THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 2, Number 1, March 1994

CONTENT

DETAILS

Fuzzy Real Numbers in Fuzzy Space

K.A.Dib

Faculty of science, Fyoum Branch

zcairo University, Egypt

Abstract: Using the concept of fuzzy spaces, fuzzy numbers is redefined. Fuzzy addition and fuzzy multiplication on fuzzy real numbers are studied. Introduced, on the fuzzy space of the fuzzy real numbers. A fuzzy topology, which is similar to the usual topology on the set of real numbers. The convergence of the sequences of fuzzy numbers is studied. The properties of the real fuzzy function are discussed.

Keywords: Fuzzy real number, fuzzy binary operation, fuzzy topology on the fuzzy real number, fuzzy compact, fuzzy continuity.

Fuzzy Prenearness Spaces

K.C.Chattopadhyay and U.K. Mukherjee

Department of Mathematics,Burdwan University

Burdwan-713104,W.Bengal, India

S.K.Samanta

Department of Mathematics,Visa-Bharati university

Santiniketan-731235, W.Berigal, India

e-mail:root@vbharat ernet in

Abstract: Idea of gradation of prenearness is introduced and some properties of fuzzy prenearness structure are studied

Keywords: Fuzzy topology , topology of fuzzy nearness, fuzzy contiguity and fuzzy proximity.

On fuzzy R-subgroups of Near-rings

Kyung Ho Kim

Department of Mathematics,Chungju National University

Changju 380-702,Korea

E-mail:ghkim@gukwon.chungju.ac.kr

Young Bae Jun

Department of Mathematics Education

Gyeongsang National University,

Chinju 660-701,Korea

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

Abstract: S. Abou-Z-aid (cf.[1]) introduced the concept of fuzzy R-subgroups of a near-ring. In this paper, some fundamental properties concerning this notion are discussed.

Keywords: Near-ring, fuzzy subnear-ring, (fuzzy) right (resp. left ), R-subgroup, level right (resp. left )R-subgroup.

Fuzzy Ordered Filters of A Negatively Partially Ordered ImplicativeSemigroups

Y.H.Kim, Y.M.Namkoong and T.E.Jeong

Department of Mathematics, chungbook national University,

Chongju 360-763, Korea

Abstract£º In this paper we introduce the notion of a fuzzy ordered filter, a fuzzy weakly ordered filter, a fuzzy implicative ordered filter on a negatively partially ordered implicative semigroup and properties.

Keywords: Fuzzy ordered filter, fuzzy weakly ordered filter, fuzzy implicative ordered filter, negatively partially ordered implicative semigroup.

On Fuzzy Semiprime Ideals of A Semiring

T.K. Dutta and B.K.Biswas

Department of Pure Mathematics, University of Calcutta

35, ballygunge Circular Road, Calcutta-700019,India.

Abstract: In this paper we introduce the fuzzy semiprime ideal in semiring and study it. We give some characterizations of fuzzy semiprime ideals of semiring.

Keywords: Fuzzy ideal, fuzzy semiprime ideal, regular semiring ,f-invariant, fuzzy semi k-ideal, semiprime ideal.

On Completely Fuzzy Semiprime Ideals of A Semiring

T.K. Dutta and B.K.Biswas

Department of Pure Mathematics, University of Calcutta

35, ballygunge Circular Road, Calcutta-700019,India.

Abstract: In this paper we introduce the notion of a completely fuzzy semiprime ideal of a semiring and study it. We also characterize completely semiprime k-ideal and completel fuzzy semiprime k-ideal of the semiring of non-negative integers.

Keywords: Fuzzy ideal, fuzzy semiprime ideal, completely fuzzy semiprime k-ideal, completely semiprime k-ideal, completely fuzzy semiprime k-ideal.

How To Solve A Fuzzy Relation Equation in A Complete Brouwerian lattice

Xue-ping Wang

Department of Mathematics, Sichuan Normal university

Chengdu 610064,Sichuan, P.R.China

Abstract : Up to now , how to solve a fuzzy relation equation in a complete Brouwerian lattice is still an open problem as A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez point out. In this paper, we investigate some conditions under which we can distinguish whether a fuzzy relation equation has a solution, and give the method how to solve it in a complete Brouwerian lattice when it has a solution.

Keywords: Fuzzy relations equation, Brouwerian lattice, linear lattice, fuzzy elementary equation.

Common Fixed Point for Sequence of Mappings

P.Vijayaraju and M.Marudai

Department of Mathematics

Anna University, Chennai-600 025 INDIA.

Abstract: The purpose of this paper is to prove common fixed point theorem for sequence of mappings in fuzzy metric spaces which offers a generalization of Kannan¡¯s fixed point theorem.

Keywords: Fuzzy set, Fuzzy metric space, Periodic point, Common fixed point.

( )-fuzzy Cyclic Subgroup

Sandeep Kumar Bhakat

Siksha-Satra, Visa-Bharati;

P.0-Sriniketetan-731236, Dist-Birbhum(W.B),India

Abstract: The notion of ( )-fuzzy cyclic subgroup for an ( )-fuzzy subgroup of a group is introduced. Inter-relation with ( )-fuzzy normal subgroup, and q-fuzzy commutativity obtained. Some of the fundamental results in fuzzy algebra are also obtained.

Keywords: Fuzzy algebra, ( )-fuzzy subgroup, ( )-fuzzy normal subgroup, ( )-level subset, q-fuzzy commutative subgroup, q-fuzzy cyclic subgroup.

Raw Sets

Yuxi Chen

Department of Computer Science

Binghamton University, New York

Abstract: Raw set is a new concept different from classical set or fuzzy set . A raw set consists of a class of objects with relationshipre the relationships of the higher levels. Such a set is characterized by a cluster of digraphs, which has two kinds of edges to represent four comparison relationships between any two objects . Different from the traditional operation itself is a raw set so that the close-form evaluation can be achieved.

Keywords: raw set, digraph, raw operation, ordering, adjacency matrices.

A Fuzzy interactive Approach to Indentify the Best Compromise Solution for Multiobjetive Programming Problems

Mohamed S.Osan

Department of Mathematics, higher Technological Institute,

Ramadan 10th City Cairo, Egypt.

Omar M.Saad

Department of Mathematics, Faculty of Science Helwan University, Cairo, Egypt.

Ali A. Yahia Faculty of Engineering, Alexandria university,Egypt.

Abstract: In general, there is no single optimal solution in multiobjective problems, but rather a set of non- inferior(Pareto optimal)Solutions from which the decision maker must select the most preferred or best compromise solution set is not practical for most real world problems. The purpose of this paper is to present a method to assist the decision maker in identifying the best compromise solution from the set of non-inferior solutions.

Keywords: Multiobjective programming, fuzzy multiobjective programming, -pareto optimality, interactive programming.

Some More Results on Induced Fuzzy Supra Topological Spaces

R.N.Bhaumik

Department of Matliematic Tripura University

Agartala-799 004, Tripura, India

Anjan Mukherjee

Department of Matliematics, Khowai Govt. Dcgree College

Khowai-799 201, Tripura, India

Abstract: The concept of induced fuzzy supra topological space, introduced by the authors[2], was defined with the ofs-lower semi continuous functions. The object of this paper is to present some more results on induced fuzzy supra topological space ( ) and that of its corresponding induced fuzzy supra topological space ).

Keywords: s-lower semi continuity fuzzy supra topology, fuzzy semi-open, fuzzy regular supra open fuzzy regular surprasemi-open, s-continuity, s-regular space, a-closd space.

Cooperative Fuzzy Game Theoretic Approach to Some Multi-Objective Linear Programming Problems

S.K. Roy, M. P. Biswal and R.N.Tiwari

Department of Mathematics, Indian Institute of Technology

Kharagpur 721302,India

Abstract: In this paper, we apply cooperative game theoretic approach to solve some multiobjective linear programming problems. This has been done by introducing the concept of utility function. Pareto optimality, and the fuzzy set theory. The cooperative game theory and fuzzy set theory together give a new optimization technique namely, single and multiple objective problems, but even the preliminary decision making and other problems where only a feasible solution is sought with no objective function being specified. It is shown that the method for solving crisp and fuzzy optimization problems are special cases of the fuzzy game theoretic formulation. Numerical example is also presented to illustrate the methodology.

Keywords: Multi-criteria decision making, Cooperative game theory , Fuzzy programming.

Axiomatic Construction for Intuitionistic Fuzzy Sets

Liu Huawen

Department of mathematics and Physics, Shandong University of technology, Jinan 250061, Shandong, China

Abstract: Based on the concept of the elementary intuitionistic fuzzy point introduced in this this paper a definition of the intuitionistic fuzzy set theory is established.

Keywords: Elementary intuitionistic fuzzy point, intuitionistic fuzzy point, intuitionistic fuzzy set, Expression function, Degree of expression.

Connectedness in Fuzzy Bitopological Spaces-II

S.Sampath Kumar

Department of Mathematics, Anna university

Chennai 600 025, India

e-mail:ssamk@hotmail-com

Abstract: In this paper we introduce four notions of connectedness called pairwise -connectedness (k-5,6,7,8) for an arbitrary fuzzy set with the help of quasi-coincidence Due to Pu and Liu[4] and discuss the implications that exist between them and pairwise -connectedness(k=1,2,3,4)[10] in fuzzy bitopological setting..

Keywords: Fuzzy topology , pairwise -connectedness, quasicoincident, pairwise separation.

Metacompactness in the L-Fuzzy Context

Sunil Jacob John

Department of mathematics, Cochin University of Science and Technology

Kochi-682022, INDIA

E-mail Johnsunil@ cuast ac in

Abstract: Some extensions of metacompactness which are good extensions in the sense of Liu and Luo are introduced in L-Fuzzy topological spaces (L-Fts) using quasi coincidence relation and -Q-covers. Also a characterization of metacompactness in weakly induced L-Fts is obtained.

Keywords: Fuzzy topology, metacompactness, Point finite families.

The convergence Criterion and the Form of Representation of Limit of Fuzzy Numbers Sequence about -mean Symmetric Difference Metric

Li Fachao and Qiu jiqing

Department of Basic Sciences,Hebei university of Science and Technology,

Shijiazhuang, 050018, P.R.China

Wu Congxin

Department of mathematics,Harbin Institute of Technology,

Harbin, 150006,P.R.China

Abstract: In this paper, We show, for the uniformly support bounded fuzzy numbers sequence ,there exists a fuzzy number A such that converges to A about p-mean symmetric metric if and only if there exists a real number such that and almost everywhere converge on [0,h] and ( )(where =[ ] is the -cut of ).Further, we give the form of representation of fuzzy number A when the fuzzy numbers sequence converges to A about mean symmetric difference metric.

Keywords: Fuzzy numbers, platform type fuzzy numbers, mean symmetric difference metric.

Fuzzy strongly Semiopen Sets and Fuzzy Strong Semicontinuity in Fuzzy Bitopological Spaces

F.S.Mahmoud, M.A. Fath Alla and A.A.Abd-Ellatif

Mathematics Department, Faculty of Science, Sohag, Egypt

Abstract: In this paper, the concepts of fuzzy strong serni-interior (resp. fuzzy strong semi-closer )of a fuzzy set due to Bai Shi Zhong in [2],have been generalized to fuzzy bitopological spaces and some of its basic properties are studied. Using these concepts, we investigate several characterizing theorems of fuzzy pairwise strongly semicontinuous (semiopen, semiclosed) mappings [8].

Keywords: Fuzzy bitopological spaces; ( )-fuzzy strongly semiopen (semiclosed) sets; ( )-fuzzy strong semi-interior(semi-closure) of a fuzzy set; fuzzy pairwise strongly semicontinuous(semi-open, semiclosed) mappings.

Closure Operators on Families of Pretopological Spaces Part1: Main properties And Links to Continuity .

R.Badard

Lab.L31 dept informatique, bat.502 INSA Lyon

69621 Villeurbanne, France.

Abstract: In this paper we define closure operators on families of fuzzy pretopological spaces. We study their compatibility with continuity of maps. In this way we are able to assert that continuity of a map can be kept when we apply some closure operators on the two spaces. This result reinforces the concept of continuity which, in certain circumstances, is kept when impose more properties to the fuzzy pretopological structures.

Keywords: Fuzzy pretopology, initial and final structure, product space, quotient space.

Closure Operators on Families of Pretopological Spaces Part2: Application to Final And Initial Structures

R.Badard

Lab.L31 dept informatique, bat.502 INSA Lyon

69621 Villeurbanne, France.

Abstract: In a preceding part we defined closure operators on families of fuzzy pretopological spaces. We studied their co4mpatibility with the notion of continuity. From results we obtained about these closure operators we now define different notions of initial and final structures. These notions are very linked to the concept of continuity. From preceding results, about the compatibility of continuity and the closure operators, we are able to relate these different structures. Finally we apply these constructions to the study of product and quotient of fuzzy pretopological spaces.

Keywords: Fuzzy pretopology, initial and final structure, product space, quotient space.

Fuzzy Localization of Modules

F.I.Sidky

Department of Mathematics,Faculty of Science,

Zagazig University, Zagazig-Egypt.

Abstract: We introduce the concepts of t-localization and -closure of modules and fuzzy submodules and prove some of their properties.

Keywords: Fuzzy multiplicative sets, fuzzy submodules, t-localization of fuzzy submodules, -closure of fuzzy submodules.

Fuzzy Pairwise Semiregularization Spaces and mappings

S.Sampath Kumar

Department of Mathematics, Anna University

Chennai 600 025, India

Email:ssamk@hotmail.com

Abstract: In this paper we introduce fuzzy pairwise semi regulation spaces corresponding to a fuzzy bitopolopical space and deduce certain characterization theorems for a fuzzy possessd semiregular space. It is observed that there are certain properties which, when semiregularization space and vise-versa. Finally, We consider certain mappings between fuzzy bitopological spaces and study their behaviour when either or both the domain and codomain spaces are replaced by their fuzzy pairwise semiregularization spaces.

Keywords: Fuzzy topology, fuzzy pairwise semiregularization space, semiregular property.

Imaginable Fuzzy Ideals of BCK-Algebras with Respect to A-norm

Young Bae Jun

Department of Mathematics Education, Gyongsang

National University,Chinju 660-701,Korea

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

Kyung Ho Kim

Department of Mathematics, Chungju

National University, Chungju 380-702,Korea

E-mail:ghkim@gukwon.chungju.ac.kr

Abstract: Using a t-norm T, we introduce the notion of (imaginable) T-fuzzy subalgebras and(imaginable)fuzzy ideals wrt T, and obtain some related results. We give relations between an imaginable T-fuzzy subalgebra and an imaginable fuzzy ideal wrt T.

Keywords: t-norm, (imaginable)T-fuzzy subalgebra, (imaginable) fuzzy ideal wrt a t-norm.

A Fuzzy Interactive Approach to Generate A Finite Set of Efficient Solutions for Multiobjective Programming Problems with Their Degree of Efficiency

Mohamed S.Osman

Department of Mathematics,High Technological Institute

Ramadan 10th City, Cairo, Egypt.

Omar M.Saad

Department of Mathematics, Faculty of Science, Helwan University, Cario, Egypt.

Ali A.Yahia

Faculty of Engineering, Alexandria university, Egypt.

Abstract: In general, there is no single optimal solution in multiobjective problem, but rather a set of noninferior (Pareto optimal) solutions from which the decision maker must select the most preferred or best compromise solutions as the one to implement. The generation of the entire noninferior solution set is not practical for most real world problems. The purpose of this paper is to present a method to assist the decision maker in generating finite set of efficient solutions with their degree of efficiency for multiobjective problems.

Keywords: Multiobjective programming, Fuzzy multiobjective programming, Pareto optimality.

Fuzzy Relations in BCK-skeleton Pogroupoids

J.Neggers

Department of Mathematics, University of Alabama

Tuscaloosa, AL35487-0350, U.S.A.

jneggerOgp.as.ua.edu

Hee Silk kim

Department of Mathematics, Hanyang university Seoul 133-791,korea Heekm(gemail.hanyang.ac.kr Abstract: In this paper we define a BCK-skeleton pogroupoid as a hybrid of BCK-algebra and a progroupiod, both algebra types with associated posets. In addition we show that certain natural fuzzy relations are in fact the empty relation for BCK-skeleton pogroupiods, while another relation defines the order precisely in the case that the operation defines a BCK-algebra.

Keywords: fuzzy relation BCK-skeleton, pogroudpoid.

A certain Strong Fuzzy paracompactness

Jie Zhang, Fu-Gui Shi and Chong-You Zheng

Department of Mathematics, Captial Normal University

Beijing, 100037, China

Abstract: In this paper a new definition of strong fuzzy paracompactness is presented by means of the concept of -locally finite family of fuzzy sets. The concept of -locally finite family can be used to characterize strong fuzzy countable compactness. Strong fuzzy paxacompactness is a direct generalization of strong fuzzy compactness. A strong countably compact set is strongly fuzzy paracompact if and only if it is strongly fuzzy compact. A topological space( ) is paxacompact if and if only if its induced space) is strongly fuzzy paracompact . when ,the fuzzy enhanced real line (L) is strongly fuzzy paracompact. The product of a strong fuzzy compact set and a strong fuzzy paracompact set is strongly fuzzy paracompact. A strongly strong fuzzy paxacompact set isstrongly -regular and strongly -normal. A strongly -regular strong fuzzy Lindeloef space is strongly fuzzy paracompact.

Keywords: Fuzzy topology, strong fuzzy (countable) compactness, strong fuzzy paracompactness, strongly space ,strong -regularity, strong -normality, strong Lindeloef property.

Subcategories of Fuzzy Convergence Spaces

Dexue Zhang

Department of mathematics, Sichuan University

Chengdu 60064, P.R.china

Abstract: In this paper several subcategories of the category FCS of fuzzy convergence spaces are characterized by properties of the limit function and relations between these subcategories are investigated. The main results are: (1) The category FTS of fuzzy topological spaces is FCS, and every fuzzy convergence space is the quotient space of some fuzzy topological space. (2) The category of probabilistic pseudotopological spaces, fuzzy topological spaces, resp.)as a simultaneously bireflective and bicoreflective full subcategory.

Keywords: Fuzzy convergence spaces, Fuzzy pretopological spaces, Fuzzy topological spaces, Probabilistic Convergence spaces, Well-fibred topological category.