THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 9, Number 4, December 2001

CONTENT

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On L-homomorphism of Groups

Jizhong Shen

Department of Mathmatics,Jiangxi Normal University, Nanchang 330027,China

Abstract: We introduce the concepts of L homomorphism of groups based on complete resituated lattice-valued logic,and establish some fundamental properties.

Keywords: Logic,group,L-homomorphism.

Remarks on the Fuzzy Subgroups in Fuzzy Spaced

M.Alghamdi

Dept.of Math,Faculty of Science, King Abdulaziz Unimersity,Saudi Arabia,

Abstract: This study concerns the rich class of fuzzy subgroups introduced by Dib as a generalization of Rosenfeld fuzzy subgroups to get out some convergent and some divergent points with ordinary case.Lagrange theorem for the relation between the order of the fuzzy group and its fuzzy subgroups was discussed.Obtained fuzzy group with prime oreder,having nontrivial fuzzy subgroups.Constructed an example of a fuzzy group having continuum fuzzy subgroups.

Keywords: Fuzzy subgroups-order of fuzzy subgroup-Lagrange Theorem.

Duality in Fuzzy Vector Valued Sequence Space

N.R.Das and Pankaja Das

Department of Mathematics,Gauhati University-781014,Assam,India

Abstract: In this paaer the notion of duality in fuzzy normed linear space is introduced.As an application,it induces a fuzzy seminorm and consequently a fuzzy locally convex topology.

Keywords: Fuzzy vector valued sequence space,Fuzzy locally convex topology,Bilinear form,Dual pair.

On Permutation Representations of Hypergrorups

Zhang Chengyi

Department.of Mathematics,Hainan Teacher's College Haikou 571158,China

Dang Pingan

Department.of Mathematics,Hainan Teacher's College Henan 463000,China

Abstract: In this paper,the operational properties of the hypergroups on a finite group are discussed.By the action of the group G* on the hypergroup,the permutation representations and the matrix representations of a hypergroup are given,and the condition of a nonempty sub-powerset forming a hypergroup is studied.

Keywords: Hypergroup,group action,permutation representation,matrix representation

Local Behaviour of Fuzzy Automata

M.K.Sen and G.Chowdhury

Departntent of Pure Mathematics,University of Calcutta 35,Ballygunge circular road,Calcutta-700 019,India.

Abstract: In this paper we study fuzzy regular languages and the local behavior of fuzzy automata.

Keywords: Fuzzy automata,Fuzzy regular languages,Behavior of Fuzzy Automata,Local behaviour of Fuzzy automata.

On an Algebra of Fuzzy Points on Modules

B.B.Makamba

Department of Mathematics University of Fort Hare,Alice 5700 South Africa

V.Murali

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

Abstract: The paper deals with the algebra of fuzzy points viz.addition and scalar multiplication of fuzzy points on modules.Finitely generated fuzzy submodules,integral powers of fuzzy submodules,and product of fuzzy ideals with fuzzy submodules are characterised using fuzzy points.We show that every finite-valued fuzzy ideal of a Noetherian ring is finitely generated.

Keywords: Fuzzy points,fuzzy modules,finitely generated fuzzy submodules,sums and products.

Interval-valued Fuzzy Ideals in BCI-algebras

Young Bae Jun

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

Abstract: The purpose of this paper is to define the notion of a (closed)interval-valued fuzzy ideal (briefly,a (closed) i-v fuzzy ideal) of a BCI-algebra. We discuss between i-v fuzzy BCI-subalgebras and (closed) i-v fuzzy ideals.We show that,in a p-semisimple BCI-algebra,the notions of both a closed i-v fuzzy ideal and an i-v fuzzy BCI-algebra coincide.We state a condition for an i-v fuzzy set to be an i-v fuzzy ideal.

Keywords: Refined minimum, (closed) i-v fuzzy ideal,i-v fuzzy BCI-algebra,weakly BCK-algebra,p-semisimple BIC-algebra.

Polygroups with Hyperoperators

B.Davvaz

Department of Mathematics,University of Yazd,Yazd,lran Institute for Studies in Theoretical Physics and Mathematics,Tehran,lran

Abstract: In this paper,we further develop the polygroup theory and introduced some concepts such as polygroup with hyperoperator,fuzzy subpolygroup with hyperoperator,fuzzy normal subpolygroup with hyperoperator,etc.

Keywords: Polygroup,subpolygroup,fuzzy set,fuzzy subpolygroup,

On the L-Fuzzy Prime Ideal Theorem of Distributive Lattices

Mohammad M.Atallah

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

Abstract: The purpose of this paper is to establish some results related to the L-fuzzy prime ideal theorem of distributive lattices,by means,a construction theorem of an L-fuzzy prime ideal of a distributive lattices,we show that it is not possible to characterize an L-fuzzy prime ideal in terms of its level ideals.We introduce a definition and a characterization theorem of a completely L-fuzzy prime ideal and the implication between the different definitions of prime ideals and the implication between the different definitions of prime ideals are discussed.The reformulation of L-fuzzy prime ideal theorem is obtained.

Keywords: Algebra,Distributive lattices,Fuzzy prime ideal.

Separation Axioms on Fuzzy Bitopological Spaces

M.Zahran*and M.A.Abd Allah**

*Department of Mathematics,Faculty of Education for Girls,

Bisha P.O.Box No 2,Saudi Arabia

**Department of Mathematics,Faculty of Education for Girls, Gizan,Ssudi Arabia.

Abstract: Let (resp ) be a bitopological (resp.fuzzy bitopological) space and be a fuzzy topology (resp.topology) on X for each and for each j=1,2 which will be called t-topologically generated fuzzy topology (resp.level topology).In thes paper we introduce and study the notions of separation axioms in fuzzy bitopological space and bitopological space .Also the relation between the above notions and the notion of separation axioms is fuzzy bitopological space is discussed.

Keywords: Fuzzy bitopological spaces,separation axioms, level topology,fuzzy semiopen sets.

Bariation of Condition Extremum in interval and Fuzzy Valued Functional

Cao Bing,-yuan

Department & Institute of Mathematics, Shantou University,Guangdong.ZIP 515063,China Email:bycao@stu.edu.cn

Abstract: Extending a variation problem in condition extremum of classical functional [1],developing that of an interval and fuzzy valued functional and verifying an effectiveness of the extension with a numerical example.

Keywords: Interval-valued,fuzzy-valued,functional.variation,condition ectremum

Smooth Topological Spaces on A Fuzzy Space

K.A.Dib*,M.Alghamdi and A.Al Ahmadi

Department of Mathematics,Faculty,of Science. King Abdulaziz University,Egypt

Abstract: The smooth topology on fuzzy space is defined as a function on the collection of all fuzzy subspaces (of a given fuzzy space),satisfying certain conditions.The interior and the closure of fuzzy subspaces and the continuity of fuzzy functions are studied.A comparison with the classical case is done.

Keywords: Smooth topology on fuzzy space,-smooth interior,-smooth closure,-smooth continuous map.

Fuzzy Irresolute Mappings in Smooth Fuzzy Topological spaces

A.Ramadan

Department of Mathematics,Faculty of Science,Beni-Suef,Egypt.

S.E.Abbas

Department of Mathematics,Faculty of Science.Sohag 82524,Egypt.

Yong Chan Kim

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

Abstract: We introduce r-fuzzy semi-open and r-fuzzy semi-closed sets in a smooth fuzzy topological space in view of the definition of A.P.Sostak[12].We investigate some properties of them.Moreover,we investigate the relationship among fuzzy continuity,fuzzy semi-continuity and fuzzy irresolute mappings.

Keywords: R-fuzzy semi-open,fuzzy continuity,fuzzy semi-continuity,fuzzy irresolute,fuzzy almost open.

R-fuzzy Strongly Preopen Sets in Fuzzy Bitopological Spaces

Y.C.Kim

Department of Mathematics.Kangnung National University,Kangwondo,Korea

A.Ramadan

Department of Mathematics.Faculty of Science,Beni-Suef,Egypt

S.E.Abbas

Department of Mathematics.Faculty of Science,Sohag,Egypt

Abstract: We introduce r-fuzzy strongly preopen, an r-fuzzy strongly preclosed sets in a fuzzy bitopological spaces in view of the definition of Sostak[8]. We investigate some properties of them. FP strong precontinuity, FP strongly propend and FP strongly precluded mappings between fuzzy bitopological spaces are defined. Their proper-ties and the relationships between these mappings and other mappings introduced previously are investigated.

Keywords: -fuzzy strongly preopen,-fuzzy strongly preclosed,FP strong precontinuity,FP strongly preopen mapping,FP strongly preclosed mapping.

Generalized Strongly Nonlinear Complementarity Problems for Fuzzy Mappings

Eisa A.Al-Said

Department of Mathematics,College of Science,King Saud University, P.O.Box 2455,Riyadh 11451,Saudi Arabia.

Abstract: We introduce and study a new class of generalized strongly nonlinear complementarity problems for fuzzy mappings.Using the change of variables technique,we established the equivalence between the generalized strongly nonlinear complementarity problems and the Wiener-Hopf equations for fuzzy mappings.This equivalence is used to suggest and analyze a number of iterative algorithms for solving the generalized strongly nonlinear complementarity problems.

Keywords: Complementarity problems,Wiener-Hopf equations,change of variables,fuzzy operators,optimization.

-degree solution of variational inequalities for fuzzy mappings*

Wang Guixiang and Wu Congxin**

Department of Mathematics,Harbin Institute of Technology Harbin 150001,P.R.China

Abstract: In this paper, we introduce the concepts of fuzzy -degree strong monotone mapping, none-less r-degree strong monotone mapping,r-degree strong montone mapping and fuzzy -degree Lipschitz continuous mapping,none-less r-degree Lipschitz continuous mapping, r-degree Lipschitz continuous mapping, and the concepts of -degree solution,none-less r-degree solution, r-degree solution of variational inequalities for fuzzy mapping. We also consider the existence problem of these solutions of one kind of variational inequalities for corresponding kinds of strong monotone and fuzzy Lipschitz continuous mappings.

Keywords: Variational inequalities,fuzzy mappings, -degree strong monotone, -degree Lipschitz continuity,-degree solution,projection.

A Fuzzy Vector Space Based on the Theory of Falling Shadows

Cheng Zhang and Zun-Quan Xia

CORA,Department of Applied Mathemalics,Dalian University of Technoiogy Dalian 116024,P.R.China E-mail:zqxiazhh@dlut.edu.cn,zqxia@mail,dlptt.ln.cn

Del Popolo

Istituto di Astronomia dell'Universita di Catania, Viale A.Doria,6 1-95125 Catania,Italy

Abstract: A definition of fuzzy vector subspaces based on the theory of falling shadows is presented. This definition and related theorems provide a theoretical approach for further studying the t-fuzzy subspace.

Keywords: Fuzzy subset, t-norm,fuzzy subspace,random set,falling shadows.

An Application of Fuzzy Bag in Decision Analysis

Supriya Kumar De

Institute for Development and Research in Banking technology IDRBT,Castle Hills,Road No.-l,Masab Tank, Hyderaabad,Andhra Pradesh,Pin-500 057,India E-mail:supriya@idrbt.ac.in

Abstract: In this paper,we define the weighted measure of similarity between two fuzzy bags which is applied in decision analysis. We consider a problem of selecting most suitable action out of n alternatives on the basis of k factors where each factor has its corresponding weight depending on the basis of assessment of m judges.Our assumption is that the criteria values are not crisp,rather fuzzy to each and every judge and thus forms a fuzzy bag.An algorithm for the method is presented along with a hypothetical case-study.

Keywords: Bag, fuzzy bag,measure of similarity,criteria-value,criteria-matrix,criteria-value set,weighted criteria,weighted measure of similarity.

Some Properties of Fuzzy n-fold Positive Implicative Ideals in A BCK-algebras

C.Lele*and Wu Congxin**

*Department of Mathematics,Faculty of Sciences University of Yaounde I,Box 812

Yaounde,Cameroon

**Department of Mathematics Harbin Institute of Technology Harbin 150001,P.R.China

Abstract: In this paper,we use the notion of fuzzy point to study ideal and n-fold positive implicative ideal in BCK-algebras,then we clarify the links between the fuzzy point approach,the classical fuzzy approach and the ordinary case.

Keywords: BCK-algebra,fuzzy point,fuzzy n-fold positive implicative ideal,n-fold positive implicative weak ideal

Fuzzy Hyper K-subalgebras of Hyper K-algebras with Respect to A Triangular Norm

Young Bae Jun and Wook Hwan Shim

Department of Mathematics Education Gyeongsang National University,Chinju 660-701,Korea E-mail:ybjun@nongae.gsnu.an.kr

Abstract: By means of the use of a triangular norm,the fuzzification of the notion of hyper K-subalgebras in K-algebras is stated,and then their properties are investigated.

Keywords: Hyper K-algebra,hyper K-subalgebra,fuzzy hyper K-subalgebra with respect to a tnorm,weak homomorphism.

On Fuzzy Fredholm and Volterra Integral Equations

D.N.Georgiou and I.E.Kougias

Department of Mathematics,Faculty of Sciences, University of Patras,265 00 Patras,Greece

Abstract: Existence and uniqueness theorems are proved for certain fuzzy integral equations.

Keywords: Fuzzy sets,Fuzzy differentiable and fuzzy integral equations

Palmer-based Fuzzy Flexible Flow-shop Scheduling

Tzung-Pei Hong**

Department of Information Management I-Shou Uniersity,Kaohsiung, 84008,Taiwan,R.O.C. E-mail:tphong@isu.edu.cn

Tzu-Ting Wang

Institute of Information Engineering I-Shou University,Kaohsiung,84008,Taiwan,R.O.C. E-mail:ttwang@chttl.com.tw

Abstract: Scheduling is an important process widely used in manufacturing, production, management, computer science, and so on, In simple flow shop problems, each machine operation center includes just one machine. If at least one machine center includes more than one machine, the scheduling problem becomes a flexible flow-shop problem. Flexible flow shops can thus be thought of as generalizations of simple flow shops. In the past, the processing time for each job was usually assigned or estimated as a fixed value. In many real-word applications, however, job processing time may vary dynamically with the situation. In the past, we demonstrated how fuzzy concepts can easily be used in the Palmer algorithm for managing uncertain scheduling on flow-shop problems. This paper extends application to fuzzy flexible flow-shops with more than two machine centers. A heuristic fuzzy flexible flow-shop scheduling is then designed since optimal solutions seem unnecessary for uncertain environments.

Keywords: Scheduling, Flexible Flow Shop, Fuzzy Set, Membership Function, LPT scheduling, Palmer scheduling.

A First Order Lattice Valued Logic System I: Semantics

Xu Yang*,Qin Keyun* and Eun Hwan Rob**

*Southwest Jiaotong University,Chengdu,Sichuan,610031,P.R.China

**Chinju National University of Education,Chinju 660-756,Korea

Abstract: In this paper ,a first order lattice valued logic system corresponding to the lattice implication algebra is established and the semantics of this system is discussed.

Keywords: Lattice implication algebra,lattice valued logic system,interpretation.

A First Order Lattice Valued Logic System II:Semantics

Qin Keyun*,Xu Yang* and Eun Hwan Rob**

*Southwest Jiaotong University,Chengdu,Sichuan,610031,P.R.China

**Chinju National University of Education,Chinju 660-756,Korea

Abstract: This paper is devoted to the discussion of the syntax of first order lattice valued logic system FM. The basic definitions about the syntax were given,the soundness theorem and the deduction theorem were proved.

Keywords: Lattice implication algebra,formal proof,a-theorem.

Some Properties of Fuzzy Lie Algebras

Keyun Qin*,Quanxi Qiao** and Chaoping Chen**

*Southwest Jiaotong University,Chengdu,610031,Sichuan,P.R.China

**Jiaozuo Institute of Technology,Jiaozuo,454000,Henan,P.R.China

Abstract: In this paper,we discuss the properties of quotient Lie algebras,which is defined by fuzzy Lie ideals,we also discuss the images of fuzzy Lie subalgebra and fuzzy Lie ideal under generalized extension principle.

Keywords: Fuzzy Lie subalgebras,fuzzy Lie ideals,quotient Lie algebras,generalized extension principle.

Fuzzy Implicative Algebras over s-Norms

Young Bea Jun

Department of Mathematics Education Gyeongsang National University,Chinju 660-701,Korea E-mail:ybjun@nongae.genu.ac.kr

Sun Shin Ahn

Department of Mathematics Education Dongguk University,Seoul 100-715,Korea E-mail:sunshine@dgu.ac.kr

Hee Sik Kim

Department of Mathematics Hanyang University,Seoul 133-791,Korea E-mail:heekim@hanyang,ac.kr

Abstract: Using an s-norm S,the notion of S-fuzzy implicative algebras is introduced, and then some related properties are investigated. The direct product and S-product of S-fuzzy implicative algebras are discussed.

Keywords: (continuous)s-norm,S-fuzzy implicative algebra,direct product,S-product.