THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 16, Number 2, June 2008

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Fuzzy Inverse Semigroup Congruence on Semigroups

Pengge Zhang, Sanyang Liu and Shuping Gao

Department of Applied Mathematics, Xidian University, Xian 710071, China

Abstract: The aim of this paper is to study a class of fuzzy congruence on semigroup, that is, fuzzy inverse semigroup congruence, after introducing the concept of fuzzy weak semi-normal subsemigroup of semigroup, we show that there is an one-to-one correspondence between the fuzzy weak seminormal subsemigroup and the fuzzy inverse semigroup congruence on a semigroup.

Keywords: Fuzzy relation; Fuzzy inverse semigroup congruence; Fuzzy weak seminormal subsemigroup.

A Sufficiency and Necessity Conditions for Metrization of The CII-fuzzy Topological Spaces

Lan Yaoyao and Li Qinguo

Department of Applied Mathematics Hunan University, Changsha, 410082, P. R. China

Abstract: Two sorts of sufficiency conditions for metrization of the fuzzy topological space have been introduced in papers [2,6].In this paper, based on the concept of the ¦Ò-fuzzy discrete base, the necessity conditions for metrization of the fuzzy topological space is obtained. Moreover, one sufficiency condition for metrization of the fuzzy topological space is also proved by the Embedding Lemma [7]. Finally, some properties of the fuzzy unit interval FI are investigated.

Keywords: Fuzzy discrete, ¦Ò-fuzzy discrete, refinement, fuzzy subspace, fuzzy unit interval FI.

Order in Probability of Fuzzy Random Variables

J. Earnest Lazarus Piriyakumar and R. Helen

Department of Mathematics, Tranquebar Bishop Manickam Lutheran College, Porayar-Tami Nadu. S. INDIA

Abstract: Stochastic ordering is highly applicable in Reliability, Decision making and Life Testing Problems .In this paper we introduce a new notion of order in probability such as at most of order and derive their properties.

Keywords: Fuzzy Random Variables, Fuzzy Numbers, Fuzzy Probability and Stochastic ordering.

Generalized Triangular Norms

Bablu Samanta

Department of Engineering Science Haldia Institute of Technology, Haldia, Midnapore (East)-721657, West Bengal, India. E-mail: bablus@rediffmail.com

Tapan Kumer Roy

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India. E-mail: roy_t _k@yahoo.co.in

Abstract: Here we define four well-known extended n-ary t-norms from binary t-norms and then incorporate the weights into the construction of generalized t-norm operators. Next, we discussed some properties of generalized t-norm and then represent some particular family of t-norms (i.e. Hamacher, Frank, Yager, Dombi, Dubois-Prade etc.) in generalized form and calculate their limiting cases. Finally we described fuzzy decision making method based on generalized t-norms operator.

Keywords: Fuzzy set, Extended t-norm, Optimization.

Intuitionistic Fuzzy Prime Ideals of Semirings(I)

T. K. Dutta

Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Kolkata-700019, India. E-mail: duttatapankumar@yahoo.co.in

A. Ghosh

Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Kolkata-700019, India. E-mail: anuradhaghosh2006@yahoo.com

Abstract: In this paper we have introduced the notions of intuitionstic fuzzy ideal, intuitionstic fuzzy prime ideal in a semiring and studied them. We have obtained a number of characterizations of intuitionistic fuzzy prime ideal of a semiring.

Keywords: Intuitionstic fuzzy set, Intuitionstic fuzzy ideal,Intuitionstic fuzzy prime ideal, Intuitionstic fuzzy k-ideal, Intuitionstic fuzzy prime k-ideal.

Time-varying Failure Rates in The Reliability Analysis of Parallel Systems

M. A. El-Damcese

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

M. M. Khalifa

Department of Mathematics, Faculty of Education, Tanta University Kafr-El-Sheikh Branch, Tanta, Egypt

Abstract: This paper discuss the two repairable Markov models to perform reliability analysis of two unit parallel system with time dependent hardware failure rate and constant human errors, common-cause failure rates.

Keywords: Markov models, time dependent hardware failure rate, reliability.

Lattice of Weakly Induced L-topologies

V. K. Jose

Department of Mathematics, St. Thomas College Pala Arunapuram P. O. 686 574, Kottayam (Dt), Kerala, India E-mail: vkjstc@rediffmail.com

T. P. Johnson

School of Engineering, Cochin University of Science and Technology, Kochi-682022, Kerala, India

Abstract: We study the lattice structure of the set of all weakly induced L-topologies on given set X. It is proved that this lattice is complete, not atomic, not complemented and not dually atomic. Some other properties of the lattice W??, the set of all weakly induced L-topologies defined by families of completely Scott continuous functions on X are also discussed.

Keywords: Weakly induced L-topology, induced L-topology, complete lattice, atoms, dual atoms, join complements, meet complements semi complements, complements.

More General Defuzzifications Are Needed

James J. Buckley

Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama, 35924, USA E-mail:buckley@math.uab.edu

Esfandiar Eslami

Department of Mathematics, Shahid Bahonar University of Kerman, Iran E-mail: eeslami@mail.uk.ir

Abstract: We show that there are linear ordering of the fuzzy numbers which are equivalent to a defuzzification method of ordering fuzzy numbers, but the defuzzification method needs to be a mapping of the fuzzy numbers into Rn, n¡Ý2, using the lexicographic ordering on the vectors in Rn, A linear ordering of fuzzy numbers is always important in fuzzy optimization problems.

Keywords: Ordering, ranking, fuzzy numbers, defuzzification.

On Some Fuzzy Mappings

B. Ahmad and A. Kharal

Centre of Advanced Studies in Pure and Applied Mathematics Bahauddin Zalsaria University, Multen, Pakistan E-mail: drbashir_9@gmail.com kharal9@hotmail.com

Abstract: Some basic results concerning closure, interior, semi-closure and semi-interior of fuzzy sets are contributed. Fuzzy s-open and fuzzy s-closed mappings have been studied further.

Keywords: Fuzzy topology, fuzzy semi-open set, fuzzy semi-interior, fuzzy semi-closure, fuzzy s-open mapping, fuzzy s-closed mapping.

Equivalence and Isomorphism of Fuzzy Subgroups

V. Murali

Department of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 South Africa E-mail: v.murali@ru.ac.za

B.B.Makamba

Department of Mathematics University of Fort Hare Alice 5700 South Africa E-mail: bmakamba@ufh.ac.za

Abstract: In this note we clarify the connection between various notions of fuzzy isomorphisms, and the more recent concept of equivalence of fuzzy subgroups. We observe here that equivalence is more general than isomorphism of fuzzy subgroups, in the sense that whenever two fuzzy subgroups are equivalent, then they are fuzzy isomorphic and not vice versa. Further, the lattice of subgroups is not an invariant for either of the notions of equivalence or isomorphism. The one and the same keychain may give rise to fewer non-isomorphic fuzzy subgroups than non-equivalent fuzzy subgroups.

Keywords: Equivalence, isomorphism, fuzzy subgroups, keychain, pinned-flag.

On L-fuzzy Closure Spaces

A. M. Zahran

Department of Mathematics, Faculty of Science, Al Azhar University, Assuit, Egypt E-mail: zahran15@hotmail.com

M. Azab Abd-Allah

Department of Mathematics, Faculty of Science, Assuit University, Assuit, Egypt E-mail:Mazab57@yahoo.com

Adb El-Nasser G. Abd El-Rahman

Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt E-mail: ghareeb_Nasser@yahoo.com

Abstract: In this paper we introduce and characterize r¡ªLc open, r¡ªLc semiopen sets (resp. r¡ªLc closed, r¡ªLc semiclosed sets) and Lc-open, Lc-semiopen and Lc-semi maps (resp. Lc-closed, Lc-semiclosed maps) in L-fuzzy closed space.

Keywords: Lc-open map, Lc-closed map, Lc-map, Lc-semiopen map, Lc-semiclosed map, Lc-semi map.

On Fuzzy Inner Product Spaces

Pinaki Majumdar

Department of Mathematics, M. U. C Wowen¡¯s College, Burdwan, West-Bengal, INDIA E-mail:pmajumdar2@rediffmail.com

S. K. Samanta

Department of Mathematics, Visva-Bharati, Santiniketan, West-Bengal, INDIA E-mail: syamal_123@yahoo.co.in

Abstract: In this paper we introduce a definition of fuzzy inner product in a linear space and investigate certain properties of fuzzy inner product function. The concept of minimizing vector is introduced in fuzzy inner product space. The parallelogram law has been proved in this setting. The notion of fuzzy orthogonality is introduced and a projection theorem in fuzzy Hilbert space is proved.

Keywords: Fuzzy inner product; ¦Á-fuzzy orthogonal; Fuzzy Hilbert space.

On Generalized Integrals Representation of Interval Valued Fuzzy Degree and Similar Degree

Wang Guijun

School of Mathematics Science, Tianjin Normal University, Tianjin, 300387, P. R. China

Li Xiaoping

School of Management, Tianjin Normal University, Tianjin, 300387, P. R. China

Abstract: In this paper, on the basis of the axiomatize definitions of the interval valued fuzzy degree and the interval valued similar degree having been given, by introducing the parameter and integral operations, give the integral representation each other and method describing the interval valued fuzzy degree and interval valued similar degree.

Keywords: Interval valued fuzzy sets; interval valued fuzzy degree; interval valued similar degree.

Generalized Quasi-variational-like Inclusions with Fuzzy Mappings

Jae Ug Jeong

Department of Mathematics, Dong Eui University, Pusan 614-714, South Korea E-mail: jujeong@dongeui.ac.kr

Abstract: In this paper, we introduce and study a class of generalized quasi-variational-like inclusions with fuzzy mappings. A new iterative algorithms for finding the approximate solutions and the convergence criteria of the iterative sequences generated by the algorithm are also given .These algorithm and existence result generalize many known results.

Keywords: Generalized quasi-variational-like inclusion; ¦Ç-proximal mapping; iterative algorithm; fuzzy mapping.

Fuzzy Programming with Piecewise Linear Membership Functions for Stackelberg Game

Sankar Kumar Roy

Department of Mathematics Raja Narendra Lal Khan Women¡¯s College, Gope Palace, Paschim Midnapore, West Bengal, Pin-721102, India E-mail: roysank@yahoo.co.in

Abstract: This paper presents the study of Stackelberg Game as a bi-level programming problem. In Stackelberg Game, I introduce the concept of leader at higher-level and the followers at lower-level of the problem. To solve the problem, I apply the piecewise linear membership function of fuzzy programming technique to solve the problem and compare the solution with the Kuhn-Tucker Transformation technique. Also I establish the Stackelberg Equilibrium solution by using both the techniques and compare the he satisfactory solution obtained from both the techniques .Lastly a numerical example is presented to illustrate the methodology.

Keywords: Stackelberg Games, Bi-level Programming Problem, Kuhn-Tucker Conditions, Fuzzy Programming.

On The Solution of Multi-objective Cutting Stock Problem in The Aluminumindustry under Fuzzy Environment

Omar M. Saad

Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Postal Code 11795, Cairo, Egypt E-mail: omarsd@hotmail.com

Abstract: A continuous production aluminum smelter produces S tons/day of molten metal which is cast as long cylindrical rods and swan into logs according to customer¡¯s specification for the dimensions and the composition of the logs. Orders for each month are known at the beginning of the month and have to be met by the end of that month. The rods about L feet long, are cast simultaneously, T rods/unit time, and then swan into logs of a variety of order lengths. Meeting demand leads to the over production of order lengths (inventory),and sawing yields unwanted offcuts (scrap) which have to be recycled. The price of aluminum and the competitiveness of the industry mean that inventory and scrap costs have to be minimized while meeting demand. This leads to a multi-objective cutting stock problem which, in its general form, is an integer nonlinear problem. It is considered that the scrap is a fuzzy parameter and a technique for the solution is presented. A practical problem example of the method implementation for the solution algorithm is proposed.

Keywords: Multi-objective mixed-integer nonlinear programming; Cutting stock problem; Fuzzy numbers; ¦Á-level set.

Common Fixed Point Theorems for Compatible Mappings in Fuzzy Metric Spaces

Shobha Jain

Shri Vaishnav Institute of Management, Gumasta Nagar, Indore (M. P.). India. E-mail:shobajainl@yahoo.com

Shishir Jain

Shri Vaishnav Institute of Technology and Science, Gram Baroli, Post Alwasa, P. O. Palia Indore (M. P.). India. E-mail: Jainshishir11@rediffmail.com Lal Bahadar Retired Principal Arts and Commerce College, Indor (M. P.). India.

Abstract: The object of this paper is to establish a unique common fixed point theorem for six self-mappings satisfying a contractive condition through compatibility and weak compatibility in a fuzzy metric space.

Keywords: Fuzzy metric space, common fixed points, t-norm, compatible maps, weak compatible maps.

The Prime Boolean Filters and Their Operations in MV-algebras

Jia-Lu Zhang

Department of Mathematics, Xiangnan University, Chenzhou 423000, Hunan, China

Abstract: This paper focuses on the operations on the filter set of MV-algebras. The lattice operations and the adjoint pair on filter sets of MV-algebras are defined. The filter sets is a residuated lattice under these operations. The order-reversing involution correspondence on prime Boolean filter sets is introduced. It is proved that the prime Boolean filter sets is a quasi-Boolean algebra under these operations.

Keywords: MV-algebras; Residuated lattice; Filter; Prime Boolean filter; Quasi-Boolean algebra.

Fuzzy Semi-prime Ideals of ??-rings

T. K. Dutta and Tanusree Chand

Department of Pure Mathematics University of Calcutta 35, Ballygunge Circular Road Kolkata-700019 INDIA E-mail: tanusree_chanda@yahoo.co.in

Abstract: In this paper we introduce the notions of fuzzy semi-prime ideal and fuzzy irreducible ideal in a ¦£-ring and study them via operator rings of ¦£-rings.

Keywords: Fuzzy (semi-prime, irreducible) ideal, left (right) operator ring, ¦£-ring, regular ¦£-ring.

¦©-bitopological Space Generated by Fuzzy Norm

T. Tamizh Chelvam

Department of Mathematics, Manonmaniam Sundaranar University Tirunelveli 627012, Tamil Nadu, INDIA E-mail: tamche_59@yahoo.co.in

A. Singadurai

Department of Mathematics, TDMNS College, T. Kallikulam 627113 Tamilnadu, INDIA

Abstract: In this paper we define ¦©-bitopological space (X, ?p, ?Q) where ?p, ?Q are ¦©-bitopologies generated by the fuzzy quasi pseudo norms P and Q. Further the characterization of a pairwise Hausdorff ¦©-bitopological space is also established.

Keywords: Fuzzy norm, ¦©-bitopology, pairwise Hausdorff ¦©-bitopological space.

On Regularity of Linear Combinations of Fuzzy Matrices

AR. Meenakshi

Emeritus Professor, Faculty of Engineering and Technology, Annamalai University, Annamalainagar-608002. India E-mail: arm_meenakshi@yahoo.co.in

Abstract: Regularity of fuzzy linear combinations of a finite set of regular matrices and their ranks are discussed. It is shown that the set of regular fuzzy matrices under minus ordering forms a fuzzy vector space. This leads to a characterization of fuzzy matrices in minus order with a give idempotent matrix.

Keywords: Regular fuzzy matrix, fuzzy vector space, minus ordering.