THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 15, Number 3, September 2011

CONTENT

DETAILS

On -level Ideals of -rings

Kyung Ho Kim

Department of Mathematics, Chungju National University, Chungju 380-702, Korea.

Abstract: Using -norm ,we introduce the notion of -level ideals,and some related properties are investigated.

Keywords: -ring,-norm,-fuzzy ideal,idempotent -fuzzy ideal,-level subset,-level ideal.

On Locally Semi-convex -topological Vector Spaces

Hui Zhang

Department of Mathematics,Nanjing Normal University, Nanjing, Jiangsu 210097,People'sRepublic of China

Department of Mathematics,Anhui Normal University, Wuhu,Anhui241000,People'sRepublic of China

Jin-xuan Fang

Department of Mathematics,Nanjing Normal University, Nanjing, Jiangsu 210097,People'sRepublic of China

Abstract: In this paper,a new difinition of locally Semi-convex -topological vector spaces is given and the concept of generalized fuzzy pseudo semi-norm is introduced.We give a characterization of the locally Semi-convex -topological vector spaces by means of a family of generalized fuzzy pseudo semi-norms.

Keywords: Semi-convex fuzzy sets; Locally Semi-convex -topological vector spaces; Generalized fuzzy pseudo semi-norms

Some Aspects of Fuzzy Operations

B.Kalita

Department of Mathematics,Dudhnoi college,Dudhnoi,Goalpara,Assam,India

N.R.Das

Department of Mathematics,Gauhati University-14,Assam,India

Abstract: By Using an operation on topological spaces ,H.Ogata [3] defined the concept of -open sets and investigated some properties of -open sets.In this paper,we have developed the concepts of fuzzy -open sets and established some properties of fuzzy -open sets.We have also studied some properties of fuzzy -continuous mapping.

Keywords: Fuzzy -open set, fuzzy -closure, fuzzy -continuous mapping.

Solving Least Time Transportation Problem under Fuzzy Environment

D.Stephen Dinagar

PG & Research Department of Mathematics,T.B.M.L.College,Porayar-609307,TamilNadu,India

K.Palanivel

School of Advanced Sciences-Mathematics,VIT University Vellore-632014,Tamil Nadu,India

Abstract: In this article,we discuss least time fuzzy transportation problem with the aid of trapezoidal fuzzy numbers and its arithmetic operations.To find initial solution of this problem we have preferred fuzzy North west corner rule,next to find the optimal solution in terms of fuzzy numbers,Finally its solution of the each shipment with least time is verified in the naturee of fuzzymembership functions.A new relevant numerical example is also included.

Keywords: Fuzzy numbers, Trapezoidal Fuzzy numbers, Fuzzy North West Corner rule Method, Least time Fuzzy transportation Method.

Solving Assignment Problem in Fuzzy Environment

D.Stephen Dinagar

PG & Research Department of Mathematics,T.B.M.L.College,Porayar-609307,TamilNadu,India

K.Palanivel

School of Advanced Sciences-Mathematics,VIT University Vellore-632014,Tamil Nadu,India

Abstract: In this article, we investigate fuzzy assignment problem with the aid of trapezoidal fuzzy numbers and its new arithmetic operations. Fuzzy Hungarian method is proposed to find the optimal solution in terms of fuzzy numbers and verified its solution in the nature of fuzzy membership functions. The fuzzified version of the problem has been discussed with the aid of a numerical example.

Keywords: Fuzzy Numbers, Trapezoidal Fuzzy Numbers, Trapezoidal Membership Functions, Fuzzy Hungarian method.

On Fuzzy -continuity and -compactness

B.Kalita **and N.R.Das*

*Department of Mathematics, Gauhati University,Gauhati-14,Assam,India.

**Department of Mathematics, Dudhnoi college, Goal para, Assam,India.

Abstract: Let be a fuzzy topological space.An operation on [2] is a mapping from into such that for every .In this paper,the concept of -continuity and -compactness in fuzzy setting is defined.Also we have investigated some basic properties of these of these notions.

Keywords: Fuzzy -continuity,fuzzy -compact,fuzzy -convergences,fuzzy -accumulation.

On Regular Intuitionistic Fuzzy Matrices

AR.Meenakshi and T.Gandhimathi

Department of Mathematics, Karpagam University-642021,India.

Abstract: In this paper, the row space and column space of an intuitionistic fuzzy matrix are determined, interms of the row and column spaces of its membership and non-membership parts.By using the Cartesian product representation of an intuitionistic fuzzy matrix,invertibility and -inverse of intuitionistic fuzzy matrix are discussed.

Keywords: Intuitionistic fuzzy matrix, regular, invertible, generalised inverse.

Fuzzy Fractional Calculus and Ostrowski Inequality

George A.Anastassiou

Department of Mathematics Sciences,The University of Memphis Memphis, TN 38152,U.S.A

Abstract: Here we introduce and study the right and left fuzzy fractional Riemann-Liouville integrals and the right and left fuzzy fractional Caputo derivatives. Then we present right and left fuzzy fractional Taylor formulate. Based on these we establish a fuzzy fractional Ostrowski type inequality with applications. The last inequality provides an estimate for the deviation of a fuzzy real number valued function from its fuzzy average, and the related upper bounds are given in terms of the right and left fuzzy fractional derivatives of the involved function. The purpose of embedding fuzziness into fractional calculus and have them act together, is to better understand, explain and describe the imprecise, uncertain and chaotic phenomena of the real world and them derive useful concludions.

Keywords: Fuzzy Real Analysis, Fractional Calculus, Taylor formula, Ostrowski inequality.

Fuzzy Global Smoothness and Fuzzy Uniform Convergence of Fuzzy Singular Integral Operators

George A.Anastassiou

Department of Mathematics Sciences, The University of Memphis Memphis, TN 38152,U.S.A.

Abstract: In this article, we study the fuzzy global smoothness and fuzzy uniform convergence of fuzzy Picard, Gauss-Weierstrass and Poisson-Cauchy singular fuzzy integral operators to the fuzzy unit operator. These are given with rates involved function. The produced fuzzy Jackdon type inequalities are tight, containing elegant constants, and they reflect the order of the fuzzy differentiability of the engaged fuzzy function.

Keywords: Fuzzy global smoothness, Fuzzy approximation, fuzzy singular integral, fuzzy modulus of continuity, rate of fuzzy convergence.

Higher Order Multivariate Fuzzy Approximation by Multivariate Fuzzy Wavelet Type and Neural Network Operators

George A.Anastassiou

Department of Mathematics Sciences, The University of Memphis Memphis, TN 38152,U.S.A.

Abstract: Here are studied in terms of multivariate fuzzy high approximation to the multivariate unit several basic sequences of multivariate fuzzy wavelet type operators and multivariate fuzzy neural network operators. These operators are multivariate fuzzy analogs of earlier studied multivariate real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order multivariate fuzzy pointwise convergence with rates to the multivariate fuzzy unit operator is established through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the th order -fuzzy partial derivatives, of the engaged multivariate fuzzy number valued function. The purpose of embedding fuzziness into multivariate classical analysis is to better understand ,explain and describe the imprecise, uncertain and chaotic phenomena of real world and then derive useful conclusions.

Keywords: Multivariate fuzzy real analysis, multivariate fuzzy wavelet type operators, multivariate fuzzy neural network operators, high order multivariate fuzzy approximation, multivariate fuzzy moduli of continuity and multivariate Jackson type inequalities.

A Note on Difference Sequences of Fuzzy Numbers

Hemen Dutta

Department of Mathematics, A.D.P.College,Nagaon-782002,Assam,INDIA.

B.Surender Reddy

Department of Mathematics, PGCS,Saifabad, Osmania University, Hyderabad-5000004,AP,INDIA.

Abstract: In this article, we generalize the notion of convergent and bounded sequences of fuzzy numbers by introducing the notion of -convergent,-null and -bounded sequences, where are non-negative integers and be a sequences of non-zero reals. We study the spaces of these sequences for completeness. We also computed their isometric spaces. Further we forward a new approach to construct difference sequence spaces and extend these notions to double sequences of fuzzy numbers.

Keywords: Sequences of fuzzy numbers; Double sequences of fuzzy number; Difference Sequence; Completeness; isometry.

On Intuitionistic Fuzzy Contractive Mapping in Non-archimedean Intuitionistic Fuzzy Metric Spaces

Sushil Sharma

Department of Mathematics, Madhav Science College, Ujjain (M.P.) India

Bhavana Deshpande

Department of Mathematics, Govt.Arts and Science P.G.College, Ratlam (M.P.) India

Abstract: We define an intuitionistic fuzzy contractive mapping on an intuitionistic fuzzy metric space. We prove a Banach contraction theorem in complete non-Archimedean intuitionistic fuzzy metric space. We also give examples to validate our results.

Keywords: Non-Archimedean intuitionistic fuzzy metric space, intuitionistic fuzzy contractive mapping, fixed point.

On Fuzzy Almost Semi -pre Generalized Continuous Mappings

Anjan Mukherjee

Department of Mathematics, Tripura University Suryamaninagar,Agartala,799130,TRIPURA,INDIA

Pranab Kumar Deb

Ramkrishna Mahavidyalaya, Kailashahar,North Tripura 799277,TRIPURA,INDIA

Abstract: In this paper ,we introduce and characterize the concept of fuzzy almost semi -pre generalized continuous mappings. Several interesting properties of these mappings are also given. Examples and couter examples are also given to illustrate the concepts introduced in the paper. We also introduce the concept of fuzzy semi -pre-space, fuzzyspace, fuzzy regular space and semi -pre generalized compact space. It is seen that a fuzzy almost semi -pre generalized continuous mappings from a fuzzy semi -pre-space to another fuzzy topological space becomes fuzzy almost semi -pre continuous mapping.

Keywords: Fuzzy pre open set, fuzzy -pre open set, fuzzy semi -pre open set, fuzzy almost continuous mapping, fuzzy almost continuous mapping, fuzzy almost semi -pre continuous mapping, fuzzy semi -pre-space. Semi-generalized

Continuous Mappings in Sostak's Fuzzy Topological Spaces

S.E.Abbas *and M.Taha**

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

Abstract: In this paper, we introduce and study the concepts of -semi generalized fuzzy closed sets and -generalized fuzzy semi-closed sets in Sostak's fuzzy topological spaces. Fuzzy-connectedness is introduced and studied with help of -semi generalized fuzzy closed sets. Fuzzy semi-generalized continuity, generalized fuzzy semi-continuity, fuzzy semi-generalized irresolute and generalized fuzzy semi-irresolute mappings are introduced and the relationship between these mappings and other mappings introduced previously are investigated. Also, some separation axioms of -generalized fuzzy semi-closed sets are introduced and studied.

Keywords: -semi generalized fuzzy closed sets,-generalized fuzzy semi-closed sets,Fuzzy-connected (-connected) sets,Fuzzy semi-generalized continuity,Generalized fuzzy semi-continuity, -irresolute,-irresolute and -regular (-normal) spaces.

The Coreflective Hull of in -Top

Veena Singh

Department of Mathematics,M.L.K.(P.G.)College,Balrampur-271201,India

Arun K.Srivastava

Department of Mathematics, and Centre for Interdisciplinary Mathematics Sciences, Faculty of Sciences, Banaras Hindu University,Varanasi-221005,India

Abstract: In this paper, we determine the coreflective hull of ,a complete lattice with an order-reversing involution, endowed with the -topology, generated by and ,in the category -Top of -topological spaces.

Keywords: Coreflective subcategory; Coreflective hull.

The Various Adjoint Algebras

Yong Chan Kim and Sung Bong Hong

Department of Mathematics, Gangneung-Wonju University,Gangneung,201-702,Korea

Abstract: In this paper, we introduce the various adjoint algebras induced by implications and conjunctions and give their examples.

Keywords: Implication, Conjunction, Forcing implication, Adjoint algebra.

Lower and Upper Minimal Weakly -semi Continuous Fuzzy Multifunctions

Anjan Mukherjee

Department of Mathematics, Tripura University,Surjyamaninagar,Agartala-799130,Tripura,India.

Abstract: Fuzzy minimal structure and fuzzy minimal space were introduced by m.Alimohammady and M.Roohi in 2006.Recently we introduced the lower and upper minimal continuous fuzzy multifunctions in the sense of Papageorgiou and studied some of their basic properties and characterization theorems.The purpose of this paper is to introduce the lower and upper minimal weakly and weakly-semi continuous fuzzy multifunctions between a general topological space and a fuzzy topological space .Using the mappings and several conditions of the lower and upper minimal weakly and weakly-semi continuous fuzzy multifunctions are also established.For a fuzzy multifunction .The upper inverse and lower inverse are defined as follows: For any fuzzy set , and . For a fuzzy multifunction . for any fuzzy set in . A family of fuzzy sets in is called a fuzzy minimal structure (briefly -structure) on if and and belong to and is a fuzzy minimal space.The members of are called fuzzy -open sets and their complements are called fuzzy -closed sets.Let be an ordiary topological space and be a fuzzy -space.A fuzzy multifunction is said to be (a)fuzzy lower minimal weakly-semi continuous (briefly flmwsc) at a point if and only if for every fuzzy minimal open (fuzzy -open) set in with there exists a -semi open neighbourhood of in X such that (-scl) i.e. for all . (b)fuzzy upper minimal weakly-semi continuous (briefly fumwsc) at a point if and only if for every fuzzy -open set in with there exists a -semi open neighbourhood of in X such that (-scl) i.e. for all . (c)fuzzy lower(upper) minimal weakly-semi continuous on if and only if it is fuzzy lower(upper) minimal weakly-semi continuous at each point .

Keywords: Fuzzy minimal structure, fuzzy -open(closed) functions, fuzzy -continuity, fuzzy multifunction.

On Solutions of Generalized Multiobjective Quadratic with Fuzzy Random Environments

E.E.Ammar

Dept. of Mathematics and Statistic, Faculty of Sciences, Taif University, Saudi Arabia

Abstract: In this paper, a generalized multiobjective quadratic programming problem having fuzzy random coefficient matrix in the objective and constaints and the decision vector are fuzzy pseudo random variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.

Keywords: Multiobjective fractional quadratic programming, Fuzzy programming, Fuzzy random variables, Interval analysis.

Fuzzy Ideals of Zero Square Rings

Satyanarayana Bhavanari

Department of Mathematics,Acharya Nagarjuna University, Nagarjuna Nagar-522510,A,P,India.

Godloza Lungisile

Department of Mathematics, Walter Sisulu University, Umtata, South Africa

Dasari Nagaraju

Department of Mathematics, Periyar Maniammai University, Vallam, Tamilnadu, India.

Abstract: We consider associative ring (not necessarily commutative) R. We deal with zero square rings and zero square ideals. In this paper we introduce the concept zero square fuzzy ideal.The aim of the paper is to prove that there exists a bijection between the set of all zero square ideals of and the set of all zero square ideals of ,where the image of is equal to two element set consisting of two numbers in the closed interval with .

Keywords: Zero square ring, Zero square ideal, Fuzzy ideal, Zero square fuzzy ideal.

Relation between Trace Systems and Compactnesses of Principal Fuzzy Extensions of A -space of Fuzzy Sets

K.C.Chattopadhyay

Department of Mathematics, The University of Burdwan-713104,W.bengal,INDIA

H.Hazra

Department of Mathematics, Bolpur College Bolpur,Birhum,W.bengal,INDIA

S.K.Samanta

Department of Mathematics, Visva Bharati Santiniketan-7131235,W.bengal,INDIA

Abstract: In this paper conditions on trace systems of principal fuzzy extensions of -topological spaces of fuzzy sets have been established which would ensure fuzzy conjoint compactness and fuzzy linkage compactness of principal fuzzy extensions.

Keywords: Topology of fuzzy sets, principal fuzzy extensions,-space, fuzzy conjoint grill, fuzzy linked grill, fuzzy conjointly compact space, fuzzy linkage compact space, finitely determined collection of fuzzy grills, binary collection of fuzzy grills.

Nilpotent -subgroups Prasenjit

Mandal and A.S.Randive

Bhalukdungri Junior High School, Purulia (W,B.), India Department of pure and applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur (C.G.).INDIA

Abstract: In this paper the -centralizer of an -subgroup is defined and its properties are discussed. Using this idea, the -central series of an -subgroup is introduced. A definition of a nilpotent -subgroup is proposed and the known results fot the crisp set-up are tested.

Keywords: Algebra; Centralizer; -centralizer;-central series; Nilpotent -subgroup

Fuzzy Pseudo Nearly Compact Space and Continuous Function

Deb Ray

Department of Mathematics, West Bengal State University Berunanpukuria,Malikapur,Dist.North 24 Parganas,PIN-700126,INDIA

Pankaj Chettri

Department of Mathematics, Sikim Manipal Institute of Technology Majitar,Rangpoo,East Sikkim,PIN-737132,INDIA

Abstract: In this article, a fuzzy compact like property called fuzzy pseudo near compactness, in terms of pseudo regular open fuzzy sets is introduced. Fuzzy pseudo near compactness is indeed different from fuzzy near compactness. Several characterization of fuzzy pseudo near compactness is studied here by means of fuzzy nets, fuzzy filter-base and finite intersection property ().In the last section of this paper, a type of fuzzy continuity is studied which is responsible for preserving fuzzy pseudo near compactness.

Keywords: Fuzzy pseudo near compact, open fuzzy sets,-closure, fuzzy -cluster point,-continuous function.

-fuzzy Bi-ideals in Ordered Semigroups

Asghar Khan

Department of Mathematics, COMSATS Institute of Information Technology Abbottabad 22062,Pakistan

Young Bae Jun

Department of Mathematics, Educations and RINS Gyeongsang National University Chinju 660-701,Korea

Muhammad Shabir

Department of Mathematics, Quaid-I-Azam University,Islamabad-45320,Pakistan

Abstract: The purpose of this paper is to initiate and study the new sort of fuzzy bi-ideals,called -fuzzy bi-ideals of ordered semigroups and investigate the basic theorem of ordered semigroups in terms of -fuzzy bi-ideals. We provide characterizations of certain classed (of regular and intra-regular) ordered semigroups in terms of -fuzzy bi-ideals and discuss the characterizations of left (resp.right and completely regular) ordered semigroups in terms of -fuzzy bi-ideals.

Keywords: -fuzzy subsets;-fuzzy bi-ideals of ordered semigroups; left (right regular),completely regular ordered semigroups; intra-regular ordered semigroups; left (right) semilattices of ordered semigroups.