THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 7,Number 2,June 1999

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On Fuzzy Continuous Static Games (FCSG) (Stackelberg Leader with min-max Followers)

Mohamed S. Osman

Department of Mathematics, Higher Technological Institute, Ramadan 10th city, Egypt

Abou-Zaid H. El-Banna

Department of Pure Mathematics, Faculty of Science, Tanata University

Mervat M.Kamel

Department of Pure Mathematics, Faculty of Science, Helwan University

Abstract: This paper present study of continuous static games with fuzzy parameters in both the cost functions and the constraints. The fuzzy parameter are characterized by fuzzy numbers. We use the concept of stackelberg leader with min-max followers solutions to solve this problems. Parametric analysis of fuzzy continuous static games are presented. An alogorithm for obtaining the stability set of first kind is also presented. Numerical example to clarify this algorithm is given.

Keywords: Continuous static games, fuzzy numbers, Kuhn Tucker conditions.

Study on Large Scale Fuzzy Nash Equilibrium Solutions

Mohamed S. Osman

Department of Mathematics, Higher Technological Institute, Ramadan 10th city, Egypt

Abou-Zaid H. El-Banna

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

Azza H. Amer

Department of Mathematics, Faculty of Science, Helwan University,Cairo,Egypt

Abstract: This paper deals with large scale continuous static games with fuzzy parameters in both the cost functions and the constraints. Here the decomposition technique is used to decompose the original problem into subproblems, each having smaller and independent subproblems. Also, the concept of Nash Equilibrium Solution is used to solve these subproblems. Finally, a numerical example to explain this method is presented.

Keywords: Continuous static games, fuzzy numbers, Kuhn Tueker conditions.

General Multiobjective Fuzzy Linear Programming Problems with Fuzzy Variables

Li Xiaozhong

Department of Computer Science, Liaocheng Teachers University, Liaocheng, Shandong, 252059 P.R.Chian

Shi Kaiquan

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

Abstract: A general model for multiobjective fuzzy linear programming problem with fuzzy variables (GMFPV) is discussed in this paper and is turned to two-stage multiobjective linear programming problem by applying a partial order method defining the order of bounded closed fuzzy numbers. Then the algorithm to GMFPV is given.

Keywords: Fuzzy variable, fuzzy linear programming, quasi-triangular fuzzy number.

The Limit of The Folding of Some Types of Fuzzy Manifolds

M.El-Ghoul

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

H.M.Shamara

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

Abstract: In this paper we introduce a new definition for the limit of the folding of fuzzy manifolds. The relation between the limit of the folding of fuzzy manifolds and their retractions are discussed. Theorems governing these relations are deduced.

Keywords: Limit of fuzzy folding, manifolds

On Fuzzy Rings and Fuzzy Homomorphisms

A.A.M.Hassan

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

Abstract: Using the concept of fuzzy spaces, we study fuzzy rings, fuzzy ideals and fuzzy ring homomorphisms. The fuzzy quotient ring is introduced by using fuzzy ideals. The concept of standard fuzzy subring is introduced.

Keywords: Fuzzy algebra.

Fuzzy Pointwise Complete Regularity and Imbedding Theorem

Shi Fugui

Department of Mathematics, Yantai Teachers College, Yantai, Shandong, 264025, P.R.Chian

Abstract: In this paper, an equivalent characterization of pointwise complete regularity is presented in an L-fuzzy topological space by means of the L-fuzzy unit interval. The pointwise complete regularity has hereditary and multiplicative properties. A pointwise completely regular space is regular and a normal space is pointwise completely regular. Finally fuzzy imbedding theorem is presented.

Keywords: Fuzzy topology, pointwise complete regularity, normality, imbedding theorem.

Restudy of Fuzzy Factor Groups and Fuzzy Solvable Groups

M.Atif A.Mishref Department of Mathematics, Faculty of Science, Zagazig University, Egypt

Abstract: In this paper we study the relation between the two definitions of the normal fuzzy subgroup of a fuzzy subgroup which aare mentioned in [4] and [9], and make clear that the definition in [4] is more general than in [9]. We give here some counter examples of some results in [9] about the definition and the structure of fuzzy factor groups which is necessary to define and study fuzzy solvable groups. We also redefine the fuzzy factor groups by using the concept of normal fuzzy subgroup of a fuzzy subgroup which was defined in [4] and hence redefine and restudy the fuzzy solvable group. Also we orove that a fuzzy subgroup is solvable iff its t-levels are solvable groups. Usin the concept of t-level subgroups we give simple proofs for most results about fuzzy solvable groups which were studied in [9]. Other results are also proved.

Keywords: Fuzzy Algebra-fuzzy factor groups-fuzzy solvable groups.

Fuzzy Orders Structure Groups

O.A.Tantawy

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

Abstract: In this paper we introduce the fuzzy orders structures as a generalization of the fuzzy syntopogenous structures, also the fuzzy orders structures restricted to a fuzzy subset was studied. The interaction between the concepts of restriction of fuzzy orders structure on a fuzzy subset of a group and the fuzzy subgroup is investigated. This leads to introducing the concept of fuzzy orders structure groups as a generalization of the fuzzy topological groups. We include this paper by illustrating the image and the inverse image of fuzzy orders structure groups under a homomorphism of groups.

Keywords: Fuzzy orders structure, fuzzy syntopogenous space, fuzzy orders structure group.

Finding Dependency of Attributes in An Information System

Supriya Kumar De, Ranjit Biswas and Akhil Ranjan Roy

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

Abstract: In this paper we generalize Pawlak¡¯s rough approach on finding dependency of attributes in an information system. We consider an information table where attribute values are not always quantitive, rather subjective having vague or imprecise meaning. Some objects may have such values which are almost identical i.e., they can not be distinguished clearly by the attributes. This observation is analyzed hereg being based on proximity relations on different domain of attributes as well as the corresponding dependency functions.

Keywords: Indiscernible relation, almost indiscernible relation ,¦Á-identical, ¦Á-dependency, ¦Á-reduct, ¦Á-core.

Coincidence Point Theorems in F-type Topological Spaces with Applications

Nan-jing Huang, Min-ru Bai, Ya-ping Liu and Ya-yong Tang

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064£¬P.R.China

Abstract: The main purpose of this paper is to establish some new versions of coincidence point theorems and a minimization theorem in F-type topological spaces. As applications, we utilize our main theorem to obtain some coincidence point theorems, fixed point theorems and a minimization theorem in fuzzy metric space. Our results improve and extend the corresponding results in [1-9,11,13].

Keywords: Coincidence point , F-type topological space, set-valued mapping , fuzzy metric space.

Fuzzy Ideals in Semigroups

Xiang Yun Xie

Department of Mathematics and Physics, Wuyi University, Jiangmen, Guangdong, 529020£¬ People ¡®s Republic of China.

e-mail:xyxie @letterbox.wyu.edu.cn

Abstract: In this paper, we give the definitions of fuzzy ideals (left ideals, right ideals and biideals ) generated by fuzzy sets in a usual semigroup. As an application, the corresponding results on semigroups with a unity element [3] are obtained by an modification.

Keywords: Semigroup, fuzzy ideal, fuzzy principle ideal .

L-fuzzy Spectrum

Nistala V.E.S.Murthy and Peruru G.Prasad

MIG-B-171, Sagernagar, VIZAG-530042£¬AP State, India

Abstract: The aim of this paper is to show that the Zariski topological space on fuzzy prime ideals of a commutative ring with unity can be regarded as a subspace of the Zariski topological space on prime ideals of a commutative d-semiring with unity. Further we give a pair of examples in such a way that in the first example, the fuzzy Zariski topological space is not compact and in the second example, the fuzzy Zariski topological space is compact.

Keywords: Fuzzy set, fuzzy prime ideal, topological space.

Minimal Fuzzy Recognizers

D.S.Malik and John N.Mordeson

Department of Mathematics and Computer Science, Creighton University, Omaha, Nebraska 68178£¬USA

Abstract: In this paper we show for every recognizer there is complete accessible deterministic fuzzy recognizer with the same behavior as and which is minimal.

Keywords: Fuzzy finite state machine, recognizer, accessible, language, recognizable, minimal recognizer.

Congruence Interval of A Soft Algebra Wang Zun-quan

Hubei Sanxia University, Teacher¡¯s College , Yichang, Hubei 44300£¬ P.R.China

Abstract: In this paper, we define congruence interval of soft algebra F discuss the properties of congruence interval. Further more, we give a necessary and sufficient condition of which a interval is a congruence interval.

Keywords: Soft algebra, congruence interval, congruence ideal, quotient soft algebra

Fixed Points of Expansive Mappings on Fuzzy Preordered Sets

Ismat Beg

Department of Mathematics, Kuwait University, P.O.Box 5969, Safat 13060£¬Kuwait

e-mail:IBEG@MATH-1.SCI.KUNIV.EDU.KW

Abstract: We study existence of maximal fixed points of expansive mapping on fuzzy preordered sets under suitable conditions. Keywords: Fuzzy preordered set, fixed point, expansive mapping, analysis Study on Reservoir¡¯s Long Term Runoff Prediction by Fuzzy Mathematical Method Ruan Yun, Haojie Qiu and Zhongjing Yang Civil Engineering Department, Shandong University of Technology, Shandong, Jinan 250061£¬P.R.China Abstract: The reservoir¡¯s long term runoff prediction during flood season by fuzzy mathematical method was studied in this paper. Firstly, business using maximal tree method and soft partition cluster method, the fuzzy cluster analysis was carried out against the observed data of each calendar year during flood season ;and several prediction models for coming water during flood season were also built . then the discharges of coming water during flood season and their realizing probabilities were predicted by means of Gamma distribution and fuzzy probability method . general programs were written for the whole prediction procedure.

Keywords: Fuzzy mathematics, Runoff prediction, maximal tree method ,soft partition cluster method , sample .

Fuzzy Ideals and Fuzzy Subalgebras of BCK-algebras

Young Bae Jun, Sung Min Hong, Seon Jeong Kim

Department of Mathematics Education, Gyeongsang National University Chinju 660-701, Korea

e-mail:ybjun@nongae.gsnu.ac.kr

Seok Zun Song

Department of Mathematics,Cheju National University ,Cheju 690-756£¬Korea

Abstract: We state the fuzzification of *-subalgebras ,additive ideals and Varlet ideals,and investigate their properties . we give an equivalent condition for a fuzzy set to be a fuzzy (implicative) ideal. We show that , in a BCK-algebra with condition (S) ,the notion of fuzzy ideals and fuzzy additive ideals coincide, and that, in a positive implicative BCK-algebra with condition (S) ,the notion of fuzzy ideals, fuzzy Varlet ideals and fuzzy additive ideals coincide.

Keywords: (fuzzy) *- subalgebras, (fuzzy) o- subalgebras, (fuzzy) additive ideals, (fuzzy) Varlet ideal, (fuzzy) implicative ideal , (fuzzy) commutative ideal , (fuzzy) positive implicative.

Fuzzy ¦Ã-open Sets and Fuzzy ¦Ã-continuity

I.M.Hanafy

Department of Mathematics,Faculty of Education,Suez Canal University ,El-Arish ,Egypt

Abstract: In this paper, we first introduce the concepts of a fuzzy ¦Ã-open (¦Ã-closed) sets which is weaker than each of the concepts of fuzzy semiopen (semiclosed)set or fuzzy preopen (preclosed )set. We study their properties and discuss the relationships between these concepts and relevant concepts in fuzzy topological space. Using the new notion, we define the fuzzy ¦Ã-continuous functions as a generalization of fuzzy semicontinuity [1] and fuzzy precontinuity [2], and establish their various characteristic properties, finally, we introduce and study the concepts of fuzzy ¦Ã-open (¦Ã-closed ) functions.

Keywords: Fuzzy topological space, fuzzy ¦Ã-open(¦Ã-closed ) sets, fuzzy ¦Ã-continuity, fuzzy ¦Ã-open(¦Ã-closed ) function .

Fuzzy Closed Ideals and Fuzzy Filter in BCH-algebras

Young Bae Jun

Department of Mathematics Education, Gyeongsang National University Chinju 660-701, Korea

e-mail:ybjun@nongae.gsnu.ac.kr

Abstract: We consider the fuzzification of ideals and filters in BCH-algebras. The relation among fuzzy subalgebras, fuzzy closed ideals and fuzzy filters will be described. We also state the fuzzy regularity of fuzzy sets in BCH-algebras.

Keywords: BCH-algebra, (quasi ) closed filter, fuzzy subalgebra, fuzzy closed ideals, fuzzy (quasi ) closed filter, (fuzzy ) regularity

Fuzzy Medial Ideals in BCI-algebras

Samy M.Mostafa, Amany El-menshawy

Mathematics Department, Faculty of Education ,AIN shams University, Roxy, Cairo, Egypt

Young Bae Jun

Department of Mathematics Education, Gyeongsang National University Chinju 660-701, Korea

e-mail:ybjun@nongae.gsnu.ac.kr

Abstract: We introduce the notion of medial ideals in BCI-algebras, and consider its fuzzification . we study fuzzy relations on BCI-algebras.

Keywords: Medial BCI-algebras, (fuzzy ) medial ideal , level medial ideal

Quotient Group of A Group Generated by A Subgroup and A Fuzzy Subset

A.K.Ray

Department of Mathematics,Dibrugrarh University,Dibrugrarh -786004£¬Assam,India

Abstract: In this short communication , we have defined left and right cosets of a subgroup generated by an element and a fuzzy subset of the group . Finally we have defined the quotient group generated by a subgroup and a fuzzy subset of the group.

Keywords: Commutative fuzzy subset, left and right cosets, left and right translational invariant fuzzy subset, quotient group .

On Quotient BCK-algebras via Fuzzy BCK-filters

Young Bae Jun

Department of Mathematics Education, Gyeongsang National University Chinju 660-701, Korea

e-mail:ybjun@nongae.gsnu.ac.kr

Sun Shin Ahn

Department of Mathematics Education,Dongguk University Seoul 100-715£¬Korea

Hee Sik Kim

Department of Mathematics Hanyang University Seoul 133-791£¬Korea

e-mail:heekim@email.hanyang.ac.kr

Abstract: In this paper , we construct a quotient BCK-algebra which is induced by (fuzzy ) BCK-filters. We investigate the relation between the quotient BCK-algebra induced by BCK-filters and the quotient BCK-algebra induced by fuzzy BCK-filters.

Keywords: (Fuzzy ) BCK-filters, bounded BCK-algebra , level BCK-filters, quotient BCK-algebra .

Fuzzy p£¦ -ideals in IS-algebras

Young Bae Ju

Department of Mathematics Education, Gyeongsang National University Chinju 660-701, Korea

e-mail:ybjun@nongae.gsnu.ac.kr

Eun Hwan Roh

Department of Mathematics Education,Chinju National University of Education ,Chinju 660-756£¬Korea

Abstract: In this paper, we consider the fuzzification of a p£¦ -ideals in an IS-algebra . we prove that in an IS-algebra every fuzzy p£¦ -ideals is a fuzzy -ideals. We give some conditions for a fuzzy -ideals to be a fuzzy p£¦ -ideals. Finally we show that in a PS-algebra every fuzzy -ideals is a fuzzy p£¦ -ideals.

Keywords: IS(resp.PS)-algebra ,(fuzzy) -ideals, (fuzzy) p£¦ -ideals.

The Colimits in The Category of Completely Distributive Complete Lattices

Bin Zhao

Department of Mathematics, Shaanxi Normal University,Xi¡¯an 710062£¬China

Abstract: With a binary relation on complete lattices we give the definitions of -complete lattices and -lower sets, and obtain a pair contravarian adjoint functors between the category of completely distributive complete lattices and the category of - complete lattices. Using the concepts of preserving and creating limits in category theory , we first transform the limit structures of set category into the limit structures of the category of - complete lattices ,then construct the colimit structures in the category of completely distributive complete lattices by the above contravariant adjoint functors. Finally, we prove that the coproduct object of a family of pointed lattices in the category of completely distributive complete lattices is again a pointed lattice .

Keywords: Complete lattice, completely distributive complete lattice, adjoint ,colimit

Oscillation and Nonoscillation Theorems for Second Order Fuzzy Differential Equations

P.C.Nayak

Department of Mathematics, M.P.C.College ,Baripada Mayurbhanja ,Orissa, 757,001£¬India

Abstract: The purpose of this paper is time discuss the oscillatory and non-oscillatory behavior of the solutions of (*) We have actually extended the results obtained by P.C.Nayak and S.Nanda [8] for the second order fuzzy differential equation in the light of the results obtained by O.Kaleva [7] for the existence and uniqueness problems associated with first order fuzzy differential equation . Such qualitative behavior of the solutions of second order fuzzy differential equation have been studied in comparison with the similar properties of the solutions of its corresponding imbedded differential equation .

Keywords: Fuzzy differential equation, Oscillation and ono-oscillation .

A New Blueprint for Fuzzification: Application to Lattices of Fuzzy Congruences

Naseem Ajmal and K.V.Thomas

Department of Mathematics, Zakir Husain College ,University of Delhi , J.L.Nehru Marg, New Delhi-110002£¬India

Abstract: In this paper, we discuss certain properties of fuzzy congruences and fuzzy normal subgroups of a group in relation to strong level subsets . We provide generating techniques for fuzzy congruences and fuzzy subgroups of a group . Also ,with our new technique , we prove that the lattices of fuzzy congruences and fuzzy normal subgroup of a group are isomorphic , which is , in fact , a generalization of our earlier result that , the lattices of t-fuzzy congruences and fuzzy normal subgroup with tip ¡®t¡¯ are isomorphic. In section 3£¬we establish this isomorphism by following the metatheorem approach developed by Tom Head. The fundamental tool used is a function that represents the algebra of fuzzy sets of a set as a subdirect product of coppices of the algebra of crisp subsets. The comoparison of these two approaches in obtaining the isomorphism theorem is a very interesting aspect of this paper .

Keywords: Fuzzy congruences, fuzzy normal subgroup

Fuzzy Variants of The Poincare Recurrence Theorem

A.I.Ban

Department of Mathematics,University of Oradea Str.Armatei Romane 5,3700 Oradea,Romania

Abstract: Using results of the fuzzy measure theory, we propose fuzzy variants of the Poincare Recurrence Theorem for different operations between fuzzy sets induced by triangular norms.

Keywords: Fuzzy point, fuzzy measure, measure preserving transformation ,fuzzy dynamical system , triangular norm.

q-fuzzy Cyclic Subgroup

Sandeep Kumar Bhakat

Siksha ¨CSatra , Visual ¨CBharati, Sriniketan-731236, Dist-Birbhum (W,B) INDIA

Abstract: The notion of q-fuzzy cyclic subgroup for an -fuzzy subgroup of a group is introduced. Inter-relation with -fuzzy normal subgroup and q-fuzzy commutativity are obtained. Some of the fundamental results in fuzzy algebra are also obtained.

Keywords: Fuzzy algebra, -fuzzy subgroup, -fuzzy normal subgroup, -level subset , q-fuzzy commutative subgroup , q-fuzzy cyclic subgroup