The Journal of Fuzzy Mathematics
Volume 2, Number 2, June 1994
CONTENT
DETAILS
On Compatibility of Two Extension Algorithms with the Extension Principle of Zadeh
Abdel Wahab M. El-Abyad, Said Anwar Gouda, Nehad N.Morsi
Department of Mathematics, Military Technical College, Cairo, Egypt
Samy E.Yehia
Department of Mathematics, Faculty of Engineering, Fayoum University, Fayoum, Egypt
Abstract: The Extension Principle of Zadeh aims at extending a binary operation *, on a set X, to a binary operation¡ð*T (=the sup-T extension of *) on IX; where T is a lower semicontinuous triangular norm. On the other hand, a fuzzy (or ordinary) binary relation in X can be extended to a fuzzy binary relation in IX, and a metric d on X can be extended to a ¡°Hausdorff ¡± probabilistic T-metric on IX; in canonical wayside by means of two algorithms introduced earlier by N.N.Morsi. These new algorithms employ the T-residuated implication operator in defining fuzzy containment of fuzzy sets.
We show that if satisfies the subsititution property for * , then will satisfy the substitution property for ¡ð*T; while if d is nonexpansive under the action of *, then will be nonexpansive under the action of ¡ð*T.
Keywords: Fuzzy relation; triangular norm, implication operator; probabilistic metric.
Fuzzy Categories of Sets, Fuzzy Homomorphisms of Modules and Fuzzy Categories of Modules
Zhao JianLi
Institute of Fuzzy System, Liaocheng Teacher¡¯s college, Shandong 252000£¬ P.R.China
Abstract:
In this paper we introduce the concepts of fuzzy homomorphisms of modules and fuzzy categories of modules (sets). We prove for fuzzy homomorphisms, analogous of homomorphisms theorems of ordinary module and introduce the concepts of fuzzy categories of modules (sets).
Keywords: Fuzzy module, fuzzy function, fuzzy homomorphism, fuzzy natural homomorphism, category.
On the N-topological Right Quasi Regularity in N-rings
T. M. G. Ahsanullah
Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh
Marouf A. Samhan
Department of Mathematics, King Saudi University, Riyadh 11451£¬ Saudi Arabia
Abstract: Introducing the concept of fuzzy neighborhood topological right quasi-regularity in fuzzy neighborhood ringside (FNR or N-rings), we produce several interesting results in the presence of the notion of boundedness of fuzzy neighborhood rings; also, considering fuzzy neighborhood modules M over ring R, we take into account the notions of R bounded modules and M-bounded rings.
Keywords: Fuzzy N-topological (FNS), bounded fuzzy neighborhood rings and modules, N-topo right quasi-regular elements.
On The Lipschitzian Solution Of A Fuzzy Functional Equation
P. V. Subrahmanyam and S. K. Sudarsanam
Department of Mathematics , Indian Institute of Technology Madras 600 036£¬ India
Abstract: The theory of functional equations of the form has been studied extensively by Kuczma, Baron, Matkowski and others. This type of functional equation has found applications in various branches of mathematics. In this paper we provide an existence theorem for the Lipschitzian solution to such a type of functional equation in the fuzzy setting with requisite assumptions. The proof is based on Schauder¡¯s fixed point theorem applied to a suitable compact, convex set in an appropriately chosen function space.
Keywords: Fuzzy numbers, Fuzzy solution, Fuzzy functional equation, Lipschitzian , Hausdorff metric, Embedding , Zadeh¡¯s extension principle, Isometry and Baron¡¯s extension theorem.
Fuzzy Almost ¨Ccontinuous Functions
S.Dang
Department of Mathematics Government College, Rourkela-769004, India
A. Behera
Department of Mathematics Regional Engineering College, Rourkela-769008, India
Abstract: The concept of fuzzy almost ¨Ccontinuous function has been introduced and studied. Several characterizations of such functions have been established by using the concepts of fuzzy ¨Ccontinuous functions, fuzzy weakly semicontinuous functions and some other weaker forms of fuzzy continuous functions.
Keywords: Fuzzy point, q-coincidence, q-neighborhood, fuzzy semiopen set, fuzzy preopen set,fuzzy -open set, fuzzy regular open set, fuzzy ¨Ccontinuous function, fuzzy almost ¨Ccontinuous function, fuzzy ¨Ccontinuous function.
Fuzzy Supra Proximity Spaces and Fuzzy Biproximity Spaces
A.Kandil
Mathematics Department, Faculty of Science, Kuwait University, Kuwait
A.A.Nouh
Mathematics Department, Faculty of Science, Mansoura University, Egypt
S.A.El-Sheikh
Mathematics Department, Faculty of Education, Ain Shams University , Egypt
Abstract: In this paper we introduce and study the concept of fuzzy supra proximity spaces. Then we introduce the functor from the category of ordinary supra proximities SP to the category of fuzzy supra proximities FSP and the functors , from FSP to SP Also, we study the relationship between fuzzy biproximity space and its associated fuzzy supra proximity space. We prove that, for every fuzzy bitopological space which is both FP*-compact and -space, the associated fuzzy supra topological space has a unique compatible fuzzy supra proximity.
Keywords: Fuzzy topological spaces, fuzzy supra proximity space, induced fuzzy supra proximity spaces,-level supra proximities, fuzzy bitopological spaces, fuzzy biproximity spaces.
L-fuzzy Continuous, Semicontinuous and Weakly Semicontinuous Multifunctions in Fuzzy Lattices
Cheng Ji-shu
Department of Mathematics, Qinghai Junior Teachers¡¯ College Xining, Qinghai 810007£¬ P.R.China
Abstract: In this paper, we introduce the concept of an L-fuzzy multifunction(LFM for short) from a general topological space (X,T) into a topological molecular lattice (L(M),Z), and prove the relationship between and . Next we introduce the notions of L-fuzzy lower (upper ) continuous, lower (upper) weakly semicontinuous multifunctions. Their various properties are systematically studied, and more equivalent theorem of these multifunctions are obtained.
Keywords: L-fuzzy multifunction, lower (upper) continuity, lower (upper) semicontinuity, lower (upper) weak semicontinuity, fuzzy lattice, topological space.
New Characterizations of Fuzzy Almost Compact Spaces
R.P.Chakraborty and S.Malakar
Department of Pure Mathematics, University of Calcutta, 35£¬ Ballygunge Circular Road Calcutta 700019 India
Abstract: The present note is aimed at presenting two new characterizations of fuzzy almost compact spaces in terms of two typeset of fuzzy sets, introduced here.
Keywords: Fuzzy ¨Cclosure, fuzzy ¨Cinterior, q-neihborhood, fuzzy almost compact space.
Bifuzzy Topology Redefined
Pratulananda Das
Department of Mathematics, University of Kalyani Dt.-Nadia, West Bengal, India
P.Das
Department of Mathematics, Visva-Bharati University Santiniketan, Weather Bengal, India
Abstract: The idea of quasi-coincidence of a fuzzy point with a fuzzy set is used to introduce twelve new concepts of a bifuzzy topology. Acceptability of the new concepts are examined.
Keywords: Fuzzy topology, bifuzzy topology, gradation of openness, quasi-coincidence.
A New Product Structure in the Category of Molecular Lattices
Zhao Bin
Department of Mathematics, Shaanxi Normal University Xi¡¯an 710062£¬ P.R.China
Abstract:
The relation on the usual direct product of a family of molecular lattices is introduced by means of the relation on complete lattices. Some fundamental properties of the relation are discussed. Based on the relation , the concept of the -lower sets on the direct product is given, and it is proved that the set of all -lower sets on the direct product forms a molecular lattice. Hence, the product structure of a family of molecular lattices in the category of molecular lattices is obtained, and all molecules and completely molecules in the product object are given. It is finally proved that the product object of a family of strong molecular lattices in the category of molecular lattices is also a strong molecular lattice.
Keywords:
Complete lattice, molecular lattice, category, product.
Commentary: Definition of Prime and Primary Fuzzy Ideals
John N. Mordeson
Department of Mathematics and Computer Science Creighton University, Omaha, Nebraska 68178 USA
Abstract: We examine two wayside of fuzzifying the concepts of prime ideals and primary ideals. We initiate the examination of these concept the idea of primary representations. We also suggest a direction of study which may lead to a meaningful analysis.
Keywords: Prime and primary fuzzy ideals, l-prime and l-primary fuzzy ideals, radical, primary representation, l-primary representation.
On Fuzzy Quasi-Implications
T. Calvo
Dept. de Mathematics e Infomatica Univ. de las Islas Baleares. Palma de Mallorca. Spain
Abstract: In this paper we study a class of implication functions on [0,1] called ¡°quasi-implications¡± which are generated by mean of an aggregation Department Morgan triplet. From this consideration we obtain a few families of fuzzy quasi-implications. Finally we consider the generalized Modus Ponens, which is defined by these triplets.
Keywords: Fuzzy quasi-implication function; aggregation function; strong negation; generalized modus ponens
Boolean Matrix Representations of Final Choice functions
Jin Bai Kim
Department of Mathematics Weather Virginia University Morgantown, WV 26506 U.S.A.
Abstract:
We obtain a Boolean matrix from of a final choice function, find some nice identities for Boolean idempotent matrices, and prove a theorem.
Keywords: Final choice functions, Boolean idempotent matrices.
Fuzzy Preference Structures and Their Characterization
Bernard De Baets* and Etienne Kerre
Department of Applied Mathematics and Computer Science University of Gent, Krijgslaan 281(S9) B-9000 Gent, Belgium
Bartel Van definitions Walle**
Nuclear Research Center SCK/CEN Boeretang 200£¬ B-2400 Mol, Belgium
Abstract: In this paper we introduce the concept of a fuzzy preference structure. By imposing a non-trivial completeness condition on a triplet consisting of a fuzzy preference relation, a fuzzy indifference relation and a fuzzy incomparability relation, constructed from a fuzzy large preference relation as prescribed by Ovchinnikov and Roubens, we are able to establish a set of conditions similar to the oneself that have to be satisfied by classical preference structures. Moreover, and far more important, we prove that this construction can now be interpreted as a characterization of the fuzzy preference structures introduced here, similar to the characterization of classical preference structures.
Keywords: Completeness, fuzzy preference structure, incomparability, indifference, preference
Weak Convergence of Fuzzy Measure Sequences
Ha Minghu and Wang Xizhao
Department of Mathematics, Hebei University, Baoding, Hebei, P.R.China
Liu Jicai
Tangshan Worker University, Tangshan, Hebei, P.R.China
Abstract: In this paper, we shall introduce the concept of weak convergence of a type of fuzzy measure sequences and discuss some properties of this weak convergence over a class of convex sets on of the fuzzy measure sequences.
Keywords: Fuzzy measure, weak convergence, convex set.
Consonant Belief Function Induced by a L-fuzzy Set Function on an Arbitrary Nonempty class of L-fuzzy Set
Zhang Guang-Quan
Department of Mathematics, Hebei University, Baoding, Hebei071002, P.R.China
Wang Jian-Ping
Department of Basic Science, Hebei Engineering and Technical College, Cangzhou, Hebei 061001, P.R.China
Abstract: In this paper, the concept of consonant belief function on the L-fuzzy set is introduced, a consonant belief function induced can by a L-fuzzy set function on an arbitrary nonempty class of the L-fuzzy set is proved, extension theorem of consonant belief function is obtained, lower semi-lattice structure of all extensions of consonant belief function is given.
Keywords: L-fuzzy set, consonant belief function, B-consistent of L-fuzzy set function, lower semi-lattice
Possibility Measure Induced by a L-fuzzy Set Function on an Arbitrary Non-empty Class of L-fuzzy Set
Zhang Guang-Quan
Department of Mathematics, Hebei University, Baoding, Hebei071002, P.R.China
Wang Jian-Ping
Department of Basic Science, Hebei Engineering and Technical College, Cangzhou, Hebei 061001, P.R.China
Abstract: In this paper, the concept of possibility measure on the L-fuzzy set is introduced, a possibility measure induced can by a l-fuzzy set function on an arbitrary nonempty class of the L-fuzzy set is proved, extension theorem of possibility measure is obtained,upper semi-lattice structure of all extension of possibility measure is given.
Keywords: L-fuzzy set, possibility measure, P-consistent of L-fuzzy set function, upper semi-lattice.
Fuzzy Vector Lattices
M. S. Samuel
Department of Mathematics, Baselius College, Kottayam 686001£¬ Kerala State, INDIA.
Abstract: Fuzzy vector lattice and fuzzy Danielle Functional are defined.
Keywords:
Fuzzy vector space, fuzzy vector lattice, fuzzy Daniell functional.
An categorical topology approach to N-compactness and H( )-complete regularity
Sheng-Gang Li
Department of Mathematics, Shannnxi Normal University, Xi¡¯an, 710062£¬ P.R.China
Abstract: From the point of view of categorical topology, this paper examines the rationality of the definition of N-compactness. It proves that the -Stone ¨CCech compactification functor is an epireflector, and the weakly induced, and -completely regular spaces can be defined, rely on and -completely regularity, in terms of category theory; it also point that the N-compactness will be a good definition.
Keywords: Fuzzy topology; categorical topology; N-compactness; -completely regular
On the Existence of Bases for Fuzzy Vector Spaces
Godfrey C. Muganda
Department of Computer Science, North Central College Naperville, IL60566, USA
Abstract: A number of families of fuzzy vector spaces, recently introduced in [9], are studied with the objective of establishing, for each family, conditions under which membership of that family can be expected to have a fuzzy basis. In addition, it is shown that the supproperty is not necessary for the existence of a basis for a fuzzy vector space with a linearly ordered lattice of membership values.
Keywords: Fuzzy vector spaces, basis.
Fuzzy Q*-normal Semigroups
Nobuaki Kuroki
Department of Mathematics, Joetsu University of Education, Joetsu-shi, Niigate-ken, Japan 943
H. V. Kumbhojkar
Department of Mathematics, Shivaji University Kolhapur 416004£¬ Maharashtra, Indian
Abstract: In this note we shall study a fuzzy Q-normal quasi-ideal of a semigroup, which is an analogous notion of a fuzzy normal subgroup of a group, and give a characterization of a semigroup that is a semilattice of groups in terms of a fuzzy Q-normality.
Keywords: Fuzzy Q*-normal quasi-ideal, fuzzy Q*-normal semigroup, semilattice of group
Fuzzy Interior Ideals in Semigroups
Nobuaki Kuroki
Department of Mathematics, Joetsu University of Education, Joetsu-shi, Niigate-ken, Japan 943
Abstract:
In this paper we shall given some properties of fuzzy interior ideals of a semigroup, and characterize a regular semigroup, a semisimple semigroup and a semigroup that is a semilattice of left groups in terms of fuzzy interior ideals.
Keywords: Fuzzy ideal, fuzzy interior ideal, fuzzy bi-ideal, regular semigroup, semi-simple semigroup, semilattice of left (right) groups.
Continuous Approximation of Fuzzy Sets
I.L. Collings and P. E. Kloeden
School of Computing and Mathematics Deakin University, Geelong Campus Geelong Victoria 3217
Australia
Abstract: It is shown constructively that the fuzzy sets with continuous membership functions are dense in the metric space of normal, fuzzy vex fuzzy sets on the base space . The approximation of fuzzy set valued functions of a real variable is also considered.
Keywords: Fuzzy sets, level sets, Hausdorff metric, upper semicontinuity
Transitivity of Fuzzy Orderings Based on Pairwise Comparisons
Xuzhu Wang*, E. E. Kerre, B. Cappelle**
Department of Applied Mathematics and Computer Science University of Gent, Krijgslaan 281(S9) B-9000 Gent, Belgium
D.Ruan
Nuclear Research Center Boeretang 200B-2400 Mol, Belgium
Abstract: In this paper several definitions of the transitivity property of fuzzy orderings that can be found in current literature, and the relationship between them are investigated. Afterwards, we discuss Ovchinnikov¡¯s transitivity condition [12] in more detail.
Keywords: Fuzzy numbers; transitivity; fuzzy ordering.
Fuzzy Symmetic Groups
Jin Bai Kim
Department of Mathematics, West Virginia University Morgantown., WV26506 USA
Kyu Hyuck Choi
Department of Mathematics Education Wonkwang University, IRI 570-749 Korea
Abstract: We study fuzzy symmetric subgroups and obtain some properties of fuzzy symmetric subgroups of symmetric groups.
Keywords: Fuzzy symmetric groups. |