THE JOURNAL OF FUZZY MATHEMATICS
  Introduction of The Journal   Instructions to Authors   Archive

The Journal of Fuzzy Mathematics

volume 5,Number 3, September 1997

CONTENT

DETAILS

Is the Axiom of choice True for Fuzzy Sets?

Nistala V. E.S. Murthy

Mig-B-171 Sagar Nagar Visakhapatnam-530043 A.p.State,India

Abstract: The definition of fuzzy sets is generalized in such a way that the product of nonempty fuzzy sets is nonempty .The definition of fuzzy map is generalized in such a way that they can be considered between fuzzy sets with truth values in a different lattices. Further, the images and inverse images of these maps are so generalized that they have several of the usual properties that their crisp counterparts have.

Keywords: Fuzzy sets, fuzzy map, axiom of choice, category, functor, monomorphism, epimorphism, equalizer.

G-compatible Fuzzy Matrices

F.I. Sidky

Department of Mathematics, Faculty of Science University of Zagazing, Zagazing, Egypt

Abstract: In this paper we define left (right) G-compatible and G-min compatible fuzzy matrices and prove some of their fundamental properties.

Keywords: Fuzzy matrix, left (right) G-compatible fuzzy matrix, G-min compatible fuzzy matrix.

On Fixed Fuzzy Set Theorems in Fuzzy Metric Spaces

Ranjit Biswas

Department of Mathematics, India Insititute of Technology Kharagpur-721302, West Bengal, India

Abstract: This paper is sequel to [2]. Some fixed fuzzy set theorems are proved in fuzzy metric spaces. A generalized fixed fuzzy set theorem is proved which generalized all the fixed fuzzy set theorem proved in [1] and [2].

Keywords: Fuzzy metric,fuzzy metric space, F-convergence, F-Cauchy¡¯s sequence, F-complete fuzzy Metric space, F-continuity, fixed fuzzy set, F-open sphere, F- closed sphere, F-neighborhood, Fuzzy -chain, fuzzy ( , )-uniformly locally contractive mapping, fuzzy -contractive, fuzzy local power contraction mapping.

On Stone Theorem for De Morgan Algebra

Congwen Luo

Department of Mathematics, Wuhan university Wuhan 430072, P.R.China

Xuewen Ma

Experiment Centre, Xinyang Teachers¡¯ College Xinyang 4644000,Henan, P. R. China

Abstract: In this paper, we obtain the Stone theorem in De Morgan algebras, and using this theorem, we give a necessary and sufficient condition that a De Morgan algebra becomes Kleene algebra.

Keywords: De Morgan algebra, quasi-Boolean ring, Kleene algebra, reversible element, Boolean complement.

On pairwise Hausdorff Fuzzy Bitopological Spaces

Rekha Srivastava, Manjari Srivastava

Department of applied Mathematics, Insitute of Technology Banaras Hindu University, Varanasi 221005, India

Abstract: Here we have studied eleven possible defining concepts of pairwise Hausdorffness in fuzzy bitopological spaces including four already existing in literature.

Keywords: Fuzzy bitopological spaces, fuzzy P-Hausdorffness, fuzzy P-continuity, goodness of extension.

The Properties of Congruence Ideal of Soft Algebra and the Stucture of Quotient Soft Algebra

Wang Zhuzheng-qun

Yichang Teachers¡¯ College, China,443000

Abstract: If (b] is a congruence ideal of a soft algebra F, is the smallest congruence whose kernel is (d]. In this paper, we give a necessary and sufficient condition such that[ (b] ] is a congruence ideal of a quotient soft algebra F/ . We give a structure of quotient soft algebra F/ and characterize absolutely indecomposable soft algebra.

Keywords: soft algebra; congruence ideal; quotient soft algebra; Boolean kernel; absolutely indecomposable soft algebra.

On Choosing Non-dominated Fuzzy Numbers

Michael Wagenknecht

Karl-Liebknecht-Str.15 01844 Neustadt, Germany

Abstract: For a giver set of fuzzy numbers we determine non-dominated ones by a two-stages procedure starting form partial ordering of intervals resettling in a global fuzzy preference relation which is tacked applying Orlovsky¡¯s approach.

Keywords: Fuzzy numbers, partial ordering, non-dominated elements

Fuzzy Homological Theory in -module (¢¢)

Zhao Jianli

Institute of Fuzzy System, Liaocheng Teachers College Shandong 2520359, P.R.China

Abstract: In this paper, We introduce the concept of F-tensor product of module- and -module, give concrete structure and some important properties of its.

Keywords: -module, module- , F-homomorphism, fuzzy abel group, F-tensor product

Fuzzy Analytic Hierarchy Process to Solve a Multi-objective Transportation Problem

Rakesh Veema*, M.P. Biswal and A.Biswas

Department of Mathematics Indian Institute of Technology Kharagpur-721320,India

Abstract: In this paper, fuzzy Analytic Hierarchy Process is applied to solve a multi-objective transportation problem, where the objective functions are not equally important as well as fuzzy. In fuzzy Analytic Hierarchy Process, we first formulate fuzzy pairwise comparison matrice by comparing the alternatives with the alternatives with respect to the overall satisfaction of the objectives and comparing objectives with the objectives with respect to each of the alternatives. We use triangular fuzzy numbers to to present the relative strength of the elements in the fuzzy pairwise comparison matrices. To estimate the eigenvector (corresponding to maximal eigenvalue ) of matrices, we utilize interval arithmetic, together with optimism of index .Assigning the estimated weights eigenvector, we estimate the weights of the objectives. Assigning the estimated weights to the respective objectives, we formulate a new objective, then the problem is solved by a transportation algorithm. A numerical example is presented to illustrate the methodology.

Keywords: Multi-criteria decision making, transportation problem, fuzzy number, , interval arithmetic, fuzzy analytic hierarchy process.

Random Completely Generalized Strongly Nonlinear Quasi-complementarity Problems for Random Fuzzy Mappings

Nan-jing Huang

Department of Mathematics, Sichuan University Chengdu, Sichuan 611064, P.R. of China

Abstract: In this paper, we introduce and study a new class of random completely generalized strongly nonlinear quasi-complementarity problems with non-compact valued random fuzzy mappings and construct a new iterative algotithm for this kind of random fuzzy quasi-complementarity problems. We also prove the existence of random solutions for this class of random fuzzy quasi-complementarity problems and the convergence of random iterative sequence generated by the algorithm.

Keywords: Complementarity problem, random fuzzy mapping, algorithm, existence, comvergence.

Fuzzy Mixed Integer Linear Programming With Nonequivalent Objectives

R.K.Dwivedi

Department of Mathematics, Sindri College Sindri, Dhanbad-828122,India

M.Chakraborty

Departmernt of Applied Mathematics, India School of Mimes Dhanabad-826004,India

D.Singh

Department of Mathematics,Bithar Institute of Technology Sindri,Dhanbad-828123, India

Abstract: This paper investigates a fuzzy linear mixed integer programming approach to avecor maximum problem with non-equivalent objectives. The overall achievement function is taken as a convex combination of the fuzzy objectives, where the coefficients of the membership functions serve the purpose of weights to reflect the relative importance of the objectives. The decision variables are restricted to take integer values while the membership functions are allowed to take fractional values too. The methodology is illustrated with sample vector maximum problems.

Keywords: Mixed integer, fuzzy objectives.

Research for a Geometric Programming Model with T-fuzzy Variations

Cao Bing-yuan

Department of Mathematics, Changsha University of Electric Power, Changsha,410077,China

Abstract: A geometric programming model with T-fuzzy variations is first built after the T-fuzzy variation is introduced to geometric programming in this paper. First, the model is turned into an ordinary geometric programming depending on the cone index J and a dual form with T-fuzzy variation is obtained corresponding to a prototype problem. Second, relation is developed between optimal solutions of prototype posynomial geometric programming is verified with numerical example.

Keywords: T-fuzzy variations, posynomial geometric programming, dual, cone indexn, optimal solution.

W-Ideals and W-Soft Algebras

Wang Zheng-qun

Yichang Teachers College Hubei, China 443000 Pei Li-wen Wuhan University Wuhan, China 430072

Abstract: In this paper, we define a ideal and soft algebra. Our main results are as following.; (1) Let (d], ( ] be congruence ideals of a soft algebra F and . If (d] is a ideal, then ( ] is also a ideal. (2) We get some sufficient and necessary conditions such that soft algebra and quotient soft algebras are ideal algebras respectively.

Keywords: Soft algebra, congruence, congruence ideal, ideal, soft algebra.

Mathematical Programming Problem with Fuzzy Coefficients -A General Analysis

Stefan Chanas

Institute of Industrial Engineering and Management, Technical University of Wroclaw,Poland

Bronislaw Florkiewicz

Institute of Mathematics, Technical University of Wrolaw, Poland

Abstract£º In this paper the problem of mathematical programming is analyzed, in which the coefficients are intervial fuzzy numbers. The main aim of this analysis is an answer to the question: what do we have assume about the problem so that the possibility distribution of the optimal value of the objective function is of the same type as the possibility distribution generated by the fuzzy coefficients of the problem, i.e., when the optimal value of the objective function is also an interval fuzzy number.

Keywords: Mathematical programming, fuzzy number, possibility distribution.

Category of fuzzy Semi Inner Product Spaces

T.V.Ramakrishnan School of Mathematical Sciences Cochin University of Science and Technology Cochin, 682022, India

Absrtact: In this paper the concept of the category of semi inner product spaces and that of fuzzy semi inner product spaces are introduced. Relation of the category of fuzzy semi inner product spaces with the categories of semi inner product spaces, fuzzy topological spaces and topological spaces are studied. We conclude with a more general approach to fuzzy semi inner product spaces by introducing the category of semi inner product in a given concrete category.

Keywords: Semi inner product space, fuzzy topological spaces, concrete category.

compatible Fuzzy Matrices

Jin Bai Kim

Department of Mathematics, West Virginia University Morgantown, WV26506

Kya Hyuck Choi*

Wonkwang University Iksan 570-749, Korea

Abstract: We study G-compatible fuzzy matrices, where G is a group. We also study compatible fuzzy matrices where denotes the symmetric group. We consider matrices ,where are group elements .

Keywords: G-compatible fuzzy matrices, matrices over elements.

The Expression Theorem and Extension Principle with the Form of Intersection

Liu Huawen

Departmernt of Applied Mathematics and phsics Shandong University of Technology Jinan 250061, P.R. China

Abstract£º In this paper, we establish the expression the theorem and extension principles with the form of intersection which are based on the intersection decomposition of fuzzy sets. Further, we give the corresponding results of poly-ary extension and extension of fuzzy mapping.

Keywords: Intersection decomposition, expression theorem, extension principle, poly-ary extension principle, fuzzy mapping

Existence and stability of Fuzzy Differential Equations

Zouhua Ding and Abraham Kandel

Computer Science and Engineering Department University of South Florida Tampa, FL33620-5350,USA.

Abstract: In this paper we study (1) the existence of the solutions of fuzzy differential equations with local Lipschitz condition, (2) the stability of the solutions of fuzzy differential equations in real line and (3) the relationship between fuzzy differential equations and differential inclusions.

Keywords: Fuzzy differential equation, existence, stability, differential inclusion.

FNN Optimum Seeking Decision Making Method

Dengfeng Li

Department of Basics, Dalian Naval Academy Dalian 116018,P.R. China

Shouyu Chen

Department of Civil Eengineering, Dalian University of Techology Dalian116023,P.R. China

Abstract: This paper investigates multiattribute decision making problems of the decision maker preference information (i.e.,weight) which is unknown a prior. Commethod the conventional BP learning algorithm and fuzzy optimum seeking model, a fuzzy neural networks comprehence optimum seeking decision making method is established for multiattribute problems. This method can adjust the weights of objectives automatically. Therefore,the determination of weights is more objective and more exact. The method proposed in this paper is also capable of self-learning. Making comparison and analysis between the fuzzy neural networks comprehence optimum seeking model is required to solve the inverse problem of nonlinear fuzzy relation equation systems for obtaining process is difficult and is not easy to implement. Therefore, the fuzzy neural networks comprehence optimum seeking method presents a new effective way to obtain the knowledge and experience of the decision maker. Finally, the new method is applied to the evaluation and decision making of industry economics in Dalian city. The result is more satisfying. So the method proposed in this paper is effective and may be applied to other fields in future.

Keywords: Fuzzy sets, neural network, weight, membership grade, decision making, expert system.

A Necessary and Sufficient Condition of a Discrete Soft Algebra Is a -soft Algebra

Zhueng-qun Wang

Yichang Teachers¡¯ College Hubei 443000 P.R. China

Abstract: The main results in this paper are as follows: 1) The discrete soft algebra is -soft algebra if and only if for every join irreducible element P, if P covers ( ) where is the greatest congruence whose kernel is (0]. 2) The discrete soft algebra F is a -soft algebra if and only if L(J(F)) 1 where J(F) denotes the poset of all nonzero join-irreducible elements of F. L(J(F)) denotes the length of the poset J(F).

Keywords: Discrete soft algebra, -soft algebra, congruence, congruence ideal, join-irreducible element.

The Derived Element of Element and the Derived Operator in TML

Jian-bing Shi

Nanjing Power Institute, 78 Ban Cang Cun Nanjing 210042,P.R.China

Abstract: In this paper,first we discuss the distributional rules of the accumulation points of elements in TML. Then we give the conclusion in a general TML similar to the C.T. Yang`s theorem concerning derived set in general topology. Finally,we introduce the concept of derived operator on molecular lattice and by which we characterize the TML.

Keywords: TML, accumulation point, derived element, derived operator.

An Existence Theorem for a Fuzzy Function Integral Equation

P.V. Subrahmanyan and S.K.Sudarsanam

Department of Mathematics, India Institute of Technology Madras 600 036, India

Abstract: In this paper we solve the following fuzzy functional integral equation Where h and g are given fuzzy functions and and are self-maps on [0,1]and is the unknown function to be found out. We use Krasnoselskii`s fixed point theorem to solve the above functional integral equation on a suitable subset of the space of fuzzy functions.

Keywords: Fuzzy function, fuzzy functional equation, fuzzy numbers, Zadeh`s extension principle, fuzzy solution, fixed point, embedding theorem for fuzzy sets and Krasnoselskii`s fixed point theorem.

Extending -connectedness to Fuzzy Subsets

Gunther Jager

Lessingstr. 13, D-76135 Karlsruhe, Germany

Abstract: In order to view connectedness in fuzzy topological spaces in Preuss` sense by continuous functions to certain test spaces being constant, Lowen and Srivastava introduced the concept of -connectedness. However this concept can only be applied to crisp fuzzy subsets. In this paper their definition is modified in a way that it is also applicable to non-crisp fuzzy subsets. The definition obtained moreover generalizes connectedness in the definition of Pu and Liu even for fuzzy subsets and is an absolute property.

Keywords: Fuzzy topology, connectedness, fuzzy subsets

Factorization of Fuzzy Ideals in Dedekind Domains

Kyoung Hee Lee*

Department of Liberal Arts Korea Institute of Technology and Education Byungcheon, Chungnam 333-860, Korea

John N. Mordeson

Department of Mathematics/Computer Science Creighton University Omaha, Nebraska 68178 USA

Abstract: We show that in a Dedekind domain, every fuzzy ideal can be factored uniquely as a product of fuzzy maximal ideals.

Keywords: Dedekind domain, fuzzy ideals, fuzzy maximal ideals, factorization, algebraic integer.

On Fuzzy Weak Continuity, Fuzzy Almost Continuity and Fuzzy Precontinuity

S.P. Arya

Maitreyi College, Bapudham Complex Chanakyapuri, New Delhi-110021, India

Neeru Jain

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

Abstract: The independence of fuzzy weak continuity [2]and fuzzy precontinuity [6] has been proved by El-Monsef, Hanafy and El-Deeb [4]. The purpose of the present paper is to determine conditions under which each will imply the other and to prove the independence of fuzzy almost continuity[2] and fuzzy precontinuity. We also give an example of a fuzzy non-continuity function which is both fuzzy almost continuous and fuzzy precontinuous. Some new characterizations for fuzzy weak continuity, fuzzy precontinuity and fuzzy preopen functions [7] have been establishd. We also introduce a new weaker form of fuzzy continuity which is strictly stronger than fuzzy weak continuity but is independent from fuzzy almost continuity.

Keywords: Fuzzy weak continuity, fuzzy almost continuity, fuzzy precontinuity

Model of Fuzzy Closed-loop Control System

Ouyang Xiantai

Department of Automation Engineering Wuhan Chemical Engineering Institute Wuhan, Hubei 430073, P.R.China

Shi Kaiquuan

Department of Automation Engineering Shandong University of Technology Jinan, Shangdong 250061, P.R.China

Abstract: This paper analyzes the structure of fuzzy controller and some examples of application, based on which this paper proposes the closed-loop model of fuzzy control.

Keywords: Fuzzy control, fuzzy relation matrix, model of closed-loop.