THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 6, Number 4, December 1998

CONTENT

DETAILS

On Th Measuring Problem of Fuzzyiness

Li Fachao, Zhang Guangquan, Su Liangqing and Qiu Jiqing

Department of Basic Sciences,

Hebei University of Science and Technology, Shijiazhuang, P.R.China

Abstract: In this paper, by taking classical measure spaces as foundation and geometry as background and by means of analysis, we establish several metric method of fuzzy set fuzziness. First we extend the concept of fuzzy degree, establish the total mean fuzzy degree and expound its geometric meaning; then basing on the measure space £¨X, , £©,we establish the concepts of demi-fuzzy degree and true mean fuzzy degree,and discuss the relation between true mean fuzzy degree and total mean fuzzy degree; finally we establish the concepts of -fuzzy generated set and -fuzzy degree, discuss some basic properties of them.

Keywords: Fuzzy sets; fuzzy degree; total mean fuzzy degree; demi-fuzzy degree; -fuzzy degree.

Admisssible and K-invariant Fuzzy Sungroups

M.Atif A.Mishref

Department of Mathematics, Faculty of Science

Zagazig University, Zagazig, Egypt

Abstract: Throughout this paper we give a definition for asmissible fuzzy subgroup and K-invariant fuzzy subgroup and prove that their t-level subgroups are also admissiblr and L-invariant respectively and characteristic fuzzy subgroups which were studied in our earlier work [9]. Some of the properties of the characteristic fuzzy are also studied. We also define a characteristic fuzzy subgroups of a fuzzy subgroup ana prove that the characteristic property is transitive. Other related normal property are also proved.

Keywords: Fuzzy algebra, normal fuzzy subgroup, characteristic fuzzy subgroup.

The Convergency of (DG) Fuzzy Integral on L-Fuzzy Sets*

Song Shiji Institute of Physical Oceanography,

Ocean University of Qingdao,Qingdao 266003,P.R.China

Qiao XiaoLin

Weihai College, Harbin Institue of Technology,

Weihai 264209, P.R.China

Zhang Jiewu

Department of Head and Neck, Tumorous Hospital of Heilongjiang province Harbin 150040, P.R.China

Che Zhao Dong

Center of Network, Ocean University of Qingdao

Qingdao266003, P.R.China

Abstract: T he definition of another kind of generalized fuzzy integrals on L-fuzzy sets is introduced by means of the so-called generalized triangular conorm. It is said tobe dual generalized fuzzy integrals on L-fuzzy sets and denoted by (DG) fuzzy integrals on L-fuzzy sets. In this paper, the convergence in fuzzy measure theorems and monotone convergence theorems on L-fuzzy sets are obtained..

Keywords: L-fuzzy sets, generalized triangular conorm, (DG) fuzzy integrals on L-fuzzy sets, convergence theorems.

Multicriteria Decision Making Using Intuitionstic Fuzzy Set Theory

Supriya Kumar De, Ranjit Biswas* and Akhil Ranjan Roy

Department of Mathematics, Indian Institute of Technology

Kharagpur-721302 West Bengal, India

Abstract: In this paper we study a problem of selecting most efficient action out of n alternatives on the basis of m criteria. Our assumption is that the criteria values or ratings are not crisp but intuitionistic fuzzy elements. The method uses the concept of a measure of similiarity between two intuitionstic fuzzy sets, which is defined in this paper. An algorithm for the method is presented and a hypothetical case-study is made.

Keywords: Fuzzy set, intuitionstic fuzzy set (IFS), criteria-value, criteria-matrix, criteria-value set, measure of similarity.

Klein -group Action on a Set of Fuzzy Subsets

D.Talukdar

Department of Mathematics, Nalbari College

Nalbari, Assam, India

Abstract: A set of fuzzy subsets are the mappings from a finite set X into an ordered subset L of [0,1]. It is denoted by . Under the absolute difference operation E is a groupoid, known as inexact groupoid. This inexact groupoid contains aome Klein -groups, which are isomorphic to each other. From the group action , replacing the group G and the set X by a Klein -group K and a set V of fuzzy subsets respectively, we get another group action on fuzzy subsets known as Klein -group actiong on V and V is called Klein -group space or K-space. The orbits and stabilizers of the elements of K-space, K-morphism of two K-spaces and some properties of this space are discussed in this paper.

Keywords: Klein -group action, Klein -group space.

Extension of Fuzzy Positive Linear Operators

Ismat Beg

Department of Marhematics, Kuwait University

P.O.Box 5969, Safat 13060, Kuwait

Abstract: We prove exsitence of an extension of an extension for a fuzzy positive linear operator defined on a majorizing vector subspace of a fuzzy Riesz space and taking values in a complete fuzzy Riesz space. We also obtain the fuzzy analogue of most general form of the Hahn- Banach theorems.

Keywords: Fuzzy positive linear operators, extension, fuzzy Hahn- Banach theorem, fuzzy Riesz space.

Existence of Finitely Additive Nontrivial T-measure on the Family of All Fuzzy Set on a Set

A.I.Ban

Department of Mathematics, University of Oradea,

Ste.Armatei Romane 5,3700 Oradea, Romania

Abstract: Using the Hahn- Banach theorem we prove the exsitence of finitely additive nontrivial T-measure on FS(X) (the family of fuzzy sets on X) which is the extension of the finitely additive nontrival measure on P(X). This T-measure (the t-norm T verfies a certain condition) have the supplimentary properties.

Keywords: t-norm, finitely additive nontrivial measure.

Fuzzy Multilinear Mappings

K.S.Abdukhalikov

Institute of Pure and Applied Mathematics

Pushkin Str 125, Almaty 480021, Kazakhstan

C.Kim

Department of Mathematics, Kwangwoon University

447-1 Wolgye-Dong, Nowoon-Gu, Seoul 139-701, Korea

Abstract: The concept of the fuzzy subspace of a soace of multilineat mapping is defined and investigated. Particular cases of this notion are fuzzy linear and fuzzy bilinear functions, and fuzzy linear maps. Also, fuzzy bases of these subspaces are determined.

Keywords: Fuzzy subspace, fuzzy basis, fuzzy linear functions, fuzzy bilinear functions, fuzzy linear maps, fuzzy multilinear maps.

Linear Fractional Programming Problem: a Fuzzy Programming Approach

Sandipan Gupta and M.Ghakraborty

Department of Applied Mathematics, Inidian School of Mines

Dhanbad-826 004, India

Abstract: The present investigation aims to solve Linear Fractional Programming (LFP) by solving either of the fuzzy models proposed in the paper depending on the sign of the numerator under the assumption that the denominator is non vanishing in the feasible region. Using fuzzy approach two related theorems have been proved. Numerical examples have been presented to support the model.

Keywords: Fractional programming , fuzzy sets, membership function, min operator, optimal solution.

A Dual Concept of Subsethood Measure: Supersethood Measure

Jiu-Lun Fan

Department of Compute Science, Xi¡¯an Institute of Posts and Telecommunications

Xi¡¯an 710061, P.R.China

Wei-Xin Xie

Shenzhen University, Shenzhen 518060, P.R.China

Abstract: In this paper, supersethood measure, a dual concept of subsethood measure, is intriduced. Some properties of supersethood measure are studied and fuzzy entropies induced by supersethood measure are obtained.

Keywords: Subsethood measure, fuzzy entropy, supersethood measure.

Stability of Multi-objective Dynamic Programming Problems with Fuzzy Parameters

Mahmoud A.Abo-Sinna*

Department of Physics&Engineering Mathematics

Faculty of Engineering, EL-Menoufiya University,

Shenbin EL-Kom, Egypt

Abstract: The aim of this paper is to study the stability of a multi-objective dynamic programming (MODP) problems with fuzzy parameters in the objective functions (FMODP). These fuzzy parameters are characterized by fuzzy numbers. To study the stability of such problem under the concept of -pareto optimality, some basic notions in parametric nonlinear programming problems are redefined and analyzed qualitatively. Also, an algorithm for obtaining any subset of the parametric space which has the same corresponding -pareto optimal solution is presented. A numerical example is given for the sake of illustration.

Keywords: Fuzzy multi-objective dynamic programming ; Fuzzy parameters; Fuzzy numbers; -Pareto optimality; Stability.

Fuzzy Symmetric Subgroups and Conjugate Fuzzy Subgroups of a Groups

W.B.Vansantha kandasamy and D.Meiyappan

Department of Mathematics, Indian Institute of Technology

Madras 600 036, India

Abstract: In this paper we study some of the properties of conjugate fuzzy subgroups of a group, introduced by N.P.Mukherjee and P. Bhattacharya in 1986. Mainly we prove the conjugate fuzzy subgroups of a group have same order. Further ,we introduce the notion of conjugate fuzzy relations of a group. Jin Bai Kim and Kyu Hyuck Choi in 1995 introduced the concepts of fuzzy symmteric subgroups and proved some of their properties. Here we prove the order of image of fuzzy symmetric subgroup of (i) is 4 and (ii) is 3 when given by them are not true.

Keywords: Conjugate fuzzy subgroups, fuzzy normal subgroups, conjugate fuzzy relations, fuzzy symmetric subgroups.

On Fuzzy Commutativity

Sandeep Kumar Bhakat

Siksha-Satra; Visv-Bharati;

P.O.-Sriniketan-731236; Dist. Birbhum (W.B) India

Abstract: The notion of q-fuzzy commutativity for an ( ) ¨Cfuzzy subgroup is introduced with some fundamental properties.

Keywords: Fuzzy algebra, ( ) ¨Cfuzzy subgroup, fuzzy normal subgroup, q-fuzzy commutative subgroup.

Remark on Approximate Solutions, Existence, and Uniqueness of the Cauchy Problem of Fuzzy Differential Equations

Song Shiji

Institute of Physical Oceanography,

Ocean University of Qingdao, Qingdao 266003, China

Wu Congxin

Department of Mathematics,

Harbin Institue of Technology, Harbin 150001, China

Abstract: The Cauchy problem of fuzzy deiffererntial equations are investigated in [12], for the fuzzy valued mappings of real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in . In this paper, by using the existence and uniqueness theorem of the solution of the Cauchy problem in [12], the existence and uniqueness theorem in [5] is rewritten, and the extension thorem of solution, and continuous dependence theorem on the initial value of solution are given ,for fuzzy differential equtions.

Keywords: Fuzzy valued mappings, H-differential; fuzzy differential equations.

Function Space in FTS

Gunther Jager*

Lessingetr. L3 D-76135 Karlsruhe Germany

e-mail: gunther. jaeger@dlr.de

Abstract: In this paper we take an idea of Peng as starting poing to examine function spaces in FTS, the category of fuzzy topological spaces. We cite Peng¡¯s construction which covers fuzzy pointwise convergernce and fuzzy compact open topology, prove that both are good extensions of the related classical concepts and discuss the connections to function space structures in FCS, the category of fuzzy convergence spaces. We especially show thar for C(X,Y) there always exists a finest splitting fuzzy topology.

Keywords: Fuzzy topology, fuzzy convergence space, fuzzy function space, spliting structure, conjoining structure, poingcise convergence, continuous convergence, compact open topology.

Boundedness and Fuzzy Sets*

D.N.Georgiou

Department of Planning and Regional Development,

Department of Civil Engineering, Faculty of Technological Science

University of Thessaly, 383 34 Volos, Greece

B.K.Papadopolos

Department of Mathematics, Democritus University

67100 Xanthi, Greece

Abstract: In this paper we introduce the notion of fuzzybounded, fuzzy bounded, fuzzy ( )-bounded sets and fuzzy compact, fuzzy ( )- compact spaces. In examing these aforementioned notions in the present paper we find on the one hand many properties of them whilst on the other , the following applications take place£º£¨ £©the characterization of fuzzy compact spaces through the contribution of fuzzy upper limit and ( ) the characterization of fuzzy bounded through the assistance of fuzzy upper limit. Finally using the notion of fuzzy bounded sets we give separation axioms in fuzzy topological spaces.

Keywords: Fuzzy sets , fuzzy topology, fuzzy upper limit, fuzzy compact spaces, fuzzy bounded sets.

Multiobjective Decision Making Theory and Application of Neyral Network with Fuzzy Optimum Selection

Chen Shouyu

School of Civil Engineer and Architechture

Dalian University of Technology

Ganjingzi District of Dalian, Liaoning 116024, China

Abstract: Combined fuzzy optimum selection theory proposed by the author with neural network theory, this paper provides a rational pattern of determing topologic structure of nerwork: number of hidden layers, number of nodes in hidden layer and stimulation function of nodes. It also provides a weight-adjusted BP model of neural network with fuzzy optimum selection. The capacity of the proposed network is good and the physics of stimulation function of neural cells is also definite. To some extent, this paper develops the theory of neural network.

Keywords: Fuzzy optimum selection, neural network, topological structure of network, stimulation function ,adjusment of weights, multiobjective decision making.

Minimal Topological Moclecular Lattices

Shenggang Li

Department of Mathematics, Shaanxi Normal University,

Xian, 710062, P.R.China

Lining Yang

Department of Mathematics, Yulin College

Yulin, 719000, P.R.China

Abstract: In this paper, we introduced the concept of minimal topological molecular lattice. Using ideal method we characterize this concept and show a method for constrcting a co-topology strictly weaker then a given nonminimal co-topology on a complete and completely distrbutive lattice L.

Keywords: Topological molecular lattice, minimal topological molecular lattice, ideal.

Fixed Point Properties of Fuzzy Negations

Micheal Wagentknecht

University of Applied Sciences Zittau/Goerlitz

IPM, Theodor-Koerner-Allee 16

012754 Zittau, Germany

Ildar Bayrahim Kazan State Technological University

K. Marx st. 68, Kazan, 420015, Russia

E-mail: batyr@emntu.Kcn.ru

Abstract: The convergence properties ( to a fixed point) of negations on [0,1] are studied. For various kinds of negations(weak, contractive,bijective) we derive conditions to ensure convergence of itearated neagations to the fixed point.

Keywords: Fuzzy negations, fixed point convergence.

Minimal Regular Topological Molecular Lattices

Fang Jinming

Department of Applied Mathematics

University of Qing Dao, Qing Dao 266003, China

Abstract: In this paper, the author introduced the concept of minimal regular topological molecular lattice, and investigate characteristic properties of its by regular closed ideal bases. In addition, using the concept of upper layer space, the author answer the rationality of the definition of minimal regular topological molecular lattices.

Keywords£º Topology, co-topology, minimal topological molecular lattice, remote neighborhood, regular closed ideal base, the upper layer space.

On T-convergences of Lattices

F.I.Sidky

Department of Mathematics, Faculty of Science

Zagazig University, Zagazig-Egypt

M.M.Atallah

Department of Mathematics, Faculty of Science

Tanta University, Tanta-Egypt

Abstract: The T-congruences and T convergences of lattices are defined and some of their properties are investigated. In particular a charcaterization of a -convergences is proved.

Keywords: Fuzzy relation, T-equivalence relation, T-compatible, fuzzy relation of a lattice, T-congtuence of lattice.

On Prime Filters and Decomposition Theorm of Lattice Implication Algebras

Liu Jun and Xu Yang

Department of Applied Mathematics, Southwest Jiaotong University,

Chengdu 610031, China

Abstract: Lattice implication algebra is an algebraic syetem which is established by combing lattice with implication algebra. In this paper, properties about the filters of lattice implication algebra were discussed, especially about the prime filter. Applying it ,a decomposition theorem of lattice implication algebras was given.

Keywords: Lattice implication algebra, prime filter, quotient lattice implication algebra, subdirectly decomposition.

Open Mapping Theorem In Fuzzy Normed Spaces

D.Antony Xavier, Magie Jose and M.L.Santigo

Racine Research Centre, Loyola College

Madras-600 034, India

Abstract£º In this paper fuzzy norm on a real or complex vector space is introdued. Fuzzy normed space (called F-normed space) and fuzzy Banach space(called F-Banach space) are defined. Open mapping theorem for fuzzy Banach space is proved.

Keywords: Fuzzy normed space, fuzzy Banach space, open mapping theorem.

Properties of Regular Sets

Shu Lan

Department of Applied Mathematics University of Electronic Science and Technology of China

Mo Zhien

Department of Mathematics Sichuan Normal University P.R.China

Abstract: Regular sets play important role in language theory. In this paper, properties of regular set are provided, and the theory of regular sets is perfected.

Keywords: regular set; finite-state grammer; closure.

Paracompactness on Topological Molecular Lattices*

Yixiang Chen

Department of Mathematics, Shanghai Normal University

Shanghai 200234, P.R.China

Abstract: The aim of this paper is to establish the theory of paracompactness on topological moclecular lattices. The author introduces a kind of paracompactness, called asterisk-paracompactness, by using the pseudo-complement operation on co-Heyting algebra. It has a little of good behaviour, such as being hereditary with respect to closed elements and strenghening the separation axioms.

Keywords: Topological molecular lattices, paracompactness, separation axioms.

The Autocontinuity of Non-additive Lattice-valued Measure on the Lattice

Guangquan Zhang and Yong ¨CHong Wu

School of Mathematics and Statistics ,Curtin University of Technology

GPO Box U1987, Perth, Western Australia, 6845, Australia

{Zhangg, yhwu}@cs.curtin.edu.cn

Abstract: In this paper, the property (SA) and the property (SB) of non-additive lattice-valued measures are introduced. A number of theorems characterising the autocontinuity of non-additive lattice-valued measures are developed and the equivalent condition of autocontinuious form above and autocontinuous form below are given.

Keywords: Non-additive measure, fuzzy measure, autocontinuity from above, autocontinuity from below.