THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 8, Number 1, March 2000

CONTENT

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F-spectrum of a BCK-algebra

A. Hasankhani

Department of Mathematics

Sistan and Baluchestan University, Zahedan,iran

Abstract;A set of all fuzzy prine ideals of a commutative BCK-algebra is topologized and the resulting space is studied.

Keywords: Fuzzy(prime)ideal,fuzzy prime spectrum,fuzzy point

Fuzzy Order and ( )-fuzzy Subgroup

Sandeep Kumar Bhakat

Siksha-Satra,visva-Bharati,P.O-Sriniketan-731236

Dist-Birbhum(WB),India

Abstract: The notion of q-fuzzy order of an element of a group is introduced. Fuzzy order of an ( )-fuzzy subgroup is defined and fuzzy version of Lagrange¡¯s the orem is derived. Relation between q-fuzzy order ,ordinary order of an element of a group and order of a group are also obtained.Characterization of a cyclic group in terms of q-fuzzy cyclic subgroup and fuzzy version of some group theoreties are also derived.

Keywords: ( )-fuzzy subgroup, ( )-fuzzy normal subgroup,q-fuzzy commutative subgroup. q-fuzzy cyclic subgroup, q-fuzzy oder, fuzzy Lagange¡¯s theorem.

On Some Weaker Forms of L-fuzzy Compactness

H.Ayg

Department of Mathematics,Kocaell University,

Anitpark Yani,41300-Izmit,Turkey

e.mail:halis@kou.edu.tr

Department of MatMatematica,Universidade Federal do Parana,

Centro Politecnico,Caixa,Postal 19081,Curitiba-Parana,815311-990,Brasil.

Abstract: In the present paper, R-compactness, weak compactness and -rigidity are introduced in L-fuzzy topological spaces, where L is a fuzzy lattice. We define these weak forms of L-fuzzy compactness on arbitrary L-fuzzy sets, obtain different characterizations and study some their basic properties.

Keywords: L-fuzzy topology, fuzzy lattice, fuzzy R-compactness, fuzzy weak compactness, fuzzy -rigidity.

Product of Fuzzy -subgroups

B. Davvaz

Department of Mathematics, Yazd University, Yazd, Iran

And Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

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

Abstract: In this paper we define T-product between two fuzzy H -subgroups of a given H -group. We consider the fundamental relation defined on an H -group and prove some interesting results in this connection.

Keywords: t-norm, H -groups, fuzzy H -subgroups, H -product, fundamental relation, fundamental group

On smoothTopological Spaces ¢ó

A.A. Ramadan

Department of Mathematics, Faculty of Science,

Cairo University, Beni-Suef, Egypt.

Abstract: This paper is devoted to study the concept of smooth topological spaces and some related concepts. Also, the functors between the category of smooth topological spaces, fuzzifying topological spaces and ordinary topological spaces are given.

Keywords: Smooth topological spaces, fuzzifying topological spaces, ordinary topological spaces and smooth closure operator.

Applications of Fuzzy Cognitive Maps to Determine the Maximum Utility Of a Route

W.B. Vasantha,V.Indra

Department of Mathematics

Indian Institute of Technology, Mardras 600036, India

indra@olmaii.com

Sreenivas Mandalam

Talarian Corporation

333,Distel Circle, Los Altos, CA 94022, USA

mandala@olmaiicom

Abstract: In this paper, we identify thej maximum utilization time period of a route and estimate its overall utility rate by initiating the study of raw data analysis. The effective tools that we are using to analyze the Fuzzy Cognitive Maps(FCMs). In this paper, we obtain simple signed FCMs as expert opinion and form the corresponding connection matrix. For a specific problem, we obtain more than one expert opinion. Each of these connection matrices is then combined together to form the cumulative connection matrix. We make model implications on the individual ad well as the cumulative connection matrices. We have developed a Java program to study the model inplications. This is available over the internet at the address http://members.tripod..com/-mandalam/javal-0/triplet.html. Model implications help in analyzing the pros and cons of a problem. In this paper, we examine the passenger transportation problem and illustrate our research by assessing the utility rate of a local route in the metropolitan city of Madras, India. FCMs play a vital role in indentifying the maxmum utilization of a time period in a day, in predicting the overall utility of the routes and in the stability analysis.

Keywords: Fuzzy cognitive maps (FCMs), conceptual nodes, hidden pattern, fixed point, limit cycle.

On fuzzifying Filters,Grills and Basic Proximities

M.H. Ghanim

Department of Mathematics, Faculty of Science,

Zagzaig University,Zagazig, Egypt

Fayza A.Ibrahim and Mervat A. Sakr,

Department of Mathematics, Faculty of science,

Ain Shann University, Cairo Egypt

Abstract: In this paper wu introduce the concepts of a fuzzifying grill and of a fuzzifying basic proximity. We present some relations between fuzzifying filters, fuzzifying grills and fuzzifying basic proximities.

Keywords: Fuzzifying filter, grill and ultrafilter, fuzzifying proximity, fuzzifying proximal neigh-bourhood

On ordered L-fuzzy Bitopological Spaces

G.Balasubramanian

Department of Mathematics, Periyar University

Salem-636 011 Tamil Nadu, India.

Abstract: In this paper we introduce the concept of ordered L-fuzzy bitopological spaces. Besides giving some interesting properties of these spaces we also prove analogues of Urysohn¡¯s Lemma and Tietze extension theorem.

Keywords: Ordered L-fuzzy bitopological space, Lower pairwise L-fuzzy -ordered, Upper pairwise L-fuzzy -ordered, Pairwise L-fuzzy -ordered, Weakly pairwise L-fuzzy -ordered, Almost pairwise L-fuzzy -ordered, Strongly paierwise normally ordered.

Fuzzy Uniform Spaces-covering Approach

G.K. Chandrika and K.N. Meenakshi

Department of Mathematics, Avinashilingam Deemed University

Coimbatore-641 043. India

Abstract: In this classical theory of uniform spaces there are three important different approaches to the definition of uniform spaces. The first approach is in terms of entourages, the second in terms of collection of pseudo-metrics and the third in terms of coverings. The entourage approach and the pseudo-metric approach have been generalized to fuzzy situation by Lowen[6] and Hohle[1] respectively. The study presented in thin paper is a generalization of the covering approach by Isbell, in his classic work ¡°Uniform spaces¡±[3].

Keywords: F-star function, F-star refinement, fuzzy uniform structure.

Fuzzy Congruence Relation on Implicative BCK-algebra

Samy M.Mostafa

Mathematics department, Faculty of Education

Ain Shams University, Roxy Cairo, Egypt

Abstract: In this paper we introduce the relationship between fuzzy ideal and fuzzy congruence relation on implicative BCK-algebra.

Keywords: Inplicative BCK-algebra-fuzzy implicative ideal-fuzzy congruecce relation.

Fixed Point for Pair of Maps in Fuzzy Metric Spaces.

P.Vijayaraju and M.Marudai

Department of Mathematics, Anna University

Chennai (Madras)-600025,India

Abstract:Many authors defined the of fuzzy metric spaces in different way (see [2,4,6,8]), Based on the definition of fuzzy metric spaces given by Kramosil and Michalek [6] in 1988, Grabiec [3] proved the fuzzy version of Banach¡¯s contraction principle and Edelsteins¡¯s fixed point theorem and in 1992, Fang [1] established some fixed point theorems for contractive maps and in 1994, Mishra et al. [7] obtained common fixed point theorems for contractive maps and asymptotically commuting maps. Subsequently in 1995, Subrahmanyam [9] generalized Jungck¡¯s fixed point theorem to fuzzy metric spaces. We prove common fixed point theorem for a pair of self-maps in a fuzzy metric space. This result offers a generalization of Theorem 5 of Grabiec [3].

Keywords: Fuzzy metric space, complete, common fixed point.

Fuzzy d-ideals of d-algebras

Young Bae Jun ,

Department of Mathematics Eduction

Gyeongsang National University, Chinju 660-701, Korea

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

J.Neggers

Department of Mathematics, University of

Alabama, Tuscaloosa, AL 35476-0350, U.S.A.

e-mail: jneggers@gp.as.ua.edu

Hee Sik Kim

Department of Mathematics, Hanyang University, Seoul 133-791, Korea

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

Abstract: We introduce the notion of fuzzy d-subalgebra, fuzzy d-ideal, fuzzy do-ideal and fuzzy d -ideal, and investigate relations among them. In particular, we are able to specialize results obtained to fuzzy BCK-algebras as well.

Keywords: BCK-algebras, d-(d -) algebras,(fuzzy) d-(d , d -) ideals.

Role of Fuzzy Clans in the Fuzzy Proximities

K.C. Chattopadhyay,S.K. Samanta

Department of Mathematics

Visva Bharati, Santiniketan-731235 India

U.K. mukherjee

Department of Mathematics

Burdwan University, Burdwan-713104 India

Abstract:In this paper fuzzy filters, fuzzy grills, fuzzy clans are introduced and their properties are studied, Using these properties we establish the result ¡®every fuzzy preproximity satisfying some condition is a clan generated structure¡¯.

Keywords: Fuzzy preproximity, preproximity of fuzzy sets, fuzzy filters, fuzzy grills, fuzzy prime filters, fuzzy clans

On A Generalization of The Absorption Equation

Tomasa Calvo

Department of Mathematics and Computer Science, University of the Balearic Islands

Ctra. De Valldemossa, km 7.5, Palma de Mallorca, Spain

Bernard De Baets

Department of Applied Mathematics, Biometrics and Process Control,

Gent University, Coupure Links 653), B-9000 Gent, Belgium

Abstract: The aim of this work is to solve the absorption equation for two classes of commutative, associative and increasing binary operators: uninorms and nullnorms. Both uninorms and nullnorms are a generalization of t-norms and t-conorms. The first ones have a neutral element and the second ones an absorbing element lying anywhere in the unit interval. Since the t-norm and t-conorm solutions of the absorption equation are well known, it is a natural question to look for uninorm and nullnorm solutions as well. In the case of two uninorms or two nullnorms, no new solutions arise. Howerver, when considering the combination of a uninorm and a nullnorm, one combination also leads to the known solutions, while the other one has as solutions any nullnorm whith absorbing element identical to the neutral element of a left-continuous, conjunctive, idempotent uninorm or a right-continuous, disjunctive, idempotent uninorm.

Keywords: Absorbing element, absorption equation, idempotence, neutral element, nullnorm, t-conorm, t-norm, uninorm.

Completely Fuzzy Prime Ideals of Distributive Lattices

M.Atallah

Mathematics Department,

Faculty of Science

Tanta University, Tanta, Egypt.

Abstract: We introduced the notion of completely fuzzy prime , some properties and a characterization theorem of it are given.

Keywords: L-fuzzy ideals, fuzzy prime ideals, completely fuzzy prime ideals, fuzzy distributive lattices.

Reduced Fields,Primitive and Fuzzy Galois Theory

Yousef Alkhamees and John N.Mordeson

Department of Applied Mathematics/Computer Science

Creighton University, Omaha, Nebraska 68178, USA

Abstract: We introduce two concepts for crisp field extensions which have not appeared before in the literature, namely, that of a reduced field and a primitive field. We not only study these fields in their own right, but we also use these concepts to characterize field extensions whose intermediate fields are chained. We also introduce a concept for fuzzy intermediate fields which has no nontrivial counterpart in crisp field theory, namely, that of a fuzzy intermediate field of maximal chain. We then apply our results to further develop fuzzy Galois theory. In particular, we examine the connection of the notion of minimal generating sets for fuzzy subgroups and fuzzy subfields via the Galois correspondence.

Keywords: Intermediate field, fuzzy intermediate field, Galois theory, purely inseparable, separable algebraic, automorphism

Some Properties of Generalized Approximation Spaces

Keyun Qin and Yang xu

Department of Applied Mathematics, Southwest Jiaotong University

Chengdu, Sichuan,61003 1, P.R.China

Abstract: This note is devoted to studying properties of the generalized approximation space based on properties of the corresponding binary relation. Some equivalent conditions are diven.

Keywords: Approximation space, rough set, interval structure

On Connectedness in Fuzzy Convergence Spaces

Gunther J?ger

Deutsches Zentrum fur Luft-und Raumfahrt DLR

Institut fur Hochfrequenztechnik; Postfach 11 16;D-82230 We ling; Germany

Abstract:In a former paper we extended a notion of connectedness in fuzzy topological spaces to fuzzy subsets. Here we generalize this definition to fuzzy convergence spaces and prove its basic characteristics. To prove its productivity we follow an idea of Vainio[15] which proved useful in the realm of classical convergence spaces.

Keywords: Fuzzy topology, fuzzy convergence space, connectedness.

S-fuzzy Subgroups

Yuan Xuehai and Shen Zhengwei

Department of Mathematics, Liaoning Normal University

Dalian 116029,P.R.China

Abstract: Concept of S-fuzzy subgroups of a group G is given. A fuzzy subgroup of a group G can be seen as a S-fuzzy subgroup. It is proved that every S-fuzzy subgroup can be generate a S-fuzzy subgroup of G.

Keywords: Groups, fuzzy subgroups, S-fuzzy subgroups, generated S-fuzzy subgroups.

A Fast Algorithm for Choquet-intergral-based Nonlinear Multiregression Used in Data Mining

Kebin Xu,Youan Ke

Department of Electronic Engineering

Beijing Institute of Technology, Beijing 100081, P.R.China

Zhengyuan Wang

Department of Mathematics, Hebei University

Baoding, Hebei 071002, P.R.China

Abstract: Multiregression is one of the most common description and prediction tools in data mining. The nonlinear relation between the target attribute and predictive attributes, for which the nonlinearity is incurred by the interaction among the predictive attributes, can be approximately described by a nonlinear multiregression based on the Choquet integral with respect to a nonadditive set function that is also called a fuzzy measure. By using a direct local-revising strategy and a double optimization technique, this paper presents a fast iterative algorithm to determine regression coefficients including the values of the set function when suitable data are available. A comparision of this algorithm is also given in the paper.

Keywords: Algorithm, data mining, fuzzy measures, nonlinear integrals, multiregression.

On the Controllability of Fuzzy Dynamical Systems(¢ñ)

Zouhua Ding and Abraham Kandel

Computer Science and Engineering Department

University of South Florida, Tampa, Fl 33620-5350 USA

Abstract: In this paper, we privade a way to combine differential equations with fuzzy sets to form a fuzzy logic system. called Fuzzy Dynamical System(FDS),controllability, will be studied.

Keywords: Fuzzy dynamical system, controllability, differential Equation, fuzzy Set.

Fuzzy Connection Networks Related to max-min Fuzzy Bidirectional Associative Memories and Their Properties

Puyin liu,Hao Wang,Hanjiang Zhang

Department of System Engineering and Mathematics,

National University of Defense Technology, Changsha,410073, P.R.China

Abstract:In the paper, we introduce the fuzzy connection networks related to max-min fuzzy bidirectional associative memories(FBAMs) to study the attractors and attractive cycles of FBAMs. And define the elementary memories of FBAMs by the fuzzy connection networks. We show some sufficient and necessary conditions that an attractor of FBAMs is the elementary memory. And utilize Homomorphism operators on [0,1] ¡Á[0,1] to analyze the transition trajectories of the state, attractors and attractive cycles of FBAMs. By the conclusions obtained here, a novel way for designing FBAM with better faulttolerance may be constructed. Finally a simulation example illustrates our conclusions.

Keywords: Fuzzy connection network, attractor, elementary memory, attractor, elementary memory, attractive cycle, attractive neighborhood.

On Pime Fuzzy Ideals of a Semigroup

Xiang-Yun Xie

Department of Mathematics and Physics, Wuyi University,

Jiangmen, Guangdong, 529020, China.

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

Abstract: In this paper a characterization of prime fuzzy ideals of a semigroup S given. We show that a non-constant fuzzy ideal f of a semigroup S is prime if and only if f is two-valued, and there exists an element x in S such that f(x )=1, f ={x } is a prime ideal of S. We also introduce the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals and weakly completely prime fuzzy ideals of a semigroup S, and establish the relations among four classes prime ideals. Furthermore, we characterize weakly prime fuzzy ideals of S by its level ideals.

Keywords: Semigoup, ideal, fuzzy ideal, prime fuzzy ideal, completely prime fuzzy ideal, weakly prime fuzzy ideal, weakly completely fuzzy ideal.

Extension of Fuzzy Number-valued Possibility Measures on a CompleteLattice

Guangquan Zhang,Yonghong Wu

School of Mathematics and Statistics, Curtin University of Technology

GPO Box U 1987, Perth WA 6845, Australia

E-mail:{zhangg, yhwu}@cs.curtin.edu.au

Kok-lay Teo

Department of Applied Mathematics, The Hong Kong Polytechnic University

Hung Hom, Kowloon, Hong Kong

E-mail: meteokl@polyu.Edu.nk

Abstract: In this paper, we extend the concept of fuzzy number-valued possibility measure to complete lattices and obtain the necessary and sufficient conditions for the extendability,

Keywords: Fuzzy number, possibility measure, extension, P-consistency, complete lattice.

Extension of Fuzzy Number-valued Necessity measures on a Complete Lattice

Guangquan Zhang,Yonghong Wu

School of Mathematics and Statistics, Curtin University of Technology

GPO Box U 1987, Perth WA 6845, Australia

E-mail:{zhangg, yhwu}@cs.curtin.edu.au

Kok-lay Teo

Department of Applied Mathematics, The Hong Kong Polytechnic University

Hung Hom, Kowloon, Hong Kong

E-mail: meteokl@polyu.Edu.nk

Abstract: In this paper, we extend the concept of fuzzy number-valued necessity measure to complete lattices and obtain the necessary and sufficient conditions for the exetendability.

Keywords: Fuzzy number, necessity measure, extension, N-consistency, complete lattice

Implication Filter Spaces

Song Zhenming

Department .of applied. Mathematics, Southwest Jiaotong University

Chengdu, Sichuan,61003 1, P.R.China

Abstract: In paper, based on lattice implication algebras, the implication filter space was established with it topological properties being discussed. The continuous of lattice implication homomorphism was also discussed. This provided a theoretical foundation for applying topological transformation to uncertain inference.

Keywords: Lattice implication algebras, implication filters, topology