THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 7, Number 3, September 1999

CONTENT

DETAILS

The Notes on The Induced Mapping

Xiong Fenglan

Applied Mathematics Department,

Ocean University of Qingdao, P. R. China

Liu Cuiying

Department of computer

Qingdao University, P. R. China

Abstract: This paper indicated and corrected the errors in the paper [1] and generalized the concept of the double-induced mapping.

Keywords: Complete distributive lattice, LF-set, induced maping

Fuzzifying Syntopogenous Structures

A. A Ramadan

Department of Mathematics, Faculty of Science

Cairo University, Beni Swef, Egypt

S.N. EI-Deeb, M-S. Saif and M. EI-Dardery

Department of Mathematics, Faculty of Science

Cairo University, Faiyum, Egypt

Abstract: In this paper we introduce the concept of fuzzifying syntopogenous structures in the framework of fuzzifying topology (uniformity and proximity). Some fundamental properties are established. Finally, a natural links between fuzzifying syntopogenous structures, fuzzifying topology, fuzzifying proximity and fuzzifying uniformity are given.

Keywords: Fuzzifying topology, fuzzifying proximity and fuzzifying uniformity, topogenous.

-continuity and D -continuity in Fuzzifying Topology

K. M. Abd EI-Hakeim, F. M. Zeyada and O. R. Sayed

Department of Mathematics, Faculty of Science

Assiut University, Assiut 71516, Egypt

Abstract: The concepts of -continuity and D -continuity are considered and studied in fuzzifying topology and by making use of these concepts, some decompositions of fuzzy cotinuity are introducd.

Keywords: Fuzzifying topology, fuzzy continuity, -continuity, - continuity.

Variation of Functional Condition Extremum with Fuzzy Variable* Cao Bing-yuan

Department of Mathematics, Shantou University

Guangdong 515063,P.R.China

Abstract: By Definition "nest of set" in this paper, a condition extremum variation problem on ordinary points of an ordinary and fuzzy functionals are extended into variables being a fuzzy state. A method to condition extremum is obtained for them containing fuzzy variables by the aid of the variational methods.

Keywords: Functional, fuzzy-valued, fuzzy variable, condition extremum, variation.

On Fuzzifying Filters and Uitrafilters

M.H. Ghanim

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

Fayza A. Ibrahim And Mervat A. Sakr

Department of Mathematics, Faculty of Science

Ain Shams University, Cairo, Egypt

Abstract: In this paper we introduce the concept of fuzzyifying filter and a fuzzyifying net. We explain how to obtain a fuzzyifying filter from a fuzzyifying net. The paper is included by introducing the concepts of base (subbase) of a fuzzyifying filter and the fuzzyifying ultrafilter.

Keywords: Fuzzyifying topology, filter, net, ultrafilter.

Generalized Semi-continuous Mappings in Fuzzy Topological Spaces

K. M. Abd EI-Hakeim

Mathematics Department , Faculty of Science,

Assiut University, Assiut, Egypt.

Abstract: We introduce the concepts of generalized fuzzy semiclosed set, generalized fuzzy weakly semiclosed sets, generalized fuzzy semicontinuous mappings, generalized fuzzy semiconnectedness and generalized fuzzy S-closed space, and we establish some fundamental of these concepts.

Keywords: Generalized fuzzy semiclosed set, generalized fuzzy weakly semiclosed, generalized fuzzy semicontinuity; generalized fuzzy S-closed space; generalized fuzzy semiconnectedness.

A New Approach to Fuzzy Separation Axioms

G.. K. Chandrika

Department of Mathematics

Avinashilingam Deemed University, Coimbatore, India.

K. N. Meenakshi

Department of Mathematics

Avinashilingam Deemed University, Coimbatore, India.

Abstract: A new approach is followed in defining fuzzy separation axioms. In this approach fuzzy separation axioms are introduced in terms of fuzzy sets whereas in the earlier ones [1,2,4,5,7,8] they are given in terms of fuzzy points. These axioms are defined in a manner similar to the separation axioms defined in the case of topological Boolean Algebras by Nobeling [6]. The new definitions prove to be good extensions of separation axioms for topological spaces.

Keywords: Fuzzy topology, fuzzy separation axioms , , , , subspace and sum fuzzy topological space.

Higher Fuzzy Derivations

Boniface 1. Eke

Department of Mathematics, Morgan State University

Baltimore, Maryland .21251, U.S.A.

Abstract: This paper studies the concept of higher fuzzy derivations, analogous to the concept of higher derivations of the crisp theory. A crucial link is established between the concept of modularity and that of higher fuzzy derivations in the case of fuzzy extension fields A/B of bounded exponent, thus fuzzyifying a classical result of the crisp theory.

Keywords: Higher fuzzy derivations, discrete fuzzy derivations, fuzzy relative -basis.

On the Appoximation of Fuzzy Numbers

Michael Wagenknecht

University of Applied Sciences Zittau/Gorlitz

IPM

Theodor-Korner-Allee 16

02763 Zittau, Gerrnany

Abstract: We consider the approximation of functions of (L,R)-numbers by powers of given basic functions leading to corresponding optimization problems. It is shown that espe-cially the use of L resp. R may lead to rather simple and good approximations.

Keywords: Membership approximations.

A Fuzzy Extension of Hahn Banach Theorem

T. V Ramakrishnan

Department of Mathematics, S. E. S. College

Sreekandapuram Kannur-670631, India.

Abstract: In this paper we establish the norm preserving fuzzy completion of the fuzzy normed algebra . Using we give a fuzzy extension of Hahn banach theorem.

Keywords: Fuzzy extension of Hahn banach theorem, fuzzy completion of .

Algebraic Structure of Lower Approximation Spaces

John N. Mordeson

Department of MathematicsIcomputer Science

Creighton University, Omaha, Nebraska 68178, USA

Abstract: We examine the algebraic structure of lower approximation spaces. We introduce the concepts of generating sets and freeness for lower approximation spaces. We use these concepts to determine bases for lower approximation spaces. This gives a notion of dimension for lower approximation spaces. We apply these concepts to rough set theory.

Keywords: Rough set, lower approximation space, nerating set, free subset, basis

Order Structure of Proximities of Fuzzy Sets

K. C. Chattopadhyay, U.K. Mukherjee

Department of Mathematics University of Burdwan

Burdwan 713104, W.B. India.

S. K. Samanta

Department of Mathematics Visvs-Bharati

Santiniketan 731235, W.B. India

e-mail: root£Àvbhart.ernet.in

Abstract: In this paper order structure of proximities of fuzzy sets is studied.

Keywords: Fuzzy proximity, proximity of fuzzy sets, fuzzy topology.

Wreath Absolute Difference of Klein -Groups

D. Talukdar

Dept. of Mathematics Nalbari College

Nalbari, Assam, India

Abstract: The Cartesian power of a Klein -group by another Klein -group is discussed here. Then wreath absolute difference of two Klein -group is defined in the line of wreath product of groups (Tamuli 1972) and some results are established in this paper.

Keywords: Klein -group, wreath product.

The Uniform Autocotinuity of Non-additive Lattice-valued Measures on Lattice

Guangquan Zhang And Yong-Hong Wu

School of Mathematics and Statistics, Curtin University of Technology

GPO Box U 1987, Perth, Western Australia

6845, Australia

Abstract: In this paper, the uniform autocontinuity and uniform pseudoautocontinuity of non-additive lattice-valued measures are introduced. The equivalent conditions of null-additivity, autocontinuity and uniform autocontinuity and then the equivalent conditions of pseudo-nulladditivity, pseudo-autocontinuity and uniform pseudo-autocontinuity are developed and presented.

Keywords: Non-additive measure, fuzzy measure, lattice, uniform autocontinuity, uniform pseudo-autocontinuity.

Determination of Membership Values Using Adjectivalmeter

Ranjit Biswas

Department of Mathematics, Indian Institute of Technology

Kharagpur-721302, West Bengal, INDIA

ranjit@maths. Iitkgp.ernet.in

Abstract: In this paper a method of determining membership values of the elements of a fuzzy set in a finite set X is proposed. C orresponding to each fuzzy set (or membership function) there exists an ¡°adjective¡±. An Abstract instrument called by adjectivalmeter is introduced to measure the amount of the ¡°adjective¡± present in each element of fuzzy sets. At end, an algorithm is framed to summarize the proposed method.

Keywords: ¡°Adjective¡±, adjectivalmeter, meter-line, meter-point, super fuzzyset, null fuzzy set, SFS, SFS point, nearest SFS, concentrate(dilated) SFS, fum, square zero (¡õ),rough set.

Super Continuity and Some Strong Forms of Fuzzy Pairwise Continuity on Fuzzy Bitopological Spaces

S. Sampath Kumar

Department of Mathematics Anna University,

Crennai-600 025,India

Abstract: The concept of super continuous mappings and some strong forms fuzzy pairwise continuous mappings are introduced and studied in the light of the concepts of quasi-coincidence due to Pu and Liu [6] in fuzzy bitopological setting . These maps have been characterized and investigated mainly in the light of the notions of quasi-coincidence, quasi-neighbourhoods of Pu and Liu [6] , -closure and -neighbourhoods of [8]. Finally, a comparative study regarding the mutual interrelations among these maps along with fuzzy pairwise continuous maps is made.

Keywords: Fuzzy pairwise super continuous, completely continuous; strongly continuous and strongly -continuous mappings, -closure, -neighbourhoods.

Characterizations of Some Fuzzy Bitopological Separation Axioms in Terms of -closure and -closure Operators

S. Sampath Kumar

Department of Mathematics, Anna University

Crennai-600 025,India

Abstract: The aim of this paper is to introduce and characterize some bitopological separation axioms in terms of quasicoincidence , quasi-neighbourhood of Pu and Liu [5], -closure and -closure operators as initiated in [6] in fuzzy setting.

Keywords: Fuzzy pairwise regular; fuzzy pairwise almost regular; -closure; -closure; fuzzy pairwise Urysohn; fuzzy pairwise .

Some Properties of -Convex Fuzzy Processes

Marian Matloka

Department of Mathematics, University of Economics,

al.Niepodleglosci 10, 60-967 poznan, poland

Abstract: Convexity of a fuzzy set and of a fuzzy process are generalized. First the definition of -convex fuzzy process is presented and next its properties are considered.

Keywords: -convex fuzzy set, -convex fuzzy process.

On Fuzzy SP-Continuity and -closedness

A.M.Zahran

Department of Mathematics, Faculty of Science,

Al Azhar University, Assuit,71511, Egypt.

Abstract: In this paper first, we show by counterexamples that some results, which appeared in the article [10] are incorrect. Secondly we introduce many classes of functions by using the concepts of fuzzy semi-preopen sets and fuzzy semipre -open sets, called fuzzy weakly semi-precontinuous, fuzzy semi-pre -continuous, fuzzy weakly semi-pre -continuous, fuzzy completely semi- precontinuous, fuzzy weakly completely semi-studied. Finally, we give some characterization of the fuzzy -closed space in terms of fuzzy semi-preopen sets and fuzzy -filterbasis.

Keywords: Fuzzy semi-preopen sets, fuzzy semi-precontinuity, fuzzy semi-pre -continuity, fuzzy -closed space, fuzzy -filterbasis.

Fuzzy Programming Technique to Solve A Non-linear Multi-objective Transportation Problem

M.P. Biswal, Rakesh Verma

Department of Mathematics, Indian Institute of Technology

Kharagpur-721302,India.

Abstract: In this paper a mathematical model of a multi-objective non-linear transportation problem is presented. The main aim of the model is to locate the warehouses from which goods can be transported to the retailers which will minimize the transportation costs for different modes of transportation. Due to the presence of the conflicting objectives, fuzzy programming technique with some non-linear membership functions is used to find the optimal compromise solution.

Keywords: Multi-objective decision making, non-linear transportation problem non-linear membership function , fuzzy programming.

The Convergent Problems for p-MeanSymmetric Difference Metric*

Li Fachao, Qiu Jiqing, Su Lianqing

Department of Basic Science, Hebei University of Sciences and Technology

Shijiazhuang, 050018, P. R.China

Wu Congxin, Ha Minghu

Department of Basic Science, Harbin Institute of Technology

Harbin 150006, P. R.China

Abstract: In this paper, we discuss the basic properties of convergent sequence in , and show, for monotone fuzzy numbers sequence , that there exists such that if and only if is uniformly support bounded. Further, we show, for monotone increasing fuzzy numbers sequence , (where ) if and only if , and show the squeezing principle about .

Keywords: Fuzzy number, platform type fuzzy number, p-mean, p-mean symmetric difference metric.

On Hypernear-Rings and Fuzzy Hyperideals

B. Davvaz

Institute for Studies in Theoretical Physics and Mathematics,

P.O.Box 19395-5746, Tehran, Iran

And Department of Mathematics, Yazd University, Yazd,Iran

e-mail:davvaz@vax.ipm.ac.ir

Abstract: In this paper, we introduce the concept of a fuzzy hyperideal of a hypernear-ring and obtain a few results in this connection.

Keywords: Hypergroup, hypernear-ring, fuzzy set, hyperideal, fuzzy hyperideal.

Fuzzy Norm of A Linear Operator and Spaces ofFuzzy Bounded Linear Operators Fang Jin-Xuan, Li Jun-Hua

Department of Mathematics, Nanjing Normal University

Nanjing, Jiangsu 210097 China

Abstract: The purpose of this paper is to introduce the notion of fuzzy norm of a linear operator from a fuzzy normed space to another and to study the space of fuzzy bounded linear operators.

Keywords: Fuzzy normed space. fuzzy norm of a linear operator; space of fuzzy bounded linear operators.

L-fuzzy Locally Convex Topological Vector Spaces*

Yan Cong-Hua

Department of Basic, Nanjing Power Institute

78 Bancang Nanjing Jiangsu,210042. China

Fang Jin-xuan

Department of Mathematics, Nanjing Normal University

Nanjing, Jiangsu 210097 China

Abstract: The purpose of this paper is to introduce the concept of L-fuzzy locally convex topological vector spaces, and to prove that every locally convex L-fuzzy vector topology on X can be determined by a family of L-probabilistic seminorms on X.

Keywords: L-fuzzy locally convex topological vector spaces, R-neighborhood base, L-probabilistic seminorms.

Connectedness on Fuzzy Topological Spaces (¢ò)

Zhao Xiaodong

School of economics& Management, Yanshan University,

Qinhuangdao, Hebei, 066004, China

Abstract: In this paper we tie up some of the loose ends left from our first paper [1]. Three new equivalent conditions for connectedness [1] on completely distributive topological lattices are proved. Some new interesting-properties of connectedness are studied. All the elements in the pointness fuzzy unit interval [2] must be connected ones w.r.t any topology on it. The product theorem of Lfuzzy sets on fuzzy connectedness is given.

Keywords: Completely distributive topological lattices, L-fuzzy set, Connectedness, Separation, Pointness fuzzy unit interval.

Fuzzy Programming Approach to Multi-objective Stochastic Linear Programming Problems with A Known Distribution

W. Mohamed

Department of Mathematics, Faculty Of Science

Zagazing University, Zagazing, Egypt.

Abstract: This paper presents an application of fuzzy programming approach to the multi-objective stochastic linear programming problem. After converting the proposed stochastic programming problem to a deterministic problem (which may be linear or non-linear), fuzzy programming problem approach is applied to find the compromise solution. Assuming the coefficients of the decision variables in the objective functions and in the constraints and the right-hand-side parameters in the constraints to be uniform random variables, a methodology is presented to convert the probabilistic problem to a deterministic problem. Fuzzy programming is applied using linear as well as non-linear membership Functions. The method leads to an efficient solution as well as an optimal compromise solution. Numerical examples are also presented to illustrate the methodology.

Keywords: Chance constrained programming; Two-stage programming; Fuzzy programming; Linear and non-linear membership functions