THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 2, Number 4, December 1994

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On Polygroups and Fuzzy Subpolygroups

M. M. Zahedi , M.bolurian and A.Hasankhani.

Department of Mathematics. Kerman University, Kerman, Iran

Abstract: In this note by using a definition of the right coset of a subpologroups, it is shown that there is a polygroup structure on the set of all right cosets of a given normal sub-polygroup. Then the fuzzy(normal) subpolygroup is defined and it is proved that the inverse image (homomorphic image) of a fuzzy (nomal) subpolygroup. Finally fuzzy -cosets of a polygroup and then a polygroup structure induced on the set of all fuzzy -cosets (which is called the polygroup of fuzzy -cosets) are defined. It is shown that there is an isomorphism (strong) of polygroups between the polygroups of fuzzy -cosets of two homomorphic (strong) polygroups.

Keywords: Polygroup, (Normal) subpolygroup, Fuzzy (Normal) subpolygroup.

Structure and morphisms of Fuzzy Algebraic Extensions

A.C.Eke

Department.of Computer Science, university of Nigeria Nsukka, Nigeria

B.I.Eke

Dept.of Computer Science, OSISATHECH Enugu, Nigeria

Abstract: The structure and morphisms of fuzzy algebraic extensions are examined. Among the traditional concepts fuzzified are those of minimal polynomials, normal extensions, splitting fields, and Galois groups and correspondence.

Keywords: Splitting fuzzy field, fuzzy normal, homomorphism, automorphism, ideal, coeset, quotient ring.

Dual Fuzzy Neighborhood Spaces II

Nehad N.Morsi

Department of Mathematics,Military Techical College, Cairo, Egypt

Abstract: A lowen fuzzy topological space is said to be localizable (at points) if it is completely determined, in some prescribed manner, by means of a system of fuzzy neighborhoods of its points. In the literature, two theories provide two types of such spaces. The first hteoy, due to U.H hle and R.lowen, associates with each continuous triangular norm T a full subcategory of the category FTS of topological spaces and continuous functions; the category T-FNS of fuzzy T-neighborhood spaces. The second theory , due to the author ,associates with each T another full subcategory of FTS; the category T-FLS of fuzzy T-locality spaces. These are topological categories. We explain the relationship between these tow theories, by means of naturally defined functors (one for each T) from T-FLS into T-FNS. We supply a right ad joint and a left adjoint of .Those two adjoints embed T-FNS into T-FLS, respectively as a brieflective full subcategory (whose objects we call T-lower dual fuzzy neighborhood spaces), and as a coreflective full subcategory(whose objects are called T-dual fuzzy neighborhood spaces) .Their objects are characterized within T-FLS. In the special case T=min , the functor has been introduced separately in the first part of this two-part work (Fuzzy Sets and Systems 44(1991)245-263);where it has been denoted by t. In the last section of this second part, we return to and to its two adjoints. We endow them with new properties and characterizations ;obtained as direct applications of the general results of this paper.

Keywords: Fuzzy topology, topological category, triangular norm, multiple-valuad logic.

Fuzzy Group Subalgebras

John N.Mordeson

Department of Mathematics/ Computer Science Creighton University, Omaha, Nebraska 68178, U.S.A

Abstract: In this paper, we introduce the concept of a fuzzy group subalgebra of a group algebra. We show fuzzy group subalgebras can be used to make a connection between fuzzy subgroups and fuzzy subfields. The concept of modularity is introduced and examined for fuzzy field extensions. It is shown that if is a fuzzy field extension and is the unique maximal pure inseparable intermediate fuzzy field of , then is modular if and only if is separable and is modular. The structure of modular pure inseparable fuzzy field extensions is also determined.

Keywords: Group algebra, fuzzy group subalgebra, fuzzy field extension modular, composite, linear disjoint, tensor product, pure inseparable, regular, separable, torsion, torsion-free.

Some Comments on Fuzzifying a Concept

H.V. Kumbhojkar

Department of Mathematics Shivaji University Kolhapur-416 004, india

N.Kuroki

Department of Mathematics Joetsu university of Education, Joetsu-shi Niifata-ken 943, Japan Yasusshi Suzusi of Mathematics Yokosuka High School, Kugo-cyo Yokosuka-shi kanagawa-ken 238, Japan

Abstract: In this paper we propose some rules which should be observed, while fuzzifying a mathematical concept. We use fuzzy ideals and fuzzy prime ideals to substantiate our view point.

Keywords: Fuzzy set, fuzzy ideal, fuzzy prime ideal.

Fuzzy -continuity, Fuzzy Almost Regularity (Normality) on Fuzzy Topology on Fuzzy Sets

A.M.Zahran

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71511, Egypt.

Abstract: The concept of a fuzzy topology on a fuzzy set has been introduced in [2]. The aim of this work is to introduce fuzzy -continuity, fuzzy almost -space, fuzzy almost regularity and fuzzy almost normality in this new situation and the validity of some standard characterizations of these concepts have been examined.

Keywords: Fuzzy topology, proper function, fuzzy subspace, fuzzy -continuity, fuzzy separation axiom.

Strongly Compact and P-closed Fuzzy Topological Spaces

A.M.Zahran

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Abstract: The purpose of this paper is to discuss strong compactness in fuzzy topology and to introduce fuzzy -closed spaces. Also we give some characterizations of strong compactness and -closed in terms of fuzzy filterbasis and fuzzy pre- -cluster points.

Keywords: Fuzzy topological spaces, fuzzy strongly compact spaces, fuzzy -closed spaces, fuzzy filterbasis, fuzzy pre- -cluster points.

The Min- Compact of Fuzzy Relations

Tian Yuying

Chemical and Industrial Department Shijiazhuang Managing Cadres¡¯College, Hebei, P.R.China

Abstract In this paper, under the condition of the composition of fuzzy relation min- (generalized fuzzy ¡°OR¡± operator), we give the concept of transitive closure t(R) of the composition of a fuzzy matrix min- . The properties of transitive closure t(R) and its simple computing method are studied.

Keywords: Generalized fuzzy ¡°OR¡± operator, fuzzy relation, transitive closure,

-structure of FTS(IV) -compactness

Hu Cheng-ming

International Fuzzy Mathematics Institute P.O.Box 629, San Gabriel, CA 91778, U.S.A

Abstract: In paper [1], [2], we introduced a kind of fuzzy compactness, which is defined by -cover and charactized by -filters and F-nets. Here we will discuss some properties of -compactness and give a product theorem.

Keywords£º -cover , -compact -ultrafilters, F-universal nets.

On Some Types of Fuzzy Supraconnectedness in Fuzzy Supratopological Spaces

N.Turanli and D.Coker N.Turanli and D. Coker

Department of Science Education, Hacettepe University, Beytepe 06532-Ankara,Turkey

Abstract: The purpose of this paper is to investigate some types of fuzzy supraconnectedness in fuzzy subpratopological spaces. Here we introduce several types of fuzzy supraconnectedness in fuzzy supratopological spaces. Furthermore, we present some counterexamples related to the product of fuzzy supaconnected sets, and show that these types of fuzzy supraconnectedness are not preserved under product of spaces.

Keywords: Fuzzy sets, fuzzy supratopology; fuzzy supratopological space, fuzzy -supraconnectedness (i=1,2,3,4); fuzzy -supraconnectedness, fuzzy -supraconnectedness, fuzzy products.

Fuzzy H-continuous Functions

S.Dang andf A.Behera

Department of Mathematics, Government College, Rourkela-769004, India

Department of Mathematics, Regional Engineering College, Rourkela-769004, India

Abstract: The concept of fuzzy H-continuity of a function is introduced. Several characterizations of fuzzy H-continuous functions are explored.

Keywords: Fuzzy point, fuzzy H-closed set, fuzzy H-continuous function, fuzzy -continuous function, q-coincidence.

Applications of Fuzzy Decision-Making Theory to Planting of Municipal Wastewater Treatment System

Xiong Deqi and Chen Shouyu

College of Marine Engineering, Dalian Martine University Dalian 116024, China

Chen Shouyu

Department of Civil Engineering, Dalian University of Technology

Abstract: The old methods for the panning of municipal wasterwater treatment systems are to minimize the treatment costs only. In this paper, we develop the multiobjective programming method by which the water quality and treatment costs can be coordinated to find the optimal comprehensive schemes. The fuzzy decision-making theoretic model for the method is also proposed. By adjusting the objective weights we can obtain all the effective solutions used to make decisions. And finally, a case study is given.

Keywords: Fuzzy decision-making, multiobjective, Planning, Wasterwater treatment system.

Fuzzy Set Theories and Methods for municipal Environmental Quality Assessments

Xiong Deqi and Chen Shouyu

College of Marine Engineering, Dalian Martine University Dalian 116024, China

Chen Shouyu

Department of Civil Engineering, Dalian University of Technology

Abstract: The municipal environment is a complex system which consists of multi-index and multi-subsystem. In this paper, the fuzzy set theories and methods for the municipal environmental Quality assessments are proposed, which mainly include: 1)the fuzzy set method for determining pollution index weights in the subsystem; 2)the fuzzy set theory and method for determining environmental subsystem weights; 3)the fuzzy set theory and model for multi-index and multi-stage municipal environmental quality assessment.

Keywords: municipal Environment Quality,Fuzzy Sets, Asssessment

A Notion of Fuzzy Nearess

E.Soetens and P.wuyts

Universite du Burundi, Bujumbura, Burndi.

P.Wuyts

Universiteit Antwerpen, R.U.C.A.,Belgium

Abstact: It is tried to develop a theory of fuzzy nearness spaces by means of coverings, thus in the mean time extending to the fuzzy situation one aspect of the concept of nearness introduced by H.Herrlich [3], and generalizing the theory of covering uniform spaces established by authors[10].starting from a notion of fuzzy senmi nearness three subcategories of idempotency of what ought to be a closure operator. Amongst these four categories, that of saturated semineaness spaces shows to be topological, while that of seminearness spaces is not. The status of the two remaining ones is not yet clear. However, the embedding of sysmmetric fuzzy neighborhood spaces and fuzzy uniform spaces in the category of saturated fuzzy nearness spaces can be performed in much the same way as that TOP en UNIF in NEAR in crisp case.

Keywords: Fuzzy topological spaces, fuzzy neighborhood spaces, fuzzy unifom spaces, -spaces.

On Fuzzy Projective and injective Modules

M.M.zahedi, R.Ameri

Department of Mathematics, Kerman University Kerman, Iran

Abstract: First several equivalent conditions for objects in the category of fuzzy -modules to be injective or projective are given. Then fuzzy projective and injective presentations of a fuzzy -modules are defined, and finally by using these definitions tow functors - from the category of fuzzy -modules to the category of Abelian groups are presented, and related results are obtained.

Keywords: fuzzy -module, fuzzy projective and injective modules.

A Detailed Study of Direct and Inverse Images under Fuzzy Multivalued Mappings

Elena Tsiporkova, Bernard De Beats and Etienne kerre

Department of Applied Mathematics and Computer Science, University of Gent, Krijgslaan 281(s9), B-9000 Gent, Belgium

Abstract: In this paper, we expose fuzzy multivalued mappings, as introduced by Ottoy and Kerre, to a scrutinous study. In particular, the direct image, and the lower and upper inverse images of fuzzy sets under fuzzy multivalued mappings are investigated, including an overview of momotnicity and complementation properties and intersection with union and intersection. Special attention is paid to the behavior of fuzzy constants .Finally, we introduce cuts of fuzzy multivalued mappings, leading to remarkable decompositions of the images, and other interesting relationships. This paper will serve as a reference for further studies of fuzzy multivalued mappings, such as a continuity study.

Keywords: Cuts, decomposition, fuzzy multivalued mapping, images.

On Some Near Fuzzy Continuous Functions on Fuzzy Topological Spaces

A.M.Zahran

Department of Mathematics, Faculty of Science, Al-Azhar university, Assiut Branch, Assiut, Egypt

Abstract: The concepts of fuzzy near continuous functions such as fuzzy almost continuity, fuzzy -continuity, fuzzy almost strong -continuity, fuzzy weak -continuity, fuzzy -continuity and fuzzy weak -continuity have been introduced. In this paper we investigate some additional properties of these functions and we explain the converse of some previous theorems on these functions are generally not true. Also we give conditions under which the converse of these functions are true. Finally we give an improved version of some results of these functions.

Keywords: Fuzzy -neighborhood, fuzzy --neighborhood, fuzzy almost continuity, fuzzy -continuity, fuzzy weak -continuity, fuzzy weak -continuity, fuzzy almost strong -continuity, fuzzy -closure, fuzzy -closure.

Compact Fuzzy Closure Spaces

M.H.Ghanim, O.A.Tantawy and M.A/Saker

Department of Mathematics, Zagazing University, Zagazing 44519 Egypt.

Abstract: The concepts of -compactness, weak forms of -compactness, -continuous functions, -Hausdorffness, -regularity and -normality are extended to fuzzy closure spaces. The relations between these concepts are investigated.

Keywords: Fuzzy topological spaces, fuzzy closure spaces.

The Preservation of Structural Characteristics of Monotone Set Functions Defined by Fuzzy Integral

Zhenyuan Wang, Geoge J.Klir, and David Harmanec

Department of Systems Science and Industrial Engineering

Tomas£Ê£®

Watson School of Engineering and Applied Science Binghamton University-SUNY, Binghamton, New York 13902-6000, U.S.A.

Abstract: Given a measurable space ( F ), a nonnegative monotone set function on ,and a nonnegative measurable function f on ( F ), the fuzzy integral of f with respect to determines another monotone set function on ¡¡( F ). Such a new set function preserves some structural characteristics, such as continuity, subadditivity, fuzzy additivity, fuzzy multiplicativity, null-additivity, autocontinuity, and uniform autocontinuity, of the original set function . Determining a monotone set function by the fuzzy integral is thus an effective strategy to construct sound fuzzy measures in expert systems.

Keywords : Monotone set function, fuzzy measure, possibility measure, necessity measure, structrural characteristic of set function, fuzzy integral.