THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 12, Number 3, September 2004

CONTENT

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Fuzzy Positive Implicative Hyper -ideals of Hyper -algebras

Xiao Long Xin

Department of Mathematics, Northwest University Xian 710069, P.R. China e-mail: xlxin@nwu.edu.cn

Abstract: We introduce the concept of fuzzy positive implicative hyper -ideals and investigate some related properties. We give a relation between a fuzzy positive implicative hyper -ideals. We also state the characterization for a fuzzy positive implicative hyper -ideals in term of it¡¯s level hyper -ideals. We give some equivalent conditions for a fuzzy subset becoming a fuzzy positive implicative hyper -ideals.

Keywords: hyper -algebras, (Fuzzy) hyper -ideals, (Fuzzy) positive implicative hyper -ideals.

Hausdorffness in Intuitionistic Fuzzy Topological Spaces

Francisco Gallego Lupianez

Department of Geometriay Topologia, Facultad de Mathematics, Universidad Complutense de Madridl, 28040, Madrid, Spain E-mail: fg_lupianez@mat.ucm.es

Abstract: The basic concepts of the theory of intuitionistic fuzzy topological spaces have been defined by D. Coker and co-workers. In this paper, we define new notions of Hausdorffiness in the intuitionistic fuzzy sense, and obtain some new properties, in particular on convergence.

Keywords: Intuitionistic topology, separation, Hausdorffiness.

A Parametric Approach for solving the Multicriteria Linear Fractional Programming Problem

M. L. Hassian and H.A.K halifa

Mathematics Department, Faculty of Education, Kafr El-Sheikh Branch, Tanta University

Kafr EI-Sheikh, Egypt E.Ammar Mathematics Department, Faculty of Science, Tanta University Tanta, Egypt Email: Ammar_al_Saeed@Hotmail.com

Abstract: In the present paper, we will apply the method of Barros [2] which is proposed to reduce the multicriteria linear fractional programming problem into a single linear fractional one, and hence use the Wolf[5] parametric approach to solve it. An illustrative is included in support of the approach developed.

Keywords: Multicriteria linear fractional programming problem, efficient solutions, weighting problem, parametric approach.

Vagus Sets, Fuzzy Bags and L-Fuzzy Sets

Liu Zhenyi

Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 610031, P.R. China

Tu Wenbiao

Department of Mathematics, Nantong Teachers College, Nantong, Jiangsu,226007, P.R. China

Keyun Qin

Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 610031, P.R. China

Abstract: This note is devoted to the discussion of the relationship between the concepts of vague sets, fuzzy bags and the concepts of -fuzzy sets respectively. We proved that vague sets and fuzzy bags are special -fuzzy set with being defined appropriately.

Keywords: Fuzzy sets, -fuzzy sets, vague sets, bags, fuzzy bags.

The Uniform Boundedness Principles for Fuzzy Normed Spaces

Xiao Jian-zhong and Zhu Xing-hua

Department of Mathematics, Nanjing University of Information Sciences and Technology Jiangsu 210044, P.R. China

Abstract: In this paper, we introduce the notions of bounded set, semi-bounded set and non-unbounded set in a fuzzy normed linear space. Using these notions we describe and study the boundedness of linear operators from one fuzzy normed linear space into another. Moreover, we establish some uniform boundedness principles relating to bounded operator, non-unbounded operator and semi-bounded operators and obtain their united forms including the cases of classical normed spaces and Menger probabilistic normed spaces.

Keywords: Fuzzy analysis, fuzzy normed linear space, bounded operator, semi-bounded operator, uniform boundedness principle.

The Semigroup of Fuzzy Points

Yong Ho Yon

Department of Mathematics Chungbuk National University Cheongju 361-763, Korea

Kyung Ho Kim

Department of Mathematics Chungbuk National University Chungju 380-702, Korea

Abstract: The set of all fuzzy points in a (regular) semigroup is a semigroup with the produc¡°¡£¡±of fuzzy sets. Here we will show that is a fuzzy ideal (bi-ideal) of if and only if is an ideal (bi-ideal) of , where is the set of all fuzzy points contained in , and introduced the relation between a fuzzy ideal generated by a fuzzy point in and an ideal generated by a point in .

Keywords: Fuzzy set, Fuzzy subsemigroup, Fuzzy point, Fuzzy ideal, Regular semigroup, Fuzzy bi-ideal

The Numerical Solutions for the Rotational Motion of a Gyrostat in a Newtonian Field of Force

A.I. Ismail

Mathematics Department. Faculty of Science, Tanta, Egypt

Abstract: The scope of the present paper is to provide numerical solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian field when the potential function is . We assume that the centre of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following. and . The required solutions are investigated when the third component of the total angular momentum is different from zero . If we cancel the third component of the gyrostatic momentum , the obtained solutions are valid for rigid bodies.

Keywords: Dynamics of rigid bodies, Periodic orbits, Spinning tops and gyroscopic motion, Mechanics of the gyroscope.

On Simple Extensions of Fuzzy Topologies

Sunil C. Mathew

Department of Mathematics St. Thomas College Pala Arunapuram P.O. 686574 Kottayam (Dt.), Kerala India

T.P. Johnson

Cochin University College of Engineering Kuttanad, Pulincunnu P. O. Alleppey 688504 Kerala, India

Abstract: In this paper we study simple extensions of fuzzy topologies and elucidate some of their properties. We also investigate on conditions under which certain properties of a fuzzy topologies spaces are carried over to its simple extensions. Certain characteristics of the lattice of all fuzzy topologies on a given set are an application of the concept of simple extensions.

Keywords: Simple extensions, R-open.

Closedness and Compactness in Characterized Spaces

A.S. Abd-Allah and M. El-Essawy

Department of Mathematics, El-Mathematics, Faculty of Science, El-Mansoura University, El-Mansoura, Egppt

Abstract: Contrary to the characterizing notions for characterized spaces develop in [2], we are investigated the notions which have the same invariance properties and which, as is shown, only depend on -converging fuzzy filter. Examples are the notions of -adherence point, of projective -closedness and of -closedness of a fuzzy filter. By restricting to valued and superior principle fuzzy filter we get analogous notions for fuzzy sets. The -closures of a fuzzy filter are defined in some sence complementarily to the notion of -fuzzy neighborhood of a fuzzy filter. A further notion considered here which has the same invariance properties and only depend on -converging fuzzy filter is that of -compactness. In general, -compactness is defined for fuzzy filters. A fuzzy set is called -compact if the related superior principle fuzzy filter is -compact. The -compactness of the whole space means the -compactness of the constant fuzzy set with value 1. In the classical case of a topological space this is the usual compactness. There are generalized some classical results on compactness. For instance, a generalized Tychonoff Theorem is presented.

Keywords: Fuzzy filters, characterized spaces, -fuzzy neighborhood, -converging, -adherence points of a fuzzy filter and fuzzy sets, projective -closedness and -closedness of a fuzzy filter and fuzzy sets, -compactness for fuzzy filters, for fuzzy sets and for characterized spaces.

On Rectangular Fuzzy Games

Shobana Devi, C.K.

Department of Mathematics, D.B .College, Talayolaparambu, Kerala, India.

A.Kuriakose

Department of Mathematics, U.C. College, Alwaye, Kerala, India.

Abstract: In this paper, we propose a different definition of a fuzzy number that can be more effectively used for defining payoff value in a two-person zero sum game. With respect to this payoff value of a necessary and sufficient condition for the existence of a saddle point is proved.

Keywords: Two-person zero sum game. Payoff value. Fuzzy rectangular game. Saddle point. Fuzzy number.

More about Fuzzy Uniform Spaces-Covering Approach

G. K. Chandrika

Department of Mathematics, Avinashilingam Deemed University, Coimbatore, India.

Abstract: The covering approach to the definition of uniform spaces by Isbell [2] was generalized to fuzzy situation in the paper entitled¡°Fuzzy Uniform Spaces-Covering Approach¡± [1]. The present paper is a continuation of the study of the concept of fuzzy uniform spaces. It deals with properties of normal family of -coverings and construction of coarsest and finest fuzzy uniformities.

Keywords: -star refinement, normal family, normal sequence, coarest and finest fuzzy uniformities.

On An Equivalence of Fuzzy Functions

V. Murali Department of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 South Africa

Abstract: This paper considers an equivalence relation on the set of all fuzzy functions, from a given non-empty set to another non-empty set. All fuzzy subsets are taken to have a finite number of degrees of membership values being taken from the unit interval. We study the effects such an equivalence have on images and pre-images of fuzzy functions, and characterize equivalence in term of alpha-cuts of such fuzzy subsets.

Keywords: Fuzzy functions; Fuzzy equivalence relation; Equivalence relation

Fuzzy Ideals in Subcommutative Semirings

P.Dheena and S. Coumaressane

Department of Mathematics

Annamalai University, Annamalainagar-608002, Tamilnadu, India

Abstract: In this paper we define fuzzy nilpotent ideal and we have shown that characteristic function is fuzzy nilpotent ideal of iff is nilpotent ideal of . We have shown that a semiring is regular and subcommutative iff (i) Every fuzzy ideal of is idempotent and (ii) For every , is a fuzzy ideal and for and , for some . Finally we have shown that any fuzzy ideal of a strongly regular ring is fuzzy maximal ideal iff it is a fuzzy prime ideal.

Keywords: Idempotent, regular, subcommutative semiring, strongly regular, fuzzy ideal, fuzzy ideal, fuzzy prime ideal, fuzzy maximal ideal.

On Intuitionistic Fuzzy Soft Sets

Pabitra Kumar Maji, Akhil Ranjan Roy

Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, West Bengal, India

Ranjit Biswas

Department of Computer Science and Engineering Hamdard University Hamdard Nagar, New Dhlhi-62, India

Abstract: In this paper we define some new operations on intuitionistic fuzzy soft sets and establish some results on them.

Keywords: Soft set, intuitionistic fuzzy soft sets.

The Dynamical Fuzzy Topological Space and Its Folding

M. EI-Ghoul

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

H.I. Attiya

Basic Science Department, College of Industrial Education, Ministry of Higher Education, Benisuef, Egypt

Abstract: In this paper we will introduce the effect of growth and declination on the dynamical fuzzy topological space. Folding the dynamical fuzzy topological space is duduced.

Keywords: Growth, declination, folding.

The Minus Partial Order in Fuzzy Matrices Matrices

AR. Meenakshi and C.Inbam

Department of Mathematics, Annamalai University Annamalai Nagar-608 002, India

Abstract: In this paper, we study the minus ordering for fuzzy matrices, analogous to that of minus partial ordering for complex matrices and prove that the minus ordering is a partial ordering in the set of all regular fuzzy matrices. Some properties of minus ordering are derived.

Keywords: Fuzzy matrices; partial ordering; generalized inverse.

On The COVERS in The Lattice of Lattice of Fuzzy Topologies

Sunil C. Mathew

Department of Mathematics, St.Thomas College Pala Arunapuram P.O. 686 574, Kottayam (Dt.), Kerala, India e-mail: sunilcmathew@rediffmail.com

T. P. Johnson

Cochin University College of Engineering Kuttana,Pulincunnu P.O., Alleppey 688 504, Kerala, India

Abstract: In this paper we study the concept of covers in the lattice of fuzzy topologies and obtain certain necessary and sufficient conditions for a fuzzy topology to have a cover in the lattice of fuzzy topologies. We also obtain certain family of fuzzy topologies in which each member is a cover some other fuzzy topology.

Keywords: Well-closed, nearly crisp, quasi-generalized closed, quasi-separated, static fts.

Fuzzy Vector, Bases and Matrices

Zun-Quan Xia and Fang-Fang Guo

CORA, Dalian 116024, China E-mail: zqxiazh@dlut.edu.cn, gracewuo@163.com

Abstract: In this paper, fuzzy vectors and fuzzy bases are bases are redefined in terms of fuzzy points. It is proved that the different definitions of fuzzy bases given in the literature are equivalent. Orthogonal fuzzy bases and fuzzy matrices are defined and some related properties and operations are presented.

Keywords: Fuzzy vector, fuzzy linear space, fuzzy basis, fuzzy matrix.

Minimality and Homogeneity in Fuzzy Spaces

Samer AI Ghour

Department of Mathematics and Statistics Jordan University of Science and Technology Irbid 22110, Jordan

Ali Fora

Department of Mathematics Yarmouk University Irbid 56, Jordan

Abstract: In this paper we are going to discuss the concept of minimality in topological spaces and concentrate on fuzzy topological spaces. We shall discuss the direct image and the inverse image of open (fuzzy open) sets under continuous(fuzzy continous) functions. We shall show that homogeneity in fuzzy topologies spaces forces the shape of minimal fuzzy open sets. We shall compare minimal fuzzy open sets and minimal open sets in a topology generated by given fuzzy topological space.

Keywords: Homogeneity, minimality, fuzzy topological spaces.

The Most General Fuzzy Dynamical Manifold and Its Folding

M.EL-Ghoul

Mathematics Department, Faculty of science, Tanta University, Tanta, Egypt

H.EI-Zhony and S.I. Abo-EL-Fotooh

Mathematics Department, Faculty of science, AL-Azhar University, for girls, Cairo, Egypt

Abstract: In this paper, we will introduced a new general type of some fuzzy manifolds, it is the fuzzy dynamical manifold. The relation between the retractions and the folding of these types of fuzzy dynamical manifolds are discussed. Theorems governing these relations are obtained.

Keywords: Fuzzy manifold, Fuzzy retractions

Remarks on Continuous Fuzzy Numbers

Taihe Fan

Dept. of Mathematics, Ningbo University Ningbo, Zhejiang, 315211, China

Abstract: First, the result of [1] is generalized to general metric space. Then it is proved that fuzzy numbers can be approximated via levelwise metric by piecewise linear fuzzy numbers to any accuracy. In both cases our method of approximation is finite in nature. Finally, relations between fuzzy numbers with Zadeh¡¯s extension principle are discussed.

Keywords: Piecewise linear fuzzy set, Hausdorff metric, approximation.