Twofold fuzzy sets and rough sets—Some issues in knowledge representation

This paper deals with the representation of sets where the membership of some elements may be ill-known rather than just a matter of degree as in a fuzzy set. The notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete. A twofold fuzzy set is made of a nested pair of fuzzy sets: the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong. Twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by Pawlak. Set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined. Twofold fuzzy relations are also introduced. Finally, various applications of twofold fuzzy sets in knowledge representation are briefly discussed.     

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh’ fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.     

多值不确定型直觉模糊识别决策方法

基于模糊环境给出多值不确定型直觉模糊集的概念,定义了集合的交、并、包含等基本运算性质。通过刻画集合间的相似程度,提出八个新的相似度公式。最后将相似度应用于识别决策,说明公式的实用性和有效性。     

PRECOMPACTNESS IN FUZZY UNIFORM SPACES

Abstract

The notion of a precompact fuzzy set in a fuzzy uniform space is defined and it is shown that this is a good extension of the standard notion. A theory of precompact fuzzy sets is developed using the previously defined notion of a Cauchy prefilter in a fuzzy uniform space and this theory generalises standard theory.     

模糊仿射集与模糊向量子空间的再定义

利用模糊直线定义模糊仿射集，研究其有关性质。在此基础上，定义模糊向量子空间，证明一个模糊集为模糊仿射集的充要条件是该模糊集是一个模糊子空间的平移。     

Fuzzy sets-the approximation of semisets

This paper deals with alternative set theory which substantially departs from classical set theory. The main notion of it -a semiset-is proposed to model the same intuitive notion as fuzzy set but it is more general. The reasonable tool for ‘grasping’ semisets could be provided by fuzzy sets whose applicability is beyond discussion. In the paper reasons for such an approximation are given and two ways how to provide it are proposed. The fact that the membership function is modeled within the theory should be stressed. At the end, some problems and an indication for further investigation are discussed.     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

Rough intuitionistic fuzzy information systems

In this paper, we consider the notion of rough intuitionistic fuzzy sets and study their properties. We present an extension of rough set theory with the concept of intuitionistic fuzzy sets with several properties given. Moreover, we discuss the knowledge reduction of the classical Pawlak information systems and the intuitionistic fuzzy information systems, respectively.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

Grasping Infinity by Finite Sets

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of (finite) wide sets.     

On the perimeter and area of fuzzy sets

The note of area comes naturally as a measure of a (fuzzy) set. Recently, A. Rosenfeld defined the permeter, diameter, height, and width of a fuzzy set, Athough these do not generalize the corresponding notions for crisp sets, they lack inner conformity in that they do not satisfy simple interrelations we are used to in the case of crisp sets. In the present note we modify definitions and show that our perimeter, diameter, and height fit more comfortably with the notion of area. We also define a projection operator that should be used along with the multiplicative conjunction in the process of fuzzy reasoning.     

R0-代数（NM-代数）的区间值模糊子代数

将区间值模糊集的概念应用于R0-代数,引入区间值模糊R0-子代数的概念并研究它的性质。给出了区间值模糊集成为区间值模糊R0-子代数的一个充要条件;讨论了区间值模糊R0-子代数和R0-子代数之间的关系;定义了区间值模糊集的象和原象,获得了区间值模糊R0-子代数的象和原象成为区间值模糊R0-子代数的条件。     

Single-Valued Neutrosophic Lie Algebras

A single-valued neutrosophic (SVN) set is a powerful general formal framework that generalizes the concept of fuzzy set and intuitionistic fuzzy set. In SVN set, indeterminacy is quantified explicitly, and truth membership, indeterminacy membership, and falsity membership are independent. In this paper, we apply the notion of SVN sets to Lie algebras. We develop the concepts of SVN Lie subalgebras and SVN Lie ideals. We describe some interesting results of SVN Lie ideals.     

Representation of the Minkowski metric as a fuzzy set

This paper proposes a representation of the family of Minkowski distances using fuzzy sets. The proposed method helps to represent human-like perceptions about distances, which can help decision making in presence of non-probabilistic uncertainties such as imprecision and ambiguity. This way we propose to define a fuzzy set regarding the concept of closeness of two elements/sets measured by a Minkowski metric. Two application examples are presented, solved, and compared to some classical approaches. Finally some concluding remarks are provided and some interpretation issues are explained.     

Fuzzy group theory: A comparison of different notions of product of fuzzy sets

The notion of penultimate subgroup of a fuzzy group is introduced. This leads to the concept of penultimate product of fuzzy subgroups. A detailed study of different notions of product of fuzzy sets in a groupoid is carried out. The ultimate superiority of Liu's set product is established. Level subsets and strong level subsets of these products are also analysed. Consequently, simplified proofs of some of the results on free product are obtained.     

集合环

本文用集环来表现模糊集，得到了表现定理的新形式，并证明了这种表现可以有限化，即用有限个经典集近似刻划模糊集并且这样的刻划是同态的。最后，证明了集对方法是集环表现的特殊情形。     

Fuzzy uniqueness of fuzzy mappings

For fuzzy mappings between fuzzy sets of higher level we study a notion of fuzzy uniqueness. i.e. of fuzzy functions, which formalizes the intuition that ‘almost identical’ images under f.Characterizations are given for such fuzzy mappings which are really (i.e. with truth value 1) fuzzy unique or fuzzy biunique, and for those fuzzy sets which can be domains of really fuzzy unique fuzzy mappings.Finally, as an application fuzzy cardinals, their ordering, sum, and product are introduced, and some properties proved.     

模糊粗糙集及粗糙模糊集的模糊度

1965年,Zadeh提出了Fuzzy集理论,1982年,Z.Pawlak提出Rough集理论。将二者结合而形成的模糊粗糙集（FR集）及粗糙模糊集（RF集）近年来越来越受到国际学术界的关注。本文所研究的FR集及RF集的模糊度,是对FR集及RF集模糊程度的一种度量,进而引进了相应的明可夫斯基距离,明可夫斯基模糊度和Shannon模糊度。     

Intuitionistic fuzzy metric spaces

Using the idea of intuitionistic fuzzy set due to Atanassov [Intuitionistic fuzzy sets. in: V. Sgurev (Ed.), VII ITKR's Session, Sofia June, 1983; Fuzzy Sets Syst. 20 (1986) 87], we define the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces due to George and Veeramani [Fuzzy Sets Syst. 64 (1994) 395] and prove some known results of metric spaces including Baire's theorem and the Uniform limit theorem for intuitionistic fuzzy metric spaces.     

Boolean duals of virtual groups

The purpose of this paper is to formulate the notion of virtual group and that of a homomorphism between two in such a way that no null sets are involved. That is, we replace the point set by the measure algebra of Borel sets modulo null sets, and the various functions involved must be described by objects associated to these measure algebras, such as σ-homomorphisms. In the process it is necessary to redefine functions satisfying algebraic identities almost everywhere to get new ones satisfying the identities in a stricter sense. Similar refinements on quasiinvariance of measures are also required.