Generalized Intuitionistic Fuzzy Ideals Topological Spaces

Transcription

1 merican Journal of Mathematics Statistics 013, 3(1: 1-5 DOI: 10593/jajms Generalized Intuitionistic Fuzzy Ideals Topological Spaces Salama 1,, S lblowi 1 Egypt, Port Said University, Faculty of Sciences Department of Mathematics Computer Science Department of Mathematics, King bdulaziz University, Gedh, Saudi rabia bstract In this paper we introduce the notion of generalized intuition istic fu zzy ideals which is considered as a generalization of fuzzy intuit ionistic ideals studies in[6], the important generalized intuitionistic fuzzy ideals has been given The concept of generalized intuitionistic fuzzy local function is also introduced for a generalized intuitionistic fuzzy topological space These concepts are discussed with a view to find new generalized intuit ionistic fuzzy topology from the original one in[5, 7] The basic structure, especially a basis for such generated generalized intuition istic fuzzy topologies several relations between different generalized intuitionistic fuzzy ideals generalized intuit ionistic fuzzy topologies are also studied here Keywords Generalized Intuitionistic Fuzzy Ideals, Intuit ionistic Fuzzy Ideals, Intuit ionistic Fuzzy ocal Function 1 Introduction The concept of fuzzy sets fuzzy set operations was first introduced by Zadeh[9] ccordingly, fuzzy topological spaces were introduced by Chang[4] Several researches were the generalizations of the notion of fuzzy set The idea of intuition istic fuzzy set (IFS, for short was first published by tanassov[1,, 3] Subsequently, Tapas et al[8] defined the notion of generalized intuitionistic fu zzy set studied the basic concept of generalized intuitionistic fu zzy topology Our aim in this paper is to extend those ideas of general topology in generalized intuitionistic fuzzy topological space (GIFTS, in short In section 3, we define generalized intuitionistic fuzzy ideal for a set Here we generalize the concept of intuitionistic fuzzy ideal topological concepts, first initiated by Salama et al[6] in the case of generalized intuitionistic fuzzy sets In section 4, we introduce the notion of the generalized intuitionistic fuzzy local function corresponding to GIFTS Recently we have deduced some characterization theorems for such concepts exactly analogous to general topology succeeded in finding out the generated new generalized intuitionistic fuzzy topologies for any GIFTS Preliminaries Deifintion1[6] nonempty collection of intuit ionistic fuzzy sets of a set X is called intuitionistic fuzzy ideal on B X iff i B (heredity, (ii Corresponding author: drsalama44@gm ailcom ( Salama Published online at Copyright 013 Scientific & cademic Publishing ll Rights Reserved B B (finite additiv ity We shall present the fundamental definitions given by Tapas: Definition [8] et X is a nonempty fixed set n generalized intuitionistic fuzzy set (IFS for short is an object having the form = { x, µ ( x, ν ( x where the function µ : X [ 0,1] ν : X [ 0,1] denote the degree of membership(namely µ ( x the degree of non membership(namely ν ( x of each element x X to the set, respectively, µ ( x ν ( x 0 5 for all x X Re mark 1 For the sake of simplicity, we shall use the symbol = x, µ, ν for the GIFS { x, µ ( x, ν ( x x X } = : O Definition3[8] 1 = { x,1,0 are empty universal generalized inuitionistic fuzzy sets Definition 4[8] generalized intuitionistic fuzzy topology (GIFT for short on a nonempty set X is a family of GIFSs in X satisfying the axioms in[8] τ 3 Basic Properties of Generalized Intuitionistic Fuzzy Ideals Definition 31 et X is non-empty set a family of GIFSs We will call is a generalized intuitionistic fuzzy ideal (GIF for short on X if { < x,0,1 > x X } = :

2 merican Journal of Mathematics Statistics 013, 3(1: 1-5 [heredity], [Finite additiv ity] generalized intuitionistic Fuzzy Ideal is called a - generalized intuitionistic fu zzy ideal if { j } j Ν, j implies (countable additivity The smallest largest generalized intuitionistic fu zzy ideals on a non -empty set X are { 0 } GIFSs on X lso, GIF f, GI F c are denoting the generalized intuitionistic fuzzy ideals (GIFS for short of fuzzy subsets having finite countable support of X respectively Moreover, if is a nonempty GIFS in X, then is an GIF on X This is called the principal GIF of all IFSs of denoted by GIF Remark 31 i If 1, then is called generalized intuitionistic fuzzy proper ideal ii If 1, then is called generalized intuitionistic fuzzy improper ideal i B B ii B B { B GIFS : B } iii Example31 et,, X = { x,1,0 O = { x,0,1 = x,0,04 B = x,05,0 6 D = x,05,03, then the family = { O,, B, D} Example33 et X = a, b, c, d, e = x, µ, ν given by : of GIFSs is an GIF on X µ ( x ν ( x µ ( x ν (x a b c d e { O } =, Then the family GIF is an GIF on X Definition 3 et 1 be two GIFs on X Then is said to be finer than 1 or 1 is coarser than if 1 If also 1 Then is said to be strictly finer than 1 or 1 is strictly coarser than Two GIFs said to be comparable, if one is finer than the other The set of all GIFs on X is ordered by the relation 1 is coarser than this relation is induced the inclusion in IFSs The next Proposition is considered as one of the useful result in this sequel, whose proof is clear Proposition 31 et j : j J be any non - empty family of generalized intuitionistic fuzzy ideals on a set X σ Then j j are generalized intuitionistic fuzzy ideal on X, where j = µ j, µ j µ {, } j ( x = inf µ i ( x : i J x X ( { ( } ν j x = sup ν x : i J, x X i In fact is the smallest upper bound of the set of the j in the ordered set of all generalized intuitionistic fuzzy ideals on X Remark3 The generalized intuitionistic fuzzy ideal by the single generalized intuitionistic fuzzy set O = x,0,1 : x X is the smallest element of the ordered set of all generalized intuitionistic fuzzy ideals on X Proposition33 GIFS in generalized intuitionistic fuzzy ideal on X is a base of iff every member of contained in Proof(Necessity Suppose is a base of Then clearly every member of contained in (Sufficiency Suppose the necessary condition holds Then the set of generalized intuitionistic fuzzy subset in X contained in coincides with by the Definition 31 Proposition34 For a generalized intuitionistic fuzzy ideal 1 with base, is finer than a fuzzy ideal with base B iff every member of B contained in Proof Immediate consequence of Definitions Corollary31 Two generalized intuitionistic fuzzy ideals bases, B, on X are equivalent iff every member of, contained in B via versa Theorem31 et η = {µ j : j J} be a non empty collection of generalized intuitionistic fuzzy subsets of X Then there exists a generalized intuitionistic fuzzy ideal (η = { IFSs : j } on X for some finite collection { j : j = 1,,, n η} Proof : Clear Remark33 ii The generalized intuitionistic fuzzy ideal (η defined above is said to be generated by η η is called subbase of (η Corollary3 et 1 be an generalized intuitionistic fuzzy ideal on X IFSs, then there is a generalized intuitionistic fu zzy ideal which is finer than 1 such that iff B for each B 1 Theorem3 If an GIFS = { O µ, ν } is an generalized intuitionistic fuzzy ideal on X, then so is _ is an generalized intuitionistic = O µ, µ fuzzy ideal on X Proof Clear Theorem33 GIFS = { O is a µ, ν } generalized intuitionistic fuzzy ideal _ on X iff the intuitionistic fuzzy sets µ, ν are generalized intuitionistic fuzzy ideals on X

3 Salama et al: Generalized Intuitionistic Fuzzy Ideals Topological Spaces 3 Proof et = µ ν be an GIF of X, = x,µ, ν, Ten clearly µ is a fuzzy ideal on X Then ( ( ( = _ ν x = 1ν x = max ν x,0 min 1, ν ( x if ν x = O Then is the smallest generalized intuitionistic fuzzy ideal, Or ν x = 1 then is the largest generalized intuitionistic fuzzy ideal on X Corollary33 GIFS = O, ν is an generalized intuitionistic fuzzy ideal on X iff = _ are O µ, µ = O, _ ν ν, generalized intuitionistic fuzzy ideals on X Proof Clear fro m the defin ition 31 Example34 et X a non empty set GIF on X given by: Then = ( x = { O }, ( x { µ } = O 04,006, 05,05 Theorem34 et = x, µ, ν 1 B = x, µ, where are generalized B, ν B 1 intuitionistic fuzzy ideals on the set X then the generalized intuitionistic fu zzy set B = µ B ( x, ν B ( x on X, 1 µ B ( x = { µ ( x µ B ( x ν B ( x = { ν ( x ν B ( x Definition34 For a GIFTS (X, τ, GIFSs Then is called i Generalized intuitionistic fuzzy dense if cl ( = 1 ii Generalized intuitionistic fuzzy nowhere dense subset if Int (cl(= O iii Generalized intuitionistic fuzzy codense subset if Int ( = O v Generalized intuitionistic fuzzy countable subset if it is a finite or has the some cardinal number iv Generalized intuitionistic fuzzy meager set if it is a generalized intuitionistic fuzzy countable union of generalized intuitionistic fuzzy nowhere dense sets The following important Examples of generalized intuitionistic fuzzy ideals on GIFTS (Xτ Example35 For a GIFTS (X, τ n = { GIFSs : Int (cl( = O is the collection of generalized intuitionistic fuzzy nowhere dense subsets of X It is a simple task to show that n is generalized intuitionistic fuzzy ideal on X Example36 For a IFTS (X,τ m = { IFSs: is a countable union of generalized intuitionistic fuzzy nowhere dense sets} the collection of generalized intuitionistic fuzzy { O 03,06, 03,05 0,05 } { O 03,07, 0,08 } meager sets on X one can deduce that m is generalized intuitionistic fu zzy σ - ideal on X Example37 For a IFTS (X, τ with generalized intuitionistic fuzzy ideal then < τ c > = { IFSs : there exists B τ c such that B} is a generalized intuitionistic fuzzy ideal on X Example38 et f: (X,τ 1 (Y, τ be a function,, J are two generalized intuitionistic fuzzy ideals on X Y respectively Then i f ( = {f (: } is an generalized intuitionistic fuzzy ideal ii If f is in jection Then f -1 (J is generalized intuitionistic fuzzy ideal on X 4 Generalized Intuitionistic Fuzzy local Functions GIFTS Definition41 et (X, τ be an generalized intuitionistic fuzzy topological spaces (GIFTS for short be generalized intuitionistic fuzzy ideal (GIF, for short on X et be any GIFS of X Then the generalized intuitionistic fuzzy local function (,τ of is the union of all generalized intuit ionistic fuzzy points ( IFP, for short C ( α, β such that if U N( C( α, β (, τ = { C( α, β X : U for every U nbd of C( α, β } (, τ is called an generalized intuitionistic fuzzy local function of with respect to τ wh ich it will be denoted by (, τ, or simply ( Example 41 One may easily verify that If = { 0 }, then (, τ = cl(, for any generalized intuitionistic fu zzy set GIFSs on X If, for any on X Theorem41 et ( X,τ be a GIFTS 1, be two generalized intuitionistic fuzzy ideals on X Then for any generalized intuitionistic fuzzy sets, B of X then the following statements are verified i ii 1 (, τ ( 1, τ iii = cl( cl( iv v, vi { all GIFSs on X} then (, τ 0 = = GIFSs B (, τ B (, τ, B = B ( B ( ( B ( ( ( vii = (, τ is generalized intuitionistic fuzzy closed set Proof

4 4 merican Journal of Mathematics Statistics 013, 3(1: 1-5 B i Since B, let p = C α, β then 1 U for every U N( p By hypothesis we get U, then β B ( 1 ii Clearly 1 implies (, τ ( 1, τ as there may be other IFSs which belong to so that for GIFP but may not be contained in ( iii Since for any GIF on X, therefore by (ii Examp le 41, ( ({ O for any } = cl( GIFS on X Suppose ( p ( 1 = C1 α, β cl( 1 So for every U N( p, 1 U O there exists p such that for every = C ( α, β ( 1 U V nbd of p N( p, U Since U V N( p then ( U V which leads to U, for every U N( C( α, β therefore p1 = C( α, β ( ( so cl ( While, the other inclusion follows directly Hence = cl( But the inequality iv The inclusion ( ( ( β C α,β p = C α, ( cl( { O } B ( B fo llo ws directly by (i To show the other implication, let β ( B then for every U N( p, B U, i e ( U ( B U then, we (, U have two cases or the converse, this means that exist U1, U N( C(α, β such that U 1, B U 1, U B U Then ( U1 U B ( U1 U this gives ( B ( U1 U, U1 U N( C(α, β which contradicts the hypothesis Hence the equality holds in various cases vi By (iii, we have ( X,τ et be a GIFTS be GIF on X et us define the generalized intuitionistic fuzzy closure operator for any GIFS of X Clearly, let cl ( = cl ( is a generalized intuitionistic fuzzy operator et be GIFT generated by τ ( cl ie c c τ ( = { : cl ( = } No w = { O } cl ( = = cl( for every generalized intuitionistic fuzzy set So, τ ({ O } = τ gain = cl( B U cl ( = = { all GIFSs on X} (, because = O, fo r every generalized intuitionistic fuzzy set so τ ( is the generalized intuitionistic fuzzy discrete topology on X So we can conclude by Theorem 41(ii τ ( { O ie, for any generalized } = τ ( τ τ intuitionistic fuzzy ideal 1 on X In particular, we have for two generalized intuitionistic fuzzy ideals 1, on X, ( ( 1 τ 1 τ Theorem4 et be two generalized intuitionistic fuzzy topologies on X Then for any generalized intuitionistic fuzzy ideal on X, implies (, τ, for every (, τ 1 τ τ 1 1 τ Proof Clear basis β (,τ for τ ( can be described as follows: β (,τ = { B : τ, B } Then we have the following theorem Theorem 43 τ 1,τ β (,τ = cl = Forms a basis for the generated GIFT of the GIFT τ { B : τ, B } ( X,τ with generalized intuitionistic fuzzy ideal on X Proof Straight forward The relationship between τ τ ( established throughout the following result which have an immediately proof Theorem 44 et τ 1,τ be two generalized intuitionistic fuzzy topologies on X Then for any generalized intuitionistic fuzzy ideal on X, τ1 τ implies τ τ 1 Theorem 45 : et ( Χ, τ be a GIFTS 1, be two generalized intuitionistic fuzzy ideals on X Then for any generalized intuitionistic fuzzy set in X, we have i ( 1, τ = ( 1 (, τ ( ii τ ( 1 = ( τ ( 1 ( ( τ ( ( 1 Proof et β ( 1, τ, this means that there exists U p N( P such that U p ( 1 ie There exists 1 1 such that U p ( 1 because of the heredity of 1, assuming 1 = O Thus we have ( U p 1 = ( U p = 1 therefore ( U p 1 = ( U p = 1 1 Hence

5 Salama et al: Generalized Intuitionistic Fuzzy Ideals Topological Spaces 5 β, τ 1 or β ( 1, τ (, because p must belong to either 1 or but not to both This gives ( 1, τ ( 1 (, τ ( To show the second inclusion, let us assume β ( 1, τ (, This implies that there e xist U p N( P such that U p By the heredity of, if we assume that define 1 Then we have U p Thus, 1, τ 1, τ ( similarly, we can get ( 1, τ ( This gives the other inclusion, which complete the proof Corollary 41 et ( (, intuitionistic fu zzy ideal on X Then i ( 1 ( = U p ( 1 1 ( ( ( ( Χ,τ be a GIFTS with generalized (, τ = (, τ τ ( = ( τ ( ( ( ( ii τ ( 1 = τ ( 1 τ ( Proof Follows by applying the previous statement REFERENCES [1] K tanassov, intuitionistic fuzzy sets, in VSgurev, ed,vii ITKRS Session, Sofia(June 1983 central Sci Techn ibrary, Bulg cademy of Sciences ( 984 [] K tanassov, intuitionistic fuzzy sets, Fuzzy Sets Systems 0( [3] K tanassov, Review new result on intuitionistic fuzzy sets, preprint IM-MFIS-1-88, Sofia, 1988 [4] C Chang, Fuzzy Topological Spaces, J Math nal ppl 4 ( [5] Dogan Coker, n introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Systems 88( [6] Salama S -Blowi, Intuitionistic Fuzzy Ideals Topological Spaces, Journal DVNCED IN FUZZY MTHEMTICS (FM ISSN X Volume (7 no(1 ( [7] Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons Fractals 7( [8] Tapas Kumar Mondal SK Samanta, Generalized intuitionistic fuzzy Sets, Visva-Bharati Santiniketan , WBengal, India [9] - Zadeh, Fuzzy Sets Inform Control 8 (