THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 2, Number 1, March 1994

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On the Construction of Free Fuzzy Groups

Yasushi Suzuki

The Graduate School of Mathematics Joetu University of Education, Japan

Abstract: In this paper, we investigate the fuzzy normal subgroup generated by a given fuzzy subset, fuzzy functions, homomorphisms (isomorphisms) of fuzzy groups and quotients of fuzzy groups by a relation with the idea of level subgroups. Also, we discuss the properties of isomorphisms of fuzzy groups and finally construct the free fuzzy groups which consist of fuzzy subgroups generated by given fuzzy subsets and an epimorphism of fuzzy groups.

Keywords and phrases: Fuzzy subgroup (fuzzy normal subgroup) generated by a fuzzy set, fuzzy function, homomorphism (epimorphism, isomorphism) of fuzzy groups, t-level left (right) coset of a fuzzy subgroup, t-level left fuzzy (right) coset of a fuzzy subgroup, t-level quotient group, generating fuzzy subset, free fuzzy group, free fuzzy basis.

On the Supercomposition of Subcontinuous and Fuzzy Subcontinuous Mappings and Relate Topics

B.De Baets and E.Kerre

Department of Applied Mathematics and Computer Science University of Gent, Krijgslaan 281-S9, 9000 Gent, Belgium

Abstract: Inspired by the new compositions of relations introduced by Bandler and Kohout, new images of a set under a relation and new images of a fuzzy set under a fuzzy relation are presented here. For functional relations, the images under the converse relation, in particular the subinverse image, suggest the introduction of a new type of continuity, subcontinuity, and a new type of fuzzy continuity, fuzzy subcontinuity. The main results of this paper are the composition laws, such as: ¡°The supercomposition of (fuzzy) subcontinuous mappings is again (fuzzy) subcontinuous¡±.s

Keywords and phrases: Subdirect image, supercomposition, subcontinuity, fuzzy subcontinuity.

Unfuzzy Non-dominated Elements for Fuzzy Interval-valued Preferences

Michael Wagenknecht

Technical University of Berlin, Institut F¨¹r Proze¦Â-und Anlagentechnik, Geb. 10.5, Rudower Chaussee 5, 12484 Berlin, Germany

Abstract: Fuzzy preference relations are a good tool to express the decision-maker¡¯s imaginations about ordering of elements. However, frequently it may happen for lack of more detailed information that the preferences are available not as numbers from [0,1], but rather as intervals. Hence we are faced with interval-valued fuzzy relations (IVFR). Therefore we introduce strong IVFR under Archimedian t-norms and generalizes the notion of non-dominated elements given by Orlovsky. Within this framework, the (crisp) set of unfuzzy non-dominated objects has drawn attention, because of its practical and theoretical importance. It turns out that for IVFR we have at least two reasonable extensions of .

Keywords and phrases: Interval-valued fuzzy preferences, Archimedian t-norms, non-dominated elements.

On an Epireflective Hull in Fuzzy Topology

Alka Choubey and Arun K.Srivastava

Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

Abstract: It is shown that the epireflective hull of , together with the fuzzy topology generated by , is the category of ¡®zero-dimensional¡¯ fuzzy -spaces. Keywords and phrases: Fuzzy-topology, - fuzzy topological space, zero-dimensional fuzzy topological space, epireflective hull. Fuzzy Neighborhood Ordered Spaces A.S.Mashhour, A.A.Allam and K.El-Saady Math. Dept. Faculty of Science at Aswan, Aswan, Egypt. Abstract: The notion of fuzzy neighborhood structure over an ordered set is introduced. Some characterization properties with fundamental results are deduced.

Keywords and phrases: Fuzzy neighborhood base, fuzzy neighborhood system, fuzzy topological ordered spaces and prefilters.

Fuzzy Uniform Structures and Order

A.S.Mashhour, A.A.Allam and K.El-Saady Math.

Dept. Faculty of Science at Aswan, Aswan, Egypt.

Abstract: The goal of this paper is to introduce and study the relationship between order and Lowen-fuzzy uniformities. Some related results are deduced.

Keywords and phrases: Fuzzy quasi-uniformities, fuzzy pesudo-metrics, preordered sets.

-Structure of FTS(II): F-mappings

Hu Cheng-ming

International Fuzzy Mathematics Institute, U.S.A.

Abstract: In paper [1], [2], the author introduced the notion of F-mapping by means of an order preserving correspondence between fuzzy points [3]. This kind of complete pointwise mappings not only extends the well know notions of fuzzy mappings but also brings some deep results. It exhibits enough the features of pointwise theory of fuzzy topology. In this paper, we will discuss the F-mappings in the -structure of FTS.

Keywords and phrases: F-mapping, interior image, total image, interior inverse image, total inverse image, dual mapping, symmetrical mapping.F-continuous mapping F-quasi-continuous mapping.

On s-Closed Fuzzy Topological Spaces

S.P.Sinha and S.Malakar

Department of Pure Mathematics, University of Calcutta, India.

Abstract: In this paper a new class of fuzzy topological spaces under the terminology fuzzy s-closed spaces is introduced and investigated. It is seen that this class properly includes the class of fuzzy S-closed spaces [11] and is independent of that of fuzzy compact spaces [2].

Keywords and phrases: Fuzzy semi-clopen set, fuzzy semi-¦È-adherent point, fuzzy s-closed space.

On s-Closedness and ¦È*-Rigidity of Fuzzy Sets

S.Malakar, R.P.Chakraborty and S.P.Sinha

Department of Pure Mathematics, University of Calcutta, India.

Abstract: The paper deals with two classes of fuzzy sets satisfying certain types of fuzzy covering properties. Such fuzzy sets are investigated mainly as to their characterizations and a few basic properties. Finally, they are subjected to a comparative study vis-¨¤-vis some allied and existing types of fuzzy sets, introduced earlier.

Keywords and phrases: Fuzzy s-closed set, fuzzy ¦È*-rigid, fuzzy semi-clousure and semi-¦È-adherence.

Fuzzy Programming Approach to Chance Constrained Multi-objective Transportation Problem

A.K.Bit M.P.Biswal and S.S.Alam

Department of Mathematics, India Institute of Technology, India.

Abstract: A method for solving a chance constrained multiobjective transportation problem has been suggested. After converting the proposed chance constrained model to a deterministic model, fuzzy programming technique is used to find the solution. Assuming the demand and availability are normal random variable the methodology has been presented. In some cases assuming the demand as log-normal or uniform random variable, methodology also has been presented. Lastly, a chance constrained multiobjective solid transportation model is presented. The method leads to efficient solutions as well as an optimal compromise solution. For this method a Fortran software has been developed on Cybel 180/840A System. Numerical examples are added to illustrate the methodology.

Keywords and phrases: Chance constrained programming, fuzzy programming, multicriteria decision making, classical transportation problem, solid transportation problem.

Another Fuzzy Connectedness Concept

Anuranwaj Singh

Department of Mathematics, Banaras Hindu University, India.

Abstract: We study a new type of fuzzy connectedness and establish that it has several desirable features.

Keywords and phrases: Fuzzy topology, fuzzy connectedness, (C 3)- connectedness.

B-valued Measure and B-valued Integral on L-fuzzy sets

Qiao Zhong

Department of Basic Sciences, Hebei Institute of Architectural Engineering, China.

Abstract: B-valued measure and B-valued integrals (i.e. measures and integrals with their values in an ordered Banach space) are defined on a fuzzy Q-algebra of L-fuzzy sets (where L is a pseudocomplenented infinitely distributive complete lattice, the ordered Banach space is a Banach space which has a partial order with some properties), and their properties are studied.Moreover this paper also obtains some interesting convergence theorems (e.g. uniform convergence theorem,mean vonvergence theorem,monotone convergence theorems,bounded convergence theorem,Fatou¡¯s Lemma) for a sequence of B-valued integrals on L-fuzzy sets.

Keywords and phrases: Functional analysis, Banach space, L-fuzzy set, B-valued measure, B-valued integral.

Theory of Fuzzy Optimum Selection for Multistage and Multiobjective Decision Making System

Chen Shouyu

Department of Civil Engineering Dalian University of Technology, Dalian, China.

Abstract: The paper presents a theory of fuzzy optimum selection for the multistage and multiobjective decision making system that may be used to select the optimum alternative for the complex system. It may be a new optimizing approach for multiobjective system. A case study is given in the paper.

Keywords and phrases: Fuzzy optimum selection, decision making, multistage, multiobjective.

De Morgan Algebra of Uniformity I

Xueyou Chen

Basic Department, Shandong Engineering College, China.

Abstract: Z. Deng initiated de Morgan algebra of topology [4] [5] [6] and -relation [7]. In this paper, we introduce de Morgan algebra of uniformity, which unifies the classical uniformity [11] and the fuzzy one [8] [10].

Keywords and phrases: De Morgan algebra, -mapping, de Morgan algebra of uniformity.

De Morgan Algebra of Uniformity II

Xueyou Chen

Basic Department, Shandong Engineering College, China.

Abstract: In [6], we introduced and investigated a uniformity of de Morgan algebra. In this paper, we will continue to study a uniform topology of de Morgan algebra.

Keywords and phrases: De Morgan algebra, de Morgan algebra of uniformity. de Morgan algebra of uniform topology.

De Morgan Algebra of Uniformity III

Xueyou Chen

Basic Department, Shandong Engineering College, China.

Abstract: In [7] [8], we introduced a uniformity and a uniform topology of de Morgan algebra. In this paper, we will continue to study the completeness and total boundeness of a de Morgan algebra of uniformity.

Keywords and phrases: Completeness, total boundedness of a de Morgan algebra of uniformity, de Morgan algebra topology, uniformly continuous, uniform topology, uniformity.

Fuzzy c-Continuous Functions

S.Dang, A.Behera and S.Nanda

Department of Mathematics, Government College, Regional Engineering College, Indian Institute of Technology, India.

Abstract: The concept of fuzzy c-continuity of a function is introduced. Several characterizations of such functions are investigated following both localized and nonlocalized approaches. Furthermore, using the concept of fuzzy c-continuity of a function some properties of fuzzy -spaces, fuzzy normal spaces, and fuzzy Hausdorff spaces are established. Finally, piecewise construction of fuzzy c-continuous function is dealt with using the concept of neighborhood finite family of fuzzy sets.

Keywords and phrases: Fuzzy point, compact fuzzy topological spaces, fuzzy continuous map, fuzzy c-continuous function, quasi coincidence, fuzzy neighborhood finite family.

Estimation of Expected Values from Probabilities of Fuzzy Events

Stefan Chanas and Bronislaw Florkiewicz

Institute of Industrial Engineering and Management, Institute of Mathematics Technical University of Wroclaw, Poland.

Abstract: Problems of an approximate estimation of the expected valued of a random variable on the basis of an incomplete and imprecise information about its probability distribution have been analyzed. It has been assumed that the probability distribution of the random variable belongs to a class of distributions fixed in advance and that the probability of a certain event connected with the random variable is know. The events as well as theirs probabilities are given in the form of fuzzy sets in respective spaces. It has been examined in which way the properties of fuzzy expected valued of the random variable depend on those of the class of probability distributions and those of fuzzy sets defining the evevt and the probability.

Keywords and phrases: Approximate reasoning; fuzzy event; expected value.

Fuzzy Riesz Spaces

Ismat Beg and Misbah U1 Islam

Department of Mathematics, Kuwait University, Kuwait Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.

Abstract: The notion of fuzzy Reisz space is defined and fundamental properties of fuzzy Reisz spaces are studied. In particular results regarding positive elements, discrete elements and fuzzy Reisz decomposition property are obtained.

Keywords and phrases: Analysis, fuzzy Reisz spaces, fuzzy ordered linear spaces, fuzzy order relation. Corrigendum to: Multiobjective Imprecise-chance Constrained Programming Problem Debjani Chakraborty, J.R.Rao and R.N.Tiari Department of Mathematics, Indian Institute of Technology, India.