THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 19, Number 1, March 2011

CONTENT

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A Study on Solutions of Vector Minimax Problem Using Fuzzy Optimization

E. E. Ammar

Department of Mathematics, Faculty of Science, Tanta University, Egypt E-mail: ammar_al_saeed@hotmail.com

Abstract: In this paper, we study a minimax vector programming problem (VOP). We use two fuzzy approaches for characterize the solution set of (VOP). For the first approach we use the minimum operator and for the product operator. We give a suitable transformation functions to reformulate the (VOP) problem in fuzzy problem in fuzzy problem (EP). We deduce the solution set of the FP problem in each approach. We discuss the relation between the set of efficient solution of VOP problem and FP problems. Illustrative is given.

Keywords: Vector minimax problem, Fuzzy programming, Minimum operator Product operator, Transformation Functions.

Intuitionistics Fuzzy Semiprime Ideals of A Semiring

T. K. Dutta

Department of Pure Mathemastics, University of Caluctta 35, Ballyunge Circular Road, Kolkata-700019, India. E-mail:duttapankumar@yahoo.co.in

A. Ghosh

Department of Pure Mathemastics, University of Caluctta 35, Ballyunge Circular Road, Kolkata-700019, India. E-mail:amuradhaghosh2006@yahoo.com

Abstract: In this paper we introduce the notions of intuitionistic fuzzy semiprime ideal in semiring and studied it. We give some characterizations of intuitionistic fuzzy semiprime ideals of semiring. Also we characterize intuitionistic fuzzy semiprime ideals of the ring of integers Z.

Keywords: Semiprime ideals, Intuitionistic fuzzy semiprime ideal, Intuitionistic fuzzy semiprime k-ideal, etxc

Critical Path Calculations for Bi-objective Imprecise parameters Project Scheduling problem

U. K. Bhattacharya

Indian Institute of Management, Indore 453331-Madhya Pradesh, India E-mail: utpalb@iimidr. ac. In

Abstract: This paper present a techniques to find the critical path when more than one objective functions are associated to the decision process and various parameters such as activity duration times aas well as the cost associated to the various activities are imprecise in nature. To calculate the critical path it is assumed some fuzzy aspiration levels for each of the goals and basing these on a comparison of flat fuzzy numbers, the original fuzzy problem is transferred into a crisp satisfactory model. To determine a suitable solution minimum of the surpluses over the aspiration levels has been maximized. For each of the objective functions membership functions are presented corresponding to the optimal solution. A numerical example has been presented to illustrate the solution methodology.

Keywords: Critical Path, L-R type fuzzy number

Fixed Points of Weakly Compatible Maps in Intuitionistic Fuzzy 2-metric Space and Intuitionistic Fuzzy 3-metric Space

Meenakshi

E-mail:mgugnani@yahoo.com

Abstract: The aim of this paper is to prove some common fixed point theorems for weakly compatible maps in intuitionistic fuzzy 2-metric space and intuitionistic fuzzy 3-metric spaces. Our results extend and fuzzify several fixed point theorems in metric spaces.

Keywords: Common fixed point, weakly compatible mappings, intuitionistic fuzzy metric space, intuitionistic fuzzy 2-metric space, intuitionistic fuzzy 3-metric space.

Double Fuzzy Closure and Topological Spaces

M. Azab Abd-Allah

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

Abstract: We will define a double fuzzy closure space and a subspace of it. We will investigate relationships between double fuzzy closure spaces and double fuzzy topological space can be obtained by the subspace of the double fuzzy closure space induced by it. Finally, we will prove the existence of initial double fuzzy closure structures. From this fact, we can define subspaces and products of double fuzzy closure spaces.

Keywords: Double fuzzy closure spaces, double fuzzy topological space, subspace.

Fuzzy stability of A Generalized Quartic Mapping

M.S. Moslehian and H.R.E Vishki

Department of pure mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91755, Iran; Center of Excellence in Analysis on Algebraic Structure(CEAAS), Ferdowsi University of Mashhad, Iran E-mail address: moslehian@ferdowsi.um.ac.ir and moslehian@ams.org

Abstract: We establish the fuzzy stability of the generalized quartic functional equation f(ax+by)+f(ax-by)=(ab)2[f(x+y)+f(x-y)]+2(b2-a2)[b2f(y)-a2f(x)], where a, b are nonzero real numbers with a¡Ù¡Àb. Our result generalize some Hyers-Ulam-Rassias stability results in the framework of classical normed spaces.

Keywords: Hyers-Ulam-Rassiasstability, fuzzy normed space, generalized quartic functional equation; normed space; fuzzy stability.

A Note on Fuzzy Annihilators in Near-rings

S. Coumaressane

Department of mathematics, PAJANCOA £¦ RI, Karaikal-609603, U. T. of Puducherry, India. E-mail address: coumaressane.s@gmail.com

Abstract: In this short note we have shown that ann (¦Ì)of Saikia and Barthakur [10] contains only two elements. They also noted that ann (¦Ìt) =[ ann (¦Ì)]t , where t ¡Ê Im(¦Ì).we show by an example that the above statement is false.

Keywords: Near ring, fuzzy ideal, ann (¦Ì), fuzzy N-subgroups.

The Most General Fuzzy Topological Space and Its Graph

M. El-Ghoul Mathematics

Department, Faculty of Science, Tanta University, Tanta, Egypt E-mail address: m.mohammed elghoul@yahoo.com (corresp.author)

A. E. El-Ahmady

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt E-mail address:a_elahmady@hotmail.com

A. M. Sokar

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt E-mail address:ashrafsokar@yahoo.com

Abstract: This paper is a continuation of El-Ghoul [Chaos Solitons and Fractals] 13 (2002) 833833. It introduces the most general fuzzy topological space. The representation of a most general fuzzy topological space by a most general graph is deduced. The matrix representation f the most general fuzzy topological spaces is also presented. The degree of the most general fuzzy topological space is obtained. Theorems governing these concepts are achieved.

Keywords: Fuzzy topology, fuzzy graph.

On Fuzzy ??-I-continuous Mapp

ing A. M. Zahran,

Department of mathematics£¬Faculty of Science, Taibah University, Assiut 71524, Eygpt E-mail address: m.ah75@yahoo.com

S. E. Abbas

Department of mathematics£¬Faculty of Science, Sohay University, Sohag 82524, Eygpt

Yaser. M. Saber

Department of mathematics£¬Faculty of Science, Al-Azhar University, Assiut 71524, Eygpt

Abstract: In this paper, we introduce a new class of function called ¨CI-continuous functions. We obtain several characterizations and some of their properties. Also, we investigate its relationship with other types of functions.

Keywords: r-¦Ä -I-cluster point, r-FRI-open set, fuzzy ¦Ä-I-continuous, fuzzy almost I-continuous, fuzzy ¦È-I-continuous, fuzzy SI-R space.

Fuzzy Soft Group

SK. Nazmul Govt.

College of Education, Burdwan, Kazirhat, Lakurdi, Burdwan-713102, West Bengal, India.

S. K. Samanta

Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India E-mail: syamal_123@yahoo.co.in

Abstract: In this paper, following Molodtsov [1], P. K. Maji, R. Biswas, A. Roy ([2],[3]), H.Aktas, N Cagman [4], Rosenfeld [12], some basic definitions and properties of soft sets, fuzzy soft sets, soft groups and fuzzy groups are introduced. Generalized Soft group, factor soft group, homomorphism and isomorphism of soft groups are defined and their properties are studied. Also fuzzy soft group, normal fuzzy soft group, fuzzy factor subgroups are defined and their properties are studied.

Keywords: Soft sets, soft groups, fuzzy soft sets, fuzzy soft groups, factor fuzzy soft group, generalized soft group, soft homomorphism.

Intuitionistic Fuzzy Inner Product Spaces

Pinaki Majumdar

Department of Mathematics M. U. C Women¡¯s College. Burdwan, West-Bengal, Pin-713104, India E-mail: pmajumdar2@rediffmail.com

S. K. Samanta

Department of mathematics, Visva-Bharati, Santiniketam, West-Bengal, Pin-731235, India E-mail:syamal_123@yahoo.co.in

Abstract: In this paper we introduce intuitionistic fuzzy inner product in a linear space and investigate certain properties of intuitionistic fuzzy inner product spaces. The parallelogram law has been proved in this setting. The notion of minimizing vector is introduced in intuitionistic fuzzy inner product settings.

Keywords: Intuitionistic fuzzy inner product, Intuitionistic fuzzy norm, Parallelogram law, minimizing vector.

A Sequential Multi-decision Procedure with Fuzzy Observations

Ruma Talukdar

Department of Statistics, Cotton College, Guwahati, Assam, India-781001 Hemanta

K. Baruah

Department of Statistics, Gauhati University, Guwahati, Assam, India-781014

Abstract: In this article a sequential multi-dicision procedure derived by Sobel and Wald (1949) is considered for fuzzification. We develop here a fuzzy sequential inference procedure for choosing one of the three hypotheses concerning the mean of a normal distribution with known variance. The corresponding fuzzy operating characteristics functions have also been found. This is followed the applications of the results to a numerical example taken from Wald (1947).

Keywords: Fuzzy triangular number, fuzzy membership function, fuzzy operating characteristics function, sequential probability ratio test.

An Effective Particle Swarm Optimization with Fuzzy Fitness Function for Network System Reliability

Wen-Zhong Guo

College of Physics and Information Engineering Fuzhou University, Fuzhou, Fujian 350108, P. R. China

Shui-Li Chen, Guo-Rong Cai

School of Science, Jimei, University, Xiamen, Fujian 361021, P. R. China

Guo-Long Chen

College of Mathematics and Computer Sciences, Fuzhou University, Fuzhou, Fujian 350108, P. R. China

Abstract: Nonlinear mixed-integer programming problem (MINLP) is a generalized problem of network system reliability, namely optimizing the system reliability by keeping the balance between component reliability and redundancy number. It is a typical NP-hard problem and difficult to be dealt with by approximate algorithms and heuristic algorithms. In this paper, an effective swarm optimization with fuzzy fitness function is developed to solve this kind of network system reliability optimization problem. To secure more objective evaluation to particles and guide the search properly, two kinds of fuzzy fitness function are used in this paper to simulate the feature of the evaluation factors changing constantly in nature evolution. Experimental result indicates that the new algorithm is efficient and can be applied to the above problems.

Keywords: Reliability optimization, nonlinear mixed integer programming, particle swarm optimization, fuzzy fitness function

Notes on ¡°C-scattered Fuzzy Topological Spaces¡±

Xiu Xin

Department of Mathematics, Tianjin University of Technology, 300384 Tianjin, P. R. China

Sheng-Gang Li

College of Mathematics and Information Science, Shaanxi Normal University, Xi¡¯an, 710062, China E-mail: shenggangli@yahoo.com.cn

Abstract: The purpose of this note is to extend the notion of C-scatteredness of fuzzy topological space and some related results in F. G. Lupianez¡¯s paper (Applied Mathematics Letters 14 (2001), 201-204) to a more general setting, to give a negative answer to the open problem in the paper, and to give a correction to the paper.

Keywords: L-topological space, Induced L-topological space, Induced I (L)-topological space, Ch-C-Scattered L-topological space, Ch-compactness, Ch-perfect mapping

Notions via r-generalised Fuzzy -open Sets

T. Nithiya

Department of Mathematics Shri Sakthi Kailassh Women¡¯s College, Salem-636003 Tamil Nadu, India

M. K. Uma and E.Roja

Department of Mathematics Sri Sarada College for Women, Salem-636016 Tamil Nadu, India. E-mail: ar.udhay@yahoo.co.in

Abstract: In this paper, the concept of r -generalized fuzzy ??-border, r ¨Cgeneralized fuzzy ??-exterior and r -generalized fuzzy ??-frontier in the sense of Ramadan [3] and Sostak [7] are introduced. Some interesting properties and characterizations of them are investigated. Interrelations among the introduced are studied with relevant examples.

Keywords: r - g f ?? ¨Copen, r - g f ?? ¨Cinterior, r - g f ?? ¨Cclosure, r - g f ?? ¨Cborder, r - g f ?? ¨Cexterior, r - g f ?? ¨Cfrontier, fuzzy g c ?? ¨Cirresolute mapping, g f ?? ¨Ccontinuous mapping and g* f ?? ¨CT1/2 space.

Fuzzy Relation Equations in Four Variables with New Composition

Prasenjit Mandal , Arpita Das and A. S. Ranadive

Department of Pure and Applied Mathematics, Guru Ghasidas University, Bilaspur (C. G) India E-mail: prasenjitmandal08@yahoo.com

Abstract: In this paper we introduce several new compositions of fuzzy relations analogous to some well-know results. We also discuss fuzzy relation equations based on various compositions introduced by us and also obtained their basic solutions.

Keywords: Fuzzy relation; Fuzzy relation equations with four variables.

On Fuzzy Maximal ¨Csemi Open and Fuzzy Minimal ¨Csemi Closed Sets

Anjan Mukherjee

Department of Mathematics, Tripura University, Suryamaninagar Agartala, PIN-799130, Tripura, India E-mail:anjan2001_m@yahoo.co.in

Bishnupada Debnath

Department of Mathematics, Tripura University, Suryamaninagar Agartala, PIN-799130, Tripura, India E-mail:debnath_bishnupada@rediffmail.com

Abstract: The purpose of this paper is to introduce the notion of fuzzy maximal ¦Ä-semi open sets, fuzzy minimal ¦Ä-semi closed sets, fuzzy maximal ¦Ä-irresolute (fuzzy maximal ¦Ä-semi-irresolute), fuzzy maximal ¦Ä-continuous (fuzzy maximal ¦Ä-semi-continuous) functions. Some basic properties and characterization theorems are also to be investigated.

Keywords: Fuzzy regular open (resp. fuzzy regular closed) set, fuzzy ¦Ä-cluster point, fuzzy ¦Ä-open (resp. fuzzy ¦Ä-closed) set, fuzzy ¦Ä-semi open (resp. fuzzy ¦Ä-semi closed) set, fuzzy maximal ¦Ä-open set, fuzzy minimal ¦Ä-closed set, fuzzy maximal ¦Ä-semi open set, fuzzy minimal ¦Ä-semi closed set, fuzzy maximal ¦Ä-irresolute (fuzzy maximal ¦Ä-semi-irresolute), fuzzy maximal ¦Ä-continuous (fuzzy maximal ¦Ä-semi-continuous) functions.

ST-quotient Subgoups

Arpita Das, prasenjit Mandal and A. S. Ranadive

Department of pure and applied mathematics, Guru Ghasidas University, Bilaspur (C. G) INDIA E-mail: prasejitmandal08@yahoo.com

Abstract: In this paper, we introduce the notions of ST -cosets and ST ¨Cquotient subgroups of a group and discuss some related results. Our main objective is to present analog of Lagrange¡¯s theorem, Caley-theorem, Poincare¡¯s theorem and the fundamental theorem of homomorphism.

Keywords: Algebra; ST ¨Ccoset; ST ¨Cquotient subgroup; Fundamental theorem of homomorphism

ST-subrings and ST-ideals Prasenjit

Mandal, Arpita Das and A. S. Ranadive

Department of pure and applied mathematics, Guru Ghasidas University, Bilaspur (C. G) INDIA E-mail: prasejitmandal08@yahoo.com

Abstract: In this paper, we will continue our work in concern with basic ideas provided in [1,2], introduce the notions of ST -subrings and ST-ideals of a ring, and study some of their basic properties.

Keywords: Algebra; ST -subring ; ST -ideal; ST-ideal of an ST ¨Csubring

Ordered Intuitionistic Fuzzy Flood Index Model

Sunny Joseph Kalayathankal

Department of Mathematics, K. E. College, Mannanam Kottayam-686561, Kerala, India E-mail: sunnyjose2000@yahoo.com

G. Suresh Singh

Department of Mathematics, University of Kerala, Kariavattom Thiruvananthpuram-695581, Kerala, India

P. B. Vinodkumar

Department of Mathematics, Rajagiri School of Engineering Technology & Kakkanad, Cochin-682039, Kerala, India

Abstract: Over the last decade, fuzzy based flood warning systems have been increasingly used in mathematical studies. An efficient flood system may significantly improve public safety, and mitigate economical damages caused by inundations. This work illustrates a fuzzy analysis that has the capability to simulate the unknown relations between a set of meteorological and hydrological parameters. In this paper, we first define ordered intuitionistic fuzzy sets and establish some results in them. Then, we define four types of similarity measures between ordered intuitionistic fuzzy sets (OIFS) and apply these similarity measures to five selected sites of Kerala, India to predict flood index under ordered intuitionistic fuzzy environment.

Keywords: Rainfall, Intuitionistic fuzzy set, Flood, Simulation.

ST-quotient Subrings

Prasenjit MMandal , Arpita Das and A. S. Ranadive

Department of pure and applied mathematics, Guru Ghasidas University, Bilaspur (C. G) India E-mail: prasejitmandal08@yahoo.com

Abstract: In this paper, we define ST -cosets and ST ¨Cquotient subrings of a ring and study the relations between the ST ¨Cquotient subrings and ST ¨Cideals. Our main objective is to present the fundamental theorem of homomorphism of ST ¨Csubrings.

Keywords: Algabra; ST-coset; ST ¨Cquotient subrings; fundamental theorem of homomorphism

Stratified L-fuzzy Topological Spaces

S. E. Abbas

Department of Mathematics, Faculty of Science, Sohag 82524, Egypt K. Gurlebeck Institute of Math. And Physics, Bauhaus Univ., Coudraystr. 13, Weimar 99421, Germany.

Abstract: In this paper, we construct stratified L-fuzzy topology from a given L-fuzzy topology, where L is a complete residuated lattice. We study the relationship lattice. We study the relationship between L-fuzzy topology and the stratification of it.

Keywords: Fuzzy sets, Residuated, L-fuzzy topology, Strstification.

On *??-closed Fuzzy Sets, Fuzzy * ?? -continuous and Fuzzy * ?? -irresolute Mappings in Fuzzy Topological Spaces

Sadanand. N. Patil and S. R. Sadugol H. C. D.

Department of Mathematics, KLSS VDRLT, HALIYAL-29, UK Dept. of E&E, B. L. D. E A¡¯s Dr. PG halakatti Engineering College, Bijapur-E-mail: srsadugol@gmail.com, sada_np@rediffmail.com

Abstract: The aim of this paper is to introduce a new class of fuzzy sets, namely *¦Ì-closed fuzzy sets for fuzzy topological spaces. This class is a super class of class of the closed fuzzy sets. We introduce and study new space namely fuzzy cT??*-spaces and ??T??*-spaces. Further, the concept of fuzzy *??-continuous, fuzzy *??-irresolute mappings , fuzzy *??-closed maps, fuzzy *??-open maps and fuzzy *??-homeomorphism in fuzzy topological spaces are also introduced, studied and some of there properties are obtained.

Keywords: *¦Ì-closed fuzzy sets, f *??-continuous, f *??-irresolute, f *??-open, f *??-closed mappings and f *??- homeomorphism.