Hausdorffness on generalized intuitionistic fuzzy filters

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INS 7000 No. of Pages 16, DTD=4.3.1

11 February 2004 Disk usedARTICLE IN PRESS

Information Sciences xxx (2004) xxx–xxx

www.elsevier.com/locate/ins

OOFHausdorffness on generalized

intuitionistic fuzzy filters

Jin Han Park *, Jin Keun Park

Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon, 3-Dong Nam-Gu,

Pusan 608-737, South Korea

Received 14 May 2003; received in revised form 28 January 2004; accepted 2 February 2004

CORR

ECTEDPRAbstract

Based on the notion of generalized intuitionistic fuzzy sets (GIF sets) given by

Mondal and Samanta, the notion of GIF filters is introduced and studied. Also the new

notion of of Hausdorffness is defined on GIF filters and their properties are studied to

some extent.

� 2004 Published by Elsevier Inc.

Keywords: GIF sets; GIF filters; Hausdorff GIF filers; Convergence GIF filterly; GIF

filter continuous; GIF filter open; Quotient GIF filters; Product GIF filters

1. Introduction

The concept of fuzzy sets was introduced by Zadeh [19]. Atanassov [1]

generalized this idea to intuitionistic fuzzy sets, and later there has been much

progress in the study of intuitionistic fuzzy sets by many authors [1–5,9,10,13].

On the other hand, Lowen [14] introduced the concept of fuzzy filter and de-

fined convergence in a fuzzy topological space which enables us to characterize

fuzzy compactness. Many results on fuzzy filter are obtained by De Prada and

Saralegui [11,12] and Ramakrishnan and Nayagam [16]. More recently,

Mondal and Samanta [13] introduced definitions of GIF sets, generalized in-

UN* Corresponding author.

E-mail address: [email protected] (J.H. Park).

0020-0255/$ - see front matter � 2004 Published by Elsevier Inc.

doi:10.1016/j.ins.2004.02.002

mail to: [email protected]

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2 J.H. Park, J.K. Park / Information Sciences xxx (2004) xxx–xxx



tuitionistic fuzzy relations and generalized intuitionistic fuzzy topology and

studied some of their properties.

In this paper, by using GIF sets, we define and study the notion of GIF

filters in Section 3. The notion of Hausdorffness on GIF filters is introducedand studied in Sections 4 and 5. Moreover, we prove that arbitrary product of

Hausdorff GIF filters is Hausdorff GIF filter.

OF

2. Preliminaries

Definition 2.1 [1]. Let X be a nonempty fixed set. An intuitionistic fuzzy set A in

X is an object having the form

A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg

ECTEDPROwhere the functions lA : X ! ½0; 1� and cA : X ! ½0; 1� denote the degree of

membership (namely lAðxÞ) and the degree of nonmembership (namely cAðxÞ)of each x 2 X to the set A, respectively, and 06 lAðxÞ þ cAðxÞ6 1 for each

x 2 X .

Obviously, every fuzzy set fhlAðxÞ; xi : x 2 Xg on X is an intuitionistic fuzzy

set of the form A ¼ fhx; lAðxÞ; 1� lAðxÞi : x 2 Xg. For an intuitionistic fuzzy

set A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg, it is observed that lAðxÞ þ cAðxÞ6 1 for eachx 2 X and hence lAðxÞ ^ cAðxÞ� 1

2for each x 2 X . As for examples, the attri-

butes (i) �beauty’ and �fatty’ (ii) �attentiveness’and �dullness’ (iii) �interior’ and�frontier’ etc. are such that both the attributes are not significant simulta-

neously, but sum of their degrees may exceed 1. Having motivated from the

observation, Mondal and Samanta [13] defined a generalized intuitionistic

fuzzy set as follows:

Definition 2.2. [13]. Let X be a nonempty fixed set. A generalized intuitionisticfuzzy set (GIF set for short) A in X is an object having the form
RA ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg
UNCO

Rwhere the functions lA : X ! ½0; 1� and cA : X ! ½0; 1� denote the degree of

membership (namely lAðxÞ) and the degree of nonmembership (namely cAðxÞ)of each x 2 X to the set A, respectively, and lAðxÞ ^ cAðxÞ6 1

2for each x 2 X .

Thus every intuitionistic fuzzy set is GIF but not conversely.

A GIF set A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg in X can identified to an ordered

pair hlAðxÞ; cAðxÞi in ½0; 1�X � ½0; 1�X or to element in ð½0; 1� � ½0; 1�ÞX . For the

sake of simplicity, we shall use the symbol A ¼ hx; lA; cAi for the GIF set

A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg.

Definition 2.3 [13]. Let A and B be GIF sets in X in the formA ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg and B ¼ fhx;lBðxÞ; cBðxÞi : x 2 Xg. Then

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J.H. Park, J.K. Park / Information Sciences xxx (2004) xxx–xxx 3



(a) A � B iff lAðxÞ6 lBðxÞ and cAðxÞP cBðxÞ for all x 2 X ;(b) A ¼ B iff A � B and B � A;(c) Ac ¼ fhx; cAðxÞ; lAðxÞi : x 2 Xg;(d) A \ B ¼ fhx;lAðxÞ ^ lBðxÞ; cAðxÞ _ cBðxÞi : x 2 Xg;(e) A [ B ¼ fhx; lAðxÞ _ lBðxÞ; cAðxÞ ^ cBðxÞi : x 2 Xg;(f) If fAi : i 2 Jg is an arbitrary family of GIF sets in X , then

\Ai ¼ fhx;^lAiðxÞ;_cAi

ðxÞi : x 2 Xg;[Ai ¼ fhx;_lAi

ðxÞ;^cAiðxÞi : x 2 Xg:

PROO

F(g) 0� ¼ fhx; 0; 1i : x 2 Xg and 1� ¼ fhx; 1; 0i : x 2 Xg.

Remark 2.4.

(a) A ¼ Ac iff lAðxÞ ¼ cAðxÞ for all x 2 X . Thus degree of membership equals

nonmembership for all x 2 X . That is A is maximal fuzzy.

(b) 0� ¼ 1c� and 1� ¼ 0c�.

Definition 2.5 [13]. Let X and Y be two nonempty sets and f : X ! Y be afunction. Let A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg be a GIF set in X and

B ¼ fhy; lBðyÞ; cBðyÞi : y 2 Y g be a GIF set in Y .

(a) The inverse image f �1ðBÞ of B under f is the GIF set in X defined by
EDf �1ðBÞ ¼ fhx; lf�1ðBÞðxÞ; cf�1ðBÞðxÞi : x 2 Xg;
where lf�1ðBÞðxÞ ¼ lBðf ðxÞÞ and cf�1ðBÞðxÞ ¼ cBðf ðxÞÞ for each x 2 X .(a) The image f ðAÞ of A under f is the GIF in Y defined by

UNCO

RRECT

f ðAÞ ¼ fhy; lf ðAÞðyÞ; cf ðAÞðyÞi : y 2 Y g;

where

lf ðAÞðyÞ ¼_x2f�1ðyÞlAðxÞ if f �1ðyÞ 6¼ /;0 otherwise;

�

cf ðAÞðyÞ ¼^x2f�1ðyÞcAðxÞ if f �1ðyÞ 6¼ /;1 otherwise:

�

Now we list the properties of images and preimages, some of which we shall

frequently use in Sections 3 and 4.

Theorem 2.6 [13].Let A and Ai ði 2 JÞ be GIF sets in X and B and Bi ði 2 JÞ beGIF sets in Y and f : X ! Y be a function. Then:

(a) If A1 � A2, then f ðA1Þ � f ðA2Þ;(b) If B1 � B2, then f �1ðB1Þ � f �1ðB2Þ;

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F

(c) A � f �1ðf ðAÞÞ. (If f is injective, then A ¼ f �1ðf ðAÞÞ);(d) f ðf �1ðBÞÞ � B (If f is surjective, then f ðf �1ðBÞÞ ¼ B);(e) f �1ð[BiÞ ¼ [f �1ðBiÞ, f �1ð\BiÞ ¼ \f �1ðBiÞ;(f) f ð[AiÞ ¼ [f ðAiÞ, f ð\AiÞ � \f ðAiÞ (If f is injective, then f ð\AiÞ ¼ \f ðAiÞ);(g) f �1ð1�Þ ¼ 1�, f �1ð0�Þ ¼ 0�;(h) f ð0�Þ ¼ 0�, f ð1�Þ ¼ 1� if f is surjective;(i) f �1ðBÞc ¼ f �1ðBcÞ, f ðAÞc � f ðAcÞ if f is surjective.

Definition 2.7 [13]. Let a; b 2 ½0; 1� and a ^ b6 12. For any x 2 X , a generalized

intuitionistic fuzzy point (GIF point for short) xða;bÞ in X is a GIF set in Xdefined by
Oxða;bÞðyÞ ¼
ða; bÞ if y ¼ x;ð0; 1Þ if y 6¼ x:

�
RR
ECTEDPROIn this case, x is called the support of xða;bÞ and a and b are called the value and

the nonvalue of xða;bÞ, respectively. An GIF point xða;bÞ is said to belong to a

GIF set A ¼ hx; lA; cAi in X , denoted by xða;bÞ 2 A, if a6 lAðxÞ and bP cAðxÞ.This is equivalent to xða;bÞ 6A.

Theorem 2.8. Let A, B be GIF sets in X and xða;bÞ be GIF point in X . Then

(a) A ¼ [fxða;bÞ : xða;bÞ 2 Ag.(b) A � B if and only if xða;bÞ 2 A implies xða;bÞ 2 B.

Proof.(a) is proved in [13].

(b) Let A � B and xða;bÞ 2 A. Then a6 lAðxÞ6 lBðxÞ and bP cAðxÞP cBðxÞ.Thus xða;bÞ 2 B. Conversely, for some x 2 X , let a ¼ lAðxÞ and b ¼ cAðxÞ.Then xða;bÞ is GIF point in X and xða;bÞ 2 A and by hypothesis xða;bÞ 2 B.Then lAðxÞ ¼ a6 lBðxÞ and cAðxÞ ¼ bP cBðxÞ and hence A � B. h

Definition 2.9. Let A ¼ fhx; lAðxÞ; cAðxÞi : x 2 Xg and B ¼ fhx; lBðxÞ; cBðxÞi :x 2 Y g be GIF set in X and Y , respectively. We define the cartesian product ofA and B as GIF set in X � Y defined as follows:

A� B ¼ fhðx; yÞ; lAðxÞ ^ lBðyÞ; cAðxÞ _ cBðyÞi : x 2 X ; y 2 Y g:

UNCO

Theorem 2.10. Let A and B be GIF sets in X and let C and D be GIF sets in Y .Then we have

(a) If A � B, then A� C � B� C and A� C � A� D;(b) ðA \ BÞ � ðC \ DÞ ¼ ðA� CÞ \ ðB� DÞ;

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(c) ðA \ BÞ � C ¼ ðA� CÞ \ ðB� CÞ and ðA [ BÞ � C ¼ ðA� CÞ [ ðB� CÞ.

Proof. Let A ¼ fhx; lA; cAi : x 2 Xg, B ¼ fhx; lB; cBi : x 2 Xg, C ¼ fhy; lC; cCi :y 2 Y g and D ¼ fhy; lD; cDi : y 2 Y g.

(a) Let A � B. Since lAðxÞ6 lBðxÞ and cAðxÞP cBðxÞ for any x 2 X , we obtain

A� C ¼ fhðx; yÞ; lAðxÞ ^ lCðyÞ; cAðxÞ _ lCðyÞi : x 2 X ; y 2 Y g� fhðx; yÞ; lBðxÞ ^ lCðyÞ; cBðxÞ _ cCðyÞi : x 2 X ; y 2 Y g ¼ B� C:
F OSimilarly, A� C � A� D.
(b)

PROðA \ BÞ � ðC \ DÞ ¼ fhx; lA\BðxÞ; cA\BðxÞi : x 2 Xg

� fhy; lC\DðyÞ; cC\DðyÞi : y 2 Y g¼ fhðx; yÞ; lA\BðxÞ ^ lC\DðyÞ; cA\BðxÞ _ cC\DðyÞi : x 2 X ; y 2 Y g¼ fhðx; yÞ; lA�Cðx; yÞ ^ lB�Dðx; yÞ; cA�Cðx; yÞ _ cB�Dðx; yÞi : x 2 X ; y 2 Y g¼ ðA� CÞ \ ðB� DÞ:
D(c) Similar to (b). h E
UNCO

RRECT3. GIF filters

Definition 3.1. A nonvoid family F of GIF sets on X is said to be a fuzzy filter

of GIF sets or a GIF filter if

(a) 0� 62 F,

(b) If A, B 2 F, then A \ B 2 F,

(c) If A 2 F and A � B, then B 2 F.

Definition 3.2. A nonvoid family B of GIF sets is called a GIF filter base if Bdoes not contain 0� and the intersection of any two element of B contains an

element of B. A family S is called a subbase of a GIF filter base if it is

nonempty and the intersection of any finite number of elements of S is not 0�.

Remark 3.3. If S is a subbase of a GIF filter, then the family BðSÞ consistingof all finite intersections of elements of S is a GIF filter base. If B is a GIF

filter base, then the family FðBÞ, consisting of all GIF sets A such that A � Bfor some B 2 B, is a GIF filter. Furthermore, BðSÞ and FðBÞ are uniquely

determined by S and B, respectively.

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OOF

Definition 3.4. The GIF filter FðBÞ and FðBðSÞÞ (or, FðSÞ) are called,

respectively, the GIF filter generated by B and the GIF filter generated by S.

A family B is called a base of the GIF filter F if B is an GIF filter base and

F ¼ FðBÞ. Similarly, S is called a subbase of the GIF filter F if S is asubbase and F ¼ FðSÞ.

Remark 3.5.

(a) Let U be any indexed family of GIF filters on X . Then

(i) \F2UF is also GIF filter.

(ii) [F2UF is also GIF filter if U is directed family of GIF filters under inclu-

sion �.

(b) Let B1 and B2 be two GIF filter bases. Then FðB1Þ � FðB2Þ if and only

if for any B 2 B1 there exists A 2 B2 such that A � B.
R
UNCO

RRECTEDPTheorem 3.6. Let F be a GIF filter on X and Y � X . Then FjY ¼

fF jY : F 2 Fg is a GIF filter on Y if AjY 6¼ 0� for any A 2 F.

Proof.

(i) Since AjY 6¼ 0� for any A 2 F, 0� 62 FjY .(ii) Let AjY , BjY 2 FjY . Then clearly AjY \ BjY ¼ ðA \ BÞjY . Since F is a

GIF filter, A \ B 2 F and hence AjY \ BjY 2 FjY .(iii) Let AjY 2 FjY and B be a GIF set in Y such that AjY � B. Choose a GIF

set C in X such that lCðzÞP lAðzÞ and cCðzÞ6 cAðzÞ for any z 62 Y and

lCðzÞ ¼ lBðzÞ and cCðzÞ ¼ cBðzÞ for any z 2 Y . Then clearly C is GIF set

in X such that C � A and CjY ¼ B. Since A 2 F and F is a GIF filter,C 2 F and hence CjY ¼ B 2 FjY .Hence FjY is a GIF filter on Y . h

Theorem 3.7. Let f : X ! Y be a function and F be a GIF filter on X . Thenf ðFÞ ¼ ff ðF Þ : F 2 Fg is a GIF filter base on Y .

Proof.

(i) Let f ðF Þ 2 f ðFÞ. Then clearly lf ðF ÞðyÞ 6¼ 0 or cf ðF ÞðyÞ 6¼ 1 for at least oney 2 Y . In fact, if lf ðF ÞðyÞ ¼ 0 and cf ðF ÞðyÞ ¼ 1 for all y 2 Y , then lF ðxÞ ¼ 0

and cF ðxÞ ¼ 1 for all x 2 f �1ðyÞ for all y 2 Y . So lF ðxÞ ¼ 0 and cF ðxÞ ¼ 1

for all x 2 X and hence 0� 2 F, which contradicts GIF filterness of F.

Hence f ðF Þ 6¼ 0� and so 0� 62 f ðFÞ.(ii) Let f ðF1Þ, f ðF2Þ 2 f ðFÞ. Since F1, F2 2 F, F1 \ F2 2 F. Then by Theorem

2.6(f), f ðF1 \ F2Þ � f ðF1Þ \ f ðF2Þ.Hence f ðFÞ is a GIF filter base on Y . h

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PROO

F

Theorem 3.8. Let f : X ! Y be a surjection and G be a GIF filter on Y . Thenf �1ðGÞ ¼ ff �1ðGÞ : G 2 Gg is a GIF filter base on X .

Proof.(i) Let f �1ðGÞ 2 f �1ðGÞ. Then clearly lf�1ðGÞðxÞ 6¼ 0 or cf�1ðGÞðxÞ 6¼ 1 for at

least one x 2 X . In fact, if lf�1ðGÞðxÞ ¼ lGðf ðxÞÞ ¼ 0 and

cf�1ðGÞðxÞ ¼ cGðf ðxÞÞ ¼ 1 for all x 2 X , then, by surjection of f , lGðyÞ ¼ 0

and cGðyÞ ¼ 1 for all y 2 Y . Hence G ¼ 0� 2 G, which contradicts G is a

GIF filter. Hence 0� 62 f �1ðGÞ.(ii) Let f �1ðG1Þ, f �1ðG2Þ 2 f �1ðGÞ. Since G is a GIF filter, G1 \ G2 2 G and

hence by Theorem 2.6(e) f �1ðG1Þ \ f �1ðG2Þ ¼ f �1ðG1 \ G2Þ 2 f �1ðGÞ.Hence f �1ðGÞ is a GIF filter base on X . h

Definition 3.9. Let ðX ;F1Þ and ðY ;F2Þ be GIF filters. A function

f : ðX ;F1Þ ! ðY ;F2Þ is called GIF filter continuous with respect to ðF1;F2Þif for every F 2 F2, f �1ðF Þ 2 F1.

Example 3.10. Let X ¼ fa; b; cg and Y ¼ fd; e; f g. Let A and B be GIF sets in

X and Y respectively defined as follows:

EDA ¼ x;aa1

;ba1

;cb1

� �;

aa2

;ba2

;cb2

� �� and

B ¼ x;da1

;eb1

;fd1

� �;

da2

;eb2

;fd2

� �� ;

RRECTwhere a1 ^ a2 6 1

2, b1 ^ b2 6

12and d1 ^ d2 6 1

2. Then clearly B1 ¼ fAg and

B2 ¼ fBg are GIF filter bases on X and Y respectively. Let F1 and F2 be the

GIF filter generated by B1 and B2 respectively. Let f : X ! Y be a function

defined by f ðaÞ ¼ f ðbÞ ¼ d and f ðcÞ ¼ e. Then by definition f �1ðB2Þ ¼ B1 and

hence f is GIF filter continuous.

However, constant function f : ðX ;F1Þ ! ðY ;F2Þ need not be GIF filtercontinuous as shown by the following example.

Example 3.11. Let X ¼ fa; b; cg and Y ¼ fd; e; f g. Let A and B be GIF sets in

X and Y respectively defined as follows:

COA ¼ x;aa1

;ba1

;ca1

� �;

aa2

;ba2

;ca2

� �� and

B ¼ x;da1

;eb1

;fd1

� �;

da2

;eb2

;fd2

� �� ;

UNwhere a1 ^ a2 6 12, b1 ^ b2 6

12, d1 ^ d2 6 1

2, b1 < a1 and b2 > a2. Clearly,

B1 ¼ fAg and B2 ¼ fBg are GIF filter bases on X and Y respectively. Let F1

and F2 be the GIF filter generated by B1 and B2 respectively. Let f : X ! Y

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NCOR

RECTED

PROO

F

be constant function defined by f ðxÞ ¼ e for all x 2 X . Choose ri ði ¼ 1; 2Þ suchthat b1 < r1 < a1 and b2 > r2 > a2. Then clearly C ¼ x; d

a1; er1; fd1

� �;

Dda2; er2; fd2

� �E2 F2 since C � B. But f �1ðCÞ ¼ x; a

r1; br1; cr1

� �; a

r2; br2; cr2

� �D E62 F1.

Hence f is not GIF filter continuous with respect to ðF1;F2Þ.The following statement is an immediate consequence of definitions.

Remark 3.12.

(i) If f : ðX ;FÞ ! ðY ;GÞ and g : ðY ;GÞ ! ðZ;HÞ are GIF filter continu-

ous, then the composition g � f : ðX ;FÞ ! ðZ;HÞ is also GIF filter con-

tinuous.

(ii) If f : ðX ;FÞ ! ðX ;FÞ is identity function, then f is GIF filter continu-

ous.(iii) Let f : ðX ;FÞ ! ðY ;GÞ be a GIF filter continuous function. If Z � X

such that F jZ 6¼ 0� for any F 2 F, then the restriction

f jZ : ðZ;FjZÞ ! ðY ;GÞ is also GIF filter continuous.

Theorem 3.13. A function f : ðX ;FÞ ! ðY ;GÞ is GIF filter continuous if andonly if for every GIF point xða;bÞ in X and every G 2 G such that f ðxða;bÞÞ 2 G,there exists a F 2 F such that xða;bÞ 2 F and f ðF Þ � G.

Proof. Let xða;bÞ be a GIF point in X and G 2 G such that f ðxða;bÞÞ 2 G. Thenf ðxða;bÞÞ ¼ f ðxÞða;bÞ. By GIF filter continuity of f and Theorem 2.6(d),

f �1ðGÞ 2 F and f ðf �1ðGÞÞ � G. Since f �1ðlGÞðxÞP a and f �1ðcGÞðxÞ6b,xða;bÞ 2 f �1ðGÞ. If F ¼ f �1ðGÞ, F satisfies our requirements.

Conversely, let G 2 G and xða;bÞ 2 f �1ðGÞ. Then clearly f ðxÞða;bÞ 2f ðf �1ðGÞÞ � G and hence there exists Fxða;bÞ 2 F such that xða;bÞ 2 Fxða;bÞ and

f ðFxða;bÞ Þ � G. By Theorem 2.6(c), Fxða;bÞ � f �1ðf ðFxða;bÞ ÞÞ � f �1ðGÞ and then

f �1ðGÞ 2 F. Hence f is GIF filter continuous. h

Recall that the characteristic set of a fuzzy filterF with respect to a fuzzy set

A is the set CAðFÞ ¼ fa 2 ½0; 1� : for all F 2 F, there exists x 2 X such that

F ðxÞ > AðxÞ þ ag and the characteristic value of F with respect to A is the

supremum of CAðFÞ [14]. Now we generalize these notions as follows:

Definition 3.14. Let F be a GIF filter on X and A be a GIF set in X . Then the

characteristic set of F with respect to A is given by CAðFÞ ¼ fða; bÞ 2½0; 1� � ½0; 1� : for all F 2 F, there exists x 2 X such that lF ðxÞ > lAðxÞ þ a and

cF ðxÞ þ b < cAðxÞg and cAðFÞ ¼ ðsup a; inf bÞ is the characteristic value of Fwith respect to A, where the supremum and infremum are taken over all

ða; bÞ 2 CAðFÞ, respectively.
U

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F

Definition 3.15. Let f : X ! Y be a function. A GIF set A in X is said to be f -invariant if f ðxÞ ¼ f ðyÞ implies lAðxÞ ¼ lAðyÞ and cAðxÞ ¼ cAðyÞ.

Theorem 3.16. Let f : ðX ;FÞ ! ðY ;GÞ be a GIF filter continuous function. IfGIF set A in X is f -invariant, then CAðFÞ � Cf ðAÞðGÞ.

Proof. Let ða; bÞ 2 CAðFÞ. We claim that ða; bÞ 2 Cf ðAÞðGÞ. Let G 2 G. Since fis GIF filter continuous, f �1ðGÞ 2 F. Then there exists x 2 X such that

lf�1ðGÞðxÞ > lAðxÞ þ a and cf�1ðGÞðxÞ þ b < cAðxÞ. That is, lGðf ðxÞÞ > lAðxÞ þ aand cGðf ðxÞÞ þ b < cAðxÞ. Since A is f -invariant, we have
Olf ðAÞðf ðxÞÞ ¼ _z2f�1ðf ðxÞÞlAðzÞ ¼ lAðxÞ;
cf ðAÞðf ðxÞÞ ¼ ^z2f�1ðf ðxÞÞcAðzÞ ¼ cAðxÞ:
TEDPRO
Then lGðf ðxÞÞ > lf ðAÞðf ðxÞÞ þ a and cGðf ðxÞÞ þ b < cf ðAÞðf ðxÞÞ. Henceða; bÞ 2 Cf ðAÞðGÞ. h

Definition 3.17. Let F and G be GIF filters on X and Y , respectively. Then afunction f : ðX ;FÞ ! ðY ;GÞ is called GIF filter open if for every F 2 F,

f ðF Þ 2 G. In addition, if f GIF filter continuous and bijective, then f is called

GIF filter homeomorphism.

Theorem 3.18. If f : ðX ;FÞ ! ðY ;GÞ is injective and GIF filter open, thenCf ðAÞðGÞ � CAðFÞ for any GIF set A in X .

Proof. Let ða; bÞ 2 Cf ðAÞðGÞ and F 2 F. Then f ðF Þ 2 G and there exists y 2 Ysuch that
C
lf ðF ÞðyÞ > lf ðAÞðyÞ þ a; cf ðF ÞðyÞ þ b < cf ðAÞðyÞ ðÞ
E
UNCO

RRClearly, f �1ðyÞ 6¼ /. In fact, if f �1ðyÞ ¼ /, then lf ðF ÞðyÞ ¼ 0 and hence there is

no a 2 ½0; 1� such that lf ðF ÞðyÞ > lf ðAÞðyÞ þ a, which is a contradiction. Since fis injective, f �1ðyÞ is singleton. Hence lf ðF ÞðyÞ ¼ lF ðf �1ðyÞÞ and so by (*),lF ðf �1ðyÞÞ > lAðf �1ðyÞÞ þ a and similarly, cF ðf �1ðyÞÞ þ b < cAðf �1ðyÞÞ. Hence

ða; bÞ 2 CAðFÞ. h

The following corollaries are immediate from Theorems 3.16 and 3.18.

Corollary 3.19. If ðX ;FÞ and ðY ;GÞ are GIF filter homeomorphic, thenCAðFÞ ¼ Cf ðAÞðGÞ for any GIF set A in X .

Corollary 3.20. If f : ðX ;FÞ ! ðY ;GÞ is an f -invariant GIF filter continuousfunction, then cAðFÞ � cf ðAÞðGÞ for any GIF set A in X .

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4. Hausdorff GIF filters

In crisp theory, it is well-known that A \ B ¼ / () A � Bc for any sets Aand B. But it is no longer valid in fuzzy setting. So Ramakrishnan and Na-yagam [16] chose the notion of fuzzy disjointness that agrees with ordinary set

theoretic disjointness in crisp case as follows: Two fuzzy set A and B in X are

said to intersect if lAðxÞ þ lBðxÞ > 1 for some x 2 X , and A and B are said to be

disjoint if these sets do not intersect. Since, in crisp theory, two disjoint

members cannot be members of a filter, one can not speak about Hausdorffness

on a filter. But in fuzzy setting, they have a previous definition of intersection

such that two disjoint members can be members of a fuzzy filter. So one can

speak about Hausdorffness on fuzzy filters and disjointness of two fuzzy setsand Hausdorffness can extended as follows:

Definition 4.1. Two GIF sets A and B in X are said to intersect at x 2 X if

lAðxÞ þ ð1� cBðxÞÞ > 1 or lBðxÞ þ ð1� cAðxÞÞ > 1. Otherwise A and B do not

intersect at x. A and B are said to be disjoint if these sets do not intersect

anywhere.

Definition 4.2. A GIF filter ðX ;FÞ is called Hausdorff if for any x; y 2 X withx 6¼ y, there exist F1; F2 2 F such that cF1ðxÞ <

12, cF2ðyÞ <

12

and

lF1ðzÞ þ ð1� cF2ðzÞÞ6 1 and lF2ðzÞ þ ð1� cF1ðzÞÞ6 1 for any z 2 X .

Example 4.3. Let X ¼ fa; b; cg and Bi ði ¼ 1; 2; 3; 4Þ be GIF sets in X defined as

follows:

ORRECTB1 ¼ x;

aa;bb;cd

� �;

a1� a

2

;b

1=4;

c1� d

2

!* +;

B2 ¼ x;aa;bb;cd

� �;

a1� a

2

;b

1� b2

;c

1=4

!* +;

B3 ¼ x;aa;bb;cd

� �;

a1=4

;b

1� b2

;c

1� d2

!* +;

B4 ¼ x;aa;bb;cd

� �;

a1� a

2

;b

1� b2

;c

1� d2

!* +:

UNCwhere a; b; d 2 ð0; 1=4Þ. Let F be a GIF filter generated by B ¼ fB1;B2;B3;

B4g. Then clearly ðX ;FÞ is a Hausdorff GIF filter.

Recall that a sequence fxng of ðX ;FÞ is said to converge filterly to x 2 X [16]

if for every F 2 F such that lF ðxÞ > 12, there exists n0 2 N such that lF ðxnÞ > 1

2

for all nP n0, equivalently 1� lF ðxnÞ < 12for all nP n0. Now we extend above

definition to GIF filter as follows:

https://www.researchgate.net/publication/265662535_Hausdorff_interval_valued_fuzzy_filters?el=1_x_8&enrichId=rgreq-bbd42f4347db629858c452235e60c758-XXX&enrichSource=Y292ZXJQYWdlOzIyMDMxMzY5MztBUzo5OTg1Mzg3NTAyMzg4NkAxNDAwODE4NDI4OTE5

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Definition 4.4. Let ðX ;FÞ be a GIF filter. A sequence fxng of X is said to

converge GIF filterly to x (denoted by fxng !gif x), and x is called a GIF limit

of F, if for every F 2 F such that cF ðxÞ < 12, there exists n0 2 N such that

lF ðxnÞ > 12for all nP n0, equivalently 1� cF cðxnÞ < 1

2for all nP n0.

Theorem 4.5. Let f : ðX ;FÞ ! ðY ;GÞ be a GIF continuous function and fxng bea sequence in X . If fxng !gif x, then ff ðxnÞg !gif f ðxÞ.

Proof. Let G 2 G such that cGðf ðxÞÞ < 12. Since f is GIF filter continuous,

f �1ðGÞ 2 F and cf�1ðGÞðxÞ < 12. Since fxng !gif x, there exists n0 2 N such that

lf�1ðGÞðxnÞ > 12for all nP n0. Then lGðf ðxnÞÞ ¼ lf�1ðGÞðxnÞ > 1

2for all nP n0.

Hence ff ðxnÞg !gif f ðxÞ. h

Theorem 4.6. In Hausdorff GIF filter ðX ;FÞ, every GIF filterly convergent se-quence of points of X has exactly one GIF limit.

Proof. Let fxng be a sequence of X . Suppose that fxng converges GIF filterly to

distinct two points x and y of X . Since ðX ;FÞ is Hausdorff, there exist

F1; F2 2 F such that cF1ðxÞ <12, cF2ðyÞ <

12and

lF1ðzÞ þ ð1� cF2ðzÞÞ6 1; lF2ðzÞ þ ð1� cF1ðzÞÞ6 1 for all z 2 X :

NCOR

RECTEDSince fxng !gif x and fxng !gif y, there exist n1; n2 2 N such that lF1ðxnÞ >

12for

all nP n1 and lF2ðxnÞ >12

for all nP n2. Put n0 ¼ maxfn1; n2g. Then

lF1ðxnÞ þ ð1� cF2ðxnÞÞ > 1 for all nP n0, a contradiction. h

Theorem 4.7. Let ðX ;FÞ be a Hausdorff GIF filter and Y � X . If AjY 6¼ 0� forany A 2 F, then ðY ;FjY Þ is also Hausdorff GIF filter.

Proof. By Theorem 3.6, ðY ;FjY Þ is an GIF filter on Y . Now we shall show that

ðY ;FjY Þ is Hausdorff. Let y1; y2 2 Y such that y1 6¼ y2. Since ðX ;FÞ is Haus-

dorff, there exist F1; F2 2 F such that cF1ðy1Þ <12, cF2ðy2Þ <

12

and

lF1ðzÞ þ ð1� cF2ðzÞÞ6 1 and lF2ðzÞ þ ð1� cF1ðzÞÞ6 1 for all z 2 X . Then

cF1jY ðy1Þ <12, cF2jY ðy2Þ <

12and lF1jY ðzÞ þ ð1� cF2jY ðzÞÞ6 1 and lF2jY ðzÞ þ ð1�

cF1jY ðzÞÞ6 1 for all z 2 Y . Hence ðY ;FjY Þ is Hausdorff. h

Theorem 4.8. Let f : ðX ;FÞ ! ðY ;GÞ be a bijective GIF filter open function. IfðX ;FÞ is a Hausdorff GIF filter, then ðY ;GÞ is Hausdorff GIF filter.

Proof. Let y1; y2 2 Y such that y1 6¼ y2. Then there exist uniquely x1; x2 2 X such

that x1 6¼ x2, f ðx1Þ ¼ y1 and f ðx2Þ ¼ y2. Since ðX ;FÞ is Hausdorff, there exist

F1; F2 2 F such that cF1ðx1Þ <12, cF2ðx2Þ <

12and
UlF1ðzÞ þ ð1� cF2ðzÞÞ6 1; lF2ðzÞ þ ð1� cF1ðzÞÞ6 1 for all z 2 X :

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ECTEDPROO

F

Since f is GIF filter open, we have f ðF1Þ; f ðF2Þ 2 G. Clearly, cf ðF1Þðy1Þ ¼cF1ðx1Þ <

12and cf ðF2Þðy2Þ ¼ cF2ðx2Þ <

12. Now suppose that there exists z 2 Y such

that lf ðF1ÞðzÞ þ ð1� cf ðF2ÞðzÞÞ > 1. By hypothesis, there exists a unique x 2 Xsuch that f ðxÞ ¼ z. Hence lF1ðxÞ þ ð1� cF2ðxÞÞ > 1, a contradiction. h

Lemma 4.9. Let F be a GIF filter on X and f : ðX ;FÞ ! Y be a surjection.Then G ¼ fG 2 ð½0; 1� � ½0; 1�ÞX jf �1ðGÞ 2 Fg is a GIF filter on Y .

Proof.

(i) Clearly 0� 62 G. Otherwise, by Theorem 2.6(g) f �1ð0�Þ ¼ 0� 2 F contra-

dicts 0� 2 F.

(ii) Let G1;G2 2 G. Then f �1ðG1Þ; f �1ðG2Þ 2 F and hence by Theorem 2.6(e)

f �1ðG1 \ G2Þ ¼ f �1ðG1Þ \ f �1ðG2Þ 2 F. Hence G1 \ G2 2 G.(iii) Let G 2 G and G � F . By definition of G and Theorem 2.6(b), f �1ðGÞ 2 F

and f �1ðGÞ � f �1ðF Þ and then f �1ðF Þ 2 F. Hence F 2 G. h

Definition 4.10. The GIF filter defined in above lemma is called Quotient GIF

filter determined by the surjective function f .Suppose ðX ;FÞ and ðY ;HÞ are GIF filters and f : ðX ;FÞ ! ðY ;HÞ is

surjection. The following theorem gives conditions on f that make H equal to

the quotient GIF filter G on Y determined by f .

Theorem 4.11. Let ðX ;FÞ and ðY ;HÞ be GIF filters, f : ðXFÞ ! ðY ;HÞ be asurjective GIF filter continuous function and let G be the quotient GIF filter on Ydetermined by f . If f is GIF filter open, then G ¼ H.

Proof. Suppose that f is GIF filter open. Since G is the finest GIF filter for

which f is GIF filter continuous, H � G. Let G 2 G. Then f �1ðGÞ 2 F by the

definition of G. Since f is GIF filter open, G ¼ f ðf �1ðGÞÞ 2 H. Hence

G � H. h

Theorem 4.12. Let f : ðX ;FÞ ! Y be bijective function and G be the quotientGIF filter on Y determined by f . If ðX ;FÞ is a Hausdorff GIF filter, then ðY ;GÞis Hausdorff GIF filter.

Proof. To prove ðY ;GÞ is a Hausdorff GIF filter, it is enough to prove f is a

GIF filter open function. Let F 2 F. Since f is bijective, f �1ðf ðF ÞÞ ¼ F 2 Fand then f ðF Þ 2 G. Hence f is GIF filter open and, by Theorem 4.8, ðY ;GÞ isHausdorff GIF filter. h

UN

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RRECTEDPROO

F

5. Product GIF filters

The definition of the product GIF filter is motivated by the fact that it is the

smallest GIF filter onQ

a2K Xa such that each projection map is GIF filtercontinuous (Theorem 5.2), and it is given in terms of a GIF filter subbase.

Definition 5.1. Let fðXa;FaÞ : a 2 Kg be an indexed family of GIF filters and

let pa :Q

a2K Xa ! Xa be projection map defined by paððxaÞÞ ¼ xa. For each

a 2 K, let Sa ¼ fp�1a ðFaÞ : Fa 2 Fag and let S ¼ [a2KSa. Then S is a subbase

of a GIF filter F onQ

a2K Xa, and F is called the product GIF filter and de-

noted byQ

Fa.

Theorem 5.2. Let fðXa;FaÞ : a 2 Kg be indexed family of GIF filters. Then theproduct GIF filter

QFa is the weakest GIF filter on

Qa2K Xa for which each

projection map pb :Q

a2K Xa ! Xb is GIF filter continuous.

Proof. Let F be the weakest GIF filter onQ

a2K Xa such that each projection

map is GIF filter continuous. We show thatQ

Fa � F by showing that each

member of the subbase S (given in the definition of the product GIF filter) ofQFa is a member of F. Let S 2 S. Then there exists b 2 K and a member Fb

ofFb such that S ¼ p�1b ðFbÞ. SinceF is an GIF filter on

Qa2K Xa such that pb is

GIF filter continuous, p�1b ðFbÞ 2 F. Hence

QFa � F.

By definition, each projection map is GIF filter continuous with respect toQFa. Since F is the weakest GIF filter such that each projection map is GIF

filter continuous, F �Q

Fa. h

Remark 5.3. Let ðX1;F1Þ and ðX2;F2Þ be GIF filters and let X ¼ X1 � X2.

Then the product GIF filter F1 �F2 is generated by B ¼ fF � G : F 2F1;G 2 F2g.

Proof. First we check B forms a GIF filter base.

(i) Let F � G 2 B. Since F 2 F1 and G 2 F2, we see F 6¼ 0� and G 6¼ 0� and

so there exist x1 2 X1 and x2 2 X2 such that lF ðx1Þ 6¼ 0 and lGðx2Þ 6¼ 0.

Then lF�Gðx1; x2Þ ¼ lF ðx1Þ ^ lGðx2Þ 6¼ 0 and thus F � G 6¼ 0�. Hence

0� 62 B.(ii) Let F1 � G1, F2 � G2 2 B. By Theorem 2.10, ðF1 � G1Þ \ ðF2 � G2Þ ¼

ðF1 \ F2Þ � ðG1 \ G2Þ and hence B forms a GIF filter base.

We know that S ¼ fp1ðF Þ : F 2 F1g [ fp2ðGÞ : G 2 F2g forms a subbase

for the product GIF filter. Let D be the family of all finite intersections of

members of S. Clearly, p�11 ðF1Þ \ p�1

1 ðF2Þ \ \ p�11 ðFnÞ ¼ p�1

1 ðF1 \ F2 \ \FnÞ 2 D, where Fi 2 F1, i ¼ 1; 2; . . . ; n. Similarly, p�1

2 ðG1Þ \ p�12 ðG2Þ \ \

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RECTED

PROO

F

p�12 ðGnÞ ¼ p�1

2 ðG1 \ G2 \ \ GnÞ 2 D, where Gi 2 F2, i ¼ 1; 2; . . . ; n. Now

we claim that p�11 ðF Þ \ p�1

2 ðGÞ ¼ F � G. Since p�11 ðF Þ ¼ F � 1� and p�1

2 ðGÞ ¼1� � G, we have p�1

1 ðF Þ ^ p�12 ðGÞ ¼ ðF � 1�Þ ^ ð1� � GÞ ¼ F � G. Hence

D ¼ B. Thus the product GIF filter F1 �F2 is generated byB ¼ fF � G : F 2 F1;G 2 F2g. h

Theorem 5.4. Let ðX1;F1Þ and ðX2;F2Þ be GIF filters, let X ¼ X1 � X2 and letF1 �F2 be the product GIF filter on X . Then for i ¼ 1; 2, the projection mappi : ðX ;FÞ ! ðXi;FiÞ is GIF filter open.

Proof. It is clear from Remark 5.3. h

Theorem 5.5. Arbitrary product of Hausdorff GIF filters is Hausdorff GIF filter.

Proof. Let fðXa;FaÞ : a 2 Kg be indexed family of GIF filters and letQ

Fa be

the product GIF filter onQ

Xa in which each pa : ðQ

Xa;Q

FaÞ ! ðXa;FaÞ isGIF filter continuous. Then we know that S ¼ fp�1

a ðFaÞ : Fa 2 Fag forms a

subbase for the product GIF filter.

To prove that ðQ

Xa;Q

FaÞ is Hausdorff GIF filter, we consider any points

x ¼ ðxaÞ, y ¼ ðyaÞ inQ

Xa with x 6¼ y. Then there exists b 2 K such that xb 6¼ yb.Since ðXb;FbÞ is Hausdorff GIF filter, there exist Fb;Gb 2 Fb such that

cFbðxbÞ <12, cGb

ðybÞ < 12, lFbðzbÞ þ ð1� cGb

ÞðzbÞ6 1 for all zb 2 Xb and

lGbðzbÞ þ ð1� cFbÞðzbÞ6 1 for all zb 2 Xb ðÞ.Clearly p�1

b ðFbÞ and p�1b ðGbÞ are elements of

QFa. Now cp�1

bðFbÞðxÞ ¼

p�1b ðcFbÞðxÞ ¼ cFbðpbðxÞÞ ¼ cFbðxbÞ <

12and similarly cp�1

bðGbÞðxÞ <

12. Now we

claim that lp�1b

ðFbÞðzÞ þ ð1� cp�1b

ðGbÞðzÞÞ6 1 for all z 2Q

Xa. Suppose that

lp�1b


ðGbÞðzÞÞ > 1 for some z 2Q

Xa. By definition,

lp�1b


ðGbÞðzÞÞ ¼ p�1b ðlFbÞðzÞ þ ð1� p�1

b ðcGbÞðzÞÞ ¼ lFbðpbðzÞÞþ ð1�

cGbðpbðzÞÞÞ ¼ lFbðzbÞ þ ð1� cGb

ðzbÞÞ > 1, where zb 2 Xb, a contradiction for

ðÞ. Similarly, lp�1b

ðGbÞðzÞ þ ð1� cp�1b

ðFbÞðzÞÞ6 1 for all z 2Q

Xa. Hence

ðQ

Xa;Q

FaÞ is Hausdorff GIF filter. h

Theorem 5.6. Let ðX ;FÞ be an GIF filter. Let R be an equivalence relation on Xand let X=R denote the collection of disjoint equivalent classes and letp : X ! X=R be the identification map and QðFÞ be the quotient GIF filter onX=R determined by p. If 1R�c 2 F�F and p is GIF filter open, thenðX=R;QðFÞÞ is Hausdorff GIF filter.

Proof. Let pðxÞ 6¼ pðyÞ 2 X=R. Then x is not related to y and by assumptionl1R�cðx; yÞ ¼ 1 and c1R�cðs; tÞ ¼ 0. Since 1R�c 2 F�F, by Remark 5.3 there

exists F � G 2 B such that F � G � 1R�c.
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We can choose F � G 2 B such that F � G � 1R�c and lF�Gðx; yÞ > 0. For,

suppose that lF�Gðx; yÞ ¼ 0 such that F � G � 1R�c. Then lF ðxÞ ¼ 0 or

lGðyÞ ¼ 0 or both are zero. Choose F1 such that lF1ðzÞ ¼ lF ðzÞ if z 6¼ x and

lF1ðxÞ > 0 and cF1ðzÞ ¼ cF ðzÞ if z 6¼ x and cF1ðxÞ < minð1� lF1ðxÞ; cF ðxÞÞ. ThenF1 � F and hence F1 2 F. Similarly, choose G1 2 F such that G � G1. Hence

F1 � G1 2 B such that lF1�G1ðx; yÞ > 0 and F1 � G1 � 1R�c.

Now we define a GIF set F2 as follows: lF2ðzÞ ¼ lF ðzÞ if lF ðzÞ > 12or

lF ðzÞ ¼ 0 and lF2ðzÞ >34otherwise and cF2ðzÞ ¼ cF ðzÞ if cF ðzÞ < 1

4or cF ðzÞ ¼ 1

and cF2ðzÞ ¼18otherwise. Then F � F2 and hence F2 2 F. Similarly, we can

define G2 2 F such that G � G2.

Since F � G � 1R�c, lF�Gðs; tÞ þ ð1� c1R�ðs; tÞÞ6 1 for all ðs; tÞ 2 X � X .

For, if s is related to t, then lF2�G2ðs; tÞ ¼ 0 and cF2�G2

ðs; tÞ ¼ 1. Also, if s is notrelated to t, then l1R�ðs; tÞ ¼ 0 and c1R�ðs; tÞ ¼ 1. Hence lF2�G2

ðs; tÞ þ ð1�c1R�ðs; tÞÞ6 1 for any ðs; tÞ 2 X � X . Similarly, l1R�ðs; tÞ þ ð1� cF2�G2

ðs; tÞÞ6 1

for any ðs; tÞ 2 X � X .

Since p is GIF filter open, pðF2Þ; pðG2Þ 2 QðFÞ. Clearly, cpðF2ÞðpðxÞÞ ¼p�ðcF2ÞðpðxÞÞ ¼ inf z2p�1ðpðxÞÞ cF2ðzÞ6 cF2ðxÞ <

12, since by our choice, cF ðxÞ < 1

2.

Similarly, cpðG2ÞðpðyÞÞ <12.

It is enough to prove that lpðF2ÞðpðzÞÞ þ ð1� cpðG2ÞðpðzÞÞÞ6 1 for all z 2 X .

Suppose that lpðF2Þðpðz0ÞÞ þ ð1� cpðG2Þðpðz0ÞÞÞ > 1 for some z0 2 X . ThenlpðF2Þðpðz0ÞÞ > 0 and cpðG2Þðpðz0ÞÞ < 1 and hence there exist z1; z2 2 p�1ðpðz0ÞÞsuch that lF2ðz1Þ > 0 and cG2

ðz2Þ < 1. Since z1 is related to z2, cF2�G2ðz1; z2Þ <

c1R�cðz1; z2Þ, a contradiction. Similarly, lpðG2ÞðpðzÞÞ þ ð1� cpðF2ÞðpðzÞÞÞ6 1 for

all z 2 X . Hence the theorem is proved. h
T OR
REC6. Conclusion

In crisp theory, one cannot speak about Hausdorffness on a filter since two

disjoint members cannot be members of a filter. But in fuzzy setup, we have a

definition of intersection such that two disjoint members can be members of a

GIF filter and hence one can speak about Hausdorffness on GIF filters. Thus

we may stress the important of Hausdorffness on GIF filters and the possible

applications in fuzzy convergence theory. Future research will continue withinvestigation of some properties of GIF filters with strong separation axioms

such as regularity.
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References

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1983 Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984).

[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. System 20 (1986) 87–96.

[3] K. Atanassov, Review and new results on intuitionistic fuzzy sets, Preprint IM-MFAIS-1-88,

Sofia, 1988.

[4] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Set. System 61

(1994) 137–142.

[5] K. Atanassov, S. Stoeva, Intuitionistic fuzzy sets, in: Polish Symp. on Interval & Fuzzy

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UNCO

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Hausdorffness on generalized intuitionistic fuzzy filters

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