The Journal of Fuzzy Mathematics
Volume 15, Number 2, June 2007
CONTENT
DETAILS
Integration with Respect to Countably Additive Fuzzy Measure
Girija Kumari R.
Department of Mathematics N. S. S. Hindu College
Changanacherry 686 102 Kerala, India
Abstract: In our paper [4], [5] we have defined fuzzy total variation using additive fuzzy measure, fuzzy simple functions, fuzzy ¦Ì-null functions, fuzzy integrability, fuzzy integrable functions. In this paper, we define ¦Ì is a real valued or complex valued or Banach valued countably additive fuzzy measure on a fuzzy set C. Then we prove fuzzy total variation is countably additive. Also we have obtained analogous forms of Egoroff¡¯s theorem, Fatou¡¯s lemma and dominated convergence theorem for the fuzzy integrable functions.
A Comparison between Intuitionistic Fuzzy Sets and Generalized Intuitionistic Fuzzy Sets
M. Panigrahi and S. Nanda
Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, West Bengal, INDIA
E-mail: motilal@maths.iitkgp.ernet.in, snanda@ maths.iitkgp.ernet.in
Abstract: In this paper we have shown that some operations that are valid for intuitionistic fuzzy sets are not valid for generalized intuitionistic fuzzy sets. Also we have found that some results which appeared in the paper by Krassimir Atanassov are not correct and we have presented the correct version of the results. We have further introduced some new operations over generalized intuitionistic fuzzy sets.
Keywords: Fuzzy sets, intuitionistic fuzzy sets, generalized intuitionistic fuzzy sets, operations on GIFSs.
Fuzzy Multiple Attribute Decision Making Ranking Alternatives and Control
Jibin Lan
School of Economics and Management, Southwest Jiaotong University,
Chengdu, Sichuan, P. R. China, 610031.
School of Mathematics and Information Science, Guangxi University.
Nanning, Gaungxi, P. R. China, 530004
Yang Xu
School of Economics and Management, Southwest Jiaotong University,
Chengdu, Sichuan, P. R. China, 610031.
Jiazhong Liu
School of Economics and Management, Southwest Jiaotong University,
Chengdu, Sichuan, P. R. China, 610031.
Abstract: A conception of the left and right projection which one fuzzy vector projects on another is introduced in this paper. The purpose is to propose a new criterion called (FMADMPT) to rank alternatives and control ordering in fuzzy multi-attribute decision making environment. Using the conception of the left and right projection, each alternative is projected on the fuzzy weight vector, the size of the combination projection coefficient that combines the left projection coefficient with the right one is used as a judgement standard to measure each alternative. According to (FMADMPT) the control variable ¦Ë which combines the left projection coefficient with the right one is used to control ordering.
Keywords: Fuzzy weight vector, fuzzy vector projection, combination projection coefficient, control variable.
Separation Axioms in Characterized Fuzzy Spaces
A. S. Abd-Allah
Department of Mathematics, Faculty of Science,
El-Mansoura University, El-Mansoura, Egypt
Abstract: Basic results on fuzzy separation axioms in characterized fuzzy spaces related to the fixed L-fuzzy topology are presented which are obtained in applying the general theory of fuzzy filters and the operations defined on the set of all L-fuzzy subsets endowed with an L-fuzzy topology. In particular, we use the notion of ¦Õ1,2-fuzzy neighborhood filter at point of¦Õ1,2-fuzzy convergence to introduce and study the notions of characterized FTi-spaces and F¦Õ1,2-Ti spaces for i¡Ê{0,1,2}. These axioms are related only the usual points and the ordinary subsets. In the classical case of L={1,2}, ¦Õ1=int and ¦Õ2=iLx these axioms are the usual ones. Our axioms are generalized a lot of the weaker and stronger forms of the existing fuzzy separation axioms in fuzzy topological spaces. So many new special cases from these classes of fuzzy separation axioms are obtained by special choices for the operations ¦Õ1 and ¦Õ2 in Table (1).
Keywords: Fuzzy filters, fuzzy topologies, operations, ¦Õ1,2-open fuzzy sets, characterized fuzzy spaces, ¦Õ1,2-fuzzy neighborhood filters, valued and superior ¦Õ1,2-fuzzy neighborhoods, fuzzy filter pretopologies, initial and final characterized fuzzy spaces, ¦Á-level characterized spaces, initial characterized spaces, induced characterized fuzzy spaces, characterized fuzzy T0 and fuzzy ¦Õ1,2T0- spaces, characterized fuzzy T1 and fuzzy ¦Õ1,2T1- spaces, and characterized fuzzy T2 and fuzzy ¦Õ1,2T2- spaces.
Correlations and Information Energy on Interval-valued Fuzzy Sets
Nishi Kanta Jana and Madhumangal Pal
Department of Applied Mathematics with Oceanology and Computer Programming,
Vidyasagar University, Midnapore-721102, INDIA
Abstract: In this paper, we define correlation of interval-valued fuzzy sets and studies some results on it. Also we studies the concept of partial and multiple correlation coefficients of interval-valued fuzzy sets and some properties. Finally, the idea to measure the fuzzyness of interval-valued fuzzy sets by the concept of information energy is introduced.
Keywords: interval-valued fuzzy sets, correlation, partial and multiple correlations, information energy.
Semisimple L-modules and Split Exact Sequences of L-modules
Paul Isaac
Department of Mathematics, Bharata Mata College,
Thrikkakara Kochi-682021, Kerala, India
E-mail:pi@cusat.ac.in
Abstract: In this paper we recall the definitions of simple L-modules, semisimple L-modules, exact sequence of L-modules and short exact sequence of L-modules. We define split exact sequence of L-modules and prove some results which includes the theorem that all short exact sequences of L-modules split if and only if all L-modules are semisimple.
Keywords:
L-modules, Semisimple L-modules, Exact sequence of L-modules , Short exact sequence of L-modules, Split exact sequence of L-modules
Essential L-submodules of an L-module
Paul Isaac
Department of Mathematics, Bharata Mata College,
Thrikkakara Kochi-682021, Kerala, India
E-mail:pi@cusat.ac.in
Abstract: The concept of an essential submodule of a module and the like form an important area in module theory. In this paper we extend this concept to the fuzzy setting and prove that if L is a regular complete distributive lattice, for , ; is an essential L-submodule of if and only if for each ,with £¾0,there exits an such that and £¾0. Also we prove that if are such that , then and . Further we prove that if , where and then and moreover if , then and .
Keywords: L-module, L-submodule, essential L-submodule.
Fuzzy Quotient Gamma-rings
Y. B. Jun
Department of Mathematics Education
Gyeongsang National University Chinju 660-701, KOREA
E-mail: ybjun@gsnu.ac.kr
M. Uckun
Department of Mathematics, Faculty of Arts and Sciences
Inonu University 44069, Malatya-TURKEY
E-mail: muckun@inonu.edu.tr
M. A. Ozturk
Department of Mathematics, Faculty of Arts and Sciences
Cumhuriyet University 58140, Sivas-TURKEY
E-mail: maozturk@cumhuriyet.edu.tr
Abstract: Fuzzy ideals with sup property is considered. The notion of fuzzy quotient -rings induced by fuzzy ideals is introduced. -isomorphism theorem and -homomorphism theorem are established.
Keywords: Fuzzy ideal, fuzzy quotient -ring, sup property, canonical -homomorphism, fuzzy point, fuzzy kernel, fuzzy image.
On product of Fuzzy Sublattices
Tazid Ali and A. K. Ray
Department of Mathematics,
Dibrugarh University, Dibrugarh-4, Assam, India
E-mail: tazid2001@yahoo.com.cn
Abstract: In this paper the Cartesian product of fuzzy sublattices of different types of lattices has been discussed intensively and some interesting results obtained.
Keywords: Cartesian product , fuzzy sublattices
Maximal Ideals of L-subrings
Anand Swaroop Prajapati
Department of Mathematics, Atma Ram Sanatan Dharma College
University of Delhi, Dhaula Kuan, New Delhi-110021, India
Naseem Ajaml
Department of Mathematics, Zakir Hussain College
University of Delhi, J. L. N. Marg, New Delhi-110021, India
Abstract: In this paper we improve certain existing results in the theory of L-subrings. In particular, we present some characterizations of ideals of an L-subring. Moreover, we introduce the concept of maximal ideal in an L-ring .Our approach is similar to that of classical ring theory. We also extend a well known result for maximal ideal of classical ring theory. Then we provide two different necessary conditions for an ideal of an L-ring to be maximal. We show by examples that these conditions are not sufficient. A partial converse of one of these conditions is also established.
Keywords:
Lattice, L-point, level subset, strong level subset, L-subring, L-ideals, maximal ideal.
Maximal Ideals of L-subrings-II
Anand Swaroop Prajapati
Department of Mathematics, Atma Ram Sanatan Dharma College
University of Delhi, Dhaula Kuan, New Delhi-110021, India
Naseem Ajaml
Department of Mathematics, Zakir Hussain College
University of Delhi, J. L. N. Marg, New Delhi-110021, India
Abstract:
The concept of maximality of an ideal in an L-ring is introduced and discussed in out paper [6]. In this paper we establish some more sufficient conditions for the maximality of an ideal in an L-ring . Moreover, we divide the class of maximal ideals into two classes of trivial maximal ideals and non-trivial maximal ideals. Finally, we also discuss the relationship of the range set of a maximal ideal with that of the range set of the L-ring.
Keywords: Lattice, level subset, L-subring, L-ideal, maximal ideal.
Some Fixed Point Theorems for Multivalued Mappings in Fuzzy Menger space
S. Kutukcu, D. Turkoglu and C. Yildiz
Department of Mathematics, Faculty of Science and Arts
University of Gazi, 06500 Ankara, Turkey
E-mail address: servet@gazi.edu.tr
Abstract: In the present work we prove some fixed point theorems on orbitally complete fuzzy Menger space or complete fuzzy Menger space.
Keywords:
Multivalued maps, complete fuzzy Menger spaces, orbitally complete fuzzy Menger spaces.
On pointwise Statistical Convergence of Sequences of Fuzzy Mappings
Yavuz Altin
Department of Mathematics, Firat University, 23119, Elazig-Turkey
E-mail: yaltin23@yahoo.com
Mikail Et
Department of Mathematics, Firat University, 23119, Elazig-Turkey
E-mail: mikailet@yahoo.com
Binod Chandra Tripathy
Mathematical Science Division; Institute of Advanced Study in Science and Technology
PASCHIM BORAGAON, GARCHUK, GUWAHATI-781035; INDIA
E-mail: tripathybc@yahoo.com
Abstract: In this paper we introduce the notion of pointwise statistical convergence of sequences of fuzzy mapping. Furthermore we introduce the concept of statistically Cauchy sequence for fuzzy mapping sequences and prove that it is equivalent to pointwise statistical convergence of sequence of fuzzy mapping.
Keywords: Sequences of fuzzy numbers, Statistical convergence, Pointwise fuzzy convergent sequence.
D. Turkoglu, I. Altun
Department of Mathematics, Faculty of Science and Arts
University of Gazi. 06500-Teknikokullar, Ankara, TURKEY
E-mail: dturkoglu@gazi.edu.tr
E-mail: ialtun@gazi.edu.tr
Y. J. Cho
Department of Mathematics Education and the RINS, College of Education,
Gyeongsang National University, Chinju 660-701, KOREA
E-mail:yjcho@gsnu.ac.kr
Abstract: In this paper, we give some new definitions of compatible mappings in fuzzy metric spaces and we prove a common fixed point theorem for four mappings under the condition of compatible mappings of type (I) and of type (II) in complete fuzzy metric spaces. We get some improved versions of several fixed point theorems in complete fuzzy metric spaces.
Keywords: Fuzzy metric space, common fixed point, compatible mappings and compatible mappings of types (¦Á), (¦Â), (I) and (II).
A Fuzzy Extension of Generalized Vector Version of Minty¡¯s Lemma and Applications
Y. J. Cho
Department of Mathematics Education and the RINS,
Gyengsang National University, Chinju 660-701, KOREA
E-mail:yjcho@gsnu.ac.kr
Salahuddin and M. K. Ahmad
Department of Mathematics, Aligarh Muslim University, Aligarh-202 002(U. P.), India
E-mail: salahuddin12@mailcity.com; mk_ahmad@postmark.net
Abstract: In this paper, we consider a generalized vector version of Minty¡¯s Lemma which is studied by Lee et al. for fuzzy set-valued mappings and establish the existence of solutions to generalized some vector variational-type inequality problems for fuzzy set-valued mappings in topological vector spaces. An application for fuzzy mappings is also given.
Keywords: Fuzzy mappings, generalized vector version of Minty¡¯s Lemma, vector variational-type inequalities, -pseudomonotone, -pseudomonotone type, -hemi-continuous, KKM-theorem, closed set, Fan geometrical lemma.
One Direction S-rough Extension Communication and Its Heredity-variation Characterisstics
Shoufeng Yin, Haiqing Hu and Kaiquan Shi
School of Mathematics and Systems Science, Shandong University, Jinan, Shandong 250100, PRC
Abstract: On the basis of S-rough sets theory in [1], this paper extends rough communication model [10], presents one direction S-rough extension communication model; by using of one direction S-rough extension communication model, this paper presents concepts of one direction S-rough extension communication F-heredity, one direction S-rough extension communication model F-variation, and give research on their characteristics. Rough communication is a new research direction of rough sets theory, therefore, research on rough communication heredity and variation characteristics has important theoretical and application value.
Keywords: S-rough sets, one direction S-rough sets, rough communication, one direction S-rough extension communication.
One Flou (Intuitionistic) Topological Spaces
A. Kandil
Mathematics department, Faculty of Science, Helwan University, Egypt
E-mail: Ali_Kandial@hotmial.com
O. Tantawy
Mathematics department, Faculty of Science, Zagazig University, Egypt
E-mail: Osama_ltantawi@hotmial.com
M. Wafaie
Mathematics department, Faculty of Science, Helwan University, Egypt
E-mail: m_wafaie@hotmial.com
Abstract:
The aim of this paper is to introduce the concept of flou set and flou topological spaces investigating many of its properties. We also investigate some interrelations between the flou topological spaces and the ordinary topological spaces.
Keywords: Flou set, flou point, flou topology-intuitionistic set
On Operations in L-fuzzy Topological Spaces
M. E. El-Shafei
Mathematics
Department, Faculty of Science, Mansoura University, Mansoura, Egypt.
A. I. Aggour
Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt
Abstract: In this paper, we introduce the concept of -operation on the class of all -fuzzy sets on a universe endowed with an -fuzzy topology where ¡¯ is a completely complete distributive lattice. Then we apply this operation to generalize and unify several characterizations and properties of different kinds of -fuzzy compactness, -fuzzy paracompactness, -fuzzy openness and -fuzzy continuity.
Keywords: -fuzzy topological space, -open -fuzzy set, -interior, -closure, -compact spaces, -paracompact spaces and -continuous mappings. |