THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 15, Number 4, December 2009

CONTENT

DETAILS

Folding of Chaotic Minkowski Space

M. El Choul

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt.

S. I. Nada Mathematics Department, Faculty of Science, Minoufya University, Shebeen El-Kohom, Egypt.

S. I. Abo EI Fotooh

Mathematics Department, Faculty of Science, Al-Azhar University for girls, Gairo, Egypt.

Abstract: In this paper we will introduce a new type of chaotic Minkowski space, it is called Dynamical Chaotic Minkowski space. The folding of chaotic Minkowski space into itself is defined. The relations between folding of chaotic Minkowski space and Riemannian manifold are discussed. Theorems governing these relations are achieved.

Keywords: chaotic manifolds, Minkowski space.

A Note on Common Fixed Points of Four Mapping in A Fuzzy Metric Space

D.Gopal

Dept. of Mathematics, S. V. National Institute of Technology, Surat, Gujrat, India

Urmila Mishra

Dept. of Appled Mathematics, Lakhmi Chand Institute of Technology, Bodari, Bilaspur, (C. G). India

E-mail:mishra.urmila@gmail.com A. S.

Ranadive Dept. of Pure and Appled Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, (C. G). India

Abstract: In this note, we remark that the Theorem 2.1 given by Balasubramainiam et al[1]appears to be wrong. This is so as the contractive condition involved in this theorem is self-contradictory at fixed point. However, we can get rid of this problem by altering the contractive condition of the proposed Theorem 2.1 of Balasubramainiam et al[1].

Keywords: Fuzzy metric space, R-weakly commuting map, compatible map, common fixed point.

A Note on Almost Compactness in terms of -Sets and -Sets

Anjana Bhattacharryya and M. N. Mukherjee

Department of Pure Mathematics University Calcutta

35, Ballygunge Circular Road Kolkata-700019 INDIA

Abstract: The concept of compact fuzzy topological spaces was introduced by Gantner er al.[5] while that of almost compact fuzzy topological space was initiated in [11]. The purpose of this article is to characterize almost compact fuzzy topological spaces in terms of two types of fuzzy sets viz. fuzzy -Sets and -Sets, and certain allied concepts.

Keywords: almost compact space, -Sets, -Sets, -adherent point.

The Category FTOPB and The Fuzzy Function Space

A. I. Aggour

Department of Mathematics, Faculty of Scienc Al-Azhar University, Nasr City (11884), Cairo, Egypt.

H. Z. Al-Zumi

Department of Mathematics, College of Education Girls Colleges, B. O. (55002), Jeddah, Saudi Arabia.

Abstract: The aim of this paper is to define the category FTOPB and we introduce more of its interesting properties. Also, we define the fuzzy function space Mf(XY) and we study some of its basic properties. We gave the sufficient conditions for the fuzzy evaluation map to be fuzzy continuous and the fuzzy exponential map to be fuzzy homeomorphism.

Keywords: Fuzzy fibred product, Fuzzy evaluation map, Fuzzy exponential map, admissible map, -product.

The Fuzy Topological Category over B

A. I. Aggour

Department of Mathematics, Faculty of Scienc Al-Azhar University, Nasr City (11884), Cairo, Egypt.

H. Z. Al-Zumi

Department of Mathematics, College of Education Girls Colleges, B. O. (55002), Jeddah, Saudi Arabia.

Abstract: In this paper, we define for two objects: in FTOPB, the category over B. We define the -topology on to get fuzzy topological cateory over B, and we study some of its basic properties. We gave the sufficient conditions for the fuzzy evaluation map to be fuzzy continuous. Also, we prove that the first exponential map and the second exponential map are fuzzy homeomorphism, under some sufficient conditions.

Keywords: Fuzzy fibred product, -admissible map, -topology on , the first exponential map, the second exponential map, -product.

The Fuzy Topological Category over B

A. I. Aggour

Department of Mathematics, Faculty of Scienc Al-Azhar University, Nasr City (11884), Cairo, Egypt.

H. Z. Al-Zumi

Department of Mathematics, College of Education Girls Colleges, B. O. (55002), Jeddah, Saudi Arabia.

Abstract: In this paper, we define for two objects: in FTOPB, the category over B. We define the -topology on to get fuzzy topological category over B, under some sufficient conditions. In fact, one of the important results in this paper is that under some sufficient conditions we have that the fuzzy evaluation map is fuzzy continuous.

Keywords: Fuzzy fibred product, -admissible map, -topology on , Fuzzy final topology, -product.

Some Aspects of Pairwise Fuzzy Preopen Sets in Fuzzy Bi-Topological Spaces

N. R. Das

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

Jonali Sharma

Laban Assamese Girls¡¯Secondary school, Laban Shillong 793004, Meghalaya, India

e-mail:jonalisharma2007@rediffmail.com

Abstract: In this paper, considering a bitopological space, the concepts of pairwise fuzzy preopen sets, pairwise fuzzy pre-T1 space, pairwise fuzzy pre-T2 space and pairwise fuzzy pre-continuity are introduced. Using different conditions on two fuzzy topologies and their mix fuzzy topology, some results are established on pairwise separation axioms and continuity.

Keywords: fuzzy pre-T1 space, pairwise fuzzy pre-T2 space, pairwise fuzzy preopen sets, pairwise fuzzy pre-T1 space, pairwise fuzzy pre-T2 space, pairwise fuzzy pre-continuity, fuzzy regular, fuzzy bitopological space, mixed fuzzy topology, quasi-coincidence, quasi-neighbourhood.

Application of Fuzzy Programming to Chance Constraints Multiobjective Solid Transportantion Problems

A. K. Bit

Department of Mathematics, Faculty of Civil Engineering, College of Military, Engineering, Pune-411031 (M. S.), India

E-mail: amalkbit@yahoo.com

Abstract: A method for solving a chance constrained multiobjective solid (there-dimensional) transportation problem has been presented. After converting the proposed chance constrained model to a deterministic model, fuzzy programming technique is used to find methodology has been presented. In some cases assuming the demand as log-normal or uniform random variable, methodology also has been presented. The method leads to efficient solutions as well as the best compromise solution. For this method software has been developed on Cyber 180/840A System. An example is included to illustrate the methodology.

Keywords: Multi criteria decision making, solid transportant problem, fuzzy programming, efficient solution, comprmise solution, membership function.

On The Product of Intuitionistic Fuzzy Subframes

Rajesh K. Thumbakara

Department of Mathematics, Mar Athanasius College, Kothamangalam-686666 Kerala, India

E-mail: rthumbakara@yahoo.com

Abstract: Frame theory is the study of topology based on its open set lattice and it was studied extensively by various authors. In this paper we introduce products of Intuitionistic fuzzy subframes and obtain some related results.

Keywords: Intuitionistic fuzzy subframes, Frame homomorphism, Morphism, Equalizers, Complete

Discussion for Fuzzy Automaton and Its Language

Qinge Wu, Tuo Wang, Yongxuan Huang and Jisheng Li

School of electronic and Information Engineering Xi¡¯an Jiao Tong University, Xi¡¯an, shaanxi, 710049, P. R. China

E-mail: wqe969699@163.com

Abstract: For processing and recognizing better different fuzzy automatic language, it requires an equal device. So, this paper presents the different fuzzy automata, and describes the relationship of the different fuzzy automata and fuzzy language. However, with the brand-new theory methods, the fuzzy Turing machine and fuzzy pushdown automation and their corresponding fuzzy grammar that generate the appropriate fuzzy language are mainly discussed. These researched will not only impulse further the development of the theories of the fuzzy automata hierarchy, but also can enhance the theoretical researches on the state fusion of the fuzzy automata. Thus, the development of theories in this paper in turn can quicken the applications of the fuzzy automata in various fields. It will also be a theoretic base for extraction and application of the fuzzy automata. Finally, some problems on the fuzzy automata and their language to be solved and development trends are discussed.

Keywords: Regular fuzzy language; content-free fuzzy language; non-restrictive fuzzy language; finite-state deterministic fuzzy automaton; fuzzy pushdown automaton; fuzzy Turing machine.

Intuitionistic Fuzzy Interior Ideals in Ordered Semigroups R.Ezhilarasi

Department of Mathematics, Annamalai University Annamalainagar-608002, Tamil Nadu, India.

E-mail: rearasi@gmail.com

S. Sriram

Mathematics Section, Faculty of Engineering and Technology, Annamalai University, Annamalainagar-608002, Tamil Nadu, India.

E-mail: ssm_3096@yahoo.co.in

Abstract: In this paper, we introduce the notion of intuitionistic fuzzy interior ideals in ordered semigroups. We give conditions for an ordered semigroup to be an intra-regular in terms of intuitionistic fuzzy set. Also we obtain the characterizations of intuitionistic fuzzy interior ideals in ordered semigroups. By using the collection of interior ideal and with some additional conditions, we construct an intuitionistic fuzzy ideal and we obtain some related properties of it.

Keywords: Ordered semigroup, interior ideal, fuzzy interior ideal, ntuitionistic fuzzy interior ideals, homomorphism.

A Note on Compactness in Intuitionistic Fuzzy Topological Spaces

M. N. Mukherjee

Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road,

Kolkata-700019, INDIA

Sumita Das

Department of Pure Mathematics, Sammilani Mahavidyalaya E. M. Bypass, Kolkata-700075, INDIA

Abstract: In this paper we develop the concept of Compactness in Intuitionistic fuzzy topological spaces. We characterize this fuzzy topological idea by sheer use of ordinary nets and power-set filterbases on the ambient set.

Keywords: Intuitionistic fuzzy set, Intuitionistic fuzzy topological spaces, Intuitionistic fuzzy compact space.

Some Paranormed Type Difference Sequence Spaces of Fuzzy Numbers and Statistically Convergent Sequences Defined by Orlicz Function

Binod Chandra Tripathy

Mathematical Sciences Division; Institute of Advanced Study in Science and Technology Paschim Boragaon; Garchuk; Guwahati-781035; INDIA.

E-mail: tripathybc@yahoo.com;

tripathybc@rediffmail.com Achyutananda Baruah

Department of Pure Mathematics; North Gauhati College Nagar; Guwahati-781031; INDIA.

E-mail: achuta_math@rediffmail.com.

Abstract: In this paper we introduce some new paranorm type difference sequence spaces of fuzzy real numbers defined by Orlicz function and statistical convergence spaces of fuzzy real numbers. We study their algebraic and topological properties.

Keywords: Orlicz function, Paranorm, Completeness, Solidness, Symmetricity Convergence free and Sequence algebra.

Interval-Valued Fuzzy Structures of K-Algebras

M. Akram

Punjab University College of Information Technology, University of the Punjab Old Campus, P. O. B OX 54000, Lahore, Pakistan.

E-mail address: m.akram@pucit.edu.pk

Karamat H. Dar

Department of Mathematics, G. C. University Lahore Katchery Road, Lahore-54000, Pakistan.

E-mail address: prof_khdar@yahoo.com

Abstract: The notion of interval-valued fuzzy sets was first introduced by Zadh in 1975 as a generalization of fuzzy sets. In this paper we apply the concept of interval-valued fuzzy sets to K-algebras. We introduce the notions of interval- valued fuzzy subalgebras and interval-valued fuzzy ideals in K-algebras and investigate some their properties. Using interval-valued fuzzy ideals, characterizations of Noetherian K-algebras are established. Some properties of interval-valued fuzzy relations are also discussed.

Keywords: K-algebras; interval-valued fuzzy sets; Noetherian K-algebras; Fuzzy relation.

Generalized Cartesian Product of Intuitionistic Fuzzy n-Inner Product Spaces

S.Vijayabalaji

Lecturer in Mathematics, Anna University, Tiruchirappalli, Panruti Campus Cuddalore District 607308, Tamilnadu, India.

E-mail: balaji_nandini@rediffmail.com

N. Thillaigovindan

Department of Mathematics, Annamalai University, Annamalainagar-608002, Tamilnadu, India.

E-mail: thillai_n@sify.com

Abstract: The aim of this paper is to present a new and interesting notion of generalized Cartesian product of intuitionistic fuzzy n-inner product spaces and to establish some results on it.

Keywords: n-inner product, fuzzy n-inner product, intuitionistic fuzzy n-inner product.

Fixed Points in Intuitionistic Generalized Fuzzy Metric Spaces

Renu Chugh

Department of Mathematics, M. D. University, Rohtak-124001, INDIA

E-mail: chugrenu@yahoo.com

Seema Mehra

Department of Mathematics, M. D. University, Rohtak-124001, INDIA

E-mail: smehra2007@yahoo.co.in

Abstract: Using the idea of intuitionistic fuzzy sets, due to Atananssov[5], Alaca, Turkoglu and Yildiz [4] defined the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek [7] and proved the intuitionistic fuzzy version of Jungck¡¯s common fixed point theorem. The aim of this paper is to introduce the concept of intuitionistic t-conorms as a generalization of generalized fuzzy metric spaces and fuzzy metric spaces due to Chugh and Kumar [8] and Kramosil and Michlek [7]. Also we prove Banach contraction principle in this newly defined space.

Keywords: Generalized fuzzy metric spaces, intuitionistic fuzzy metric spaces and intuitionistic fuzzy generalized fuzzy metric spaces.

Pre-Separation Axioms in Ordered Fuzzy Topological Spaces

E. Roja, M. K. Uma

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

G. Balasubramanian

Department of Mathematics Periyar University Salem-636016 Tamil Nadu, India.

Abstract: Pre-separation axioms are introduced in ordered fuzzy topologicalspaces and some of their basic properties are investigated besides establishing an analogue of Urysohn¡¯s lemma.

Keywords: Fuzzy pre-neighbourhood, fuzzy pre-T1-ordered spaces and fuzzy pre-T2-ordered spaces.

Almost Lacunary Statistical and Strongly Almost Lacunary Convergence of Generalized Difference Sequences of Fuzzy Numbers

Hifsi Altinok

Department of Mathematics, Firat University, 23119, Elazig-TURKEY

E-mail: hifsialtinok@yahoo.com

Rifat Colak

Department of Mathematics, Firat University, 23119, Elazig-TURKEY

E-mail: rcolak@firat.edu.tr

Abstract: The purpose of this paper is to introduce the concepts of almost lacunary statistical convergence of generalized difference sequences of fuzzy numbers. We give some relations related to these concepts.

Keywords: Fuzzy numbers, statistical convergence, lacunary statistical, difference sequences, Orlicz function.

Implicative Interior Operators

Yong Chan Kim

Department of Mathematics, Kangnung National University, Gangnenug, 201-702, Korea

Abstract: In this paper, we define the notion of implicative open systems and implicative interior spaces on stsc-quantale L. We study their properties.

Keywords: stsc-quantales, implicative interior spaces, implicative open systems

A Study on Category of mx Induced Fuzzy Minimal Structure

Sharmistha Bhattacharya (Halder)

Department of Mathematics, Tripura University, Tripura-799130, India

E-mail: halder_731@rediffinail.com

Abstract: The aim of this paper is to introduce the concept of mx-induced fuzzy minimal structure and the category formed by the fuzzy minimal structure. The corresponding functors are also studied in this paper and a relation between induced fuzzy topological spaces, induced fuzzy supra topological space and mx induced fuzzy minimal structure is created.

Keywords: Induced fuzzy topological space, supra topological space, minimal structure etc.

A Quadratic Programming Method for Fuzzy Ranking

Ji-Hui Yang

Staff Room of Preparetory Courses, Shenyang Artillery Academy Shenyang Liaoning, ZIP 110161, P. R. China

E-mail: yangjihui@163.com

Bing-Yuan Cao

School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, ZIP 510006, P. R. China

Summary: New method is presented for ranking the alternative based on 0.1-0.9 scale and fuzzy complementary judgement matrix about the scheme set provided by the decision maker. The structure of the fuzzy complementary judgement matrix and its meanings are explained. A quadratic programming model is proposed based on the fuzzy complementary judgement matrix and the model is solved Lingo software to obtain ranking values of the alternatives. A numberical example is given to illustrate the feasibility of the proposed method.

Keywords: Multiple attribute decision making; fuzzy complenentary judgement matrix; quadratic programming; ranking.

Application of Interactive Fuzzy Algorithm to Multiobjective Solid Transpor- tation Problems A. K. Bit Department of Mathematics, Faculty of Civil Engineering, College of Military Engineerin, Pune-411031 (M. S.), India

Abstract: The linear multiobjective solid (three-dimensional) transportation problem in while the supply, demand and capacity constraints are all equality type and the objectives are conflicting in nature. An interactive fuzzy algorithm for obtaining the best compromise solution of a multiobjective solid transportation problem has been presented in this paper. This algorithm has two phases such as computation and decision. The concept of fuzzy membership function is used where the solution procedure. The value of membership function of an objective represents the satisfaction level of the objective. A software has been developed based on simple additive algorithms. An example is included to illustrate the methodoloty.

Keywords: Multi criteria decision making, solid transportation problem, interactive algorithm, fuzzy programming, efficient solution, compromise solution.

Fuzzy Linear Programming with Fuzzy Variables and Its Application in Capacitated Transportation Model

Bablu Jana Tapan Kumar Roy

Department of Mathematics, Bengal Engineering and Science University, Shipur, Howrah, West-Bengal, Pin-711103, India.

E-mail: roybj_t_k@yahoo.co.in

Abstract: The purpose of this paper is to present the solution procedure of fuzzy liner programming with fuzzy variables. It is reduced to a crisp multi-objective linear programming problem. Them it has been solved by weighted sum method. Here, different weights are determined through game theory. It is also applied to fuzzy capacitated transportation problem. Numerical examples are presented in order to illustrate our method and application.

Keywords: Fuzzy variable, Symmetric fuzzy number, Game theory, Capacitated Transportation.