THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 10, Number 3, September 2002

CONTENT

DETAILS

Starcompactness and Fuzzy Function Spaces

J. K. Kohil

Department of Mathematics, Hindu College, University of Delhi Delhi-110007,India

A.R. Prasannan

Department of Mathematics, Maharaja Agrasen College, University of Delhi Delhi-110091,India

Abstract:A notion of compactness for an arbitrary fuzzy set, called stracomactness, in fuzzy topological spaces is introduced and its basic properties are studied, Analogues of Wallace theorem and Tychonoff theorem for stracompact fuzzy sets are obtained. A necessary and sufficient condition for a fuzzy set to be stracompact, n a function space relative to the pointwise fuzzy topology is obtained. Moreover, the notion of tracompact open fuzzy topology on a function space is introduced and its properties and interrelations with the pointwise fuzzy topology and the fuzzy topology of joint fuzzy continuity are investigated.

Keywords: Psudo closed fuzzy set, Psudo open fuzzy set, stracompact fuzzy set, pointwise fuzzy topology, N-compact open fuzzy topology, jointly fuzzy continuous fuzzy topology, jointly fuzzy continuous fuzzy topology on stracomacta.

Fuzzy Direct Sums and Free Submodules

V. Murali

Department of Mathematics Ipure and Applied, Rhodes University Grahamstown 6140, South Africa

B. B. Makamba

Department of Mathematics, University of Fort Hare Alice 5700, South Africa

Abstract: Sums of fuzzy submodules of -modules are discussed and a notion of uniqueness in fuzzy setting is used to characterize fuzzy direct sums. Various quotients of fuzzy modules are presented and two of them are shown to be equivalent. Fuzzy versions of isomorphism theorems in-modul ¨Ces are presented using only one of the versions of fuzzy quotient.The notion of a fuzzy submodul ¨Ce generated by a fuzzy subset plays a major role in this paper and elsezhere. We characterize this idea of fuzzy generation using products of fuzzy ideals with fuzzy subsets of fuzzy modules, and connecting it with sums of cyclic submodules.We also show that finitely generated fuzzy submodules behave like their crisp counterparts . We present two versions of freeness and then characterize them with fuzzy direct sums. We discuss various relationships between the crisp and the fuzzy notions of freeness.

Keywords: Fuzzy submodules, fuzzy point, quotient, finitely generated, support, strongly freely generated, freely generated, fuzzy direct sum.

Fuzzy Order Ideals and Torsion Submodules*

V. Murali

Department of Mathematics Ipure and Applied, Rhodes University Grahamstown 6140, South Africa

B. B. Makamba

Department of Mathematics, University of Fort Hare Alice 5700, South Africa

Abstract: This paper is a sequel to our paper on direct sums and free submodules[3]. The notions of torsion and torsion-free in fuzzy modules are presented here. We use the notions of torsion-free to characterize freely generated submodules and then establish some analogues of the crisp theoretic result of modules. We round off with discussions on order ideals and link their relation to freeness.

Keywords: Fuzzy submodule, fuzzy point, finitely generated, support, strong freely generated, freely generated,fuzzy direct sum, torsion, torsion-free, order ideal.

On Fuzzy Filters and Fuzzy Order Ideals

Min Xia Luo

Department of Mathematic£¬Yuncheng Advanced Training College Yuncheng, Shanxi,044000, P. R. China

Abstract: In tise paper, we introduce the notion of order ideals, fuzzy order ideals of the adjoint semigroup M(X) of BCK-algebra X and fuzzy filters of X, Some characterizations of fuzzy order ideals and fuzzy filters are discussed. Moreover, we show that there is a bijection from the set of all fuzzy filters of BCK-algebra X to the set of all fuzzy order ideals of the adjoint semigroup M(X).

Keywords: Fuzzy filter, order ideal, fuzzy order ideal.

The Application of Grey Prediction Method in Multi-order Nonlinear Control System

Shu Hong Zhang, Mian Yun Chen and Kai Qi Zou

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074,China E-mail:shzhanghust@yahoo.com

Abstract: The system with partial unknown structure, parameters, and characteristics is called a grey system. The grey system theory can be employed to improve the control performance of a system without sufficient information or with highly nonlinear property. In this paper, grey system theory is introduced into sliding mode control of high order nonlinear system and a novel grey fuzzy sliding mode controller is developed. In this controller design grey prediction model combined with fuzzy resoning is proposed to obtain sliding mode control strategy . The controller can drive the state from an arbitray initial point of state to the origin along a sliding mode hyper plane quickly an reduce the vibration phenomenon that is caused by switch delay brought by system inertial. Grey system theory is efficient tool in the analysis an control of nonlinear, uncertain systems.

Keywords: Grey system theory, grey prediction method, grey fuzzy sliding mode controller, fuzzy reasoning.

Fuzzy 2-Inner Product Spaces

AR.Meenakshi and R. Cokilavany

Department of Mathematics, Annamalai University Annmalainager-608 002, India

Abstract: In this paper, the notion of Fuzzy 2-inner product spaces is introduced. Standard results in inner product spaces are extended to fuzzy 2- inner product spaces, The well known Riesz respresentation theorem is proved in the realm of fuzzy 2-inner product spaces.

Keywords: Fuzzy 2-norm, fuzzy pseudo-norm, fuzzy inner product, 2-functional, uniformly convex, strictly convex.

Fuzzy Topological Convergence for Sequence in Bounded Posets and Related Structures

Debasis Sarkar

Department of Applied Mathematics, University of Calcutta, 92 A. P. C. Road Calcutta-700009, West Bengal, India

Abstract: The notion of fuzzy topological convergence and weak fuzzy topological convergence for sequences in bounded posets are introduced here, We have defined some new fuzzy topologies on bounded posets and also on orthocomplemented bounded posets. Aconvergence theory is developed on bounded posets as well as orthocomplemented bounded posets and quantum logic structures through the fuzzy topological ideals.

Keywords: Fuzzy topology, fuzzy topological convergence, weak fuzzy topological convergence,bounded posets quantum logic .

Unification of Some Functions between Fuzzy Topological Spaces

N. Mukherjee

Department of Pure Mathematics, University of Calcutta 35, Ballygunge CircularRoad Calcutta-700019, India

R. P. Chakraborty1

Department of Mathematics, Hijli College, Hijli Co-operative Society, Kharagpur-721306, India

Abstract: In this paper, we present a unified study of different types of functions including the continuous ones, that have already been investigated as individual enitites, between fuzzy topological spaces. Starting with a very generalized premise, we have established some results. We observe that the definitions of almost all the existing types of functions between fuzzy topological spaces and many results concerning them follow as particular cases from our corresponding unified versions.

Keywords: Fuzzy operator system, fuzzy topological space, fuzzy continuity.

Fuzzily Compact Space

P. V. Ramakrishnan and V. Lakshmana Gomathi Nnyagam*

School of Mathematics M. K. University, Madurai-625021, Tamil Nadu, India

Abstract: The notion of fuzzy L-fuzzy was introduced in [2] and the notion of fuzzy compactness was studied in [2] and the notion of fuzzy compactness was studied in [2], [5], [6], etal. In this paper a new notion of fuzzily compactness is introduced and some results analogous to the results in crisp topological spaces are discussed in fuzzy setup.

Keywords: Fuzzily compactness, fuzzily closed, nearly fuzzy continuous, fuzzy points, fuzzy singletons.

Lattice-valued Zp-Pan-integrals ¢ó: Relationships of Pan-integral with Fuzzy Integral, N-integral Labesgue Integral

Guang Quan Zhang1 and Yong Hong Wu2

1Faculty of Information Technology, University of Technology, Sydey P. O. Box 123, Broadway, NSW2007, Australia E-mail: Zhangg@it.uts.edu.au 2School of Mathematics and Statistics, Curtin University of Technology GPO Box U1987, Perth, Western Australia,6845, Australia E-mail: yhwu@cs.curtin.edu.au

Abstract: Based on the study of the Zp-Pan-integrals for lattice-valued functions, we here study the correlation of the integral with various other types of integrals. A number of theorems describing the relationships of the Zp-Pan-integral with the fuzzy integral, the N-integral and the Labesgue integral are developed and presented is this paper.

Keywords: Non-additive measure, fuzzy integral, pan-integral , Labesgue integral, N-integral.

L-fuzzy Theory in The Case L=P(I)

Nasser Boroojerdian

Department of Mathematics, Amikabir University Tehran, Iran broojerd@aut.ac.ir

Abstract: We investigate and devlop L-fuzzy theory in the case L=P(I), ,when I is a nonempty set. We show that this fuzzy theory is a natural extension of ordinary mathematics and extension of ordinary concepts are natural and unique. So, this fuzzy theory is parallel to and is the same as ordinary mathematics. We also show that, by considering a measure on I, L-fuzzy theory ( L=P(I)) can be reduced to ordinary fuzzy theory (L=[0,1]). This reduction is the main like between this fuzzy theory and ordinary fuzzy theory. This fuzzy theory has a natural interpretation which makes fuzziness meaningful and understandable. In final section of the paper we show that techniques of fuzzy control are applicable in the same way in this fuzzy but the results may be somewhat different and better.

Keywords: L-fuzzy proposition, fuzzy sets, fuzzy points, fuzzy relations, measure.

Characterizations of NC-continuous Order-homomorphisms*

Shui-Li Chen

Department of Mathematics,Teacher¡¯s College,Jimei University Xiamen, Fujian 361021,P. R.China

Abstract: In this paper, the concepts of NC-continuity of order-homomorphisms, NC-convergence of molecular nets and fuzzy ideals, and NC-closue operator on L-fuzzy topological spaces are introduced by means of the N-compactness. The characterizations of NC-continuous order-homomorphisms and the relationships between these notions are systematically discussed. Some interesting results are obtained.

Keywords: L-fuzzy topological space, ordered-homomorphism, N¡ªcompactness, NC-continuity, molecular net, fuzzy ideal.

Remarks on Vague Groups

\ Mustafa Demirci and Dogan Coker

Akdenzi University, Department of Mathematics 07058-Antalya/TURKEY E-mail: demirci@pascal.sci.akdeniz.edu.tr, coker@ pascal.sci.akdeniz.edu.tr

Abstract: The purpose of this paper is give some characterizations and examples concerning vague groups developed by Demirci[4].

Keywords: Fuzzy equality, fuzzy function, vague binary operation, vague semigroup, vague monoid, vague group, vague subgroup, vague homomorphism.

A New Characterizations of L-fuzzy Locally Convex Topological Vector Spaces And Its Applications

Cong Hua Yan

Department of Mathematics,Nanjing Normal University, Nanjing Jiang su,210097, China

Abstract: The purpose of this paper is to give a new characterization of L-fuzzy locally convex topological vector spaces [5], and to prove that each L-fuzzy locally convex topological vector spaces can be determined by a family of L-fuzzy psendo-norms. As an application, we discuss some interesting properties of L-fuzzy locally convex topological vector spaces.

Keywords: L-fuzzy Minkowshi function, L-fuzzy convex prebase, L-fuzzy psendo-norm, L-fuzzy locally convex topological vector spaces.

Extension of Fuzzy Zorn¡¯s Lemma to Nontransitive Fuzzy Relations

Smat Beg

Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, PAKISTAN

Abstract: We proved existence of maximal element for a nontransitive fuzzy relation defined on a nonempty set. A necessary and sufficient condition has been established to completely characterize the existence of maximal element for general irreflexive nontransitive fuzzy relations.

Keywords: Fuzzy relation, fuzzy order,fuzzy Zorn¡¯s Lemma, maximal element, fuzzy chain dominant property.

On Fuzzy Submetrizability

R. Sreekumar

Department of Mathematics,Cochin University of science and Technology Cochin 682022, Kerala, India E- mail: sreekumarr@cusat.ac.in

Abstract: Submetrizability introduced by Grey Gruenhage, in hand book of set theoretic topology, edited by K. Kunen and J. E. Vaughan, Elsevier science [pp 423-500] 1984, is less restrictive but retains some of the nice properties of metrizability. In this paper we introduce the fuzzy analogue of submetrizability, study some of its properties, including its relations to fuzzy paracompactness.

Keywords: Fuzzy topology, fuzzy paracompactness, fuzzy submetrizability.

Fuzzy Pariwise -continuous Mappings on Fuzzy Bitopological Spaces

Young Bin Im, Eun Pyo Lee and Se Won Park\

Department of Mathematics,Seonam University, Chonbuk 590-711, Korea

Abstract: In this paper, we define the concept of --open (--closed) set on fuzzy bitopological spaces and study ome of their properties.And we introduce the--interior (--closures) and discuss the characteristic properties of fuzzy pairwise -continuous (fuzzy pairwise -open, fuzzy pairwise -closed) mapping on fuzzy bitopological spaces.

Keywords: -fuzzy -open set, -fuzzy -closed set, --interior, --closure, fuzzy pairwise -continuous mapping, fuzzy pairwise -open mapping, fuzzy pairwise -closed mapping.

On the Relationship Between Intuitionsitic Fuzzy Sets and Some Other Extensions of Fuzzy Set Theory

Glad Deschrijver and Etienne E. Kerre

Department of Applied Mathematics and Comput Science Ghent University, Krijgslaan 281(S9),B-9000 Gent ,Belgium E-mail: Glad.Deschrijver@rug.ac.be and Etienne. Kerre@rug.ac.be

Abstract: In a previous paper [6] we presented the relationship between intuitionistic fuzzy sets (Atanassov [1]) and some other extensions of fuzzy set theory, such as L-fuzzy sets (Goguen[9]), interval valued fuzzy sets (Sambuc [15]),interval valued intuitionistic fuzzy sets Atanassov [2]). In this paper we will continue this work and investigate the interrelationship between intuitionistic fuzzy sets, rough sets (Pawlak [13]), fuzzy rough sets (Dubois and Prade [7], Nanda and Majumdar [12]), rough fuzzy sets (Dubois and Prade [7]), probabilistic sets (Hirota [1]), L-fuzzy sets, soft sets (Basu,Deb and Pattanaik [3]) and type 2 fuzzy sets (Zadeh [17]).

Keywords: Links between different models, intuitionistic fuzzy set, L-fuzzy set, interval valued fuzzy set, probabilistic set, soft set, rough set, fuzzy rough set, rough fuzzy set, type 2 fuzzy set.

Fubini Theorem of Fuzzy-valued Integral of Fuzzy-valued Functions with Respect to Fuzzy-valued Measure

Ai Bing Ji

Hebei Medical College for Continuing Education Baoding 071000, P. R. China

Abstract: In this paper, we construct the product interval-valued measure and product fuzzy-valued measure on Cartesian product space , and prove the Fubini Theorem of fuzzy -valued integrals with respect to interval-valued measure and fuzzy-valued measure.

Keywords: Fuzzy-valued measure, fuzzy ¨Cvalued integral, product fuzzy-valued measure, Fubini theorem.

Derived Operator in L-fuzzy Topological Spaces

Jin Ming Fang

Department of Mathematics£¬Ocean University of QingDao, QingDao 266003,P. R. China

Abstract: For a crisp topological space and a completely distributive lattice L, we give an useful way to concertely construt the derived operator.When an L-fuzzy topological space is induced from the background space , the derived operator on is exactly produced from the background space in this way. In addition, we prove some of characterization theorems for full stratified, weekly induced and induced spaces by derived operators. These theorems are every different from the result in the published papers.

Keywords: L-fuzzy topology, N-cluster point, derived operator, induced space, background space.

The Cardinality and Representation of the Fuzzy Powergroups

Cheng Yi Zhang

Department of Math, Hainan Teachers¡¯ College Haikou, 571158, P. R. China

Ping An Dang

Department of Mathematics, Zhumadian Teachers¡¯ College, Henan, 463000, P. R. China

Abstract: Inthis paper, the cardinality of the element in a fuzzy powergroup is discussed. It is pointe that this kind of fuzzy powergroup is not the Zadehs¡¯ fuzzy subgroup, and it can be represented by a crisp group. For example, a regular representation of a fuzzy powergroup on a finite group is given. Furthmore, the homomorphic properties of the fuzzy powergroups are discussed.

Keywords: Powergroup, fuzzy powergroup, group action, regular representation, homomorphism.

Rate of Convergence of Fuzzy Netural Network Operators, Univariate Case

George A. Anastassiou

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

Abstract: In this article we study the rate of convergence to the unit operator of very specific well described univariate fuzzy neural network operators of Cardaliguet-Euvrard and ¡°Squashing ¡±types. These rates are arise in a very natural and common way among fuzzy neural networks. These rates are given through Jackson type inequalities involving the fuzzy sense. Also several interesting new results in fuzzy real analysis are presented to be used in the proofs of the main results.

Keywords: fuzzy real analysis, fuzzy neural networks and operators, fuzzy modulus of continuity, fuzzy real number valued functions, Jackson type inequalities.