of the attributes, the system is called an incomplete fuzzy information system. Learning from incomplete fuzzy data sets is usually more difficult than learning from complete data sets and incomplete data sets. In this paper, we deal with the problem of producing a set of certain and possible rules from incomplete fuzzy data sets based on rough sets. The notions of lower and upper generalized fuzzy rough approximations are introduced. By using the fuzzy rough upper approximation operator, we transform each fuzzy subset of the domain of every attribute in an incomplete fuzzy information system into a fuzzy subset of the universe, from which fuzzy similarity neighbourhoods of objects in the system are derived. The fuzzy lower and upper approximations for any subset of the universe are then calculated and the knowledge hidden in the information system is unravelled and expressed in the form of decision rules.     

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.     

双论域下多粒度模糊粗糙集上下近似的包含关系

针对双论域上集合的多粒度乐观与悲观上下近似不具有包含关系的问题,本文给出了双论域上集合的多粒度上下近似具有包含关系的一个充分条件,进而采用标准化的方法将不具有包含关系的上下近似转化为具有包含关系的上下近似。通过实例验证,该方法能有效解决双论域下多粒度模糊粗糙集上下近似具有包含关系的问题。     

多粒度模糊粗糙集的表示与相应的信任结构

利用多粒度粗糙集的上、下近似及其性质,结合模糊集的分解定理,研究多粒度模糊粗糙集的上、下近似的表示及性质,根据多粒度模糊粗糙集的上、下近似构造信任函数与似然函数。     

基于概念格的粗糙集近似

粗糙集理论的一个重要研究方面是用已定义的概念来近似未定义的概念,而如何构建可定义概念以及如何确定近似运算是这一工作的基础.利用粗糙集这一工具,从概念格的角度来确定可定义概念,并在此基础上研究了概念的粗糙近似.根据粗糙集上下近似的包含关系,得到概念的一种新的上下近似的运算的定义.粗糙集近似理论利用两种不同的近似运算,产生两种不同的近似来描述概念格背景下的对象集合.     

粗糙集的矩阵定义

提出了关系矩阵和布尔列向量重量上乘法和下乘法的概念,证明了上乘法就是上近似,下乘法就是下近似,同时研究了上下近似的性质,最后给出了计算上下近似的算法。     

基于矩阵取小乘法的粗糙集

提出划分矩阵和布尔列向量取小乘法的概念;证明了下矩阵和上矩阵的行并向量分别是下近似和上近似;研究了上下近似的性质;给出了计算上下近似的算法。     

序粒度标记结构及其粗糙近似

粒计算是知识表示和数据挖掘的一个重要方法.它模拟人类思考模式,以粒为基本计算单位,以处理大规模复杂数据和信息等建立有效的计算模型为目标.针对具有多粒度标记的序信息系统的知识获取问题,提出了基于序粒度标记结构的粗糙近似.首先,介绍了序标记结构的概念,并在序标记结构的对象集中定义了一个优势关系,同时给出了由优势关系导出的优势标记块,并进一步定义了基于优势关系的集合的序下近似与序上近似和序标记下近似与序标记上近似的概念,给出了近似算子的一些性质.证明了由序标记结构导出的集合的下近似质量与上近似质量是一对对偶的必然性测度与可能性测度.最后,定义了多粒度序标记结构的概念,并讨论了多粒度序标记结构中不同粒度下近似集之间的关系.