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The Journal of Fuzzy Mathematics

Volume 5, Number 2, June 1997



Fuzzifying Compactifications

Daowen Qiu

Department of Basic Course, Guang Dong Institute of Technology, Guang Zhou 510090, P.R. China

Abstract: We shall introduce the concept of compactification, and establish some of its fundamental properties as well as obtain the Alexandorff compactification theorem in the framework of fuzzifying topology.

Keywords: Fuzzifying topology, compactification.

DE Morgan Algebra of Bitopology

Xueyou Chen

Basic Department, Shandong Institute of Technology Zibo, Shandong 255012, P.R. China

Abstract: In this paper, we introduce de Morgan algebra of bitopogy, and some separation axioms, the Urgsohns Lemma is established.

Keywords: De Morgan algebra of topology, de Morgan algebra of bitopogy.

Fuzzy Inference in an Expert System of Runoff Forecasts: Fuzzy Theoretic Forecast Pattern Method

Shouyu Chen, Huicheng Zhou and Bende Wang

Department of Civil Engineering, Dalian University of Technology Dalin 116024, P.R. China

Abstract: Patterns of mid-and long-term runoff forecasts are developed according to the experts idea of runoff inference forecasting by synthesizing the forecast factors affecting runoff. The patterns take into consideration a weighting vector for forecast factors so that they are more suitable and may provide more information for decision making. When applied to the expert system of mid-and long-term runoff forecasts, the fuzzy inference method presented here can summarize the expertise into patterns, and increase the learning ability of the expert system.

Keywords: Fuzzy inference, runoff forecast, expert system, forecast pattern.

-compactness in Fuzzy Topological Spaces

I.M. Hanafy

Department of mathematics, Faculty of Education Suez Canal University, El-Arish, Egypt

Abstract: In this paper, we introduce and study the concept of-compactness [6] in fuzzy topological spaces. The goodness of our definition is of interest. We give some characterizations of -compactness, and in a semiregular fuzzy topological spaces, the concept of-compactness, -compactness, -near compactness and-almost compactness are equivalent. Also, we investigate the image and the inverse image of-compactness under some types of continuity.

Keywords: Fuzzy topological spaces, fuzzy -compactness, fuzzy -compactness, fuzzy -continuity, fuzzy super continuity, fuzzy -open function.

Product Fuzzy Number Measures of Product Spaces

Hou Renen

Department of mathematics, Shangrao Teachers College, Jiangxi, 334001 China

Abstract: This paper is the continuation of the paper [1]. In this paper, we try to introduce the concept of the product fuzzy number measures on product spaces. Now we have the result of the product fuzzy number measure theorem. And we have discussed some problems about the Fubini theorem on the product fuzzy number measure spaces.

Keywords: Random set, Fuzzy number measure, product fuzzy number measure, measurable space, product set-valued measure.

Smooth Filter Structures

A.A. Ramadan

Department of mathematics, Faculty of Science Cairo University Beni-Suef, Egypt

Abstract: We introduce the concept of smooth filter structures in the framework of smooth topology and we establish some fundamental properties of smooth filter structures.

Keywords: Fuzzy logic, smooth filter, smooth topology, convergence.

Measures of Noncompactness for Fuzzy Sets

S.G. Gal

Department of mathematics, Universityof Oradea Str. Armatei Romane 5, 3700 Oradea, Romania

Abstract: In this note we continue the study of the measures of noncompactness introduced in [5-6]. Also, by using the level sets method, other measures of noncompactness are considered and some interesting open questions are pointed out. Finally, fuzzy measures of noncompactness for fuzzy sets are introduced.

Keywords: Measures of noncompactness of Kuratowski-type and of Hausdorff-type for fuzzy sets, fuzzy measures of noncompactness for fuzzy sets.

On Smooth L-fuzzy Topological Spaces

Halis Aygun

Department of mathematics, City University Northhampton Square, London ECIV OHB, U.K.And Department of mathematics, Kocaeli University Ataturk Bulvari, Izmit, Turkey

m.w. Warner

Department of mathematics, City University Northhampton Square, London ECIV OHB, U.K.

S.R.T. Kudri

Department of mathematics, City University Northhampton Square, London ECIV OHB, U.K. and Department of mathematics, Federal University of Parana, Curutiba, Brazil

Abstract: `-Scott continuous functionsfrom an ordinary topological space to a fuzzy lattice L with its Scott topology are introduced. These functions turn out to be the natural tool to set up a goodness of extension criterion for smooth L-fuzzy topological properties. Good definitions of smooth Hausdorffness, smooth compactness and smooth local compactness in smooth L-fuzzy topological spaces are presented and some of their properties are studied.

Keywords: Fuzzy lattice, smooth fuzzy topology, -Scott continuous function, good extension, smooth Hausdorffness, smooth compactness, smooth local compactness, prime level space.

Improvement of the Fixed Point Theorem for Fuzzy Mappings in Menger PM-spaces

Xie Jin

Department of Basic Science, Wuhan Urban Construction Institute Wuhan 430074, P.R. China

Liu Zuo Shu

Department of mathematics and Physics, Wuhan University of Technology Wuhan 430070, P.R. China

Abstract: In this paper the notion family of fuzzy mappings defined on probabilistic metric spaces is introduced and its properties are investigated. A common fixed point thorem for fuzzy mappings defined on PM-spaces is proved. These mappings are assumed to satisfy some generalizations of the contraction condition.

Keywords: Menger PM-spaces, fuzzy mapping, fixed point.

Neighborhoods in De Morgan Proximity Algebra

Xueyou Chen

Basic Department, Shandong Institute of Technology Zibo 255012, Shandong, P.R. China

Abstract: In [7], De Morgan algebra of proximity was introduced, based on [1], [7], using the notion of ab, the proximity neighborhood relation is investigated, and given a straight proof of the proposition: every induced by completely regular.

Keywords: Proximity, proximity neighborhood relation.

On Additivity and Pseudo-additivity of a Pseudo-additive Measure Based Integral

Radko Mesiar

S T U,81368 Bratislava, Slovakia and UTIA AV CR, 18200 Prague, Czech Republic

Endre Pap

Institute of Mathematics, University of Novi Sad, 21000 Novi Sad, Yugoslavia

Abstract: Different kinds of additivity of an integral F+ (with respect to a +-measure m) introduced by Sugeno and Murofushi are studied. As corollaries we obtain the additivity (and linearity) of the Weber integral and the pseudo-additivity (and pseudo-linearity) of the integrals of Pap and Ichihashi et al.

Keywords: Pseudo-addition, pseudo-multiplication, decomposable measure, integral.

On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces

Hasmet Gurcay Dogan Coker

Department of Mathematics Education, Hacettepe University Beytepe, 06532-Ankara,Turkey

A. Haydar Es

Department of Mathematics , Hacettepe University Beytepe, 06532-Ankara,Turkey

Abstract: The purpose of this paper is to introduce several types of fuzzy continuity between intuitionistic fuzzy topological spaces; namely, fuzzy semicontinuity, fuzzy almost continuity, fuzzy weak continuity, fuzzy precontinuity, fuzzy -continuity, fuzzy - continuity.

Keywords: Intuitionistic fuzzy set, intuitionistic fuzzy topological space, intuitionistic fuzzy semiopen set, intuitionistic fuzzy regular open set, intuitionistic fuzzy -open set, intuitionistic fuzzy preopen set, intuitionistic fuzzy -open set, fuzzy continuity, fuzzy semicontinuity, fuzzy almost continuity, fuzzy weak continuity, fuzzy precontinuity, fuzzy-continuity, fuzzy-continuity.

Fuzzy Homological Theory in FRA -modules (I)

Jianli Zhao

Abstract: In this paper, we first introduced the concept of the kernels, cokernels in FRA -module category and some important properties are given. Then we define the concept of fuzzy exact sequences in FRA -module category and some results are studied. Finally, we introduce the concept of free FRA -module, give concrete structures and some properties of free FRA -module.

Keywords: FRA -module,kernel, cokernel, fuzzy exact sequence, free R-module, free FRA -module.

Relative Membership Function and New Frame of Fuzzy Sets Theory for Pattern Recongnition

Shouyu Chen

Department of Civil Engineering, University of Technology Dalian 116023, P.R. China

Abstract: Based on generalization of definition for classical membership function, this paper establishes a new frame of fuzzy sets theory for pattern recognition. Its theorical basis is definition of relative membership function. It includes membership functions of recongnition indexes and of ranks for samples belonging to fuzzy subset A, new indexes of recongnition for sample set, feature value vectors and so on. A real application example is given: cluster recognition for unknown chemical compound.

Keywords: Relative membership function, pattern recognition, fuzzy sets, rank featurevalue.

Fuzzy Archimedean Spaces

Ismat Beg

Department of Mathematics, Kuwait University P.O. Box 5969, Safat 13060, Kuwait

Misbah Ui Islam

Department of Mathematics, Indiana University Bloomington, Indiana, U.S.A.

Abstract: The notions of directed fuzzy ordered linear space and fuzzy Archimedean space are defined and studied.

Keywords: Fuzzy ordered linear space, fuzzy Archimedean space.

Possibilistic Linear Programming Problems with Fuzzy Constraints

Xiaozhong Li

Department of Computer Science, Liaocheng Teachers College Liaocheng, Shandong 252059, P.R. China

Abstract: In this paper possibilistic linear programming problems with fuzzy constraints are considered. The solving method is presented. Solutions to them are obtained.

Keywords: Possibilistic linear programming, fuzzy constraints, fuzzy numbers, fuzzy variables, possibility distribution.

On the convergence of the Fuzzy Valued Functional

Wang Gui-jun and Li Xiao-ping

Department of Mathematics , Tong hua Teachers college, Tong hua,Jilin, 134002, China

Abstract: In this paper, we define a fuzzy valued functional on fuzzy measure spaces, discuss its important properties and give its convergence theorem.

Keywords: Measurable fuzzy valued functions; integrable fuzzy valued functions; fuzzy valued functional.

Use of Projective and Scaling Algorithm to Solve Multi-Objective Fuzzy Linear Programming Problems

M.P. Biswal

Department of Mathematics , Indian Institute of Technology Kharagpur-721302, India E-mail:

Abstract: In this paper projective and scaling algorithm have been used to solve multi-objective fuzzy linear programming problems. The main steps of both the algorithms to solve a linear programming problem are presented and the advantages of the algorithms are analyzed. Then the fuzzy programming algorithm to solve a multi-objective linear programming problem is presented. By considering a numerical example solution procedure is discussed. Finally, a comparison with simplex algorithm and interior point algorithm is presented.

Keywords: Projective algorithm, scaling algorithm, multi-objective linear programming, fuzzy programming, simplex algorithm.

A Note on Obstinate Fuzzy Ideals in BCI-algebras

Young Bae Jun

Department of Mathematics Education Gyeongsang National University, Chinju 660-701, Korea e-mail:

Abstract: In this paper, we show that the Theorem 6 of [8] holds without the sup property. For the homomorphic image f(u) and preimage f-1(v) of obstinate fuzzy ideals u and v, respectively, we establish the chains of level obstinate ideals of f(u) and f-1(v) , respectively.

Keywords: Obstinate ideal, obstinate fuzzy ideal, level obstinate ideal, sup property.

Some Weakly Forms of Fuzzy Continuous Mappingsin Fuzzy Bitopological Spaces

K.M. Abd El-Hakeim

Department of Mathematics Faculty of Science Assiut University, Assiut, Egypt

Abstract: The aim of this paper, deals with the concepts of fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings due to Park and Lee [10] have been generalized to the fuzzy bitopological setting and some of its basic properties are discussed . Characterizations of these mappings along with their relationship with certain other types of mappings are investigated.

Keywords: Fuzzy Pairwise weakly semicontinuous (semiopen, semiclosed) mappings, fuzzy pairwise weakly continuous(open, closed) mappings

Pseudo-metrics and -equivalences

Bernard De Baets

Department of applied Mathematics and Computer Science University of Gent, Krijslaan 281 (S9) B-9000 Gent, Blegium

Radko Mesiar

Department of Mathematics Slovak Technical University, Radlinskeho 11 81368 Bratislava, Slovakia

Abstract: In this paper, we discuss the relationship between pseudo-metrics and-equivalences; the latter are a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of pseudo-metrics from-equivalences triangular norms with an additive generator play a key role. Conversely, in the construction of-equivalences from pseudo-metrics this role is played by triangular norms with a continuous additive generator, or equivalently, Archimedean triangular norms.

Keywords: Additive generator, Archimedean property, pseudo-metric, -equivalence, triangular norms.

Decomposition Theorems and Representation Theorems of L-fuzzy Fields

Guangqing Wu, Ping Du

Department of Mathematics, Hebei Vocation and Technology Teachers College Shijiazhuang 050031, Hebei, P.R. China

Fugui Shi

Department of Mathematics, Yantain Teachers College Yantai 264025, Shandong, P.R. China

Abstract: In this paper, first the concept of L-fuzzy fields is introduced when L is a completely distributed lattice. Then the equivalent characterizations, decomposition thorems and representation theorems of L-fuzzy fields are given.

Keywords: L-fuzzy algebra, L-fuzzy fields.

Monotonic Intersection Decomposition of Fuzzy Sets

Liu Huawen, Chen Guangdong

Department of Mathematics and Physics Shandong University of Technology, Jinan 250061, P.R. China

Abstract: In this paper, themonotonic decomposition theorems are introduced. It shows that any fuzzy set can be described as the product or limit of some simple fuzzy sets which are countable and montonic decreasing. At the same time, the monotonic decomposition theorems of 2-fuzzy sets are given.

Keywords: Fuzzy set, monotonic intersection decomposition, intersection decomposition theorem.