THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 2, Number 3, September 1994

CONTENT

DETAILS

Strong Fuzzy Inner Product Spaces

Wu Congxin and Li Ronglu

Department of Mathematics, Harbin Institute of Technology, Harbin, China

Min-Hyung Cho

Department of Mathematics, Kum-Oh University of Technology, Kumi, Korea

Abstract: We introduce a concept of fuzzy inner product which is based on the concept of fuzzy norm in [4], A series of basic results on fuzzy inner product spaces and strong fuzzy inner product spaces are established.

Keywords: Fuzzy norm, fuzzy inner product, strong fuzzy inner product space.

Some Properties of Fuzzy Integral

Zeng wenyi and Luo chengzhong

Department of Mathematics, Beijing Normal University, Beijing, 100875, China

Abstract: The concept of fuzzy integral was first performed in [1]. After then, Wang [2] introduced concept of auto-continuty of set function and obtained many conclusions, In yhis paper, we base on the theory of lattice, continuously research fuzzy integral of fuzzy sets on and obtain some interesting properties.

Equation Type¢òof an F-matrix and the Computation of the Content for Realizable F-matrix

Wang Hongxu

Dep. Of Basic. Liaoyang College of Petrochemistry Engineering, Liaoyang, Liaoning, 111003, P. R. China

Abstract: A definition of the equation type ¢ò of an F-matrix and its properties were given and discussed preliminarily in this paper. Also, the conditions of the solution and the solution itself was studied. We got a method of computing content for the realizable F-matrices.

Keywords: Equation type ¢ò of an F-matrix, realizable F-matrices, content for realizable F-matrices

The Endomorphism of (F(X), ?, ?)

Liu Wenbin

Department of Mathematics,Ocean University of Qingdao, Qingdao, Shandong, 266003, China

Abstract: Suppose is a transformation in an ordinary set X, we know that the extension principle given by [1] and [2] can extend to the transformation of . In this paper, we extend the results in [1] and [2], and use them to investigate another basic problem of fuzzifying, i.e.,the operation- Preserving relation. The main results are as follow: 1. The transformation and extension principle of from to is given, which is a more generalized fuzzifying metod than that in [1] and [2]. 2. The problem of relation-preserving of set operation before and after the transformation is studied (in this paper it is called Homomorphism). This is another basic problem about fuzzy set theory and is not mentioned in [1] and [2]. 3. The decomposition and composition theorem of endomorphism of corresponding to decomposition of fuzzy set are given.

Keywords: Fuzzy power set, Endomorphism.

NS-set and Its Fuzzy Images

Chen Shu-Li

Department of Mathematics, Jianghan Petroleum Institute, Jianghan Hubei,434102, China

Abstract: In this paper the concept of an NS-set in an L-fuzzy topological space is introduced. The characteriz- ations and fuzzy images of an NS-set are systematically discussed. We show that the NS-set is hereditary for the L-fuzzy regular open subset, an L-fuzzy topological space is NS-closed if and only if every proper regular open set in is an NS-set, the fuzzy almost continuous image of an NS-set is an almost F-compact set and a subset A of an L-fuzzy E space is an NS-set in if and only if A is an almost F-compact set in . We also investigate the NS-set in are finitely additive.

Keywords: , -net, -filter, -cluster point, NS-set, fuzzy mapping.

On Convergence of Nets of L-fuzzy Sets

Chen Shui-Li

Department of Mathematics, Jianghan Petroleum Institute, Jianghan Hubei,434102, China

Cheng Ji-Shu

Department of Mathematics, Qinghai Junior Teacher¡¯s College, Xining Qinghai, 810007, China.

Abstract: The concepts of a net of L-fuzzy sets and its convergence in an L-fuzzy topological space are introduced, their properties are systematically discussed, and more interesting results are obtained in this paper. By means of the convergent theory, some equivalent characterizations with respect to L-fuzzy cloaed sets, L-fuzzy open set, continuous order-homomorphisms and open order-homomorph- Isma are given.

Keywords: L-fuzzy topological space, net of L-fuzzy set, limit point, cluster point, order-homomorphism.

Fuzzy Closure Spaces

Rekha Srivastava

Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanast-221005, India

Arun K. Srivastava and Alka Choubey

Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanast-221005,India

Abstract: We introduce the notion of fuzzy closure space and show that the corresponding category FCS of fuzzy closure spaces is a topological category in which the category CS of (Birkhoff) closure spaces is embedded as a bicoreflective subcategory. T0-objects in FCS are also identified and they are shown to form an epireflective subcategory of FCS.

Keywords: Fuzzy closure space, fuzzy closure preserving map, topological category, bicoreflective subcategory, epireflective subcategory, functors and , T0-fuzzy closue space, T0-object, universal category.

Construction of Fuzzy Relations Based on Complemented t-norms

Peng Zuzeng and Sun Yunyu

Wuhan University of Hydraulic and Electrical Engineering, Wuhan, China

Abstract: In this paper by using compolemented t-norms we construct fuzzy relations for and such that , , where and denote the complement of and respectively. Furthermore,we give the upper bound and the minimal element of these R under proper conditions.

Keywords: t-norm, t-conorm, pseudo-complement, complemented t-norm, complemented t-conorm.

study on the Dual Problems in the Theory of Fuzzy Measure and Fuzzy Integral

Liu Xuecheng

Department of Mathematics, Hebei Normal College, Shijiazhuang, Hebei, 050091, P. R. China

Abstract: In this paper, we propose the dual proposition rules for fuzzy measure and fuzzy integral, and list the dual relations of the most important concepts and propositions in the theory of fuzzy measure and fuzzy integral.

Keywords: Fuzzy measure, fuzzy integral, duality.

Non-dominated Elements and Fuzzy Scalarizing Functions in Vector Optimization

A.V. Yazenin

Tver State University, Dept. of Algorithmic Languages, ul. Zhelyabova 33, 170000 Tver, Russia

M. Wagenknecht

TU Berlin, Insit.fur Proze -u. Anlagenetechnik,Projekt SVT, Rudower Chaussee 5(Geb.10.5),12484 Berlin, Germany

Abstract: In the present paper we deal with classical vector optimization problems where by additional fuzzy information (e.g. on the importance of singer criteria) is available. For it we introduce fuzzy scalarizing functions and define corresponding preference relations on the real axis. Applying Orlovsky¡¯s concept of non-dominated elements we get relations between the latter and the set of Pareto points of the original problem.

Keywords: Vector optimization, fuzzy preferences, fuzzy weights, efficient points

Fuzzy Regular Semiopen Sets and s-closed Spaces

A.M. Zahran

Department of Mathematics,Facuty of Science, Al-Azhar University, Assuiut,Egypt.

Abstract: In this paper, we discuss s-closeness in fuzzy topological spaces, and we give some characterizations of s-closeness. Also, we show that an fts (X,T) is s-closed iff is -compact, where is a fuzzy topology on X which has RSO(X) as a subbase. Finally, we investigate the image of s-closed under some types of continuity.

Keywords: Fuzzy topology, fuzzy -compactness, fuzzy s-closeness, fuzzy semicontinuity, fuzzy regular semiopen sets.

Duality on Completely Distributive Lattices

Fan Taihe

Department of Mathematics, Shanxi Normal University, Xi¡¯an,China

Abstract: In the class of completely distributive lattices, an operation ¡°*¡± is introduced. It is proved that for each element, a maximal family can be constructed by virtue of minimal families of those elements not less than nor equal to the element, hence an internal relation between minimal family and maximal family is obtained. This suggests a new way to deal with duality both in fuzzy topology and in the more general setting of topological molecular lattices. As application, two interesting results in the complete rings of sets are obtained.

Keywords: Completely distributive lattice, ¡°*¡± operation, maximal family, minimal family.

Functions with Closed FuzzyGraph

S. P. Arya

Maitreyi College, Bapu Dham Complx, Chanakypuri, New Delhi-110021, India.

Debajit Hazarika

Department of Mathematics, University of Delhi, Delhi-110007, India.

Abstract: Closed graphs of maps and related notions are widely sued in General Topology and allied fields. This paper endeavours to establish similar notions for mappings in fuzzy topological spaces. In this paper we introduce the notions of closed fuzzy graph and a stronger form of it as a natural extension of the concepts of closed graph and strongly closed graph in general topological spaces. We show that these notions can be utilized to inter-relate various concepts of fuzzy topology, which include the -axiom, -axiom, fuzzy continuity, fuzzy weakly continuity, fuzzy Urysohn space etc.

Keywords: Closed fuzzy graph, strongly closed graph, fuzzy Urysohn space.

Some Notes on the Fuzzy Pseudo-normed Spaces

Ma Ming

Department of Mathematics, Harbin Institute of Technology, Harbin 150006, China

Fan Ying

Department of Basic Science, Harbin University, Harbin 150020, China

Abstract: In 1988, N.N. Morsi initially introduced the definition of fuzzy pseudo-normed vector space(J. of Fuzzy Sets and Systems,27(1988),351-372). In this paper, we are going to investigate the criteria of bounded fuzzy set, -convergent fuzzy net, equivalent fuzzy pseudo-norms in the fuzzy pseudo-normed vector spaces.

Keywords: Fuzzy pseudo-normed vector spaces, fuzzy norm, bounded fuzzy set, -convergent net.

Separation Axioms in Fuzzy Topology Fuzzy Convexity Spaces

M. V. Rosa

Department of Mathematics and Statistics Cochin University of Science and Technology Cochin 682 022, India.

Abstract: In this paper we introduce separation axioms in fuzzy topology fuzzy convexity spaces (stfcs) as a continuation of the study of ftfcs introduced in [5].Separation involves closed convex fuzzy neighborhoods and we study subspace, product and quotient in this context.

Keywords: Convex fuzzy set, fuzzy convexity spaces, fuzzy subspace, fuzzy topology fuzzy convexity spaces.

Hausdorff Distances Between Fuzzy Sets

S. G. Gal

Department of Mathematics,University of Oradea Str. Armatei Romane 5,3700 Oradea, Romania

Abstract: In this paper we discuss some properties of the Hausdorff distance between two fuzzy sets introduced in [4]. Some generalizations also are given and an application to fixed point theory is obtained. Finally, the Kuratowski¡¯s measure of noncompactness for fuzzy sets is introduced.

Keywords: Hausdorff distance between two fuzzy sets, fuzzy sets valued mapping, fixed point, probabilistic Hausdorff distance between two fuzzy sets, Kuratowski¡¯s probabilistic function of a fuzzy set, Kuratowski¡¯s measure of noncompactness for fuzzy sets.

Fuzzy Classification Networks

Kaoru Hirota and Witold Pedrycz

Dept.of Control Systems Engineering College of Engineering, Hoset University Koganet-city, Tokyo 184, Japan

Witold Pedrycz

Dept.of Electrical&Computer Engineering University of Manitoba Winipeg, Manitoba, Canada, R3T 2N2

Abstract: The paper proposes a distributed computational structure called a fuzzy classification network. Unlike existing architectures and algorithms of pattern recognition the network allows for an explicit representation of domain knowledge about the classification problem while maintaining its learning capabilities. The network blends useful properties of knowledge-based systems (namely explicit knowledge representation) with the advantages of neural networks(viz. leaening). The network is composed of basic AND and OR neurons.

Keywords: Classification network, logic neurons, domain classification knowledge, learning

On Non Associative Binary Operations on [0,1]

G. Mayor and T. Calvo

Dept.of Mathematics and Computer Science Universitat de les Illes Balears Palma de Mallorca, Spain

Abstract: Motivated by the study of generalized logical connectives for Fuzzy Set Theory, several non associative operations have been investigated. Our aim in this paper is the study of a method to increase the associativity degree of a binary operation on [0,1].

Keywords: Unite real interval, associativite binary operation, partial order.

On Convexity of Fuzzy Sets

M. Mashinchi

Department of Mathematics, Kerman University, Kerman, Iran.

Abstract: A generalized notion of fuzzy convexity(concavity) is introduced and several generalized set-theoretic operations on fuzzy sets which preserve fuzzy convexity are introduced.

Keywords: Lt-norm, Lt-conorm, fuzzy convex(concave), fuzzy relations, n-fold fuzzy set, fuzzy knowledge.

Boltzmann Machines for Fuzzy Optimization

Armando Blanco, Miguel Delgado, Ignacio Requena, and Jose L. Verdegay Dpto.

Cienclas Computation E Inteligencia Artifical Facultad Ciencias. Universidad De Granada,18071-Granada, Spain

Abstract: In this work, we apply Artifical Neural Networks (ANN), in particular, the Boltzmann Machine (BM ) model to Fuzzy Optimization Problems. Furthermore, we present a modification of the BM model, which takes into account bivaluated variables instead of binary variables, showing that the values of the variables participate in the Energy Function of the BM. We also show two examples solved using these methodologies. On the one hand, we apply the BM model to the maximal cut problem in a graph with fuzzy weights on the arcs. On the other hand, we apply the modified BM model to the maximal independent set of nodes problem when the nodes have values which are trapezoidal fuzzy numbers.

Keywords: Boltzmann Machine, Combinatorial Optimization, Fuzzy Optimization