THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 5, Number 4, December 1997

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On the Integral Representation of Possibility Measures of Fuzzy Events

Anna Kolesarova

Department of Mathematics, Faculty of Mechanical EngineeringSlovak Technical University Nam. Slobody 17, 821 31 Bratislava,Slovakia

Abstract: In this paper the approach of Klement and Weber to the integral representation of fuzzy possibility measures is analyzed. The reason why in general this approach can fail is shown and moreover a necessary and sufficient condition, under which their representation theorem holds, is give. The paper also extends the results of Mesiar (which concern the integral representation of possibility measures in denumerable spaces) to general spaces.

Keywords: Possibility measure, fuzzy possibility measure, Markov Kernel, Sugeno integral.

The Latin Square Design Using Fuzzy Data

Pranita Goswami,

Department of Statistics, Pragjyotish College Guwahati, Assam, India 781009

Papia Dutta and Hemanta K. Baruah

Department of Statistics, Gauhati University Guwahati, Assam, India 781014

Abstract: A Latin square design is an incomplete three-way layout, where rows, columns and treatments are at the same at the same number of levels. We have studied the effects of fuzziness in a Latin square design. The calculations have been illustrated with a numerical example.

Keywords: Fuzzy data, fuzzy Latin square design.

Fuzzy Daniell Integral ¢ó

M.S. Samuel

Department of Mathematics, Baselius College Kottayam 686 001, Kerala State, India

Abstract: In our papers [6], [7], [8], we have defined a fuzzy vector lattice s, fuzzy Daniell functional , -integrable class of fuzzy point , shown that is a vector lattice, obtained analogous forms of Monotone Convergence theorem, Fatou¡¯s lemma and the Lebesgue-dominated convergence theorem for the fuzzy Daniell integral and established the uniquencess of the extension of from , the set of fuzzy points of s to . In this paper our main effort is to establish the fact the analogus form of Stone¡¯s theorem is also true for the fuzzy Daniell integral and .

Keywords: Fuzzy vector lattice, fuzzy vector space, fuzzy Daniell functional, -integrable class of fuzzy points.

An Iterative-approach to Fuzzy Chance-constrained Parametric Goal Programmin

Monhammad Lotfy Hussein

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

Abstract: This paper deals with an iterative approach to fuzzy chance-constrained parametric goal programming. The problem is considered in the case of having parametric functions and undeterministic aspiration levels. However, these aspiration levels are random variables with known probability density functions. An iterative fuzzy approach is used for characterizing the optimal solutions to the problem. In addition, the basic notions of stability are redefined and analyzed qualitatively for the problem. A numerical example is given for the sake of illustration.

Keywords: Goal programming; chance-constrained; fuzzy version; random variable; parametric analysis.

The Algebraic Properties of the Linguistic Truth-values

Luisa Cavaliere, Luigi Di Lascio and Antonio Gisolfi

Dipartimento di Informatica, Universita di Salerno, Italy

Abstract: In this work the algebraic properties for the linguistic values of the variable Truth are investigated. The basic feature of the model we are going to illustrate is that the truth value of each label is proportional to the area representing the quantify of information carried by the lable. By this assumption it is shown that four strings are sufficient to represent any linguistic truth value. Moreover one can define a function that, by means of crisp function, allows to reduce any Boolean expression to an atomic term. The model besides satisfying the Hedges Algebra axioms introduced by Cat and Wechler, also complies with those characteristic of a MV-algebra.

Keywords: Boolean linguistic variable, Hedges Algebra, MV-algebra.

The Continuity of Fuzzy Linear Order-homomorphism

Jin-xuan Fang

Department of Mathematics, Nanjing Normal University Nanjing, Jiangsu 210097, China

Abstract: In this paper, some of the properties of continuous fuzzy linear order homomorphisms from a L-fuzzy topological vector space into another are studied. The relations between continuity and boundedness of fuzzy linear order-homomorphisms are discussed.

Keywords: L-fuzzy topological vector space; fuzzy linear order-homomorphism; continuity; boundedness.

Multicriteria and Multilevel Decision Making in Fuzzy Talent Selection

Yo-Ping Huang, Mei-En Chen

Department of Information Engineering, Da-Yeh University Changhwa, Taiwan 515,R.O.C.

Kai-Quan Shi

Department of Automation Engineering, Shandong University of Technology Jinan,Shandong 250014, P.R. China

Abstract: This paper proposes a new conjunct method for multicriteria and multilevel decision making based on the concept of fuzzy majority. The model focuses on talent selection problem. The basic theoretics which bases on fuzzy talent decision tree and n-person game theory is applied to this new method. The precedence ordering criterion can help us to derive a preference ordering relational matrix which represents the relative important degrees for each candidate to others. We apply this new conjunct method to one layer decision tree, multilayer decision making problem, and extended multilevel decision tree. Simulation results from various selection examples are presented to verify the applicability of the proposed model to the real world problems.

Keywords: Fuzzy talent selection, fuzzy decision making, ordering relational matrix.

Fuzzy Flat and Faithfully Flat R-modules

M.M. Zahedi and R.A. Ameri

Department of Mathematics, Kerman University Kerman, Iran

Abstract: In this paper by considering the notion of top category, the tensor product of two fuzzy R-modules is defined. Then the fuzzy exact sequence of fuzzy R-modules is introduced, and some related results are obtained. In particular by defining the concepts of F-left exact and F-right exact functors, it is shown that two functors hom ( ) and hom (-, ) are not F-left exact, while in ordinary algebra they are left exact. Also it is seen that the functor is F-right exact. Finally the fuzzy left flat and faithfully flat R-modules are defined and some equivalent conditions are proved.

Keywords: Fuzzy R-module, fuzzy tensor product and fuzzy exact sequence.

Fuzzy Programming Approach to Multi-objective Stair-case Transportation Problems

Rakesh Verma, M.P. Biswal and A. Biswal and A. Biswas `

Department of Mathematics Indian Institute of Technology Kharagpur-721302, India

Abstract: The multi-objective stair-case transportation problem refers to a special class of transportation problem in which the unit cost matrices contain elements along the main diagonal and a band below it (called the square stair-case transportation problem), while the other elements of the cost matrices are treated as infinite. These objectives are non-commensurable and conflicting in nature. In this paper, the Pareto optimum solution approach is used to solve the multi-objective stair-case transportation problems. Fuzzy programming technique is applied by using a linear membership function. Numerical examples are added to illustrate the methodology. It is shown that the ideal solution is an optimal compromise solution for the multi-objective stair-case transportation problem.

Keywords: Multi-objective decision making, stair-case transportation problem, tridiagonal transportation problem, Pareto optimum solution, linear membership function, fuzzy programming.

Fractionary Fuzzy Ideaals and Fuzzy Invertible Fractionary Fuzzy Ideals

Kyong Hee Lee and John N. Mordeson

Department of Mathematics and Computer Science Creighton University, Omaha, Nebraska 68178, USA

Abstract: We introduce and study the concepts of fractionary fuzzy ideals and fuzzy invertible fractionary fuzzy ideals. We characterize fuzzy invertible fractionary fuzzy ideals. We also describe the algebraic structure of the set of all fuzzy invertible fractionary fuzzy ideals.

Keywords: Integral domain, fractionary fuzzy ideal, fuzzy invertible fractionary fuzzy ideal.

Fuzzy Quasi associative Ideals in BCI-algebras(¢ò)

Yong Bae Jun

Department of Mathematics Education Gyeongsang National University, Chinju 660-701, Korea

Abstract: We construct an extension of a fuzzy quasi-associative ideal in a BCI-algebra.

Keywords: Fuzzy QA-ideal, extension of fuzzy QA-ideal.

Fuzzy Commutative -Ideals in BCI ¨Csemigroups

Y.B. Jun

Department of Mathematics and Research Institute of Natural Science Gyeongsang National University Chinju 660-701, Korea

J.Y. Kim

Department of Mathematics Sogang University Seoul 121-724, Korea

Y.H. Kim and H.S. Kin

Department of Mathematics(Education), Chungbuk National University Chongju 361-763, Korea

Abstract: We introduce the concept of commutative -ideals in BCI-semigroups, and fuzzify it. We give some characterizations of fuzzy commutative -ideals. We also show that the homomorphic image (resp. preimage) of a fuzzy commutative -ideals is a fuzzy commutative -ideal.

Keywords and Phrases: BCI-semigroup, (fuzzy) commutative -ideals.

Semilattices and Fuzzy Sets

Branimir Seseljia and Andreja Tepavcevic

Institute of Mathematics, University of Novi Sad Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia

Abstract: Fuzzy sets are considered to be mappings from an arbitrary nonempty set into a (meet, or join) semilattice. For such fuzzy sets, representation theorems by collections of subsets are given. If the condomain of a fuzzy set is a join-semilattice, then this fuzzy set induces a partition of the semilattice into(also) join-semilattices. The obtained join-semilattice of classes is a homomorphic image of the former, and also dually isomorphic with the poset of levels. Moreover, this poset of classes is a lattice if and only if one of the levels of the fuzzy set is the whole domain. We also give necessary and sufficient conditions under which a meet-fuzzy set induces a partition of the corresponding meet-semilattice, so that the poset of classes(and hence also the poset of levels) has a structure of a lattice.

Keywords: Semilattice, meet, join, fuzzy set.

Separation Axioms for Fuzzy Convergence Spaces

Gunther Jager

Lessingstr. 13 D-76135 Karlsruhe Germany

Abstract: In a foregoing paper compactness in fuzzy convergence spaces was studied. However some properties of compact sets in topological resp. Limit spaces could not be extended to the theory of fuzzy convergence spaces in the absence of a suitable Hausdorff separation axiom. In this paper such an axiom is given and its properties are studied. Together with a T1-axiom it turns out to be both a good extension of strong axioms in fuzzy topology and of the corresponding axioms in the theory of limit spaces. Some results concerning separation and compactness are proved. It is shown that a compact T2 fuzzy convergence space is a Choquet limit space, which answers in part the question when a fuzzy convergence space can have a fuzzy T2 Richardson compactification.

Keywords: Fuzzy topology, fuzzy convergence space, fuzzy separation axioms, fuzzy compactness.

Generalized Fuzzy Integrals with Respect to Fuzzy Number Fuzzy Measures

Congxin Wu

Department of Mathematics, Harbin institute of Technology Harbin 150001, P.R. China

Deli Zhang

Department of Mathematics, Jilin Institute of Education Changchun, Jilin 130022, P.R. China

Caimei Guo

Department of Basic Science, Changchun University Changchun, Jilin 130022, P.R. China

Abstract: In the paper, a theory of (G)fuzzy integrals of functions with respect to fuzzy number fuzzy measures is built up. It includes definition, properties and generalized convergence theorems. These are an extension of our earlier work in [8,9].

Keywords: (G)fuzzy integral, fuzzy number fuzzy measure, generalized monotone convergence theorems.

Ideals and Minimal Ideals in L-semigroup

Jizhong Shen

Department of Mathematics, Jiangxi Normal University Nanchang 330027, P.R. China

Abstract: In this paper, we introduce the notion of ideals and minimal ideals ideals in a L-semigroup, and investigate some of their algebrac properties.

Keywords: Logic, semigroup, ideal.

Fixed Points in Fuzzy Metric Spaces

Yeol Je Cho

Department of Mathematics, Gyeongsang National University Chinju 660-701, Korea

Abstract: In this paper, we give some common fixed point theorems for five mappings satisfying some conditions in fuzzy metric spaces in the sense of Kramosil and Michalek.

Keywords: Fuzzy metric space, fixed point, mappings of type( ), asymptotically regular sequence.

Continuity of Graded Mappings

Marian Matloka

Department of Mathematics, Academy of Economics, Al. Niepodleglosci 10, 60-967 Poland University of Banking Al. Niepodleglosci 2, 61-874 Poznan, Poland

Abstract: In this paper the topological concepts connected with graded mappings theory are introduced and their properties are investigated.

Keywords: Upper and lower semicontinuous graded mapping, continuous graded mapping in the Kakutani and Hausdorff sense.

Fuzzy Linear Programming Problems with Fuzzy Interval Coefficients

Xiaozhong Li

Department of Computer Science, Liaocheng Teachers College Liaocheng, Shandong 252059, P.R. China

Kaiquan Shi

Department of Automation, Shandong University of Technology Jinan, Shandong 250061, P.R. China

Abstract: A kind of fuzzy linear programming problems with fuzzy interval coefficients has been presented in the paper. The auxiliary models of the problems are obtained with ranking fuzzy interval numbers in the setting of random sets. Fuzzy solutions of the problem are given.

Keywords: Fuzzy linear programming, fuzzy interval numbers, fuzzy preference relation, auxiliary model.

Fuzzy Less Strongly Irresolute Mappings and Fuzzy Less Strongly Semi-connected Set

Jin Han Park, Yong Beom Park

Department of Applied Mathematics, Pukyong National University Daeyeon-dong Nam-gu, Pusan 608-737, Korea

Jong Seo Park

Department of Mathematics Education, Chinju National University of Education Shinan-Dong 380, Chinju 660-756, Korea

Abstract: In this paper, we first introduce fuzzy less strongly irresolute, fuzzy pre-less strongly semiopen and fuzzy pre-less strongly semiclosed mappings on fuzzy topological space, and establish their various characteristic properties. Finally, we introduce and study fuzzy less strongly semi-separation axioms and fuzzy less strongly semi-connectedness with the help of fuzzy less strongly semi-q-neighborhoods.

Keywords: Fuzzy less strongly semiopen sets, fuzzy less strongly semi-q-nbds, fuzzy less strongly irresolute mappings, fuzzy less strongly semi-Ti spaces, fuzzy less strongly semi-connected sets.

Fuzzy Measure on Locally Compact Space

Ai Bing Ji

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

Abstract: In this paper, we study the fuzzy measure on locally compact Hausdorff space, and introduce the concept of the regularity of the fuzzy measure. On some condition, we prove several theorems on regular fuzzy measure.

Keywords: Autocontinuity of set function, uniform autocontinuity of set function, inner(outer) regular set, regular fuzzy measure.

Noninvolutive Negations on [0,1]

Ildar Batyrshin

Kazan State Technological University K.Marx St. 68, Kazan 420015, Russia

Michael Wagenknecht

University of Applied Sciences, Zittaul Gorlitz, IPM Theodor-Korner-Allee 16, 02 763 Zittan, Germany

Abstract: The properties of noninvolutive negations on [0, 1] in terms of elements are studied. The structures of objective negations and negations of rank 2 are described. It is shown that bijective negation has finite rank iff it is an involution. Moreover, the set of noninvolutive elements of bijective negation has a unique representation as union of nonintersected intervals. The properties of these intervals are studied and on the set of these intervals involutive negations are induced. Examples of noninvolutive negations with different properties are considered.

Keywords: Fuzzy logic, negation operation, involution, noninvolutive element, bijection.