THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 22, Number 1, March 2014

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On Fuzzy Multifunctions

N. Gowrisankar 

70/232 6B, Kollupettai Street, M. Chavady, Thanjavur-613001, Tamil-Nadu, India

E-mail address: gowrisankartnj@gmail.com

N. RajeshDepartment of mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India. E-mail address: Nrajesh_topology@yahoo.co.in

Abstract:In this paper we have introduce and study the concept of fuzzy g-irresolute multifunctions between topological and fuzzy topological spaces

Keywords:Fuzzy multifunctions, fuzzy -open set, fuzzy -closed set.

Fuzzy Weak Structures

N. Gowrisankar

70/232 6B, Kollupettai street, M. Chavady, Thanjavur-613001, Tamil-nadu, India.

E-mail address: gowrisankartnj@gmail.com

N. Rajesh

Department of mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India.

E-mail address: nrajesh_topology@yahoo.co.in

V. Vijayabharathi

Department of Mathematics, National Institute of Technology, Tiruchi-Rappalli, Tamilnadu, India.

E-mail address: vijayabharathi_v@yahoo.com

Abstract:In this paper we have introduce and study a new concept in fuzzy theory called fuzzy weak structures.

Keywords and phrases: Fuzzy weak structures.

Intuitionistic Fuzzy -locally Continuous Function

S. Padmapriya , M. K. Uma and E. Roja

Department of mathematics, Sri Sarada College for Women, Salem, Tamil Nadu, India, Email: priyasathi17@gmail.com

Abstract:The purpose of this paper is to introduce the concepts of an intuitionistic fuzzy -open sets, intuitionistic fuzzy -closed sets, intuitionistic fuzzy -locally  closed  sets which are the weaker forms of the class of an intuitionistic fuzzy locally closed sets. The concepts of an intuitionistic fuzzy e -locally quasi neighbourhoods,intuitionistic fuzzy -locally continuous functions, intuitionistic fuzzy -local

 spaces, intuitionistic fuzzy -local Urysohn spaces, intuitionistic fuzzy  -local connected  spaces  and intuitionistic fuzzy  -local compact spaces are introduced and interesting properties are established.

Keywords and phrases:Intuitionistic fuzzy -open set, intuitionistic fuzzy -closed set , intuitionistic fuzzy -locally closed set,intuit-ionistic fuzzy  -locally quasi neighbourhood, intuitionistic fuzzy p-b-locally continuous function, intuitionistic fuzzy -local  space, intuitionistic fuzzy -local Urysohn space, intuitionistic fuzzy p-b-local connected  space  and intuitionistic fuzzy -local compact space.

Multicriteria Decision Making under Interval Valued Intuitionistic Fuzzy Context 

Shiny Jose 

Associate Professor in Mathematics, St. George's College Aruvithura,

E-mail: shinyjosedavis@gmail.com

Sunny Kuriakose A

Principal B. P. C. College Piravom, Kerala, India.

E-mail: asunnyk@gmail.com

Abstarct: The notion of Intuitionistic Fuzzy Set (IFS for short) theory by Krassi-mir Atanassov strikes a paradigm shift in solving decision making problems, which is one of the crucial problems in our real life. Ranking of IFS and Interal Valued intuitionistic Fuzzy Set (IVIFS for short) is very ofthen required in decision making. Many Many authors studied multicriteria decision making problems based on IVIFSs. and they proposed accuracy function and novel accuracy function to solve these problems. In this paper we propose another novel accuracy function based on IVI-FS, which can rank all comparable and incomparable interval valued intuitionistic fuzzy sets. It appears to be more elegant and simple than the existing accuracy functions. The proposed model is illustrated by a numerical example.

Keywords and phrases: Intuitionistic Fuzzy Sets (IFSs), Interval valued Intuitionist-ic Fuzzy sets (IVIFSs), accuracy function, novel accuracy function.

On Statistical Convergence in Fuzzy Normed Linear Spaces

A. Bhuniya 

Department of Mathematics, Visva-Bharati, Santiniletan, India.

E-mail: anjankbhuniya@gmail.com

T. Bag

Department of Mathematics, Visva-Bharati, Santiniletan, India.

E-mail: tarapadavb@gmail.com

Abstarct: Here we introduce and characterize the notions of statistical convergenc-e, statistical limit points and statistical cluster points in a Bag and Samanta type fuzzy normed linear space .  The set of all statistically convergent sequences in is a  closed   subspace of the fuzzy normed linear space of all sequences in X.  

Keyword and phrases: Fuzzy normed linear spaces, statistical convergence, statistic-al convergence, statistical cluster points.

Analysis of The Child Development Index

Michel Mallenby

Department of Mathematics Creighton University Omaha, Nebraska 68178.

E-mail: mmall@creighton.edu

Alex Pham

Department of Mathematics Creighton University Omaha, Nebraska 68178.

E-mail: alexpham@creighton.edu

John N Mordeson

Department of Mathematics Creighton University Omaha, Nebraska 68178.

E-mail: mordes@creighton.edu

Abstract: The paper includes a re-ranking of the Child Development Index (CDI). It then compares the average ranking of countries by the CDI to their economic free-dom status, as determined by the Index of Economic Freedom (IEF). CDI rankings are also compared to those of the Human Development Index (HDI) in an attempt to find cor-relations between them.  The Child Development Index ranks the countries of the world based on three factors-child health, nutrition, and education-while the IEF and HDI use ten and three factors, respectively. This paper employs fuzzy mathem-atics techniques, specifically the Analytic Hierarchy Process and the Guiasu Met-hod, and expert opinion to assess the relative importance of the three componen-ts of child development mentioned above.  The numerical data from the CDI and the newly weighted factors are then used to re-rank the CDI. To compare the CDI and HDI, we used statistical analysis to determine the strength of the correlati-ons between the factors of each index as well as between the indexes themselve-s. We carry out the same procedure between the CDI and the IEF and their com-ponents. These evaluations confirmed the strength of several expected relationship-s. but also showed many strong unanticipated relations.

Keywords and phrases: Child Development Index, analytic hierarchy process, Guiasu m-ethod,Save the Children Foundation.

Some Properties of The Lattice of Ordered Possibility Distributions

Gigi George

Department of Mathematics, Mar Thoma College for Women, Perumbavoor, Kerala-683542 S.India.

E-mail: frpraise@yahoo.com

Sunny Kuriakose

Principal, BPC College, Piravom, Kerala, S. India.

Abstract: Klir [5] has shown that the set of all ordered possibility distributi-ons defined on a finite set forms a lattice.  In this paper, its further alg-ebraic and topological properties are investigated.  In particular it is proved that it forms a complete distributive lattice and satisfies the infinite distrib-utive laws and hence Brouwerian and dually Brouwerian. A metric structure is ind-uced on the lattice using the -uncertainty function in Possibility Theory and its properties are analysed.  Similar metric structures are also introduced on ordered  joint and marginal possibility distributions and certain relations conn-ected  these metrics are derived.

Keywords and phrases: Ordered possibility distribution, valuation induced metric, or-dered joint and marginal possibility distributions.

Some Aspects of Fuzzy -continuum 

N. R. Das 

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India.

E-mail: nrd47@yahoo.co.in

Brojen Das

Department of Mathematics, M.C.College, Barpeta, Assam, India.

E-mail: brojen.das@rediffmail.com

Abstract: This paper deals with the study of fuzzy N-continuum where the concept of fuzzy -closeness and fuzzy -connectedness is utilized. S. Ganguly and S.Jana have introduced the concept of continuous like structure called -continuum in their paper   Note on -continuum in general topological space. We have tried to study  -continuum  structure in fuzzy setting and give some of its prope-rties. Here  we shall also see  that the presence of both fuzzy N-closedness and fuzzy -connectedness give rise  to some  results which are not available in the fuzzy topological space as a consequence  of only  one of them.  The necessity and importance of studying such a space is justified in  this paper. Our endeavor will continue to explore more results in this direction.

Keywords and phrases: -connectedness, -closed, -compact, Fuzzy regular open, N-continuum, -continuous.

Intuitionistic Fuzzy Sets in A Lattice

Latha S Nair

Department of Mathematics, Mar Athanasius College Kothamangalam, Kochi-Kerala, India.

E-mail: lathavichattu@gmail.com

Abstract: In this paper we give the characterizations of intuitionistic fuzzy sub-lattices, ideals, and prime ideals in terms of their level sets. The notion of int-utionistic fuzzy convex sub lattice is introduced and discussed. Moreover various intuitionistic fuzzy analogs of the results of classical lattice theory are established.

Keywords and phrases : Intuitionistic fuzzy set (IFS), intuitionistic fuzzy sublatti-es (IFL), intuitionistic fuzzy ideals (IFI),intuitionistic fuzzy primeideals (IEP-I), levelsubsets, upper level set, lower level set, convex sublattic, epimorphism.

Decomposition of Soft Fuzzy  Open Se

T. Yogalakshmi , E. Roja and M. K. Uma

Department of Mathematics, Sri Sarada College for Women, Salem-636016, Tamil Nadu, India

E-mail: deeppayogmathi@gmail.com

Abstract: In this paper, a new definition, called the soft fuzzy local function of a soft fuzzy set is coined.  Interrelations among soft fuzzy open set and soft fuzzy ideal open set are established.  Decomposition of a soft fuzzy C1 open set is discussed and several properties of it are studied.  Further, deco-mposition of a soft fuzzy complete  continuity is established. Moreover, soft I-extremally disconnected space is introduced.  In this connection, several charac-terizations and properties are established.  Indeed, many examples and counter examples are provided wherever necessary.

Keywords and phrases: Soft fuzzy  open set, soft fuzzy  open set, soft f-uzzy  open set, soft fuzzy -extremally disconnected space.

Normal Closure of An -subgroup of An -group

Naseem Ajmal 

Department of Mathematics, Zakir Hussain College, University of Delhi, J.L.Nehru Marg New Delhi, India.

E-mail: nasajmal@yahoo.com

Iffat Jahan

Department of Mathematics, Ramjas College, University of Delhi, Delhi-110007, India.

E-mail: ij.umar@yahoo.com

Abstract: In this paper, we define the conjugate of an -subgroup by an L-group instead of a crisp point. Then we introduce the notion of normal closure of an L-subgroup by an -group instead of a crisp point.  Then we introduce the n-otion  of normal closure of an -subgroup of an -group.  This notion gives rise to ‘normal closure series’ of an -subgroup of an -group which leads to the definition of subnormal L-subgroup.

Keywords: -algebra; -subgroup; Normal -subgroup; Normal closure; Subnormal L-group.

Fixed Point Theorems on Intuitionistic Fuzzy Quasi-metric Spaces and Applications

Bhavana Deshpande 

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

E-mail: bhavnadeshpande@yahoo.com

Sushil Sharma

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

E-mail: sksharma2005@yahoo.com

Amrish Handa

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

E-mail: amrishhanda83@gmail.com

Abstract: In this paper, we prove intuitionistic fuzzy quasi-metric versions of fixed point theorems of Gregori and Sapena [9] and Mihet [15]. We give applic-ations of our result to the domain of words.

Keywords and phrases: Intuitionistic fuzzy quasi-metric space, Common fixed point, G-bicomplete, Non-Archimedean intuitionistic fuzzy quasi-metric, Intuitionistic fuzzy contractive mapping, Domain of words.

Fuzzy Fractional Approximations by Fuzzy Normalized Bell and Squashing Type Neur-al Network Operators

George A. Anastassiou

Department of Mathematics Science University of Memphis Memphis, TN 38152, USA. E-mail: ganastess@memphis.edu

Abstract: This paper deals with the determination of the fuzzy fractional rate of onvergence to the unit to some fuzzy neural network operators, namely, the fuzzy normalized bell and “squashing” type operators.  This is given through the f-uzzy moduli of continuity of the involved right and left fuzzy Caputo fraction-al derivatives of the approximated function and they appear in the right-hand side of the associated fuzzy Jackson type inequalities.

Keywords and phrases: Neural network fuzzy fractional approximation, bell function, squashing, function, fuzzy fractional derivative, fuzzy modulus of continuity.

Linear Equation of Interval-valued Trapezoidal Fuzzy Numbers

Rituparna Chutia

Department of Mathematics Gauhati University Assam-781017, India.

E-mail: Rituparnachutia7@rediffmail.com

Abstract: The basic arithmetic operations of the interval-valued trapezoidal fuzzy numbers are reviewed and defined.  A necessary and sufficient condition for the existence of solution of linear equation by Buckley [Fuzzy Sets and systems 38, (1990),43-59] are generalized on interval-valued trapezoidal fuzzy numbers.

Keywords and phrases: Quartiles, Generalised fuzzy number, cardinality.

Multivariate Fuzzy-random Quasi-interpolation Neural Network approximation operators

George A. Anastassiou

Department of Mathematics Science University of Memphis Memphis, TN 38152, U.S.A. E-mail: ganastss@memphis.edu

Abstract: In this paper we study the rate of multivariate pointwise and uniform convergence in the -mean to the Fuzzy-random unit operator of multivariate Fuzzy-Random Quasi-Interpolation neural network operators. These multivariate Fuzzy-Random operators arise in a natural and common way among multivariate Fuzzy-Random neural networks. These rates are given through multivariate Probabilistic-Jackson type inequalities involving the multivariate Fuzzy-Random modulus of continu-ity of the engaged multivariate Fuzzy-Random function.  The plain stochastic ext-reme analog of this theory is also presented.

Keywords: Fuzzy-Random analysis, Fuzzy-Random neural networks and operators, Fuzzy-Random modulus of continuity, Fuzzy-Random functions, Jackson type and probabilist-ic inequalities.

On Fuzzy -compact Spaces and Fuzzy -closed Spaces

Pradip Kumar Gain 

Department of Mathematics, Kharagpur College, India, Kharagpur, Paschim Medinipur-721305, West Bengal, India.

E-mail: pradip_gain@rediffmail.com

Ramkrishna Prasad Chakraborty

Department of Mathematics, Hijli College, Hijli, Kharagpur, Paschim Medinipur-721301, West Bengal, India.

E-mail: ramda2004@rediffmail.com

Madhumangal Pal

Department of Applied Mathematics with Oceqnology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India. E-mail: mmpalvu@gmial.com

Abstract: This paper deals with the concept of fuzzy -compactness in the gen-eralized  setting of a fuzzy topological space .  The concept of fuzzy b-compact topological  space was introduced by G. Balasubramanian in 1997.  Fuzzy-compactness is shown  to  be a generalization of the concept of fuzzy b-compactness.Fuzzy filter_bas (ffb) is  used  to characterize the said concept to some extent.  Concept of   (finite intersection  property z  modulo ) is used to characterize and  study  -compactness  for fuzzy  topological spaces. Restriction of the notion of fuzzy  -compactness to the  arbitrary  fuzzy sets in  and invarianceof the property under suitable  maps are also taken into  consideration. The concept of fuzzy -closed spaced is also introduced.

Keywords: Fuzzy ideals, Fuzz -compact space, fuzzy -closed space, fuzzy -compact space, , fuzzy filter-base, fuzzy -closed space.

On Fuzzy Boundary

J. Mahanta  and P. K. Das

Department of Mathematics NERIST, Nirjuli, Arunachal Pradesh, 791109, India.

E-mial: jm_nerist@yahoo.in( J.Mahanta), pkd_ma@yahoo.com (P.K.Das)

Abstract: In this paper, we introduce a new type fuzzy boundary and study some re-lated set theoretic identities.  Further, this new type of fuzzy boundary is co-mpared with different existing fuzzy boundaries.

Keyword: Fuzzy boundary, Fuzzy topology.

-closure Operator in Fuzzy Setting

Anjana Bhattacharyya

Department of Mathematics, Victoria Institution (College) 78 B,A.P.C.Road Kolkata-700009, India.

E-mial: anjanabhattacharyya@hotmail.com

Abstract: In this paper a new type of fuzzy closure operator, viz. fuzzy b-closure operator has been introduced and studied in fuzzy topological space and found the mutual relationship among fuzzy -closure operator, fuzzy -closure operator [1], fuzzy closure operator [6], fuzzy -closure operator [4].  In Section 2, a new type of separation axiom has been introduced and studied and shown that under this separation axiom fuzzy -closure operator is an idempotent operator. In the last section, a new type fuzzy convergence viz., fuzzy -convergence of a fuzzy net has been introduced and using this concept, fuzzy -closure operator has been characterized.

Keywords: Fuzzy -closure operator, fuzzy -open set, fuzzy -closed set, fuzzy-regular space.

On Fuzzy Magnified Translation of A Lattice

S. H. Dhanani 

Department of Mathematics, K.I.T.'s College of Engineering, Kolhapur.

E-mial: sachindhanani@rediffmail.com

Y. S. Pawa

Department of Mathematics, Shivaji University, Kolhapur.

E-mail: pawar_y_s@yahoo.com

Abstract: In this paper we introduce the concept of fuzzy magnified translation of a bounded lattice and study some of its properties.

Keywords: Fuzzy magnified translation, Fuzzy ideal.

Some Common Fixed Point Theorems for Fuzzy Mappings

S. Muralisankar 

School of Mathematics, Madurai Kamarai University, Madurai-625021, India.

E-mail: muralimku@gmail.com

Abstract: In this paper, using the concept of supremum metric on the space of fuzzy sets we proved some common fixed point theorems for fuzzy mappings which generalized some well known results in the literature.

Keywords: Hausdorff metric, Complete metric space, -metric, Fuzzy mappings, Common fixed point.

A Study on Fuzzy -sets Versus Fuzzy Locally Closed Sets

Amudhambigai 

Assistant Professor of Mathematics Department of Mathematics Sri Sarada College for women, Salem-16 TamilNadu, India.

E-mial: rbamudha@yahoo.co.in

A. Hajira

Sri Sarada College for women, Salem-16 TamilNadu, India.

Abstract:In this paper the interrelations of fuzzy -sets with other types of fuzzy locally closed sets are discussed with the suitable counter examples. Also, the interrelations of fuzzy -continuous functions with other types of fuzzy locally continuous functions are established with the required counter examples. Finally the properties and the characterizations of fuzzy separation axioms are established.

Keywords: Fuzzy locally semi-closed sets, fuzzy locally -closed sets, fuzzy local-ly -continuous functions, fuzzy locally -continuous functions, fuzzy locally semi-ultra-normal spaces and fuzzy completely locally semi-ultra-normal spaces.

The Restriction of -fuzzifying Matroid

Shou-Bin Sun , Ling-qiang Li and Guang-wu Meng

School of Mathematics, Liaocheng University, Liaocheng 252059, PR China.

E-mial: shoubinsun@126.com; lilingqiang@126.com

Abstract: In this paper, the concept of restriction of -fuzzifying matroid is introduced. Its basic properties and relationship to -fuzzifying rank function, and -fuzzifying  weak mapping are studied in detail.

Keywords: Matroid; -fuzzifying matroid; -fuzzifying rank function;M-fuzzifying weak mapping; Restriction; Direct sum.

A Note of Retracted A Paper Which Is Second Appearance in The Jounal of Fuzzy Mathematics, Vol.21, No. 2 (2013).

The Journal of Fuzzy Mathematics

E-mail: internationalfuzzymath@hotmail.com