THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 10, Number 2, June 2002

CONTENT

DETAILS

Fuzzy Lattices I

Naseem Ajmal

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

K.V.Thomas

Department of Mathematics Bharata Mata College Mahatma Gandhi University,Thrikhakara,Cochin-682021,India

Abstract: In this paper, we introduce some operations on fuzzy sets of a lattice.A detailed study of fuzzy ideals (dual ideals) under these operations are provided. Structural theorems for fuzzy sublattice (ideal, dual ideal) generated by a fuzzy set provided in [1] are modified..Using this modification,unique representation theorem for convex sublattices is extended to the fuzzy setting. Moreover, we establish that a lattice is distributive if and only if the lattice of its fuzzy ideal (dual ideals) is distributive.

Keywords: Lattice,distributive lattice,fuzzy sublattice,fuzzy ideals.

Fuzzy Lattices ¢ò

Naseem Ajmal

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

K.V.Thomas

Department of Mathematics Bharata Mata College Mahatma Gandhi University,Thrikhakara,Cochin-682021,India

Abstrsct: In this paper, we study fuzzy congruences and fuzzy ideals of a lattice. It is proved that lattlice of all fuzzy congruences of all fuzzy congruences of a given lattice is distributive. Futher,we establish a relationship of fuzzy ideals and fuzzy congruences with the help of strong level subsets,and using this, we prove an isomorphism between their lattices. Finally,we introduce and study fuzzy quotient lattice and prove a fuzzy version of fundamental theorem of homomorphism.

Keywords: Lattice, fuzzy congruence, fuzzy ideals,quotient lattice

A Soft Algebra which Every Primitive Ideal Is A Congruence Ideal

Wang Zhua-quan

Sanxia University Teacher College Hubei,443000,P.R.China

Abstrsct: In this paper,we prove that every primitive ideal of soft algebra F ia a congruence ideal if and only if F is an absolutely indecomposable or a class Boolean algebra.

Keywords: Soft algebra, primitive ideal, class Boolean algebra, congruence idea, congruence realtion,absolutely indecomposable.

Fuzzy Indicators and Multi-criteria Evaluation

Henri Gw¨¦t

Department of Computer Sciences National Polytechnic Institute,P.O.Box 8390 Yaounde,Cameroon

Abstrsct: This paper provides a new way of multi-criteria evaluation. This method is based on the use of fuzzy set connectives and fuzzy measures. At the end, we make some comparisons between new and old ways.

Keywords: Aggregation of preferences, classification,utility,multi-criteria evaluation.

General Notions Related to Fuzzy Filters

A.S.Abd-Allah

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

Abstrsct: Basic results on L-fuzzy topology are presented which are obtained in applying the general theory of fuzzy filters and operations defined on the set of all L-fuzzy subsets endowed with an L-fuzzy topology. In particular,the category CSOF of all characterized spaces and several of its bireflective subcategories are studied.

Keywords: Fuzzy filters,fuzzy stacks,fuzzy topologies,opertations, ¦Õ1,2-open fuzzy sets, characterized spaces, ¦Õ1,2-fuzzy neighborhood filters,valued and superior¦Õ1,2¨Cfuzzy neighborhoods,idempotent,stratified and strongly stratified¦Õ1,2-fuzzy interior operators, monadic topologies,Initial and final characterized spaces.

Lattice of Subalgebras in the Category of Fuzzy Groups

Naseem Ajmal and Sunil Kumar*

Department of Mathematics,University of Delhi, Delhi-110007,India

Abstrsct: In the category of fuzzy groups,we introduce the notion of a subgroup of a fuzzy group,with the help of definition of subobject of an object.Also we construct for each object ¦Ì, which is a fuzzy group,a lattice of fuzzy subgroups of ¦Ì.It is shown that these lattices are complate.Moreover,this leads to a discussion of normality of a fuzzy subgroup in a fuzzy group. We establish characterizations of normality of fuzzy subgroups in a fuzzy group. Consequently,we construct a lattice a fuzzy normal subgroups of a fuzzy group ¦Ì, which is a sublattice of the lattice of its fuzzy subgroups of ¦Ì. Finally,it is shown that for each fuzzy group ¦Ì, the lattice of its fuzzy normal subgroups is modular.

Keywords: Fuzzy algebra,fuzzy group, lattice,category,subobject,normal fuzzy subgroup.

Fuzzy Ideals with Operators in BCK/BCI-algebras

Young Bae Jun

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

Abstrsct: The notion of M-ideals in M-BCK/BCI-algebras is introduced,and then its properties are investigated.The fuzzification of M-ideals is discussed.Given an M-fuzzy ideal,a new fuzzy ideal is established.Using a collection of M-ideals,M-fuzzy ideals are constructed.

Keywords: M-BCK/BCI-algebras,closed(fuzzy)ideal,M-ideal,M-fuzzyideal,M-homomor-phism.

Common Fixed Points of Four Mappings in A Fuzzy Metric Space

P.Balasubramaniam and S.Muralisankar

Department of Mathematics,Gandhigram Rural Institute-Deemed University Gandhigram-624302,Tamil Nadu,India

R.P.Pant

Department of Mathematics,Statistics and Computer Science G.B.Pant University of Agriculture and Technology Pantnagar 263145,Distt.Udhamsinghnagar,U.P.,India

Abstrsct: In this paper,we prove some general fixed point theorems,which generalize a result of Pant [4] for fuzzy mappings in fuzzy metric space and provides an affirmative answer in the setting of fuzzy metric spaces to the open question on the existence of a contractive definition which generates a fixed point but does not force the mapping to be continuous at the fixed point.

Keywords: Fuzzy metric space,compatible,R-weakly commuting map,fixed point theorem.

Half-encryption and Full-encryption of Fuzzy Recognition

Guo Ping Han and Kai Quan Shi

School of mathematics and System Science Shandong University, Jinan,Shandong 250100,P.R.China

Abstrsct: The paper is a term of research in which fuzzy recognition theroy,information encryption theory and encryption technique are mutually engrafted. Half-encryption and full-encryption of fuzzy recognition and its general concept are introduced in the paper. It gives the mathematic structure of half-encryption and fuzzy- encryption of fuzzy recognition.The half- encryption algorithm of fuzzy recognition,the full- encryption algorithm and application of fuzzy recognition are presented in the article.

Keywords: Fuzzy recognition, recognition model, half- encryption, full- encryption, encryption algorithm,application.

L-fuzzy Set Categories Based on Generalized Extension Principle

HaiMing Li*,KeYun Qin andYang Xu

Intelligence Control Development Center,Southwest Jiaotong University, Chengdu 610031,Sichan,P.R.China

Abstract: In this paper,the L-generalized extension principle for L-fuzzy set is introduced and some of its properties are discussed,Then five L-fuzzy set categories are defined and their relations are discussed.

Keywords: Category,L-fuzzy set category,the generalized extension principle for L-fuzzy set. The extension principle is one of the main tools in zadeh fuzzy set theory. Many papers have discussed about the extension principle and its generalization.The paper [6] had proposed the generalized extension principle based on fuzzy relations and discussed some of its properties.In this paper,we proposed the generalized extension principle for L-fuzzy relations and discussed some of its properties.Based on them£¬we constructed five L-fuzzy categories and discussed relations between them.

Lattice Valued Smooth Topological Molecular Lattices

Jie Zhang and ChongYou Zheng

Department of Mathematics,Capital Normal University, Beijing,100037

Abstract: In this paper,the concept of lattices valued L-smooth topological molecular lattices was given.We have discussed the relations between L-smooth co-topologies and co-topologies on a completely distributive lattice L. we have obtained the representation theorems and the decomposition theorems of L-smooth co-topology on a completely distributive lattice L. besides,we discussed some properties of smooth continuous generalized order-homomophisms and proved that the category of L-smooth topological molecular lattices is a topological category and the category of topological molecular lattices is a bireflective full subcategory of the category of L-smooth topological molecular lattices.

Keywords£» L-STML,co-topology,continuous GOH,category.

Generating an ¦Á-Pareto Optimal Solution to Multiobjective Nonlinear Programming Problems with Fuzzy Parameters:A Decomposition Method Mahmoud

A.Abo-sinna*

Department of Basic Engineering Sciences,Faulty of Engineering. E1-menoufiya University.Shebin E1-kom,Egypt

Abstract: This paper proposes a decomposition method for generating an ¦Á-Pareto optimal solution for multiobjective nonlinear programming £¨MONLP£©problems with fuzzy parameters in the objective functions £¨FMONLP£©.These fuzzy parameters are characterized be fuzzy numbers.For such problems,the concept of ¦Á-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basic of the ¦Á-Level sets of the fuzzy numbers.The decomposition method is based on the principle of optimality in dynamic programming£¨DP£©,and on the concept of fuzzy¡°decision¡± model£¨FDM£©. Assuming the separability and monotonicity of the problem,a generalized functional equation of dynamic programming(DP) is derived. Also,an interactive fuzzy decision-making algorithm for generating ¦Á-Pareto optimal solution through the decomposition method is developed.A numerical example is given to illustrate the results developed in this paper.

Keywords: Fuzzy multiobjective nonlinear programming problems,fuzzy parameters,fuzzy numbers,¦Á-Pareto optimality,dynamic programming,fuzzy decision model, decomposition,interactive decision making.

Semigroups Characterized by Their Fuzzy Bi-ideals

Javed Ahsan and Kui Yuan Li

Department of Mathematics and Statistics University of West Florida,Pensacola,FL 32514,USA

Muhammad Shabir

Department of Mathematics, Quaid-I-Azam University Islamabad-Pakistan

Abstract: In this paper we characterize regular semigroups,semigroups that are semilattices of left groups,semigroups that are both regular and intraregular and weakly regular semigroups by the properties of their fuzzy bi-ideals.

Keywords: Ideals,quasi-ideals,regular and intraregular semigroups,fuzzy ideals,fuzzy quasi-ideals,fuzzy bi-ideals.

A Fuzzy Approach Based Goal Programming to Generate A Priority Vector in the Analytical Hierarchy Process

Waiel F.Abd EL-Wahed*

Engineering Mathematics Department,Faculty of Engineering, EL-Minufiya University,Shiben El-Kom,Egypt

Abstract£» The generation of weights to evaluate a set of alternatives from pairwise comparison judgements represents an integral part of the analytic hierarchy process£¨AHP£©, Due to ambiguity of the experts¡¯understanding of the weighting scale and inevitable fuzziness in the input judgements to the AHP,the result weight vector may be inaccurate.In this paper,these problems are surmounted by developing a fuzzy approach,which is called FABGP,that implements the attractive features of both fuzzy numbers and ¦Á-level cut to generate an interval pairwise comparison matrix. This matrix is transformed into a linear goal programming mode£¨GPM£©to determine the optimal weight vector. The mathematical background of the given approach has been discussed.The performance of the FABGP is verified by comparing its results with some common approaches.In addition, a heuristic method that isolates the independent stable intervals of the optimal weights is highlighted.The proposed approach is illustrated by means of a numerical example.

Keywords: Analytical hierarchy process,fuzzy numbers,stability analysis and goal programming.

Fuzzy Filters/Ideals in BCI-algebras

Young Bae Jun*and Wook Hwan Shim

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

Celestin Lele

Department of Mathematics Harbin Institute of Technology Harbin 15001,china E-mail:lele-clele@yahoo.com

Abstract: The fuzzificaiton of filters in BCI-algebra is considered,and then relations between such fuzzifications and fuzzy ideals are discussed.Conditions for a fuzzy ideal to be fuzzy filter are given.Using a collection of filters,a fuzzy filter is established.

Keywords: Fuzzy£¨closed£©filter,fuzzy(closed)ideal,p-radical,p-semisimple BCI-algebra.

Stronger and Weaker Forms of Fuzzy Pairwise Continuous Mappings

Y.C.Kim

Department of Mathematics,kangnung National University kangnung,210-702,korea

A.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 some stronger and weaker forms of fuzzy pairwise continuity in fuzzy bitopological spaces.We study some properties between them.

Keywords: Fuzzy pairwise continuity,fuzzy pairwise strongly continuity,fuzzy pairwise ¦È-continuity£¬fuzzy pairwise weakly ¦È-continuity fuzzy paiwise super-continuity,fuzzy pairwise almost continuity,fuzzy pairwise ¦È-continuity, fuzzy pairwise weakly ¦È-continuity.

Fuzzy Semi-preconnectedness and Fuzzy Semi-preseparation Axioms*

Bai shi-zhong

Department of Mathematics Wuyi University, 529020Jiangmen£¬Guangdong,P.R.china

Abstract: In this paper,we introduce a kind of new fuzzy connectedness and some new fuzzy separation axioms,and establish some of their fundamental properties in fuzzy topological spaces .

Keywords: Fuzzy topology, semi-preopen set, semi-pre-q-neighborhood, SP-irresolute mapping,semi-preconnectedness,semi-pre-T axioms.

Lattice-valued Zp-Pan-integrals ¢ò£ºFor Lattice-valued Borel Functions on Lattice

Guang Quan Zhang1£¬Yong Hong Wu2 and Jie Lu1

1 Faculty of Information Technology,University of Technology, Sydney P.O.Box 123,Broadway,NSW2007,Australia E-mail;jielu@it.ust.edu.au E-mail£ºzhangg@ituts.edu.au 2 School of Mathematics and Statistics,Curtin University of Technology GPO Box U1987,Perth,Western Australia E-mail£ºyhwu@cs.curtin.edu.au

Abstract: In this paper£¬ we introduce and study a new type of integral,namely the lattice-valued Zp-pan-integral,for lattice-valued Borel functions on lattice,A number of theorems characterizing the properties of the integral are developed and presented in the paper.

Keywords: Non-additive measure,fuzzy integral,lattice-valued Borel function,pan-integral.