On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of (I,T)-interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T-similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I-lower and T-upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I-lower and T-upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.     

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.     

On (⊥,?)-generalized fuzzy rough sets

In the present paper, we mainly discuss the (??,?)-generalized fuzzy rough sets introduced by B.Q. Hu and Z.H. Huang with both constructive approach and axiomatic approach considered. In the former, we started from the investigation of the properties of the ??-upper and ?-lower approximation operators generated by binary fuzzy relations. In the latter, by defining a pair of fuzzy set-theoretic operators, we show (??,?)-fuzzy rough approximation operators can be characterized by different sets of axioms.     

基于集值映射的形式概念分析

Formal concept analysis (FCA) is a discipline that studied the hierarchical structares induced by a binary relation between a pair of sets,and applies in data analysis,information retrieval,knowledge discovery,etc.In this paper,it is shown that a formal context T is equivalent to a set-valued mapping S :G →P(M),and formal concepts could be defined in the set-valued mapping S.It is known that the topology and set-valued mapping are linked.Hence,the advantage of this paper is that the conclusion make us to construct formal concept lattice based on the topology.     

Axiomatic systems for rough set-valued homomorphisms of associative rings

Axiomatic approaches to study approximation operators are one of the primary directions for the investigation of rough set theory. In this paper, we provide some axiomatic systems of lower and upper approximation operators in rough set theory. We also apply the axiomatic systems of generalized rough sets for definitions of generalized lower and upper approximations with respect to an ideal of a ring and discuss some of their significant properties.     

基于格值逻辑的模糊概念格

从逻辑的角度,将非经典逻辑之一的格值逻辑引入概念格,建立了格值模糊形式背景,通过格结构来刻画对象与属性之间的模糊关系,证明了由蕴涵算子诱导的算子对是伽罗瓦连接,并讨论了相关的一些性质,进而给出了格值模糊概念格的构造算法.格值模糊概念格的建立为模糊性与不可比较性信息的处理提供了可靠的数学工具.     

Study of a universal formal context

Studying a universal formal context, we obtain a number of properties of the context itself, its concepts, and the lattice formed by the set of these concepts. The most significant of these properties is represented by a theorem showing that there exists an embedding of the concept lattice of an arbitrary at most countable universal context into the concept lattice of a universal context under which the image of the embedding is an initial segment of the concept set of a universal formal context with infinite volumes, and the validity of the dual result. It is shown that the theorem also holds in the computable case. This theorem demonstrates the complexity of the structure of a universal formal context.     

模糊粗糙近似算子公理集的独立性

用双论域上的模糊关系定义了广义模糊粗糙近似算子,并讨论了近似算子的性质。用公理刻画了模糊集合值算子,各种公理化的近似算子可以保证找到相应的二元模糊关系,使得由模糊关系通过构造性方法定义的模糊粗糙近似算子恰好就是用公理定义的近似算子。讨论了刻画各种特殊近似算子的公理集的独立性,从而给出各种特殊模糊关系所对应的模糊粗糙近似算子的最小公理集。     

Two axiomatic approaches to decision making using possibility theory

This paper proposes a utility theory for decision making under uncertainty that is described by possibility theory. We show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by Dubois et al. The generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty, namely, pessimism and optimism. In their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize. We prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility.     

Axiomatic systems for rough sets and fuzzy rough sets

Rough set theory is an important tool for approximate reasoning about data. Axiomatic systems of rough sets are significant for using rough set theory in logical reasoning systems. In this paper, outer product method are used in rough set study for the first time. By this approach, we propose a unified lower approximation axiomatic system for Pawlak’s rough sets and fuzzy rough sets. As the dual of axiomatic systems for lower approximation, a unified upper approximation axiomatic characterization of rough sets and fuzzy rough sets without any restriction on the cardinality of universe is also given. These rough set axiomatic systems will help to understand the structural feature of various approximate operators.     

基于对象导出三支概念格的决策背景规则获取

规则获取是当前形式概念分析领域的研究热点.首先给出了基于对象导出三支概念格间的细于关系,定义了基于对象导出三支概念格的三支弱协调性,并研究了其与经典概念格下的二支弱协调性之间的关系.然后,研究了基于对象导出三支概念格的规则获取,并与经典概念格的规则获取进行了比较.最后,定义了对象导出三支概念的弱闭标记,研究了基于弱闭标记的三支弱协调决策形式背景的规则获取,剔除了冗余规则,并且得到一些新的更为精简的三支规则.     

On uniqueness of （fuzzy） relation in some generalized rough set model

In rough set theory, crisp and/or fuzzy binary relations play an important role in both constructive and axiomatic considerations of various generalized rough sets. This paper considers the uniqueness problem of the （fuzzy） relation in some generalized rough set model. Our results show that by using the axiomatic approach, the （fuzzy） relation determined by （fuzzy） approximation operators is unique in some （fuzzy） double-universe model.     

Block relations in formal fuzzy concept analysis

One of the main problems in formal concept analysis (especially in fuzzy setting) is to reduce a concept lattice of a formal context to appropriate size to make it graspable and understandable. A natural way to do it is to substitute the formal context by its block relation which is equivalent to factorization of the concept lattice by a complete tolerance. We generalize known results on the correspondence of block relations of formal contexts and complete tolerances on concept lattices to fuzzy setting and we provide an illustrative example of using block relations to reduce the size of a concept lattice.     

Set-based granular computing: A lattice model

Set-based granular computing plays an important role in human reasoning and problem solving. Its three key issues constitute information granulation, information granularity and granular operation. To address these issues, several methods have been developed in the literature, but no unified framework has been formulated for them, which could be inefficient to some extent. To facilitate further research on the topic, through consistently representing granular structures induced by information granulation, we introduce a concept of knowledge distance to differentiate any two granular structures. Based on the knowledge distance, we propose a unified framework for set-based granular computing, which is named a lattice model. Its application leads to desired answers to two key questions: (1) what is the essence of information granularity, and (2) how to perform granular operation. Through using the knowledge distance, a new axiomatic definition to information granularity, called generalized information granularity is developed and its corresponding lattice model is established, which reveal the essence of information granularity in set-based granular computing. Moreover, four operators are defined on granular structures, under which the algebraic structure of granular structures forms a complementary lattice. These operators can effectively accomplish composition, decomposition and transformation of granular structures. These results show that the knowledge distance and the lattice model are powerful mechanisms for studying set-based granular computing.     

Core–EP pre-order of Hilbert space operators

A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the (dual) core partial order between core parts of operators and the minus partial order between quasinilpotent parts of operators, new pre-orders and partial orders are also introduced and characterized.     

广义概率对偶犹豫模糊语言幂集结算子在多属性决策中的应用

在对偶犹豫模糊语言集、概率对偶犹豫模糊集和广义幂集结算子的基础上,研究了在概率对偶犹豫模糊语言环境下的广义幂集结算子问题。首先,给出了概率对偶犹豫模糊语言集的定义、运算规则、得分函数、精确函数、距离测度、熵。然后,定义了广义概率对偶犹豫模糊语言幂集结算子,并研究其具有的性质。其次,提出了一种决策方法来解决集结数据之间存在相互关系的概率对偶犹豫模糊语言多属性决策问题。最后,结合相关案例验证了该方法的有效性和可行性。     

基于粗集与概念格的属性简约讨论

概念格的属性简约是在形式背景下解决复杂问题的重要途径,通过对概念格、粗糙集的讨论,将两者有效结合,并借助粗糙集上(下)近似的方法,得出了一个对概念格属性简约的方法,方法将二维的概念格属性简约转化为一维的一种对象格的简约,避免了形式背景下的概念的计算和进一步的可辨识矩阵的计算,方法简便,算法简单易实现,是概念格属性简约有效的算法.     

一般模糊近似算子的公理化刻画

给出无限双论域上一般模糊近似算子的构造性定义,叙述一般模糊近似算子的基本性质。引入邻域有限模糊关系的概念,利用上、下模糊粗糙近似的截集性质,给出一个刻画模糊近似算子的新公理,得到不同于以往的刻画模糊近似算子的公理集。     

n-closure systems and n-closure operators

It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and vice-versa. Recent work of Wille on triadic concept analysis and subsequent work by the author on polyadic concept analysis led to the introduction of complete trilattices and complete n-lattices, respectively, that generalize complete lattices and capture the order-theoretic structure of the collection of concepts associated with polyadic formal contexts. In the present paper, polyadic closure operators and polyadic closure systems are introduced and they are shown to be in a relationship similar to the one that exists between ordinary (dyadic) closure operators and ordinary (dyadic) closure systems. Finally, the algebraic case is given some special consideration. This paper is dedicated to Walter Taylor. Received March 10, 2005; accepted in final form March 7, 2006.     

概念格的粗糙近似比较研究

形式背景产生了概念格,每个节点由外延和内涵组成.对形式背景论域中的任何一个子集,可用外延来近似,在这方面已有了4种方法.对这些方法进行了比较研究,利用粗糙集理论证明了用这些方法所求出的概念的上近似外延是相同的,并利用粗糙集理论研究了概念格属性约简后,原来方法对结果的一致性.