THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 12, Number 4, December 2004

CONTENT

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Fuzzy Random Variable: A Mathematical Tool for Combining Randomness and Fuzziness

M.K.Luhandjula

Department of Quantitative Management, University of South Africa, P.O. box 392, UNSIA, Pretoria 0003, South Africa Algorithm of -resolution Principle Based on An Intermediate Element Lattice-valued Propositional Logic

Dan Meng , Xiaoping Qiu , Yangxu and Keyun Qin

1. School of Economics Information Engineering, Southwest Jiaotong University Chengdu, Sichuan 610031, P.R. China

2. Institute of Image and Graphics, Sichuan University

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

Abstract: In this paper, algorithm of -resolution principle based on an intermediate element Iattice-valued proposition logic is focused. Concretely, some concepts and results related to and is given firstly. Then, the concrete algorithm of resolution based on is presented. It concludes four steps: First , the algorithm of transforming any formula in to a reducible form is given; second , the equivalent or the same-satisfiable transformation is mentioned; Third, method of resolution based on path searching is expounded; Finally, the method of judging if any processed formula is -false is given.

Keywords: Automated reasoning; Resolution principle; Lattice Implication Algebra; Intermediate element lattice-valued propositional loglic.

Fuzzy Reasoning Based on The

Ping Zhou, Xue-fang Wang, Ke-yun Qin and Yang Xu

Dept. of Applied Math. Southwest Jiaotong University Chengdu, 610031, Sichuan, P.R. China

Abstract: In this paper, fuzzy reasoning of modus ponens and modus tollens based on the -valued logic are discussed. The -restoration theorems of modus ponens and modus tollens are proved.

Keywords: -valued logic , fuzzy modus ponens, fuzzy modus tollens, -restoration theorem.

A Few Notes on K-quasi-additive Fuzzy Integrals

Wang Guijun , Li Xiaoping

College of Mathematics Sciences, Tianjin Normal University, Tianjin, 300074, P.R. China.

Abstract: On the basis of the definition of the K-quasi-additive fuzzy integrals and integrals transformation theorem, we prove that this kind of fuzzy integrals constitute the K-quasi-additive fuzzy measures exactly in this paper, and according to integrals transformation theorem, discuss some complemental properties of this kind of the K-quasi-additive fuzzy integrals.

Keywords: Induced operators; K-quasi-additive fuzzy measures; K-quasi-additive fuzzy integrals; countable additivity; absolute continuity.

Solution of A Linear Programming Problem with Fuzzy Data

Makani Das

Department of Mathematics, Assam Engineering College, Guwahati, Assam-781013

Hemanta K. Baurah

Department of Stastics, Gauhati University, Guwahati, Assml-781014

Abstract: The problem of fuzzy linear programming has been studied extensively by various authors who have fuzzified the augmented coefficient matrix and the cost vector. We have in this article attempted to fuzzify the simplex computational procedure and soloved a fuzzy linear programming problem fuzzifying the variable vector also.

Keywords: Fuzzy linear programming, fuzzy constrains, triangular fuzzy numbers.

Simple and Semisimple -modules

Paul Isaac

Department of Mathematics, Cochin University of Science and Techonlogy Kochi-682 022, Kerala, India e-mail: pi@cusat.ac.in

Abstract: In this paper we extend the notions simple modules and semisimple modules to the fuzzy setting and prove some related theorems. If is a regular complete Brouwerian lattice, it is proved that a semisimple -module contains a simple -submodule. Some equivalent conditions for seimisimple of -modules are also obtained. Also if is a semisimple -module, it is proved that the strict -lever subset of is semisimple for every .

Keywords: -module, proper -submodule, strict proper -submodule, simple -module, semisimple -module, direct sum.

On Several Types of -fuzzy Compactness

Y.C.Kim and S.E. Addas

Department of Mathematics, Kangnung National University, Kangnung, Kangwondo 210-702,

Korea

Department of Mathematics, Faculty of Science, Sohag, Egypt

Abstract: In this paper, we introduce the concept of -fuzzy almost compactness and -fuzzy near compactness in Sostak¡¯s fuzzy topological spaces. We study some properties of them under several types of fuzzy continuous mappings and -fuzzy(semi-) regular spaces.

Keywords: Fuzzy topological spaces, -fuzzy(semi-) regular spaces, -fuzzy (almost, near)compactness, fuzzy(semi, weakly semi, almost, weakly) continuous

Fixed Point Theorems in -spaces

S.L. Singh and Anita Tomar

Department of Mathematics Gurukula Kangri Vishwavidyalaya, Hardwar-249404, India E-mail: vedicmri@sancharnet.in

Abstract: In this paper we prove fixed point theorems for maps in -Spaces. These results are good extensions and improvement over several results on metric and probabilistic metric and probabilistic metric spaces as well. These fixed point theorems are applied to obtain a common solution of fixed point type equations on product spaces.

Keywords: -Space, common fixed point, coincidence point, reciprocal continuity, -weakly commuting maps.

Fuzzy Hyper -ideals of Hyper -algebras

Young Bae Jun and Wook Hwan Shim

Department of Mathematics Gyeongsang National University, Chinju(Jinju) 660-701, Korea E-mail: ybjun@gsnu.ac.kr

Abstract: The fuzzification of (weak, strong) hyper -ideals is considered. Relations among fuzzy hyper -subalgebras, fuzzy hyper -ideals, fuzzy weak hyper -ideals, and fuzzy strong hyper -ideals are given.

Keywords: Hyper -algebras, fuzzy (weak, strong) hyper -ideal.

On Fuzzy Probability

Guangwu Meng and Xingfang Zhang

Department of Mathematics and System science, Liaocheng, P.R. China

Abstract: The probability of fuzzy event is redefined. Basic Properties for the Probability are discussed. It is given that relation between new probability and old one defined by expectation method.

Keywords: Fuzzy events, Probability of fuzzy event, expansion method, expansion method.

Generalized Closed Fuzzy Sets in ech-fuzzy Closure Spaces

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 Kuttanad, Pulincunnu P.O. Alleppey 688 504 Kerala, India

Abstract: In this paper we introduce the concept of generalized closed fuzzy sets in ech-fuzzy closure spaces and study some of its properties in relation with certain separation axioms.

Keywords: -closed, well-closed, , regular, quasi-separated.

Fuzzy Supper Irresolute Functions

S.E. Abbas

Department of Mathematics, Faculty of Science, Sohag, Egypt

Abstract: In this paper, the concept of fuzzy super irresolute function is considered and studied in Sostak fuzzy topology[15]. A comparison between this type and other existing ones are established, several characterizations, properties and its effect on some fuzzy topological spaces[15] are studied. Also, a new class of fuzzy topological spaces under the terminology fuzzy -closed spaces in introduced and investigated.

Keywords: -fuzzy semiopen sets, fuzzy irresolute, fuzzy completely irresolute, fuzz compactness, fuzzy near(almost) compactness, fuzzy -closed.

On Fuzzy -irresolute Continuous Mappings

S.E. Abbas and M.Azab Abd-alla

Department of Mathematics, Faculty of Science, Sohag, Egypt

Department of Mathematics, Faculty of Science, Assiut, Egypt

Abstract: In this paper, The concept of fuzzy -irresolute continuous, semi- -irresolute continuous and -precontinuous mappings in fuzzy topological spaces in Sostak sense are introduced and studied. Some of their characteristic properties are considered. Also, fuzzy -connectedness is introduced and studied with the help of -fuzzy strongly semiopen seets.

Smooth Neighborhood Systems

A.A. Ramadan

Department of Mathematics, King Saud University, Saudi Arabia

M.A. Abdel-Sattar

Department of Mathematics, Faculty of Science, Cairo University, Beni-Suef, Egypt

S.E. Abbas

Department of Mathematics, Faculty of Science, Sohag, South Valley University, Egypt

Y.C. Kim

Department of Mathematics, Kangnung National University, Kangnung, Korea

Abstract: In this paper, we introduce the notion of smooth neighborhood (degree) systems in smooth topological spaces and establish their fundamental properties.

Keywords: Smooth topologies, Smooth neighborhood (degree) systems.

Fuzzy Fourier Systems for Periodic Function Mapping

Ching-Hung Lee

Department of Engineering, Yuan Ze University Taiwan, R.O.C. E-mail: chlee@saturn.yzu.edu.tw

Abstract: This paper presents a new system, called Fuzzy Fourier systems( ), for periodic function mapping. The composes a set of fuzzy rules in fuzzy Takagi-Sugeno-Kan( )models. The linear combination of the consequent part on each rule is set to be a Fourier series with adjustable Fourier coefficients(linear combination of orthogonal cosine functions). Therefore, the input/output mapping of is a periodic one. In additions, the fuzzy system with rules can be constructed to be a neural network-fuzzy Fourier network( ). Besides, for the given training pattern, we use the back-propagation learning algorithm with adaptive learning rate to update the parameters for obtaining high-speed convergence. In additions, with minor revising, the can be used to be a digital-filter for given magnitude specification. Simulation examples illustrate the effectiveness of our approach.

Key words: Function mapping, fuzzy neural network, Fourier series, periodic function

Fuzzy Pre-semi-irresolute mapping

S.E. Abbas

Department of Mathematics, Faculty of Science, Sohag, South Valley University, Egypt

M.A. Abdel-Sattar

Department of Mathematics, Faculty of Science, Cairo University, Beni-Suef, Egypt

Y.C. Kim

Department of Mathematics, Kangnung University, Kangnung,Korea

Abstract: We introduced -fuzzy pre-semiopen (semiclosed) sets in Sostak¡¯s fuzzy topological spaces. We investigate some properties of them. Moreover, we introduce and study fuzz pre-semicontinuous (fuzzy pre-semi-irresolute) mapping and fuzzy pre-semi-connectedness in Sostak¡¯s fuzzy topological spaces.

Keywords: -fuzzy pre-semiopen(closed) sets, fuzzy pre-semi-irresolute(continuous)mapping, fuzzy -(pre-)semi-connected

Static Fuzzy Closure Spaces

Sunil C. Mathew

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

T.P. Johnson

Cochin University College of Engineering Kuttanad, Punlicunnu P.O. Alleppey 688 504 Kerala, INDIA

Abstract: In this paper we introduce the concept of static ech-fuzzy closure space and study some of its properties. Further we introduce the concept of D-fuzzy closure space and establish its relation with static fcs. We also study subspaces and sums of static ech-fuzzy closure spaces.

Keywords: Static fcs, D-fcs, Nearly crisp.

On an Equivalence of Fuzzy Congruence Relations

V. Murali

Department of Mathematics (Pure&Applied), Rhodes University Grahamstown 6140 South Africa

Abstract: In this paper we consider an equivalence relation on the set of all fuzzy relations, fuzzy equivalence relations on a given non-empty set, and fuzzy congruence relations on a givens finite group, with a finite number of membership values being taken from the unit interval. We prove that such equivalence classes of fuzzy relations (equivalence relations) are in one-to-one correspondence with chains of crisp relations (equivalence relations). Further, in the case of groups, equivalence classes of fuzzy congruence relations are proved to be in one-to-one correspondence with equivalence classes of normal fuzzy subgroups which are in one-to-one correspondence with chains of normal subgroups.

Key words: Fuzzy congruence relations, fuzzy equivalence relation, normal fuzzy subgroups.

Fuzzy Convergence Spaces

G.Richardson, Q. Xue

Department of Mathematics, University of Central Florida, Orlando, FL32816USA E-mail: garyr@pegasus.cc.ucf.edu

T.-U. Jandeck

Department of Mathematics, Free University of Berlin, 14159Berlin, Germany

Abstract: The notion of a semiuniform convergence space is extended to that of a fuzzy semiuniform convergence space in the context developed by R. Lowen. The category of semiuniform convergence spaces is shown to be embedded as a bireflective and bicoreflective subcategory of the category of fuzzy semiuniform convergence spaces. Sufficient conditions are given to ensure that a fuzzy convergence space is induced by a fuzzy semiuniform convergence space.

Keywords: Semiuniform convergence, fuzzy semiuniform convergence, bireflective subcategories.

Study of Welfare-A Fuzzy Approach

Mary George

Senior Lecturer in Mathematics, Mar Ivanios College, Trivandrum, India

P.G. Thomaskutty

Reader in Economics, Mar Ivanios College, Trivandrum, India

Abstract: In Economics, the analysis of interpersonal welfare is usually made by methods of classical Mathematics. As the concept of welfare is imprecise (or non-exact), fuzzy mathematical tools seem to be more suitable to accomplish a study on welfare than that of classical Mathematics. This paper develops a measure of individual welfare using fuzzy mathematical methods.

Keywords: Economics, individual welfare, fuzzy binary relation, fuzzy income.