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The Journal of Fuzzy Mathematics

Volume 14, Number 1, March 2006



New Proof to First Dual Theorem on Fuzzy Posynomial Geometric Programming

Cao Bing-yuan

School of Mayhematics and Information Sciences,Guangzhou University, Guangzhou,ZIP 510405 ,P.R.China

Abstract: Based on the first dual theorem previously obtained on a fuzzy posynomial geometric programming, a general fuzzy posynomial geometric one is changed into a monomial fuzzy posynomial geometric one.Since the latter is equivalent to a fuzzy linear one,a new proof is given to the theorem,which will play an important role in the study of a fuzzy strong dual theorem and of a more general fuzzy posynomial geometric one as well as a directly repeated algorithm to it.

Keywords: Geometric programming, Fuzzy posynomial, First dual theorem, New proof

Minimization Algorithm of Mizumoto Type of Fuzzy Finite Automata

Wei Cheng

College of Computer Sciences and Egeneering University of Electronic Science and Technology of China Chengdu 610054 ,P.R.China

Zhi-Wen Mo

College of Mathmetics and Software Science Sichuan Normal University, Chengdu 610066 ,P.R.China

Department of Applied Mathematics Southwest Jiaotong University, Chengdu 610031 ,P.R.China

Abstract: In this paper,a calss of fuzzy finite automata with a fuzzy initial state-i.e.,so-called Mizumoto type of fuzzy finite automata-is considered,and also two type of statewise equivalence relations are introduced. Based on the statewise equivalence relations,a minimal form is difined and a minimization algorithm of Mizumoto type of fuzzy finite automata is obtained.

Keywords: Fuzzy finite automata, Mizumoto type of fuzzy finite automata, Minimization, Minimization algorithm

Principal T-ideals of Lattices


Math Dept.,Faculty of Science,Tanta Univ.,Tanta,Egypt.

F.I.Sidky and M.A.Miserif

Math Dept.,Faculty of Science,Zagazig Univ.,Zagazig,Egypt

Abstract: The purpose of this paper is to give a construction theorems of T-ideal generated by a fuzzy subset of a lattice L as a result the principal fuzzy ideal generated by a fuzzy subset is given. Also ,we prove some basic properties of the principal fuzzy ideal of a lattice L and two examples are given to show that the behavior of the principal fuzzy ideals is different from the principal ideals of a distributive lattice L.

Keywords: Algebra, Distributive lattice, T-ideals, Generators.

Fuzzy Topology on Fuzzy Sets:

Fuzzy Retracts F.S.Mahmoud,M.A.Fath Alla and S.M.Abd Ellah

Department of mathematics, Faculty of Science ,Sohag, Egypt

Abstract: The concept of fuzzy topology on a fuzzy set was introduced by Chakrabarty and Ahsanullah in 1992.The concept of fuzzy retract was introduced by Rodabaugh in 1981.The aim of this paper is to introduce various fuzzy retracts in this new situation and establish some of their properties. Also, the relations between these new concepts are discussed.

Keywords: Fuzzy topology, fuzzy proper function, fuzzy retract, fuzzy semiretract.

The Saturation and Different Notions of Fuzzy Topogenous Orders


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

Abstract: Different notions of fuzzy tpogenous orders are introduced by many authors,e.g.,Katsaras [6],Manshour [9] and Gahler [4]. These three notions are each related to other by the notion of saturation of fuzzy topogenous order .Moreover,The notion of saturation is given in different forms (cf [1,4,5,9]) .In this paper ,we show that the saturated fuzzy topogenous orders ,in the sense of Mashhour [9],are in one-to-one correspondence with KatSaras fuzzy topogenous orders ,restricted to crisp sets. Also,the saturated fuzzy topogenous orders, in the sense of Gahler [4], restricted to crisp sets can be identified with Katsaras fuzzy topogenous orders, restricted to crisp sets.We show that [1,theorem3.5] lacks to a missing axiom ,so a correction is made and the equivalence of the two notions of saturation of Chung [1] and Kandil [5] is established .Finally ,some basic properties of these different notions of saturated fuzzy topogenous orders are investigated.

Keywords: Fuzzy neighbouhood spaces; Fuzzy topogenous order ; saturation fuzzy topogenous order ;fuzzy syntopogenous spaces.

On Perturbation Properties of Fuzzy Relation Equations

Zhang Chengyi

Depart. of Math.And Compu.Sc.,Hainan Normal University ,571158,Hainan,Haikou ,P.R.of china

Depart. Of math.,Zhumadian Theachers College ,463000,Henan , P.R.of china

Lu changjiang Depart. of Compu.Sc.,Sanming College ,365000,Fujian, Depart. P.R.of china Li Dongya

Depart. Of math.,Zhumadian Theachers College ,463000,Henan , P.R.of china

Abstract: In this paper ,the solving process of two kind of fuzzy relation equations are simplified .By the fuzzy solution invariant matrices ,the perturbation properties of fuzzy relation equations are considered and the open problem in [1]is solved .

Keywords: Fuzzy relation equation ,Solution invariant matrix ,Perturbation .

On Generalized Capacities and Para-regularity Theorems for Fuzzzy Measures

Jiongqi Wu

Depart. of Math., Zhangzhou Normal Univ.,36 Xianqianzhi Street , Zhangzhou 363000,Fujian,China.

Abstract: Suppose You is a Hausdorff space and Z is a non-empty set ,W is a cone of some positive functions on Z.The generalized capacity in this paper is defined to be some non-additive set mapping from to W. The capacitability theorem and some related properties of the generalized capacities are investigated .And from them a pararegulurity theorem for finite fuzzy measure on the Borel algebra of a metric space Y is obtained without other restrictions imposed on .

Keywords: non-additive set mapping, generalized capacity, capacitability, fuzzy measure, regularity.

Some Properties of Implication Operators

Yong Chan Kim and Jung Mi Ko

Department of Mathematics ,Kangnung National University , Gangneung,Gangwondo,210-702,Korea

Abstract: In this paper ,we introduce the t-norms,t-conorms and implications introduced by the product t-norm. We investigate the relations among implication operators ,Archimedean t-conorm(t-norm)and strictly decreasing (increasing )continuous functions .

Keywords: implication operators, t-norms, t-conorms, fuzzy negations

On Choquet Integrals of Fuzzy-valued Functions

Rui-sheng Wang

Academy of Mathematics and Systems ScienceChinese Academy of Science Beijing100080Peoples Republic of China

Ming-hu Ha

College of Mathematics and Computer Science ,Hebei University , Baoding,071002, Peoples Republic of China

Abstract: In this paper ,we give a further discussion on Choquet integrals of fuzzy-valued functions, including some properties and various kinds of convergence theorems like Lebesgue convergence theorem,Fatous lemma ,a.e. convergence theorem, convergence in fuzzy measure theorem ,etc.

Keywords: Choquet integrals; fuzzy-valued functions; convergence theorems

On n-fold Fuzzy Implicative Filters of Lattice Implication Algebras with Operators

Jianming Zhan and Zhisong Tan

Department of Mathematics ,Hubei Institute for Nationalities, Enshi, Hubei Province, 445000,P.R.China Email:

Abstract: E fuzzify n-fold implicative filters of lattice implication algebras, and investigate some of their properties. Moreover, we introduce the concept of n-fold M-fuzzy implicative filters and M-lattice implication algebras and obtain some related results. We give characterizations of M-lattice implication algebras. Finally, we study the normalization of n-fold M-fuzzy implicative filters.

Keywords: n-fold fuzzy implicative filters, n-fold M-fuzzy implicative filters, M-lattice implication algebras, M- normalization.

Nilpotent Faithful Fuzzy Ideals

P.Dheena and S.Coumaressane

Department of Mathematics,Annamalai University, Annamalainagar-608 002,Tamilnadu,India. E-mail: and

Abstract: In this paper we have introduced the definitions like point insertion of factors property (PIFP), insertion of factors property (IFP) and nilpotent faithful fuzzy ideal(NFFI). We have shown that a fuzzy ideal f of R is PIFP iff it has IFP and if f is a completely fuzzy semiprime ideal of R, then f has IFP. We have proved that R is a 2-primal ring iff FPR(R) is NFFI. We have also proved that R is 2-primal ring iff FPR(R) has IFP.

Keywords: fuzzy ideal, fuzzy point, fuzzy prime ideal, completely fuzzy prime ideal, fuzzy prime radical, nilponent faithful fuzzy ideal, completely fuzzy semiprime ideal, fuzzy left symmetric, fuzzy right symmetric.

On Generalized Multivalued Co-variational Inequalities with Fuzzy Mappings in Banach Spaces

Salahuddin and M.K.Ahmad

Department of Mathematics Aligarh Musilm UniversityAligarh-202 002(U.E),India Emails:;

Abstract: In this paper we consider a new class of generalized multivalued co-variational inequality problems with fuzzy mappings in Banach spaces. We establish an equivalence relation between a class of the generalized multivalued co-variational inequality problems with fuzzy Mappings and a class of implicit Wiener-Hopf equations with f Mappings in Banach spaces. This equivalence is used to suggest and analyze some iterative algorithms for finding the approximate solutions of generalized multivalued co-variational inequality problems with fuzzy mappings. Further, we prove the existence of solutions of this class of prolems and discuss the convergence of iterative sequences generated by the iterative algorithms.

Keywords: Generalized multivalued co-variational inequalities, implicit Wiener-Hopf equations, fuzzy mappings, Hausdorff metric, normalized duality mappings, Banach spaces, strongly accretive and Relaxed accretive mppings, g- strongly accretive and g-relaxed accretive mppings.

Choquet Integrals on L-fuzzy Sets

Rui-sheng Wang

Academy of Mathematics and Systems ScienceChinese Academy of Science Beijing100080P.R. China

Li-min Wang

College of basic Science, China University of mining and technology, Beijing, 100080, P.R. China

Ming-hu Ha

College of Mathematics and Computer Science ,Hebei University , Baoding,071002, P.R. China

Abstract: In this paper, we consider Choquet Integrals with respect to real-valued fuzzy measure on L-fuzzy sets. We first give the definition of Choquet Integral on L-fuzzy sets and discuss its some properties, then establish various kinds of convergence theorems like monotone convergence theorem, Fatous lemma, bounded convergence theorem, a.e. convergence theorem, convergence in fuzzy measure theorem, etc.

Keywords: Lattice; L-fuzzy set; Choquet integrals; convergence theorems

Structure of Lattice Implication Algebra

Jun Ma

Department of Mathematics, School of Science, Southwest Jiaotong University Chengdu, 610031, P.R.China

Wenjiang Li

School of Electric Engineering, Southwest Jiaotong University Chengdu, 610031, P.R.China

Tianrui Li and Yang Xu

Department of Mathematics, School of Science, Southwest Jiaotong University Chengdu, 610031, P.R.China

Abstract: Lattice implication algeria is an important logical algebra. In this paper, we discussed its construction on distributive lattice by analyzing the structure properties of it. We have show that a no-chain and non-Boolean lattice implication algebra must has at least two dual atoms, and it is impossible to build lattice implication algebra on some kinds of distributive lattice.

Keywords: Uncertainty reasoning, non-classical logic, lattice-valued logic, lattice implication algebra

An -fuzzy Version of Amanns Fixed Point Theorem for -fuzzy Monotone Multifunctions

Abdelkader Stouti

Department of Mathematics, Faculty of Science and Techniques University Casdi Ayyad, P.O.Box 523. Beni-Mellal 23000, Morocco E-mail:

Abstract: An Cfuzzy version of Amanns fixed point theorem for Cfuzzy monotone multifunctions is given. Furthermore, we obtain the existence of greatest fixed point for Cfuzzy monotone multifunctions.

Keywords: Fuzzy set, Cfuzzy order relation, Cfuzzy monotone multifunction, least fixed point.

Fuzzy Algebras and Directions


Department of Mathematics University of Alabama Tuscaloosa, AL 35487-0350 U.S.A. Email:;

Hee Sik Kim

Department of Mathematics Hanyang University Seoul 133-791, Korea Email:

Abstract: Given an set X, a direction on X is a triple of functions from to used in modeling very general versions of (asymmetric ) order relations. Many natural examples of directions results from modeling a variety of systems including digraphs, fuzzy posets and intuitionistic fuzzy sets. Because of the naturally fuzzy from of directions there are many possible and interesting connections between directions on sets X and fuzzy subalgebras of (binary) algebras or fuzzy relations on X, i.e., fuzzy subsets of . In this paper we investigate some of the possibilities. We introduce the notion of P-proper binary operations, fuzzy coalgebras (directions), (directions) order and compare the latter to the order on BCK-algebras associated with these structures among other results.

Keywords: Directions (order), P-proper, pogroupoid, fuzzy subalgebra, fuzzy coalgebra, BCK-algebra.

Countably Paracompactness in L-topological Spaces

Meng Han and Meng Guangwu

Department of Mathematics, Liaocheng University, Shandong 252000, Peoples Republic of China

Abstract: In this paper, the concept of countably paracompactness in L-topological space is introduced. The characterizations and properties of countably paracompact spaces are systematically discussed. In order to embody fuzzy the stratiform character of L-topology, the concept of Cclosed set, which is a generalization for L-closed set, used in the paper.

Keywords: L-topology; countable paracompactness; Cclosed set; Fuzzy lattice.

On Convering Dimension and Finitistic Spaces in L-topology

Dalip Singh Jamwal and Shakeel Ahmed

Department of Mathematics, University of Jammu,Jammu-180006 J&K, India

Abstract: In this paper, we have defined Covering Dimension and Finitistic Spaces in L-topology and proved some of their basic properties. According to [7], all L-fuzzy topological spaces defined in [9] are L- topological spaces. In this paper, we have also considered L-fuzzy topological spaces of [9] as L- topological spaces. Throughout this paper, the pair means an L- topological space where X is a set and is an L- topology on X. there are many types of covers defined in [9] (page no 817 to 190) . One among these is defined as a subfamily is said to be an open cover of an L- topological space if . We have used this cover in defining Covering Dimension and Finististic Space in L-topology.

Keywords: L-topology, Covering Dimension, Finististic Space, Topology.

Fuzzy - Paracompactness in Fuzzy Box Products


Department of Mathematics, Catholicate College Pathanamthita Kerala, India. Email:

Abstract: In this paper we characterize fuzzy Cparacompactness fuzzy entourages. The preservation of fuzzy Cparacompactness in fuzzy box products is also discussed.

Keywords: Fuzzy box products, fuzzy uniformity, -shading, Cparacompact, locally finite, fuzzy entourage, fuzzy topologically complete.

Fixed Point Theorem for Six Maps in Fuzzy Metric Spaces

Bijendra Singh

S.S. in Mathematics Vikram university Ujjain (M.P.)456010

Shishir Jain

Shrink Vaishnav Institute of Technology and Science Gram Baroli, Post Alwasa, P.O.Palia,Indore Email:

Abstract: Sushil Sharma [11](Fuzzy Sets and Systems, 2002) established a fixed point theorem for six maps satisfying a contractive condition in a fuzzy metric space. The object of this paper is to correct, generalized and improve this result using compatibility of type ( ) and weak-compatibility of self-maps.

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

Fuzzy Multiplicative Ideal Theory II

Hwankoo Kim Information Security Major, Division of Computer Science and Engineering, Hoseo University, Asan 336-795, Korea E-mail:

Myeong Og Kim,Sung-Mi Park,Young Soo Park

Department of Mathematics, Kyungpook National University, Taegu 702-701,Korea E-mail:

Abstract: In this paper, we characterize UFDs valuation domains and Prufer domains as Kaplansky-type theorem. We also characterize Cdomains and Mori domains. Finally we introduce the concept of nonfactorable Fuzzy ideals and characterize Dedekind domains in terms of nonfactorable Fuzzy ideals.

Keywords: (Fuzzy) Algebra, fractionary fuzzy ideal, fuzzy invertible, nonfacterable fuzzy ideal, Dedekind domain, Prufer domain, UFD, valuation domain, Cdomains ,Mori domain.