THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 6, Number1, March 1998

CONTENT

DETAILS

A New Definition for L-fuzzy Semi-open Sets*

Meng Han and Meng Guangwu

Department of Mathematics, Liaocheng Teacher¡¯s College

Shandong 25200, P.R.China

Chen Shuili

Department of Mathematics, Jianghan Petroleum Institute

Jiangling, Hubei 434102, P.R.China

Abstract: In this paper, Azad¡¯s definition for fuzzy semi-open sets is improved, and a new definition for L-fuzzy semi-open sets in L-fuzzy topological spaces is presented. The L-fuzzy semi-open sets possess almost all properties of usual semi-open sets in crisp topology. L-fuzzy semi-closure is built, and Kuratowski¡¯s 14 sets theorem is proved. The concept of semi R-neighborhood for L-fuzzy points and L-fuzzy semi-adherence points of L-fuzzy sets are presented, and the structure of an L-fuzzy set is yielded. Some characterizations for L-fuzzy semi-open mappings are obtained.

Keywords: L-fuzzy topology, L-fuzzy semi-open sets, L-fuzzy semi-closure oprator, L-fuzzy semi-open mapping, fuzzy lattice.

General Variational Inequalities for Fuzzy Mappings

Muhammad Aslam Noor

Department of Mathematics, College of Sience

King Saud University, P.O.Box 2455, Riyadh 11451, Saudi Arabia

Abstract: In this paper, we introduce the concept of general variational inequalities for fuzzy mappings. Using the projection technique and its variant forms, we establish the equivalence between the general variational inequalities for fuzzy mappings, fuzzy fixed point problems and the Wiener-Hopf equtions.This equivalence is used to suggest and analyze a number of iterative algorithms foe computing the approximate solutions of variational inequalities and related optimization problems. The convergence analysisi is also studied. We discuss several special results, which can be obtained from our main results.

Keywords: Fuzzy mappings, variational inequalities, algorithms, optimization, mathematical programming, Wiener-Hopf equations.

Convergence for Sequence of Fuzzy Measurable Functions on Fuzzy-Number-valued Fuzzy Measure Spaces

Li Jun

Department of Mathematics and Informatics

Faculty if Science, Chiba University

Chiba 263, Japan

Abstract: The purpose of this paper is to investigate the convergence for sequence of fuzzy measurable functions on a fuzzy-number-valued fuzzy measure space. The concepts of the property (p.p.p.) and the order-continuity of a fuzzy-number-valued fuzzy measure are proposed. Some results on the convergence for sequence of real valued-number measurable functions on fuzzy measure space, such as, Egoroff¡¯s theorem, Lebesgue¡¯s theorem and Riesz¡¯s theorem, are generalized to that of fuzzy number-valued measure spaces by using the property (p.g.p.) and the order-cintinuity.

Keywords: Fuzzy numbers, fuzzy-number-valued measure, property (p.g.p.), fuzzy measurable functions, Egoroff¡¯s theorem, Lebesgue¡¯s theorem, Riesz¡¯s theorem.

Fuzzy Direct Product of Fuzzy Subgroups of Subgroups

Y.Alkhamees*

Department of Mathematics/Computer Science

Creighton University, Omaha, Nebraska 68178, USA

Abstract: We introduce the notion of a fuzzy ditrct product of fuzzy subgroups of subgroups of a group and study its properties and employ it to obtain necessary and sufficient conditions for a group G to be the direct product of its normal subgroups in terms of a fuzzy direct product of fuzzy subgruops of the subgroups. We also generalize the definition of weak direct product introduces in [8] to the non-abelian case and then establish a one-one correspondence between these fuzzy direct products.

Keywords: Fuzzy subgroups, fuzzy normal subgroups, fuzzy subnormal subgroups, fuzzy subgroups of composition length, fuzzy subgroups of pricipal length, fuzzy direct product, fuzzy weak direct product.

Fuzzy Super Irresolute Functions

I.M.Hanafy and A.A.Nasef

Department of Mathematics, Faculty of Education

Suez Canal University, El-Arish, Egypt

Abstract: In this paper , the concept of strongly semicontinuous functions [1] (which is renamed by super irresolute functions ) is considered and studied in fuzzy setting. Fuzzy super irresolute is weaker than fuzzy completely irresolute [3]. A comparison between thia type and other exsiting ones are established, several characterizations, properties and its effect on some fuzzy topological spaces are sutdied. Also we have succeeded to improvement Corollary 3.2, Theorem 3.4, and Theorem 5.2 of [3].

Keywords: Fuzzy regular open set; fuzzy semiopen set; fuzzy completely irresolute; fuzzy semicontinuous functions; fuzzy compactness; fuzzy near (almost) compactnessl; fuzzy semiconnectedness.

Fuzzy Linear Programming Problems with Interval-value Fuzzy Coefficients

Xiaozhong Li

Department of Computer Science, Liaocheng Teachers University

Liaocheng, Shandong, 252059, P.R.China

Kaiquan Shi

Department of Automation Engineering, Shandong University of Technology

Jinan, Shandong 250061, P.R.China

Abstract: In this paper, fuzzy linear programming problems with interval-value fuzzy coefficients (IFLP) are presented. Corresponding auxiliary models are obtained in the meaning of different criteria. Algorithm to solve IFLP is given.

Keywords: Interval-value fuzzy linear programming£¨IFLP£©£¬Interval-value fuzzy number£¨IFN£©£¬auxiliary model.

Stratified Fuzzy Uniformities

Fatma Bayoumi

Department of Mathematics, Faculty of Science

Benha University, Benha,Egypt

Abstract: In [6] a notion of fuzzy uniformity has been introduced and many theorems corresponding to the usual theorems have been proved. In this paper the notion of this fuzzy uniformity to be stratified and the related notion of stratification are introduced and studied. We proved that fuzzy topology and the fuzzy proximity associated to a stratified fuzzy uniformity also are stratified, moreover the stratification to coincide with the fuzzy topology and the fuzzy proximity associated to the stratification of , respectively. We also prove that every stratified fuzzy topology is stratified fuzzy quasi-uniform spaces and the category of all stratified fuzzy quasi-uniform spaces.

Keywords: Fuzzy uniformities, stratified fuzzy uniformities, fuzzy topologies, strstified fuzzy topologies, fuzzy proximities, stratified fuzzy proximities.

Entropy of Fuzzy -dynamical systems

Adrian I.Ban

Department of Mathematics, University of Oradea

Str. Armatei Romane 5, 3700 Oradea, Romania

Abstract: Entropy of a fuzzy partition and the fuzzy process [Dumitrescu, 1993] and the entropy of the fuzzy dynamical system [Dumitrescu, 1995] had been defined for operations between fuzzy sets induced by t-conorm associated. In this work same notions are introduced if the operations between fuzzy sets are induced by every triangular norm having an certain property.

Keywords: Entropy, t-norm, fuzzy -measure, fuzzy -partition, fuzzy -measure preserving transformation, fuzzy -dynamical system.

On the Variational Principle and Fixed Point Theorem in Fuzzy Metric Space

Fang Jin-xuan*

Department of Mathematics, Nanjing Normal University

Nanjing, Jiangsu 210079, P.rRChina

Abstract: The main purpose of this paper is to establish a variational priciple and a fixed point theorem in the general fuzzy metric spaces and to give a direct proof of the equivalence between these two theorems. As an application, a varitional principle and a fixed point theorem in Menger probabilistic metric spacess are obtained.

Keywords: Fuzzy metric space, Menger probabilistic metric space, variational principle, fixed point.

Prediction of Sofrware Reliability Using Fuzzy Time Serise Approach

S.Chatterjee and S.S.Alam

Department of Methematics, Indian Institute of Technology

Kharagpur 721 302, India

R.B.Misra

Department of Electric Engineering, Indian Institute of Technology

Kharagpur 721 302, India

Abstract: All the software reliability models, available in literature, are probabilistic in nature. Aserious drawback of all these models is that none of the models is able to analyze all types of software failures. This is due to various assumptions made in all these probabilistic models. Software is totally design and envirnment dependent. It is developed and tested by buman beings. As human behavior is fuzzy, therefore, the concept of fuzzy set theory is applicable in develop a software reliability models. In this paper, a fuzzy time series approach is used to develop a software reliability model expressing the time between failures ( TBF) linguistically for estimating the reliability of a software. Usefulness of the model is demonstrated using real failure data.

Keywords: Software reliability, fuzzy time series, times between failures (TBF) , fuzzy reliation.

On the Category of L-fuzzy Topological Rings

Guangwu Meng, Xingfang Zhang and Han Meng

Department of Mathematics, Liaocheng Teacher¡¯s College

Shandong 252000, P.R.China

Shuili Chen

Departmenr of Mathematics, Jianghan Petroleum Institute

Jingsha, Hubei 434102, P.R.China

Abstract: In this paper, L-fuzzy topological rings is first proposed, and it is discussed from the point of view of category. The category FTOPRNG of L-fuzzy topological rings is introduced. Some characterizations of the objects of FTOPRNG are given. Some subcategory of FTOPRNG. The relation between FROPRNG and the category RNG of crisp rings is built by introducing a forgetful functor.

Keywords: L-fuzzy topological ring, L-fuzzy topological space, ring, category, functor, fuzzy lattice.

Analysis of Software Reliability Using Fuzzy Test Effort and Learning Factor

S.Chatterjee, S.S.Alam

Department of Mathematics, Indian Institute of Technology, Kharagpur, India, 721 302

R.B.Misra

Department of Electric Engineering, Indian Institute of Technology, Kharagpur, India, 721 302

Abstract: In software reliability, most of the models so far developed are probabilistic. All these probabilistic models have various draw backs due to different assumptions made in all these models. Software development and testing is human dependent and as the human bahavior is fuzzy, it is better to use fuzzy set theory for developing software reliability models. This paper presents a methodology for software reliability estimation incorporating the concept of fuzzy test effort and learning factor. A numerical example is given for the illustration of the method.

Keywords: Software reliability, fuzzy sets, fuzzy number, expected number of errors, tear effort, learning factor, possibility distrbution.

On the Product of Two Fuzzy Measures

Xuecheng Liu

Department of Mathematiques et de Statistique, Universite de Montreal

C.P.6128, Succ.A,Montreal, Quebec, H3C 3J7, Canada

Abstract: In this paper, we consider the third of the open problems presented by Z.Wang [6]. The problem is that under what condition the product the product of two fuzzy museasures with null additivity (resp. autocontinuous and resp. uniformly autocontinuous). We obtain that the above conclusion is true if the two fuzzy measures are mutually weakly absolutely continuous.

Keywords: Fuzzy measure, null-additivity, autocontinuity, uiform autocontinuity, weakly absolute continuity, strong absolute continuity.

Prime Ideals and Congruence Relations of Fuzzy Lattices*

Yiquan Zhu

Department of Mathematics, Huanggang Teacher College

Hubei 436100, P.R.China Liwen Pei

Department of Mathematics, Wuhan University

Wuhan, Hubei 430071, P.R.China

Abstract: Let F be a fuzzy lattice. In this paper, we will provide a new method to establish a congruence relation of F by a given prime ideal, and obtain a set of -generators of the congruence lattice C(F) of F. Moreover, we will also prove that any congruence relation on a fuzzy sublattice of F can be extended to a congruence relation on F.

Keywords: Fuzzy lattice ( De Morgan algebra), congruence relation, -generators, congruence extension.

The Fuzzy Linear Maps

Chang Bum Kim

Department of Mathematics

Kookmin University, Seoul, 136-702 Korea

Abstract: We investigate some situations in connection with two exact sequences of fuzzy linear maps.

Keywords: Fuzzy R-module, Fuzzy linear maps, Quasi-monic maps.

A Note on Non-dominated Elements of Fuzzy Relations

Michael Wagenknecht

University of Applied Sciences Zittau/Gorlitz, IPM

Tjepdor-Korner-Allee 16, 02763 Zittau, Germany

Abstract£º We investigate certain relationships between generators of Archimedean t-norms, negations, and aggregations when evaluating fuzzily non-dominated elements for preference relations.

Keywords: Fuzzy preferences, non-dominated elements, group decision making.

A Fuzzy Interpretation of Linear Syetems

Boniface I.Eke

Department of Mathematics

Morgan State University

Baltimore, MD 21251

Abstract£º Properties of systems of linear equations with which we are familiar in the crisp rheory are here examined in the fuzzy theory. Important concepts of the crisp theory fuzzified include the well known Gaussian elimination process and the representation and solution of systems of linear equations by matrices.

Keywords: Fuzzy linear equation, fuzzy unity, fuzzy zero, principal solution, pseudo-consistency, fuzzy matrix.

Anti Fuzzy Ordered Filters in Adjoint Semigroups of BCK-algrbra*

S.W.Kuresh

Department of Mathematics, Xinjiang Normal University, Urumqi 830053, P.R.China

Young Bae Jun

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

E-mail: ybjun@nongae. Gsnu.ac.kr

Abstract: We introduce the notion of anti fuzzy ordered filter in an n.p.o. semigroup. By using an anti fuzzy ideal of a BCK-algebra X, we induce an anti fuzzy ordered filter of adjoint semigroup M(X) of X. Conversely, by using the anti fuzzy ordered filter of M(X) of a BCK-algebra X, we induce an anti fuzzy ideal of X. Using these results,we finally prove that there is a bijection between the set of all anti fuzzy ideals of a BCK-algebra X and the set of all anti fuzzy ordered filters of M(X).

Keywords: Anti fuzzy ideal, anti fuzzy ordered filter.

Difference and Symmetric Difference of Fuzzy Sets Based on the Theory of Falling Shadows

Xue-hai Yuan

Department of Mathematics, Liaoning Normal University

Dalian 116029 ,P.R.China

Abstract: In this paper, we establish a theortical approach to define difference and summetric difference operators for fuzzy sets based on the theory of falling shadows and obtain three kinds of operators respectively.

Keywords: Fuzzy sets, random sets, falling shadows, difference, symmetric difference.

Strongly Variational Inequalities for Fuzzy Mappings

Jong Yeoul Park

Department of Mathematics, Pusan National University

Pusan 609-735, South Korea

Jae Ug Jeong

Department of Mathematics, Dong-Eui University

Pusan 614-714, South Korea

Abstract: In this paper, we show that the completely generalized strongly inequlities for fuzzy mappings are equivalent to the strongly generalizaed Wiener-Hopf equations for fuzzy mappings. We use this equivalence to suggest an iteative algorthm for solving completely generalized strongly variational inequalities for fuzzy mappings. We also discuss the convergence criteria for these algorthms.

Keywords: Fuzzy mappings, Wiener-Hopf equations, variational inequalities, iterative algorithms.

A Characterization of Fuzzy Commutative I-ideals in BCI-semigroups

Young Bae Jun

Department of Mathematics, Gyeongsang National University

Chinju 660-701, Korea

Abstract: We give an equivalent condition in order that a closed fuzzy I-ideal would be commutative in a BCI-semigroups, and prove that, in a commutative BCI-semigroups, every closed fuzzy I-ideal is a fuzzy commutative I-ideal.

Keywords: BCI-semigroups, (closed) fuzzy commutative I-ideal.

Coincidences and Fixed Points in Fuzzy metric Spaces

S.N.Mishra

Department of Mathematics, University of Transkei

Privated Bag XI Unitra, Umtata, Transkei,Southern African

S.L.Singh

Department of Mathematics, Gurukula Kangri University

Hardwar 249404, India

V.Chadha

Department of Mathematics, University of Wisconsin

Eau Claire, WI 52702, U.S.A

Abstract: Following Grabiec¡¯s approach to Banach type fixed theorems in fuzzy metric spaces, this paper obtains some new coincidence theorems for a family of mappings on an arbitrary set with values in fuzzy metric space, and derives a few general fixed point theorems for a family of mappings on a fuzzy metric space. These fixed point theorems are applied to obtain common solutions of fixed point type equations on product spaces.

Keywords: Coincidence, fixed point, fuzzy metric space (FM-space), Menger space, comtraction, weakly/asymptotically commuting mappings, comatible mappings.

On Fuzzy Reasoning by K¨®cz¡¯s Linear Rule Interpolation

Qihao Chen

Beijing University of Post and Telecommunications

Beijing, P.R.China

Shin Kawase

Department of Information Syetems, Faculty of Informatics

Teikto Heisei University, Chiba 209-01, Japan

Abstract: For fuzzy reasoning with sparse rule base, K¨®czy and Hirota proposed a linear interpolative method, which is interesting but has some default. We give here a renovation of their method, to obtain conclusion of the triangular type when premises are so.

Keywords: Fuzzy sets, linear interpolative fuzzy reasoning .

Cartesian Composition of Transition Spaces

J.N.Mordeson and P.S.Nair

Department of Mathematics and Computer Science

Creghton University, Omaha, NE 68178-0109

Abstract: There exists a common notion among finite state machines, directed graphs, fuzzy finite state machines and fuzzy directed graphs, namely that of a next state. Information retrieval systems and fuzzy systems also share this notion of next state. The purpose of this paper is to give an axiomatic tratment of the notion of next state under Cartesian composition. The concept of successor follows directly form the concept of next state. We study differrent structural properties including retrievability, connectedness and primary subspaces.

Keywords: Transition space, Cartesion composition, retrievability, connectedness.

ERRATUM On Pairwise Hausdorff Fuzzy Bitopological Spaces

Rekha Srivastava and Manjari Srivastava

Department of Applied Mathematics, Institutw of Technology

Banaras Hindu University, Varanasi-221 005, IndiaIn the above metioned paper [1], the ¡°Counter example 3.1¡± on page 558 does not work. We replace it by the following counter example which is suitable for our purpose in the paper.