THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 4, Number 2, June 1996

CONTENT

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Minimal Hausdorff Tmls*

Shengguang Li

Department of Mathematics, Shaanix Normal University, Xi'an, 710062, P.R. China

Abstract: The paper propose the concepts of minimal Hausdorff topological molecular lattices.With the aid of some tools, such as generalized order-homomorphism, ideal, etc., it establishes some characteristic theorems of minimal Hausdorff topological molecular lattices.

Keywords: Minimal Hausdorff topological molecular lattice, ideal, generalized order-homomorphism.

Fuzzy Ring and Fuzzy Ideals

K.A. Dib* and N. Galhum

Department of Mathematics, Faculty of Education,, Cairo University, Fayoum Branch, Fayoum, Egypt.

A.A. M Hassan

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

Abstract: Liu has defined the fuzzy subring and the fuzzy ideal of a given ordinary ring. But we have defined the fuzzy ring and redefined the fuzzy shubring and the fuzzy ideal of a given fuzzy ring by using the concept of the fuzzy space introduced by Dib. Also we study its relationship with Liu definitions of fuzzy subring and fuzzy ideal.

Keywords: Fuzzy space, Fuzzy group, Fuzzy ring, Fuzzy subring, Fuzzy ideal.

Linear Multiobjective Programming under Randomness and Fuzziness

Maha z. El-Asharm

Department of Statistice, Faculty of Economics and Political Science, Cairo University, Giza, Egypt.

Nadia M. Girgis

Faculty of Economics & Political Science, Cairo University, Giza, Egypt.

Abstract: A linear Multiobjective programming problem is presented, in which the coefficients of the objective functions and the left hand side of the constraints are independent random variables with known probability distribution functions. The approach used to programming solve the problem is based on the use of the chance-constrained approach [7] of stochastic programming problems, the extension principle [25]; the possibility indices to compare fuzzy numbers proposed by Dubois & Prade [11]; and -levels of fuzzy sets [23], to get an equivalent deterministic problem that can be solved by means of any classical procedure for solving linear multiobjective programming problems.

Keywords: Fuzzy numbers, possibilistic programming, multiobjective programming, chance, constraints.

Paracompactness in Fuzzifying Topology

Jizhong Shen

Department of Mathematics, Jiangxi Normal University Nanchang 330027, China

Abstract: We introduce the concept of paracompactness in the framework of fuzzifying topology and establish some fundamental properties.

Keywords£º Fuzzifying topology; paracompactness.

De Morgan Algebra of Topology Vi

Zike Deng

Department of Mathematics, Hunan University Chansha 410082, Hunan China

Abstract: We give a necessary and sufficient condition for a De Morgan algebra of uniformity to be pseudo-merizable.

Keywords£º De Morgan Algebra of uniformity£¬

De Morgan Algebra of Metric. Fuzzy Programming Approach to Probabilistic Multiobjective Transportation Problem with Pareto Optimum Solution

Rakesh Verma,M.P. Biswal and a. Biswas

Department of Mathematics Indian Institute of Technology Kharagpur-721302, India

Abstract: In this paper, we have considered the demands variables as stochastic variables in a probabilistic multiobjective transportation problem. The demand variables follow some specific probability distribution namely uniform(discrete), exponential and gamma distribution. Methodology to convert the probabilistic problem into an equivalent deterministic problem, has been presented. In this paper we have also proposed a fuzzy programming approach to probabilistic multiobjective transportation problem, based on pareto optimum solution. It is shown that this approach gives an efficient solution to the probabilistic multiobjective transportation problem. Numerical examples are presented to illustrate the methodology.

Keywords: Probabilistic programming, fuzzy programming, multicriteria decision making, transportation problem, uniform, exponential and gamma distributions, pareto optimum solution.

Fuzzy Programming Approach to Some Multi-Objective Nonlinear Programming Problems

M. P. Biswal and S. B. Sinha

Department of Mathematics Indian Institute of Technology Kharagpur-721302, India

Abstract: In this paper fuzzy programming method is presented to solve some multi-Objective Nonlinear programming problems where the decision variables are continuous. The concept of fuzzy linear and nonlinear membership function is utilized to solve the problem. One numerical example is present to illustrate the methodology. Finally, the solution is compared with the solution obtained by weighting method.

Keywords: Fuzzy programming, nonlinear programming, linear and nonlinear membership function, elipsoid algorithm, weighting method.

On Levi-Like Theorem, Borel-Cantelli Lemma and Lusin Theorem for Fuzzy Measures and Fuzzy Integrals

Liu Xuecheng*

Department of Mathematics, Hebei Normal College, Shijiazhang, 050091, P.R. China

Abstract: In this paper, we prove that the Levi-like theorem, Borel-Cantelli lemma and Lusin theorem are also valid in the condition that the fuzzy measure is null-additive (to Levi-like theorem) or autocontinuous from above (to Borel-Cantelli lemma and Lusin theorem). These results improve the corresponding ones obtains in [1], in which the condition of a sub-additive fuzzy measure is required. Keywords: Fuzzy measure, fuzzy integral, null-additivity, autocontinuity, continuous function. New Operations on Fuzzy Topological Spaces B.A. Nouh Mathematics Department, Faculty of Science, Mansoura University, Egypt. Abstract: In this paper special types of operations on fuzzy topological spaces are defined namely: open and Q-open operations. The interrelations between these operations and the regular and Q-regular operations as introduced in [8, 12] are outlined in detail. In particular the interaction between open operations on ordinary topological spaces due to H. Ogata [15] and its corresponding q-open operation on induced fuzzy topological spaces is described in detail. Then we introduce and study the concept of .G-closed fuzzy sets and use it to introduce the concepts of . FT1/2-space. Finally, it is proved that the concepts of .G-closed fuzzy sets and .FT1/2-space are good extension in the sense of Lowen [14]. Keywords: Fuzzy topological spaces, Q-neighborhood, Q-open and Q-regular operations, .G-closed fuzzy sets, .G-closed fuzzy sets, .FT1/2-space, fuzzy -continuity. Generalized Quasi-complementarity Problems for a Pair of Fuzzy Mappings* Shih-Sen Chang and Nan-Jing Huang Department of Mathematics, Sichuan University Chengdu, Sichuan 610064,P.R. China Abstract: In the paper, a new class of generalized quasi-complementarity problems for a pair of fuzzy mappings is introduced and a new iterative algorithm is constructed. Some existence theorems for this kind of quasi-complementarity problems and some convergence results are given. The results presented in this paper unify and extend a number of known results.

Keywords: Fuzzy mapping, generalized quasi-complementarity problem, quasi-variational inequality, iterative algorithm.

A New Approach to Fuzzy Multiple Objective Programming

Li Lushu and Sheng Zhaohan

School of Economics and Management, Southeast University Nanjing, 210096, P.R. China

Abstract: In this paper, the multiple objective linear programming problem with imprecise objective and constraint coefficients is studied based on the fuzzy set theory. Using the method of ranking fuzzy numbers with total integral values, we propose an auxiliary multiple objective linear programming model to resolve the imprecise nature. Further, we develop an extended Zimmermann's approach to solve the auxiliary multiple objective linear programming problem, and the -pareto optimum solution of the original fuzzy multiple objective linear programming is derived.

Keywords: Fuzzy programming, multiple objective programming, Pareto optimum.

Generalized Uniform Spaces

M. H. Burton

Department de Mathematics, Rhodes University Grahamstown, 6140, South Africa

M. A. De Prada Vicente; J. Gutierrez Garcla

Department de Mathematics, Facultad de Ciencias. Universidad del Pais Vasco, Apartado 644, Bilbao, Spain.

Abstract: We extend the notion of a uniform space in a nature and intuitive way by defining the degree to which a subset is an element of a uniformity and we call these structures generalized uniform spaces, we show that the category of Lowen fuzzy uniform spaces is isomorphic to the category of generalized uniform spaces and that the category of generalized uniform spaces is a good extension of the category of uniform spaces. The generalized uniform space analogues of various fuzzy uniform space notions are characterized.

Keyword: Uniform spaces, fuzzy uniform space, saturated prefilter, isomorphic categories, adjoint functor, closure, adherence, convergence.

The Comparison of Two Comprehensive Evaluation Methods and The Improvement

Liu Xuecheng and Meng Linyan

Department of Mathematics, Hebei Normal College, Shijiazhang, Hebei 050091, P.R. China

Abstract: Two multi-factoral comprehensive evaluation methods¡ªmatrix method and fuzzy integral method-are outlined; the comparisons of these two methods are made; an improved comprehensive evaluation method is presented.

Keywords: Comprehensive evaluation, fuzzy relation matrix, importance measure, fuzzy integral.

Expert System Development for Convulsion Diseases

D.K. Gupta and Mayank Khanna

Department of Mathematics,I.I.T. Kharagpur Kharagpur-721302, India

Abstract: This paper describes an expert system used as a decision tool for the treatment of convulsion diseases. Convulsion diseases are concerned with brain fits. To deal with uncertainties involved in the medical knowledge, fuzzy logic concepts are used in the design.

Keywords: Knowledge representation of convulsion diseases, function of inference engine.

Fuzzy S-Irresolute Mappings

Bai Shi-zhong

Department of Mathematics, Xiangtan University Xiangtan, Hunan 411105, P.R. China

Abstract: In this paper, first some new characteristic properties of fuzzy strongly semicontinuous mapping are established, and fuzzy S-irresolute, fuzzy S-irresolute-open (closed) and fuzzy S-homoeomorphism mappings are introduced and studied.Next fuzzy S-separation axioms and fuzzy S-connectedness are introduced and studied with the help of fuzzy strongly semiopen sets and fuzzy S-q-neighborhoods.

Keywords: Fuzzy strongly semiopen sets, fuzzy S-q-neighborhood, fuzzy strongly semicontinuous mappings, fuzzy S-irresolute mappings, fuzzy S- axioms, fuzzy S-connected.

Generalized Convergence Theorem of Generalized Fuzzy Integrals

Gaimei Guo

Dept. of Basic Science, Changchun Univ. Changchun, Jilin 130022, P.R. China

Deli Zhang

Department of Mathematics, Jilin Prov. Inst. of Education Changchun, Jilin 130022, P.R. China

Abstract: In this paper, we will further investigate generalized fuzzy integrals given by Wu [4]. Our purpose is concentrated on the convergence of sequences of generalized fuzzy integrals which include the sequences of fuzzy measures. Various kinds of generalized convergence theorems are obtained, these are generalized monotone convergence theorem, generalized Fatou's lemmas, etc.

Keywords: Fuzzy measure, generalized fuzzy integrals, generalized convergence theorem.

Absolute Value Like Fuzzy Real Number and Fuzzy Real-valued SequenceSpace

N.R. Das, Pankaja Das and Ajanta Choudhury

Department of Mathematics, Gauhati University Guwahati-781 014, Assam, India

Abstract: Notion of absolute value like fuzzy real number is introduced and some of its basic properties are discussed. It has applications in the later part while studying different types of fuzzy real-valued sequence spaces.

Keywords: Fuzzy real number, fuzzy normed linear space, sequence space.

Strong Compactness of L-fuzzy Topological Spaces

Pu Yishu

Department of Mathematics, Hanzhong Teachers College Shanxi, P.R China

Abstract: In this paper, strong compactness is introduced, some results are proved and B-pairwise strong compactness is discussed in L-bifuzzy topological spaces.

Keywords: S closed remoted-neighborhood, -S closed remoted-neighborhood system, LF strong compactness, strong accumulation point of -net, B-pairwise strong compactness.

A Remark On Possibilistic and Fuzzy Programming

M. Wagenknecht

TU Berlin, Institut fiir Proze -und Anlagentechnik Rudower Chaussee 5 (Geb. 10.5), 12484 Berlin, Germany

B.V. Yazenin

Tver state University, Dept. of Informatics Ul. Zhelyabova 33, 170000 Tver, Russia

Abstract: We investigate a certain optimization task in a possibilities environment with crisp decision variables thus generalizing well-known fuzzy optimization problems.

Keywords: Possibilities programming, aspiration level optimization.

-Structure Of FTS(VI): Fuzzy Uniform Spaces

Hu Cheng-ming

International Fuzzy Mathematics Institute P.O. Box 639, San Gabriel, CA 91778, U.S.A

Abstract: In papers [1-5] we introduced and investigated a kind of fuzzy uniformities. While we discussed the complete fuzzy uniform spaces in [2], we introduced the notion of -sets firstly to define the centred family which contains small -sets and then we got a characterization of complete fuzzy uniform spaces. Actually, that is the beginning of the -structure. This paper will give a systemally relation on fuzzy uniform spaces in -structure.

Keywords: Preserve union relation, fuzzy uniformity, fuzzy uniformly continuous, fuzzy metrization, fuzzy embedding, complete fuzzy uniformity, totally bounded fuzzy uniformity.