the journal of fuzzy mathematics · the journal of fuzzy mathematics vol. 23, no. 1, 2015 1 los...

The Journal of Fuzzy Mathematics

Volume 23 Number 1 2015

MATHEMATICSMATHEMATICSFUZZYFUZZYTHE JOURNAL OFTHE JOURNAL OF

Editor-in-ChiefHu Cheng-ming

INTERNATIONAL FUZZY MATHEMATICS INSTITUTE

Los Angeles USA

THE JOURNAL OF FUZZY MATHEMATICS

Volume 23 Number 1, March 2015

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The Journal of Fuzzy Mathematics Vol. 23, No. 1, 2015 1 Los Angeles

__________________________ Received February, 2013 * This work is supported by the research award fund of Shandong province middle aged and young

scientists (BS2010SF004). 1066-8950/15 $8.50

© 2015 International Fuzzy Mathematics Institute Los Angeles

On (enriched) L -fuzzy Topologies: Decomposition Theorem *

Guopeng Wang, Lingqiang Li, GuangwuMeng and Qingxue Su

School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China

Email address: [email protected]

Abstract: To our knowledge, the existed decomposition theorems for L -fuzzy topologies are

only available in the case that the lattices to be completely distributive complete lattice. In this paper, considering L to be an arbitrary complete Heyting algebra, a decomposition theorem for (enriched) L -fuzzy topologies is presented. Said precisely, it is proved that an (enriched) L -fuzzy topology can be represented as a family of (stratified) L -topologies with some left-continuous condition.

Key words: Enriched L -fuzzy topology, stratified L -topology, decomposition theorem

1. Introduction

The notion of fuzzy topologies was first initiated by Chang [2]. He defined a fuzzy topology on a set X as a crisp subset of I -power set XI . Later, Goguen [4] generalized this notion from I to arbitrary complete lattice L . Thus this kind of fuzzy topology is often called Chang-Goguen L -topology, or L -topology for short. In a completely different direction, HÖhle [5] defined a fuzzy topology on a set X as a fuzzy subset of the power set 2X . Ying [11] studied HÖhle’s fuzzy topology from a logical point of view and called it fuzzifying topology Kubiak [7] and Šostak [9] extended HÖhle’s definition and defined a fuzzy topology on a set X as a fuzzy subset of the L -power set XL . This kind of fuzzy topology is usually called L -fuzzy topology. The decomposition theorems for L -fuzzy topologies are important content in the theory of fuzzy topologies. When L to be a completely distributive complete lattice, we obtain the satisfying results. That is, an L -fuzzy topology can decompose a family of L -topologies with some additional condition [3,12,13]. But, to our regret, these decompose theorems can not be generalized to the more general lattice contexts. In this paper, we

2 Guopeng Wang, Lingqiang Li, Guangwu Meng and Qingxue Su

shall present a decomposition theorem for L -fuzzy topology in the case of that L to be complete Heyting algebra. Said precisely, we shall prove that an (enriched) L -fuzzy topology can decompose a family of (stratified) L -topologies with some left-continuous condition. The enriched L -fuzzy topology and stratified L -topology means they contain all the constant-valued fuzzy sets.

Let L to be a complete lattice and ,a b LŒ . Then we say “ a is wedge below b ” (resp., “ a is way below b ”), in symbol a b (resp., a b ) if for each (directed) subset B LÕ , b B£ � , implies a d£ for some d BŒ . For a completely distributive complete lattice L , it possesses the following properties:

(1) a b implies that there exists c LŒ such that a c b ; a b implies that there exists c LŒ such that a c b .

(2) { } { }: :a b b a b b a= =� � .

Let d be a nonempty subset of XL , then it is said to be an L -topology on X if d satisfies the following conditions:

(T1) 0,1 dŒ .

(T2) ( ),A B A Bd dŒ fi Ÿ Œ .

(T3) t T" Œ , t t T tA Ad dŒŒ fi Œ� . The pair ( ),X d is called an L -topological space. Then d is said to be stratified if

for any a LŒ , we have a dŒ . Let ( ),X d and ( ),Y e be L -topological spaces. The mapping :f X YÆ is said to

be a continuous mapping from ( ),X d to ( ),Y e if ( ) ( )Lf fl l d¨ = Œ for any l eŒ . We use the symbol L -Top to dente the category composed by L -topological space and continuous mapping.

Let : XL Lt Æ be a fuzzy subset of XL . Then we call t an L -fuzzy topology on XL if t satisfies the following conditions:

(FT1) ( ) ( )0 1 1X Xt t= = .

(FT2) ( ) ( ) ( )t l m t l t mŸ ≥ Ÿ .

(FT3) ( ) ( )t T t t T tt l t lŒ Œ≥� � .

The pair ( ),X t is called an L -fuzzy topological space. Then t is said to be

enriched if it further satisfies: (FT4) a L" Œ , ( ) 1at = .

Let ( ),X d and ( ),Y e be L -fuzzy topological spaces. The mapping :f X YÆ is

said to be a continuous mapping from ( ),X d to ( ),Y e if ( ) ( )( )Lfs l t l¨£ for any YLl Œ . Let -FTopL denote category whose object are L -fuzzy topological spaces and

morphisms are continuous mapping.

2. The main conclusions

On (enriched) L -fuzzy topologies: Decomposition Theorem 3

Let ( )LT X (resp., ( )LFT X ) denotes all L -topologies (resp., L -fuzzy topologies)

on a nonempty set X . Then ( )LT X (resp., ( )LFT X ) constitute a complete lattice under the inclusion order (i.e., point-by-point order).

Definition 1.1. Let X be a nonempty set. Then the mapping ( ):L LT XG Æ is called a layer L -fuzzy topology if it satisfies the following conditions:

(1) ( ) ( )a b b a£ fi G Õ G ,

(2) ( )0G is discrete L -topology.

The pair ( ),X G is called a layer L -fuzzy topological space. Further, if each ( )aG is stratified L -topology then G is called stratified layer L -fuzzy topology.

A layer (stratified) L -fuzzy topological space ( ),X G is said to be left continuous if it additionally satisfies the following condition:

(3) ( )al ŒG iff there exists B LÃ such that ( )bl ŒG and B a≥� for any b BŒ .

Let ( )CLT X denote all the layer L -fuzzy topology on X .

Let ( )CLT XG Œ , ( )CLT YXŒ . A mapping :f X YÆ is said to be a continuous

mapping between layer L -fuzzy topologies ( ),X G and ( ),Y X if f is continuous

between L -topological spaces ( )( ),X aG and ( )( ),Y aX for any a LŒ . Denote -CTopL as category constituting by all the above objects and continuous mapping.

Proposition 1.1. Let ( ),X t be an L -fuzzy topological space. Then for any a LŒ ,

the set ( ){ }:Xa L at l t l= Œ ≥ is an L -topology on X . In addition, for the mapping

( ):L LT XtG Æ defined by a L" Œ , ( ) ( ){ }:Xaa L at t l t lG = = Œ ≥ , we have

( )CLT XtG Œ .

Proof. It is sufficient to check that tG satisfies the conditions (1)-(3) in Definition 1.1.

(1) Let ,a b LŒ , a b£ . By the definition of tG , we have ( ) aat tG = , ( ) bbt tG = .

Then it follows that b at tÕ and so, ( ) ( )b at tG Õ G .

(2) It is easily seen that ( ) ( ){ }00 : 0X XL Lt t l t lG = = Œ ≥ = , i.e. ( )0tG is the discrete L -topology on X .

(3) Necessity. Note that if ( )atl ŒG , then al tŒ , i.e. ( ) at l ≥ . So, the left-

continuous condition holds by taking { }B a= . Sufficiency: Suppose B LÃ satisfies the condition (3) of Definition 1.1. For all

b LŒ , ( )btl ŒG , we have ( ) bt l ≥ , then ( ) { }:b b B at l ≥ Œ ≥� . It follows that

( )atl ŒG by the definition of tG .

A combination of the above, we have ( )CLT XtG Œ .

4 Guopeng Wang, Lingqiang Li, Guangwu Meng and Qingxue Su

Proposition 1.2. If the mapping ( ) ( ): , ,f X Yt sÆ is continuous in -FTopL , then

( ) ( ): , ,f X Yt sG Æ G is continuous in -CTopL .

Proof. Note that we need to prove the following inclusion a L" Œ , ( )a fsl lŒG fi

( )atŒG . Actually, for each ( )asl ŒG , by the continuity of the mapping ( ): ,f X t Æ

( ),Y s , we obtain that ( ) ( )f at l s l≥ ≥ , i.e., afl tŒ . Then it follows that fl

( )atŒG by ( ) aat tG = .

Proposition 1.3. Let ( )CLT XG Œ . Then the mapping : XL Lt G Æ defined by

XLl" Œ , ( ) ( ){ }:a L at l lG = Œ ŒG� ,

is an L -fuzzy topology on X .

Proof. (FT1) For any a LŒ , then ( )0 ,1X X aŒG , thus ( ) ( )1 0 1X Xt tG G= = .

(FT2) Let , XLl m Œ . Because ( ) ( )a b b a£ fi G Õ G , then ( ) ( ) ( { :at l t mG GŸ = �

( )}) ( ){ }( ):a b bl mŒG Ÿ ŒG� ( ) ( ){ }: ,a b a bl m= Ÿ ŒG ŒG� { (: ,a b al m£ Ÿ ŒG�

)}bŸ ( ){ }:a b a bl m£ Ÿ Ÿ ŒG Ÿ� ( ){ } ( ):c cl m t l mG= Ÿ ŒG = Ÿ� , where, the

second inequality holds because ( )a bG Ÿ is an L -topology, thus it is closed w.r.t. finite intersections.

(FT3) For any { }: Xt t T Ll Œ Õ , let ( ) ( ){ }( ):t t t t

t T t Ta a at l lG

Œ Œ= = ŒG� � � . Then

for any t TŒ , we get ( ){ }:t t ta a al ŒG ≥� , and thus ( )t al ŒG by Definition 1.1 (3).

Furthermore, because ( )aG is an L -topology, thus ( )t T alŒ ŒG� . By the definition of

t G we obtain ( )t tt Tt T

at l t lG

ŒŒ

Ê ˆ ≥ =Á ˜Ë ¯� � .

Proposition 1.4. If the mapping ( ) ( ): , ,f X YG Æ X is continuous in -CTopL , then

( ) ( ): , ,f X Yt tG XÆ is continuous in -FTopL .

Proof. Suppose ( ) bt lX = , for any YLl Œ , then ( )bl ŒX . For f is continuous, so

( )f bl ŒG , then ( ) ( )f bt l t lG X≥ = , i.e. ( ) ( ): , ,f X Yt tG XÆ is continuous.

Proposition 1.5 (Decomposition theorem). Let ( )CLT XG Œ , ( )LFT Xt Œ , then

(1) t

t tG = ,

(2) t G

G = G , i.e. there exists a one-to-one correspondence between ( )CLT X and

( )LFT X .

On (enriched) L -fuzzy topologies: Decomposition Theorem 5

Proof. (1) For all XLl Œ , we have ( ) ( ){ } { ( ): :a L a a Lt tt l l t lG = Œ ŒG = Œ� �

} ( )a t l≥ =

(2) Note that we only need to check ( ) ( )a at G

G = G for any a LŒ . For one thing

( ) ( ) ( ){ } ( ){ } ( ): :a

a L a L a at t l t l l lG G GG = = Œ ≥ Œ ŒG = G . For another, for

any ( )a

l t GŒ , one has ( ) ( ){ }:a b L bt l lG£ = Œ ŒG� . According to Definition 1.1

(3), we have ( )al ŒG . A combination of the above we have ( ) ( )a at G

G = G .

Corollary 1.1. Categories -CTopL isomorphic to -FTopL .

( )( ) ( ), ,X X t GF G = ; ( )( ) ( )1 , ,X X tt-F = G .

Remark. Some decomposition theorems have been presented in [3,12,13] in the case of L being completely distributive lattice. It is easily seen that these theorems rely on the properties (1) and (2) posed by completely distributive lattice. Thus it seems that those decomposition theorems can not be extended to the more general lattice contexts. In addition, we can obtain a similar decomposition theorem for the enriched L -fuzzy topologies.

References

[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New York, (1990).

[2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [3] J. M. F, Categories isomorphic to -FTOPL , Fuzzy Set and Systems, 157, (2006), 820-831.

[4] J. A. Goguen, The fuzzy tychonoff theorem, J. Math. Ana. Appl., 43 (1973), 734-742. [5] U. Hohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems, 8 (1982), 63-69. [6] U. Hohle and A. P. Sostak, Mathematics of fuzzy sets, Logic, Topology and measure theory, boston:

Kluwer academic publishers, (1999). [7] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam mickiewicz, Poznan, Poland, (1985). [8] Y. Liu and M. Luo, Fuzzy topology, World scientific publishing, Singapore, (1997). [9] A. P. Sostak, On a fuzzy topological structure, Suppl. Rend. Cirecc. Mat. Palermo Ser., II, 11 (1985),

89-103. [10] G. Wang, Theory of L -fuzzy topological spaces, Shaanxi normal university press, Xi’an, (1988), (in

Chinese). [11] M. Ying, A new approach to fuzzy topology (I), Fuzzy Sets and Systems, 39 (1991), 303-321. [12] D. X. Z, On the relationship between several basil category in fuzzy topology, Questions

Mathematicae, 25 (2002), 289-301. [13] J. Z, F. G. S and C. Y. Z, On L -fuzzy topological spaces, Fuzzy Sets and Systems, 149 (2005), 473-

484.


__________________________ Received April, 2013

1066-8950/15 $8.50 © 2015 International Fuzzy Mathematics Institute

Los Angeles

On Pairwise Intuitionistic Fuzzy Resolvable (Irresolvable) Spaces

K. Biljana

Institute of Mathematics, Faculty of Mathematics and Natural Sciences, University St. Cyril and Methodius, P. O. 162, 1000 Skopje, Macedonia

E-mail address: [email protected]

N. Rajesh

Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India. E-mail address: [email protected]

V. Vijayabharathi

Department of Mathematics, National Institute of Technology, Tiruchi-Rappalli, Tamilnadu, India.


Abstract: In this paper, we introduce and study the concepts of intuitionistic fuzzy resolvability,

intuitionistic fuzzy irresolvability and intuitionistic fuzzy open hereditarily irresolvability in intuitionistic fuzzy bitopological spaces.

Key words and phrases: Intuitionistic fuzzy bitopology, pairwise intuitionistic fuzzy resolvable spaces.

1. Introduction

After the introduction of fuzzy sets by Zadeh [5], there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them. Using the notion of intuitionistic fuzzy sets, Coker [3] introducted the notion of intuitionistic fuzzy topological spaces. In this paper, we introduce and study the concepts of intuitionistic fuzzy resolvability, intuitionistidc fuzzy irresolvability and intuitionistic fuzzy open hereditarily irresolv-ability in intuitionistic fuzzy bitopological sapces.

2. Preliminaries

8 K. Biljana, N. Rajesh and V. Vijayabharathi

Definition 2.1 [1]. Let A be a nonempty fixed set. An intuitionistic fuzzy set ( IFS , for short) A is an object having the form ( ) ( ){ }, , :A AA x x x x Xm g= Œ where the

functions :A X Im Æ and :A X Ig Æ denote respectively the degree of membership (namely ( )A xm ) and the degree of non-membership (namely ( )A xg ) of each element

x XŒ to the set A , and ( ) ( )0 1A Ax xm g£ + £ for each x XŒ . Obviously, every fuzzy set A on a nonempty set X is an IFS having the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ .

Definition 2.2 [1]. Let X be a nonempty set and let the IFSs A and B in the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ and ( ) ( ){ }, , :B BB x x x x Xm g= Œ . Let { }:jA j JŒ

be an arbitrary family of IFSs in ( ),X t . Then,

(1) A B£ if and only if x X" Œ [ ( ) ( )A Bx xm m£ and ( ) ( )A Bx xg m≥ ];

(2) ( ) ( ){ }1 , , :A AA x x x x Xg m- = Œ ;

(3) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∩ � � ;

(4) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∪ � � ;

(5) { }1 ,1,0 :x x X= Œ and { }0 ,0,1 :x x X= Œ .

Definition 2.3. An intuitionistic fuzzy topology [3] ( IFT , for short) on a nonempty set X is a family t of IFSs in X satisfying the following axioms:

(i) 0,1 tŒ ;

(ii) 1 2A A t« Œ for every 1 2,A A tŒ ;

(iii) jA tŒ∪ for any { }:jA j J tŒ Õ .

In this case, the ordered pair ( ),X t is called an intuitionistic fuzzy topological space ( IFTS , for short) and each IFS in t is known as an intuitionistic fuzzy open set ( IFOS , for short) in X . The complement of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set ( IFCS , for short). The family of all IFOSs (resp. IFCSs ) of ( ),X t is denoted by ( )IFO X (resp. ( )IFC X ).

Definition 2.4. A triple ( )1 2, ,X t t , where X is a nonempty set, 1t and 2t are two arbitrary intuitionistic fuzzy topologies on X is called the intuitionistic fuzzy bitopological space (for short, IFBTS ).

Definition 2.5 [3]. Let ( ),X t be an IFTS and ( ) ( ){ }, , :A AA x x x x Xm g= Œ be

an IFS in X . Then the intuitionistic fuzzy interior and the intuitionistic fuzzy closure

On Pairwise Intuitionistic Fuzzy Resolvable (Irresolvable) Spaces 9

of A is defined by ( ) { }Int \ is an in and A G G IFOS X G At = Õ∪ and ( )Cl At =

{ }\ is an in and G G IFCS X G A ∩ .

Remark 2.6. For any IFS A in ( ),X t , we have, ( ) ( )Cl 1 1 IntA At t- = - ,

( ) ( )Int 1 1 ClA At t- = - .

3. Pairwise intuitionistic fuzzy resolvable and intuitionistic fuzzy irresolvable spaces

Definition 3.1. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is called a pairwise intuitionistic fuzzy resolvable space if there exists a 1t -intuitionistic fuzzy dense set l such that 1 l- is a 2t -intuitionistic fuzzy dense set and a 2t -fuzzy dense set m such that 1 m- is a 1t -intuitionistic fuzzy dense set. Otherwise ( )1 2, ,X t t is called a pairwise intuitionistic fuzzy irresolvable space.

Example 3.2. Let { },X a b= and A , B be intuitionistic fuzzy sets defined by

, , , ,0.3 0.4 0.6 0.5a b a b

A xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , , ,0.7 0.8 0.3 0.2a b a b

B xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

.

Let { }1 0, ,1At = and { }2 0, ,1Bt = . Then ( )1 2, ,X t t is a pairwise intuitionistic fuzzy resolvable space.


, , , ,0.6 0.5 0.4 0.5a b a b

A xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , , ,0.3 0.2 0.5 0.5a b a b

B xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

.

Let { }1 0, ,1At = and { }2 0, ,1Bt = . Then ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space.

Proposition 3.4. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space.

Proposition 3.4. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is a pairwise

intuitionistic fuzzy resolvable space if and only if ( )1 2, ,X t t has a 1t -intuitionistic fuzzy dense set 1l and 2t -intuitionistic fuzzy dense set 2l such that 1 21l l£ - .

Proof. Let ( )1 2, ,X t t be a pairwise intuitionistic fuzzy resolvable space. Suppose that for all 1t -intuitionistic fuzzy dense sets, il and 2t -intuitionistic fuzzy dense sets jl ,

1i jl l£/ - . That is 1i jl l> - . Then ( ) ( )1 1

Cl Cl 1i jt tl l> - . Now il is a 1t -intui-


tionistic fuzzy dense set in ( )1 2, ,X t t implies that ( )1

Cl 1it l = . Hence ( )1

1 Cl 1 jt l> - .

That is, ( )1

Cl 1 1jt l- π . Also we have 1j il l> - . Then ( ) ( )2 2

Cl Cl 1j it tl l> - . Since

jl is a 2t -intuitionistic fuzzy dense set in ( )1 2, ,X t t implies that ( )2

Cl 1jt l = . Hence

( )2

1 Cl 1 it l> - . Then, we have ( )2

Cl 1 1it l- π . Hence we have ( )1

Cl 1it l = , ( )2

Cl 1 it l-

1π and ( )2

Cl 1jt l = , ( )1

Cl 1 1jt l- π , which is a contradiction to ( )1 2, ,X t t being a pair-

wise intuitionistic fuzzy resolvable space. Therefore ( )1 2, ,X t t has a 1t -intuitionistic fuzzy dense set 1l and a 2t -intuitionistic fuzzy dense set 2l such that 1 21l l£ - . Con-versely, suppose that ( )1 2, ,X t t has a 1t -intuitionistic fuzzy dense set 1l and a 2t -

intui-tionistic fuzzy dense set 2l such that 1 21l l£ - . We want to show that ( )1 2, ,X t t

is a pairwise intuitionistic fuzzy resolvable space. Suppose that ( )1 2, ,X t t is a pairwise intuit-tionistic fuzzy irresolvable space. Then, for all 1t -intuitionistic fuzzy dense sets

il in ( )1 2, ,X t t we have ( )2

Cl 1 1it l- π . In particular ( )2 1Cl 1 1t l- π . That is there

exists a 2t -intuitionistic fuzzy closed set m in ( )1 2, ,X t t such that ( )11 1l m- < < . Then

( )1 m- 1 1l< < and ( ) 1 21 1m l l- < £ - implies that ( ) 2 21 1 1m l l m- < - fi < < ,

which is a contradiction to 2l being a 2t -intuitionistic fuzzy dense set in ( )1 2, ,X t t .

Hence ( )1 2, ,X t t is a pairwise intuitionistic fuzzy resolvable space.

Proposition 3.5. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space if and only if for a 1t -intuitionisitc fuzzy dense set

1l , ( )2 1Int 0t l π and for a 2t -intunitionc fuzzy dense set 2l , ( )

1 2Int 0t l π .

Proof. Let 1l be a 1t -intuitionistic fuzzy dense set and 2l a 2t -intuitionistic fuzzy dense set in ( )1 2, ,X t t . Since ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space, 11 l- is not a 2t -intuitionistic fuzzy dense set and 21 l- is not a 1t -intuitionistic fuzzy dense set in ( )1 2, ,X t t . That is, ( )

2 1Cl 1 1t l- π and ( )1 2Cl 1 1t l- π . Then, we

have ( )2 11 Int 1t l- π and ( )

1 21 Int 1t l- π . Hence we have ( )2 1Int 0t l π and ( )

1 2Intt l

0π . Conversely, let 1l be a 1t -intuitionistic fuzzy dense set and 2l a 2t -intuitionistic fuzzy dense set in ( )1 2, ,X t t . By hypothesis, ( )

2 1Int 0t l π and ( )2 1Int 0t l π . Then

( )2 11 Int 1t l- π and ( )

1 21 Int 1t l- π implies that ( )2 1Cl 1 1t l- π and ( )

1 2Cl 1 1t l- π .

That is, ( )1

Cl 1t l = implies that ( )2 1Cl 1 1t l- π and ( )

2 2Cl 1t l = implies that (1

Cl 1t -

)2 1l π . Hence ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space.

Proposition 3.6. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise intuitionistic fuzzy resolvable if and only if there exists a 1t -intuitionistic fuzzy dense set


1l and 2t -intuitionistic fuzzy dense set 2l in ( )1 2, ,X t t such that ( )2 1Int 0t l = and

( )1 2Int 0t l = .

Proof. The proof follows from Proposition 3.5.

Definition 3.7. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is called a pairwise intuitionistic fuzzy submaximal if each 1t -intuitionistic fuzzy dense set is a 2t -intuitionistic fuzzy open set and each 2t -intuitionistic fuzzy dense set is a 1t -intuitionistic fuzzy open set.


, , , ,0.3 0.4 0.6 0.5a b a b

A xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , , ,0.7 0.8 0.3 0.2a b a b

B xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , ,0.4 0.6 0.6 0.4a b a b

C xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , , ,0.1 0.3 0.3 0.5a b a b

D xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

.

Let { }1 0, , ,1A Dt = and { }2 0, , ,1B Ct = . Then ( )1 2, ,X t t is a pair-wise intuitionistic fuzzy submaximal space.

Proposition 3.9. If an intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise intuitionistic fuzzy submaximal, then it is pairwise intuitioniatic fuzzy irresolvable.

Proof. Let ( )1 2, ,X t t be a pairwise intuitionistic fuzzy submaximal space. Let l be

a 1t -intuitionistic fuzzy dense set in ( )1 2, ,X t t and m be a 2t -intuitionistic fuzzy

dense set in ( )1 2, ,X t t . Since ( )1 2, ,X t t is pairwise intuitionistic fuzzy submaximal, l is 2t -intuitionistic fuzzy open and m is 1t -intuitionistic fuzzy open. Then we have

( )2

Int 0t l l= π and ( )1

Int 0t m m= π . Hence by the Proposition 3.5, ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space.

Definition 3.10. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is called a

pairwise intuitionistic fuzzy open hereditarily irresolvable if ( )2 1

Int Cl 0t t l π for any 1t -

intuitionistic fuzzy open set l implies that ( )2

Int 0t l π and ( )1 2

Int Cl 0t t m π for any

2t -intuitionistic fuzzy open set m implies that ( )1

Int 0t m π .

Proposition 3.11. If an intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise intuitionistic fuzzy open hereditarity irresolute, then it is pairwise intuitionistic fuzzy irresolvable.

Proof. Let l be a 1t -intuitionistic fuzzy dense set and m be a 2t -intuitionistic fuzzy dense set in ( )1 2, ,X t t . Now ( )

1Cl 1t l = implies that ( ) ( )

2 1 2Int Cl Int 1 1 0t t tl = = π and


( )2

Cl 1t m = implies that ( ) ( )1 2 1

Int Cl Int 1 1 0t t tm = = π . Since ( )1 2, ,X t t is pairwise

intuitionistic fuzzy open hereditarily irresolvable, ( )2

Int 0t l π and ( )1

Int 0t m π . Hence

for a 1t -intuitionistic fuzzy dense set l , ( )2

Int 0t l π and for a 2t -intuitionistic fuzzy

dense set ( )1

Int 0tm m π . Therefore, by Proposition 3.5 ( )1 2, ,X t t is a pairise intuit-tionistic fuzzy irresolvable space.

Remark 3.12. It is clear that every pairwsie intuitionsitic fuzzy open hereditarily irresolvable space is pairwise intuitionistic fuzzy irresolvable. However the converse need not be true.


, , , ,0.6 0.5 0.4 0.5a b a b

A xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

, , , , ,0.3 0.2 0.6 0.5a b a b

B xÊ ˆ Ê ˆ= Á ˜ Á ˜Ë ¯ Ë ¯

Let { }1 0, ,1At = and { }2 0, ,1Bt = . Then the intuitionistic fuzzy bitopological space

( )1 2, ,X t t is pairwise intuitionistic fuzzy irresolvable but not pairwise intuitionistic fuzzy open hereditarily irresolvable.

Proposition 3.14. If an intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise

intuitionistic fuzzy open hereditarily irresolvable and if ( )2

Int 0t l = for a 1t -

intuitionistic fuzzy open set l and ( )1

Int 0t m = for a 2t -intuitionistic fuzzy open set m ,

then ( )( )2 1Int Cl 0t t l = and ( )( )1 2

Int Cl 0t t m = .

Proof. Let 0l π be a 1t -intuitionistic fuzzy open set in ( )1 2, ,X t t such that ( )2

Intt l

0= and 0m π be a 2t -intuitionistic fuzzy open set in ( )1 2, ,X t t such that

( )1

Int 0t m = . We claim that ( )( )2 1Int Cl 0t t l = and ( )( )1 2

Int Cl 0t t m = . Suppose that

( )( )2 1Int Clt t l 0π and ( )( )1 2

Int Cl 0t t m π . Since ( )1 2, ,X t t is pairwise intuitionistic

fuzzy open hereditarily irresolvable, we have ( )( )2 1Int Cl 0t t l π for any 1t -intuitionistic

fuzzy open set l implies that ( )2

Int 0t l π and ( )( )1 2Int Cl 0t t m π for any 2t -

intuitionistic fuzzy open set m implies that ( )1

Int 0t m π , which is a contradiction to our

hypothesis. Hence we must have ( )( )2 1Int Cl 0t t l = and ( )( )1 2

Int Cl 0t t m = .

Definition 3.15. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is called a pairwise intuitionistic fuzzy hyperconnected if for every nonzero 1t -intuitionistic fuzzy open set l and a 2t -intuitionistic fuzzy open set m such that 1l m+ > .


Proposition 3.16. An intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise intuitionistic fuzzy hyperconnected if and only if for any nonzero intuitionistic fuzzy set l in ( )1 2, ,X t t either ( )

1Cl 1t l = or ( )

2Cl 1 1t l- = .

Proof. Let ( )1 2, ,X t t be a pairwise intuitionistic fuzzy hyperconnected space. Then

for every 1t -intuitionistic fuzzy open set ( )0l π and a 2t -intuitionistic fuzzy open set

( )0m π such that 1l m+ > . Suppose that for an intuitionistic fuzzy set g , ( )1

Cl 1t g π

and ( )2

Cl 1 1t g- π . Now ( )1

Cl 1t g π implies that ( )1

1 Cl 0t g- π . Then ( )1

Int 1t g-

0π . Hence there exists a 1t -intuitionistic fuzzy open set d in ( )1 2, ,X t t such that

1d g£ - . Also ( )2

Cl 1 1t g- π implies that there exists a 2t -intuitionistic fuzzy closed

set h in ( )1 2, ,X t t such that 1 1g h- £ < . Hence 1d g h£ - £ implies that d h£ .

That is ( )1 1d h£ - - . Then ( )1 1d h+ - £ . That is, for a nonzero 1t -intuitionistic

fuzzy open set d and a 2t -intuitionistic fuzzy open set 1 h- in ( )1 2, ,X t t we have

( )1d h+ - 1£ , which is a contradiction to our hypothesis. Hence we have either

( )1

Cl 1t g = or ( )2

Cl 1 1t g- = for a nonzero intuitionistic fuzzy set g in ( )1 2, ,X t t .

Conversely, let ( )1

Cl 1t g = or ( )2

Cl 1 1t g- = for any nonzero intuitionistic fuzzy set g

in ( )1 2, ,X t t . Suppose that for every 1t -intuitionistic fuzzy open set ( )0l π and a 2t -

intuitionistic fuzzy open set ( )0m π such that 1l m+ / . Then 1l m+ £ implies that

( )2

Clt l £ ( )2

Cl 1 1t m m- = - (since m is a 2t -intuitionistic fuzzy open set 1 mfi - is

2t -intuitionistic fuzzy closed set in ( )1 2, ,X t t ). Hence ( )2

Cl 1 1t l m£ - < (since

0m π ). That is, ( )2

Cl 1t l π . Now ( ) ( )1 1

Cl 1 1 Int 1 1t tl l l- = - = - π (since 0l π ).

That is, ( )1

Cl 1 1t l- π . Therefore for the nonzero intuitionistic fuzzy set 1 l- ,

( )1

Cl 1 1t l- π and [ ]( )2

Cl 1 1 1t l- - π , which is a contradiction to our hypothesis.

Hence for every 1t -intuitionistic fuzzy open set ( )0l π and a 2t -intuitionistic fuzzy

open set ( )0m π such that 1l m+ > which implies that ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected space.

Proposition 3.17. If an intuitionistic fuzzy bitopological space ( )1 2, ,X t t is pairwise intuitionistic fuzzy hyperconnected, then it is a pairwise intuitionistic fuzzy open hereditarily irresolvable space.

Proof. Let l be a 1t -intuitionistic fuzzy open set such that ( )( )2 1Int Cl 0t t l π and m

a 2t -intuitionistic fuzzy open set such that ( )( )1 2

Int Cl 0t t m π in a pairwise intuitionistic

fuzzy hperconnected space ( )1 2, ,X t t . Then for the nonzero 1t -intuitionistic fuzzy open set l and for the 2t -intuitionistic fuzzy open set m we have 1l m+ > . Now 1l m> -

( ) ( )2 2

Int Int 1t tl mfi > - . Then ( ) ( )2 2

Int 1 Cl 0t tl m> - ≥ . That is, ( )2

Int 0t l π . Also,


1l m l+ > - . Then ( ) ( ) ( )1 1 1

Int Int 1 1 Cl 0t t tm l l> - = - ≥ . That is, ( )1

Int 0t m π .

Therefore ( )( )2 1Int Cl 0t t l π for any 1t -intuitionistic fuzzy open set l implies that

( )2

Int 0t l π and ( )( )1 2

Int Cl 0t t m π for any 2t -intuitionistic fuzzy open set m implies

that ( )1

Int 0t m π . Hence ( )1 2, ,X t t is a pairwise intuitionistic fuzzy open hereditarily irresolvable space.

Proposition 3.18. If an intuitionisitc fuzzy bitopological space ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected and pairwise intuitionistic fuzzy irresolvable space, then ( )

1Cl 1t l = implies that ( )

2Cl 1 1t l- π for any intuitionistic fuzzy set l in

( )1 2, ,X t t .

Proof. Since ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected space, by

Proposition 3.16, either ( )1

Cl 1t l = or ( )2

Cl 1 1t l- = for any non zero intuitionistic

fuzzy set l in ( )1 2, ,X t t . Suppose ( )1

Cl 1t l = . Now ( )1 2, ,X t t is a pairwise

intuitionistic fuzzy irresolvable space implies that ( )2

Int 0t l π . Then ( )2

1 Int 1t l- π .

Hence ( )2

Cl 1 1t l- π .

Proposition 3.19. If an intuitionistic fuzzy bitopological space ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected and pairwise intuitionistic fuzzy irresolvable space, then ( )1 2, ,X t t is a pairwise intuitionistic fuzzy open hereditarily irresolvable space.

Proof. Let l be a 1t -intuitionistic fuzzy open set and m be a 2t -intuitionistic fuzzy open set in ( )1 2, ,X t t . Since ( )1 2, ,X t t is pairwise intuitionistic fuzzy hyperconnected,

by Proposition 3.16, either ( )1

Cl 1t l = or ( )2

Cl 1 1t l- = . Suppose ( )1

Cl 1t l = . Then

by Proposition 3.18, ( )2

Cl 1 1t l- π . Now ( ) ( )( ) ( )1 2 1 2

Cl 1 Int Cl Int 1 0t t t tl l= fi = π

and ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable space implies that ( )2

Intt l

0π . Again since ( )1 2, ,X t t is intuitionistic fuzzy hyper-connected by Proposition 3.16,

either ( )2

Cl 1t m = or ( )1

Cl 1 1t m- = for the 2t -intuitionistic fuzzy open set m . Suppose

( )2

Cl 1t m = . Then, by Proposition 3.18 we have ( )2

Cl 1 1t m- π . Now ( )2

Cl 1t m = im-

plies that ( )( ) ( )1 2 1

Int Cl Int 1 0t t tm = π and ( )1 2, ,X t t is a pairwise intuitionistic fuzzy

irresolvable space implies that ( )1

Int 0t m π . Hence ( )2 1

Int Cl 0t t l π for any 1t -intui-

tionistic fuzzy open set l implies that ( )2

Int 0t l π and ( )( )1 2Int Cl 0t t m π for any 2t -

intuitionistic fuzzy open set m implies that ( )1

Int 0t m π . Therefore, ( )1 2, ,X t t is a pairwise intuitionistic fuzzy open hereditarily irresolvable space.


Proposition 3.20. For any intuitionistic fuzzy bitopological space ( )1 2, ,X t t the following are equivalent:

(1) ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected and pairwise intuitionistic fuzzy irresolvable space.

(2) For every nonzero intuitionistic fuzzy set l in ( )1 2, ,X t t either ( )( )1 2

Cl Int 1t t l =

or ( )( )2 1Cl Int 1 1t t l- = .

Proof. (1) fi (2) Let ( )1 2, ,X t t be a pairwise intuitionistic fuzzy hyperconnected

and pairwise intuitionistic fuzzy irresolvable space. Suppose that ( )( )1 2Cl Int 1t t l π and

( )( )2 1Cl Int 1 1t t l- π . Then ( )( )1 2

1 Cl Int 0t t l- π and ( )( )2 11 Cl Int 1 0t t l- - π . Hence

( )( )1 2Int Cl 1 0t t l- π and [ ]( )( )2 1

Int Cl 1 1 0t t l- - π . That is ( )( )1 2Int Cl 1 0t t l- π and

( )( )2 1Int Cl 0t t l π . Since ( )1 2, ,X t t is a pairwise intuitionistic fuzzy open hereditarily

irresolvable space, we have ( )1

Int 1 0t l- π and ( )2

Int 0t l π . Then, ( )1

1 Cl 0t l- π

and ( )2

1 Int 1 0 1t l- π - = . That is, ( )1

Cl 1t l π and ( )2

Cl 1 1t l- π , which is a

contradiction to ( )1 2, ,X t t being a pairwise intuitionistic fuzzy hyperconnected space.

Hence we have either ( )( )1 2Cl Int 1t t l = or ( )( )2 1

Cl Int 1 1t t l- = .

(2) fi (1) Let either ( )( )1 2Cl Int 1t t l = or ( )( )2 1

Cl Int 1 1t t l- = for every nonzero

intuitionistic fuzzy set l in ( )1 2, ,X t t . Then ( )2

Int 0t l π or ( )1

Int 1 0t l- π . There-

fore for an 1t -dense intuitionistic fuzzy set ( )2

Intt l , ( )( ) ( )2 2 21 1Int Int Int 0t t tl l= π

and for a 2t -dense intuitionistic fuzzy set ( )1

Int 1t l- , ( )( ) ( )1 1 1

Int Int 1 Int 1t t tl l- = -

0π . Hence by Proposition 3.5, ( )1 2, ,X t t is a pairwise intuitionistic fuzzy irresolvable

space. Now ( )2

Intt l l£ implies that ( ) ( )1 2 1

Cl Int Clt t tl l£ . Then ( )1

1 Clt l£ . That

is, ( )1

Cl 1t l = . Also ( ) ( )1

Int 1 1t l l- £ - implies that ( ) ( )2 1 2

Cl Int 1 Cl 1t t tl l- £ - .

Then ( )2

1 Cl 1t l£ - . That is, ( )2

Cl 1 1t l- = . Therefore, ( )1

Cl 1t l = or ( )2

Cl 1 1t l- =

for any intuitionistic fuzzy set l in ( )1 2, ,X t t . Hence by Proposition 3.16, ( )1 2, ,X t t is a pairwise intuitionistic fuzzy hyperconnected space.

References

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [2] D. Coker, An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces, J. Fuzzy

Math., 4 (2) (1996), 749-764. [3] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88

(1997), 81-89. [4] D. Coker and M. Demirci, On intuitionistic fuzzy points, Notes on Intuitionistic Fuzzy Sets, 1 (1995),

79-84.


[5] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.


__________________________ Received April, 2013


Los Angeles

On Generalized Intuitionistic Fuzzy Topology

N. Gowrisankar

70/232 6B, Kollupettai Street, M. Chavady, Thanjavur-613001, Tamil-nadu, India. E-mail address: [email protected]

N. Rajesh

Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India.


V. Vijayabharathi

Department of Mathematics, National Institute of Technology, Tiruchi-Rapalli, Tamilnadu, India. E-mail address: [email protected]

Abstract: The aim of this paper is to present a common approach allowing to obtain familes of

intuitionistic fuzzy sets in an intuitionistic fuzzy topological space.

Key words and phrases: Generalzied Intuitionistic fuzzy topology, g -intuitionistic fuzzy open set.

1. Introduction

After the introduction of fuzzy sets by Zadeh [7], there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them. Using the notion of intuitionistic fuzzy sets, Coker [3] introduced the notion of intuitionistic fuzzy topological spaces. The aim of this paper is to present a common approach allowing to obtain familes of intuitionistic fuzzy sets in an intuitionistic fuzzy topological space.

2. Preliminaries

18 N. Gowrisankar, N. Rajesh and V. Vijayabharathi


functions :A X Im Æ and :A X Ig Æ denote respectively the degree of membership (namely ( )A xm ) and the degree of non-membership (namely ( )A xg ) of each element


( ) ( ){ }, , :A AA x x x x Xm g= Œ .

Definition 2.2 [1]. Let X be a nonempty set and let the 'IFS s A and B in the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ , ( ) ( ){ }, , :B BB x x x x Xm g= Œ . Let { }:jA j JŒ be

an arbitrary family of IFS ’s in ( ),X t . Then,


(2) ( ) ( ){ }, , :A AA x x x x Xg m= Œ ;

(3) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∩ � � ;

(4) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∪ � � ;

(5) { }1 ,1,0 :X x x X= Œ and { }0 ,0,1 :X x x X= Œ .


(i) 0 ,1X X tŒ ;



In this case, the ordered pair ( ),X t is called an intuitionistic fuzzy topological space ( IFTS , for shrot) and each IFS in t is known as an intuitionistic fuzzy open set ( IFOS , for short) in X . The complement of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set ( IFCS , for shrot). The family of all IFOSs (resp. IFCSs ) of ( ),X t is denoted by ( )IFO X (resp. ( )IFC X ).

Definition 2.4 [3]. Let ( ),X t be an IFTS and let ( ) ( ){ }, , :A AA x x x x Xm g= Œ

be an IFS inX . Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A is defined by ( ) { }Int \ is an in and A G G IFOS X G A= Õ∪ and ( ) {Cl \ A G G= ∩

}is an in and IFCS X G A .

On Generalized Intuitionistic Fuzzy Topology 19

Remark 2.5. For any IFS A in ( ),X t , we have, ( ) ( )Cl 1 1 IntA A- = - , ( )Int 1 A-

( )1 Cl A= - .

Definition 2.6. An intuitionistic fuzzy point [4] ( IFP , for short), written ( ),p a b , is

defined to be an IFS in X given by

( )( )( ),

, , if 0,1 , otherwise.

x pp a b

a bÏ =Ô= ÌÔÓ

Let X be a nonempty set and [ ]{ }: 0,1Xl l= ÆF be the family of all

intuitionistic fuzzy sets defined on X . Let :g ÆF F be a function such that l m£ implies that ( ) ( )g l g m£ for every ,l m ŒF . That is, g is a monotonic function

defined on F by ( )G F or simpily G . We will defined the following subclasses of G .

(1) For every [ ]0,1a Œ , define ( ){ }a g g a aG = ŒG = where a is the

intuitionistic fuzzy set defined by ( )xa a= for every x XŒ ,

(2) ( ) ( ){ }22 for every g g l g l lG = ŒG = ŒF ,

(3) ( ){ } for every g l g l l+G = ŒG £ ŒF and

(4) ( ){ } for every g g l l-G = ŒG £ F .

If Â is a collection of some of the symbols 2, - , + and [ ]0,1a Œ , then

{ } for every ig g iÂG = ŒG ŒG ŒÂ .

3. Properties of g -intuitionistic fuzzy open sets

Definition 3.1. Let X be a nonempty set g ŒG . An intuitionistic fuzzy set l ŒF is said to be g -intuitionistic fuzzy open (for short, -IFg ) if ( )l g l£ .

Theorem 3.2. Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on X and g ŒG . Then the following hold.

(1) Arbitrary union of -IFg open sets is a -IFg open set.

(2) 1 is -IFg open if and only if 1g ŒG .

(3) If 2g ŒG , then every intuitionistic fuzzy set of the form ( )g l , lF is a -IFg open set.

(4) If g +ŒG , then every intuitionistic fuzzy subset is -IFg open.

(5) g -ŒG , then l is -IFg open if and only if ( )l g l= .


(6) 0 is always -IFg open set.

Proof. (1) Let { }:i i Jl Œ Õ F . If ( )i il g l£ for all i JŒ and il l=� , then for all

i JŒ , ( ) ( ) ( )i i il g l g l g l£ fi £ ( ) ( )ig l g lfi £� � ( )i il l g lfi = £ £� � ( )g l .

(2) 1 is -IFg open if and only if ( )1 1g£ if and only if 1g ŒG .

(3) If 2g ŒG ,then ( )( ) ( )g g l g l= for every l ŒF and so ( )g l is a -IFg open set.

(4) If g +ŒG , then ( )l g l£ for every l ŒF is -IFg open.

(5) Suppose g -ŒG . If l is -IFg open, then ( )l g l£ and so ( )l g l= ,by hypo-

thesis. If ( )l g l= , then clearly, l is -IFg open.

(6) Clear.

Let X be a nonempty set and F be the family of all intuitionistic fuzzy sets defined on X . A subfamily G of F is called generalized intuitionistic fuzzy topology if 0 ŒG and { }al a ŒD Œ� G whenever al ŒG for every a ŒD .

Remark 3.3. If g ŒG , by Theorem 3.2 (1), it follows that A , the family of all -IFg open sets is a generalized intuitionistic fuzzy topology.

Definition 3.4. For l ŒF , the g -interior of l , denote by ( )gi l , is given by

( ) { }gi l m m l= Œ £� A .

Proposition 3.5. Let g ŒG , l ŒF and ( ),X t be an intuitionistic fuzzy topological

space. Then, for the g -interior operator ( ) ( )IntInt, Inti l l= for all l ŒF .

The following Theorem 3.6 gives all the properties of ( )gi l .

Theorem 3.6. Let X be a nonempty set, F the family of all intuitionistic fuzzy sets defined on X and g ŒG . Then the following properties hold.

(1) For every l ŒF , ( )gi l is the largest -IFg open set contained in l .

(2) l is -IFg open if and only if ( )gi l l= .

(3) 02gi -ŒG for every g ŒG .

(4) 1gi ŒG if and only if 1g ŒG .

(5) If 02g l -Œ , then g i= .

(6) l is -IFgi open if and only if ( )gi l l= .

Proof. (1) By definition and Theorem 3.2 (1), ( )gi l is a -IFg open set. By

definition, ( )gi l l£ . If b is -IFg open such that b l£ , then ( )gb i l l£ £ and so

( )gi l is the largest -IFg open set contained in l .


(2) If a is -IFg open, then by definition, ( )gl i l£ and so ( )gi l l= by (1). The converse part is clear.

(3) Clearly, ( )0 0gi = and so 0gi ŒG . By (1), ( )gi l is a -IFg open set for every

l ŒF and so by (2), ( )( ) ( )g g gi i l i l= and so 2gi ŒG . By (1), it follows that gi -ŒG .

(4) 1g ŒG if and only if ( )1 1g = if and only if 1 is -IFg open, by (2). Therefore,

1g ŒG if and only if ( )1 1g = if and only if 1 is -IFg open, by (2). Therefore, 1g ŒG

if and only if ( )1 1gi = if and only if 1gi ŒG .

(5) Let l ŒF . Since 2g ŒG , ( )( ) ( )g g l g l= . Therefore, ( )g l is a -IFg open

set. Since g -ŒG , ( )g l l£ . Thus ( )g l is a -IFg open set such that ( )g l l£ . If

m l£ is -IFg open, then ( ) ( )m g m g l£ £ and so by (1), ( ) ( )gg l i l= and so gg i= .

Remark 3.7. Let X be a nonempty set and ( ),X t an intuitionistic fuzzy topological space. Let Int represent the interior operator and C1 denote the closure operator of the intuitionistic fuzzy topological space ( ),X t . Then,

(1) for Intg = , -IFg open sets coincide with intuitionisitc fuzzy open sets,

(2) for Int Clg = , -IFg open sets coincide with intuitionistic fuzzy preopen sets [5],

(3) for Cl Intg = , -IFg open sets coincide with intuitionistic fuzzy semiopen sets [5],

(4) for Int Cl Intg = , -IFg open sets coincide with intuitionistic fuzzy a -sets in [5],

(5) for Cl Int Clg = , -IFg open sets coincide with intuitionistic fuzzy semipropen sets in [5],

(6) for Int Cl Cl Intg = ⁄ , -IFg open sets coincide with intuitionistic fuzzy g -open sets [6].

Moreover, ( )Int Cli l is the intuitionistic fuzzy preinterior [6], ( )Cl Inti l is the intuit-

tionistic fuzzy semiinterior [6], ( )Cl Int Cli l is the intuitionistic fuzzy semipreinterior [5],

( )Int Cl Inti l is the intuitionistic fuzzy a -interior [5] and ( )Int Cl Cl Inti l⁄ is the intuitionistic

fuzzy g interior [6] of ( )l ŒF .

Theorem 3.8. Let 1 2,g g ŒG .

(1) 1 2g g ŒG .

(2) If 1 2, ng g ŒG for all { }0,1, ,nŒ + - , then 1 2 ng g ŒG .

Proof. One can easily prove this theorem by means of the definitions of nG for all

{ }0,1, ,nŒ + - .


Proposition 3.9. Let 2,i k ŒG and for all l ŒF , ( ) ( )ik l k l£ (*) and ( )ik l £

( )kik l (**). If g is a composition of alternating factors i and k , then 2g ŒG except for the case g ki= .

Proof. Let l ŒF . For g ik= , we have ( ) ( ) ( )kik l kk l k l£ = by (*). Hence, we

obtain ( ) ( )ikik l ik l£ . On the otherhand, by (**), we ger ( ) ( )ik l kik l£ implying

that ( ) ( )ik iik l ikik l= £ . Hence, we obtain ( )ikik l ik= , that is, 2g ik= ŒG . For

g iki= , we have ( )( )( ) ( ) ( ) ( )iki iki l ikiki l ikik i l= = . Hence, 2g iki= ŒG . For g

kik= , we have ( )( )( ) ( ) ( )( ) ( )kik kik l kikik l k ikik l kik l= = = implying that g =

2kik ŒG . For g kiki= , we have ( )( )( ) ( ) ( )kiki kiki l k ikik iki l= = ( ) ( )k ikik i l =

( )kiki l . Hence 2g kiki= ŒG . For g ikik= , we have ( )( )( ) ( )ikik ikik l ikik l= . Hence 2g kiki= ŒG . On can easily show that 2g ŒG for any g -composed of more than four factors.

Proposition 3.10. Let 2,i k ŒG , the inequality (*) and ( ) ( )i l ki l£ (***) are satisfied for all l ŒF . Then 2g ki= ŒG .

Proof. Let l ŒF . If ( ) ( )i l ki l£ , then ( ) ( )i l iki l£ implying that ( ) ( )ki l kiki l£ .

If in (*), we replace l by ( )i l , then we get ( ) ( )iki l ki l£ . Hence, we obtain ( )kiki l

( )ki l£ . Thus, ( ) ( )kiki l ki l= , that is, 2ki ŒG .

Corollary 3.11. Let ( ),X t be an intuitionistic fuzzy topological space. If i and k are chosen as the interior and closure operator respectively, since 2Int,ClŒG and (*), (**) and (***) are satisfied by Int and Cl operators, any composition of alternating factors Cl and Int is equal to one of the mappings Int , Cl , ( )Int Cl , ( )Cl Int ,

( )( )Int Cl Int or ( )( )Cl Int Cl by Proposition 3.9 and 3.10.

Corollary 3.12. If 2i -ŒG and 2k +ŒG , then any composition of alternating factors i and k is an element of 2G .

Proof. It follows from the fact that inequalities (*), (**), (***) are verified by i and k .

Proposition 3.13. Let g ŒG . Every -IFg open set is -IFng open for all n NŒ . If

2g -ŒG , then -IFg open sets coincide with -IFng open sets.

Proof. Let g ŒG and l ŒF is a -IFg open set. Then, ( ) ( )l g l gg l£ £ £ £

( ) ( )1n ng l g l- £ . Therefore, l is a -n IFg open set. For the case 2g -ŒG , assume


that l is a -n IFg open set. Then, ( ) ( ) ( ) ( )1 2n nl g l g l g l g l-£ £ £ £ £ . Hence, l is a -IFg open set.

Proposition 3.14. Let 2,i k -ŒG and i , k satisfy the inequality (*). Then, any composition of factors i and k is an element of 2-G , Moreover, if g ŒG¢ is an arbitrary composition of factors i and k , then

(1) a -IFg open set is an -IFik open set, where g ig k= ¢ .

(2) a -IFg open set is an -IFiki open set, where g ig i= ¢ .

(3) a -IFg open set is a -IFki open set, where g kg i= ¢ .

(4) a -IFg open set is a -IFkik open set, where g kg k= ¢ . The converse implications hold if no factor i is immediately followed by another

factor i .

Proof. Let l ŒF . It can be easily seen that if 2,i k -ŒG , then ( ) ( )ni l i l£ and

( ) ( )nk l k l£ and by (*), we obtain ( ) ( ) ( ) ( )n nik l k l k l£ £ for all n NŒ . Since

for all n NŒ , ( ) ( ) ( ) ( ) ( ) ( )2 1n n n n ngg l i i l ii l i i l i l g l-= £ = = = , we have 2g -ŒG

for ng i= . Now let 1 2g g kg= , where 1 2g g are nay (may be empty) compositions of factors i and k that is, g has at least one factor k . Then, ( ) ( )1 2 1 2gg l g kg g kg l= . If

i is replaced instead of each ni factors, and k is replaced instead of mk factors, and k is replaced instead of each ( )pik factors in the composition 2 1kg g k , we get ( )gg l =

( ) ( ) ( )1 2 1 2 1 2g kg g kg l g kg l g l£ = . Hence 2g ŒG - .

(1) Let g ig k= ¢ and l ŒF is a -IFg open set. For a suitable m NŒ , ( )l g l£ =

( ) ( ) ( )( ) ( ) ( ) ( ) ( )1 1m m m mig k l ik l ik ik l ikk l ik l ik l- -£ = £ = £¢ by similar substitu-tions in the above manner. Hence l is an -IFik open set. Conversely, suppose that no factor i is immediately followed by another factor i and l is -IFik open set. We have ( )l ik l£ implying that ( ) ( )nl ik l£ , where n is the number of the factors k in

g and also we get ( ) ( )l ik l k l£ £ . Then by the inequality (*) and ( )l k l£ , we

obtain ( )l g l£ . Hence, l is a -IFg open set.

(2) Let g ig i= ¢ and l is a -IFg open set. By (*), we obtain ( ) ( ) ( )ml ig i l ik i l£ £¢

( )( ) ( ) ( )( ) ( ) ( ) ( )1 1m m mik ik i l ik k i l ik i l iki l- -£ £ = £ for a suitable m NŒ . Hence, l is an -IFiki open set. Conversely, if no factor i is immediatedly followed by another factor i and l is -IFiki open set, then for m NŒ ,

( ) ( ) ( ) ( )m ml iki l ik i l ik£ fi £ ( ) ( ) ( ) ( ) ( )1m miiki l ik iki l ik i l+£ = . Therefore, by

the inequality (*), ( )l iki l£ £ ( ) ( ) ( )nik i l g l£ , where n is the number of the factors k in g . Thus l is a -IFg open set.


(3) Letg kg i= ¢ and l is a -IFg open set. For 2m ≥ and m NŒ , we have ( )l g l£

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1m m m mki l k ik i l kk i l k i l ki l- -£ = £ = £ . Hence l is a -IFki open set. Conversely, if no factor i is immediately followed by another factor i and l is

-IFki open set, then by the inequality (*), we have ( ) ( ) ( ) ( )nl ki l l ki l g l£ fi £ £ , where n is the number of factors k in g . Hence l is a -IFg open set.

(4) Let g kg k= ¢ and l is a -IFg open set. For a suitable m NŒ , we have l £

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1m m m mkg k l ki k l k ik ik l kk ik l k ik l kik l- -£ = £ £ £¢ . Thus, l is a -IFkik open set. Conversely, if no factor i is immeditately followed by another factor i and l is -IFkik open set, then we get ( ) ( ) ( ) ( )m ml kik l ki k l ki£ fi £

( ) ( ) ( ) ( ) ( )1m mkkik l ki kik l ki k l+£ = , for all m NŒ . Thus, by the inequality (*), we

obtain ( ) ( ) ( ) ( )1nl kik l ki k l g l-£ £ £ where n is the number of factors k in g . Hence, l is a -IFg open set.

Corollary 3.15. Let i -ŒG and 2k -ŒG . Then, the statements of Proposition 3.14 are valid. Furthermore

(1) Every -IFiki open set is -IFiki open set. (2) Every -IFiki open set is -IFki open set. (3) Every -IFik open set is -IFkik open set. (4) Every -IFki open set is -IFkik open set. (5) Every -IFkik open set is -IFk open set. (6) Every -IFiki open set is -IFiki open set. (7) If l ŒF is an -IFik open set and -IFki open set, then it is -IFiki open set.

Proof. Let l ŒF . It is clear from the fact that the inequality (*) and 2i -ŒG are satisfied under these conditions.

(1) Let l is an -IFiki open set. Hence, ( )( ) ( )l i ki l ki l£ £ implying that l is a

-IFki open set. If ( ) ( ) ( )l ik i l ik l£ £ , then l is a -IFik open set. Providing that

l is a -IFik open set or ki -open set, then ( ) ( ) ( )l ik l ikik l kik l£ £ £ or ( )l ki l£

( ) ( ) ( )kik i l kik l£ £ , respectively. That is, l is a -IFkik open set in both cases.

For the case ( )l kik l£ , we get ( ) ( ) ( )l kik l kk l k l£ £ £ implying that l is a -IFk open set. (2) Suppose that l is an -IFik open set and ki -open set. Hence, we obtain ( )l ik l£

( )( ) ( )ik ki l iki l£ £ . Thus, l is an -IFiki open set.

Remark 3.16. If i and k are choosen as the interior and closure operator, respectively, and Remark 3.7 considered, then we have the following

(1) An -IFik open set need not be an -IFiki open set, (2) A -IFki open set need not be an -IFiki open set,


(3) A -IFkik open set need not be a -IFki open set, (4) A -IFkik open set need not be an -IFik open set, (5) An intuitionistic fuzzy set need not be a -IFkik open set, (6) The -IFik open sets and ki -open sets are independent notions.

Let ( ),X t be an intuitionistic fuzzy topological space. We denote the family of all

( )Xg ŒG satisfying the property t"Œ , l" ŒF ; ( ) ( )m g l g m lŸ £ Ÿ by 3G . It it

clear that for an intuitionistic fuzzy topological space ( ),X t , the interior operator

3Int ŒG . On the other hand, a closure operator may not be an element of 3G .

Proposition 3.17. If 1 2 3,g g ŒG , then 1 2 3g g ŒG .

Proof. Let ( ),X t be an intuitionistic fuzzy topological space, 1 2 3g g ŒG , m tŒ , and

l ŒF . Since ( ) ( )( ) ( )( ) ( )( )1 2 1 2 1 2 1 2 1m g g l m g g l g m g l g g m l gŸ = Ÿ £ Ÿ £ Ÿ =

( )2g m lŸ , we have 1 2 3g g ŒG .

Proposition 3.18. Let ( ),X t be an intuitionistic fuzzy topological space and 3g ŒG . If m tŒ , and l ŒF is a -IFg open set, then m lŸ is a -IFg open set.

Proof. Since m m£ and ( )l g l£ , we have ( ) ( )m l m g l g m lŸ £ Ÿ £ Ÿ . Hence m lŸ is a -IFg open set.

Proposition 3.19. Let ( ),X t be an intuitionistic fuzzy topological space. If 3g ŒG

and m is an intuitionistic fuzzy open set such that ( )1m g£ , then m is a -IFg open

set.

Proof. Let m tŒ and ( )1m g£ . Since ( ) ( ) ( )1 1m m g g m g m= Ÿ £ Ÿ = , m is a -IFg open set.

Corollary 3.20. Let 13g ŒG . Then every intuitionistic fuzzy open set is a -IFg open set.

Proof. Let m tŒ . Then, ( ) ( ) ( ) ( )1 1 1m m g m g g m g m= Ÿ = Ÿ £ Ÿ = . Hence m is a -IFg open est.

Corollary 3.21. Let ( ),X t be an intuitionistic fuzzy topological space. If 3g ŒG and m is an intuitionistic fuzzy open set such that there exists a -IFg open set l containing m , then m is a -IFg open set.

Proof. Let l ŒF , ( )l g l£ and m l£ . Since ( ) ( )1m l g l g£ £ £ , m is a -IFg open set from Proposition 3.19.


Proposition 3.22. If 3g ŒG , then 3ig ŒG .

Proof. Let m tŒ and l ŒF . Since ( )igm lŸ is a -IFg open set by Proposition

3.18 and ( )igm l m lŸ £ Ÿ , we have ( ) ( )i i fg gm l mŸ £ Ÿ . Hence, we have 3ig ŒG .

Proposition 3.23. Let 1g ŒG if and only if ( ) ( )Int l g l£ for all l ŒF .

Proof. Let l ŒF . 13g ŒG , we have ( ) ( ) ( ) ( )Int 1 Int 1 Intl g l lŸ = Ÿ = £

( )( ) ( )( ) ( )Int 1 Intg l g l g lŸ = £ . Since ( ) ( )Int 1 1 1 1g= £ £ , we get 1g ŒG .

Remark 3.24. Consider the function g ¢ consisting of alternating compositions of Int and g where 23g ŒG . It is clear that 3g ŒG¢ by Proposition 3.17.

Proposition 3.25. Let 23g ŒG and g ¢ consist of alternating compositions of Int and g . Then 2g ŒG¢ except for the condition Intg g=¢ .

Proof. Since the inequalities (*) and (**) are satisfied for the cases Inti = and k g= , we have 2g ŒG¢ by Proposition 3.9.

Remark 3.26. If 23g ŒG , then the case Int , g , Intg , Intg , Int Intg , Intg g , Int Intg g are enough to be considered.

Proposition 3.27. If 23g ŒG , then

(1) Int Int Int Intg g g g£ £ .

(2) Int Int Int Int Intg g g g g£ £ .

(3) Intg g g£ .

(4) Intg g£ .

Proof. (3) and (4) are clear from the fact that 2g ŒG and Int -ŒG . We need to show

that ( ) ( )Int Intg l g g l£ for all l ŒF . Since ( ) ( ) ( )Int Int 1 1g l g g£ £ , ( )Intg l is a

-IFg open set by Proposition 3.19. Hence, we have ( ) ( )Int Intg l g g l£ which completes for the proofs of (1) and (2).

Proposition 3.28. Let ( ),X t be an intuitionistic fuzzy topological space. If 123g ŒG ,

then for all l ŒF , ( ) ( ) ( ) ( ) ( )Int Int Int Int Int Intl g l g l g g l g l£ £ = £ .

Proof. Let l ŒF . If 123g ŒG , then by Proposition 3.19, we have ( ) ( )Int Intl g l£ . Since the inequalities (*) and (**) are satisfied for the case k g= and Inti = , we get

2Intg ŒG . Thus we obtain ( ) ( ) ( ) ( ) ( )Int Int Int Int Int Int Int Intl l g l g l g g l= £ £ = £

( ) ( )gg l g l= .


Corollary 3.29. If 123g ŒG , then the implications in the following diagram hold:

Int - open

open Int Int- open Int - open

Int- open Int Int- open

IF

IF IF

IF IF

g

g g g

g g g

Æ

¤

Theorem 3.30. Let 2g -ŒG . Then, for n NŒ , l ŒF is

(1) ( )Int -n IFg open Int -IFg¤ open.

(2) ( )Int Int-nIFg open Int Int-IFg¤ open.

(3) ( )Int -n IFg open Int-IFg¤ open.

(4) ( )Int -nIFg g open Int -IFg g¤ open.

(5) Int Int-IFg open Intg¤ and Int-IFg open.

(6) Int - open,

Int- open,IF

IF

ggÏÌÓ

Int - open - openIF IFg g gfi fi .

Proof. (1), (2), (3) and (4) are clear from Proposition 3.13. (5) and (6) are obvious from Corollary 3.15.

Theorem 3.31. Let ( ),X t be an intuitionistic fuzzy topological space and 13g ŒG . Then the family O of all -IFg open sets satisfies the following properties:

(1) If l tŒ , then l ŒO . (2) If il ŒO for all i JŒ , then i J ifŒ Œ� O .

(3) If m tŒ and l ŒO , then fl Ÿ ŒO . Conversely, if O is a subset of an intuitionistic fuzzy topological spaces satisfying the

properties from (1) to (3), then there exists a function 0123g -ŒG such that O consists of all -IFg open sets.

Proof. (1), (2) and (3) are obvious from Corollary 3.20, Proposition 3.2 (1) and 3.18, respectively. Conversely, let O be a subset of an intuitionistic fuzzy topological space satisfying the properties from (1) to (3). Define :g ÆF F . ( ) {l g l mÆ = ŒO�

}fm £ . It is clear to see that 12g -ŒG . Let m ŒO and l ŒF . Since ( )m g lŸ ŒO

by (3) and ( )m g l m lŸ £ Ÿ , we have ( ) ( )m g l g m lŸ £ Ÿ . Hence, 3g ŒG . Now it is

enough to show that O consists of all -IFg open sets. Let l ŒO . Then we have ( )g l

l= implying that l is a -IFg open set. Conversely, ( )l g l£ , then ( )l g l= ŒO .


Definition 3.32. Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on X and g ŒG . An intuitionisticd fuzzy subset l ŒF is said to be a

-IFg closed set if 1 l- is a -IFg open set.

Definition 3.33. The intersection of all -IFg closed sets containing l ŒF is called the g -closure of l and is denoted by ( )cg l .

The following Theorem 3.34 gives some properties of -IFg closed set and the g -closure operator. The easy proof of the theorem is omitted.

Theorem 3.34. Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on X and g ŒG . Then the following hold.

(1) 1 is a -IFg closed set.

(2) Arbitrary intersection of -IFg closed sets is a -IFg closed set.

(3) 0 is -IFg closed if and only if 1g ŒG .

(4) If g +ŒG , then every l ŒF is -IFg closed.

(5) For every l ŒF , ( )cg l is the smallest -IFg closed subset containing l .

(6) l is -IFg closed if and only if ( )cg l l= .

(7) 12cg +ŒG for every g ŒG .

(8) 0cg ŒG if and only if 1g ŒG .

(9) l is -IFgi closed if and only if ( )cg l l= . Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on

X and g ŒG . Define :g * ÆF F by ( ) ( )1 1g l g l* = - - for every l ŒF . The

following Theorem 3.35 gives properties of g * .

Theorem 3.35. (1) g * ŒG .

(2) ( )g g** = .

(3) 0g ŒG if and only if 1g * ŒG .



(6) g +ŒG if and only if g *-ŒG .

(7) ( ) cg gi*= .

(8) ( )cg gi*= .

(9) ( ) ( )1 1 cg gi l l- = - for every l ŒF .


(10) ( ) ( )1 1cg gl i l- = - for every l ŒF

Proof. (1) Let ,l m ŒF , such that l m£ . Then ( ) ( )1 1 1 1g l g m- - £ - - and so

( ) ( )g l g m* *£ . Hence g * ŒG .

(2) For l ŒF , ( ) ( ) ( )( )( ) ( )1 1 1 1g l g g l g l** *= - - - - = .

(3) 1g * ŒG if and only if ( )0 0g = if and only if 0g ŒG .

(4) The proof follows form (3).

(5) For l ŒF , 2g * ŒG if and only if ( ) ( ) ( )1 1g l g l** *- = - if and only if 1 -

( )( ) ( )1 1 1g g l g l*- - = - if and only if ( )( ) ( )1 1g g l g l- = - and so ( )( )g g l =

( )g l if and only if 2g ŒG .

(6) For l ŒF , g *-ŒG if and only if ( )1 1g l l* - £ - if and only if

( )( )1 g l- £ 1 l- if and only if ( )l g l£ if and only if g +ŒG .

(7) ( )1gi l- , is the largest -IFg open set contnained in 1 l- . Hence ( )1 1ig l- - ,

is the smallest -IFg closed set cotnaing l and so ( ) ( ) ( )1 1 cg g gi l i l l*- - = = . Hence

cg gi*= .

(8) The proof follows from (7).

(9) For l ŒF , from (8), ( ) ( ) ( )1 1 1c cg g gi l l l*- = - = - .

(10) The proof follows from (9). By Theorem 3.35 (1) above, g * ŒG and so we can define the collection of all -IFg *

open sets and -IFg * closed sets. Hence, we can define the g * -closure for a l ŒF .

The following Theorem 3.36 gives a characterization of -IFg * closed sets and Theorem

3.37 below gives a property of g * -closure operator cg * .

Theorem 3.36. Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on X and g ŒG . Then l is a -IFg * closed set if and only if ( )g l l£ .

Proof. l is a -IFg * closed set if and if 1 l- is a -IFg * open set if and only

( )( )1 1 1 1l g l- £ - - - if and only if ( )1 1l g l- £ - if and only if ( )g l l£ .

Theorem 3.37. Let X be a nonempty set, F be the family of all intuitionistic fuzzy sets defined on X and 12g +ŒG . Then cgg *= .

Proof. Let l ŒF . Since 2g ŒG , ( )( ) ( )g g l g l= . Therefore by Theorem 3.36, ( )g l

is a -IFg * closed set. Since g +ŒG , ( )l g l£ . Thus ( )g l is a -IFg * closed set such


that ( )l g l£ . If m is -IFg closed set such that l m£ , then ( ) ( )l g l g m£ £ and

so by Theorem 3.34 (5), ( ) ( )cgg l l*= and so cgg *= .

Theorem 3.38. Let X be a nonempty set, F the family of all intuitionistic fuzzy sets defined on X and 1 2,g g ŒG . Then the following hold.

(1) 1 2g g ŒG .

(2) If 1 2, ig g ŒG for { }2i sŒ - , then 1 2 ig g ŒG .

(3) ( )1 2 1 2g g g g* * *= .

Proof. (1) Let ,l m ŒF such that l m£ . Since 2g ŒG , ( ) ( )2 2g l g m£ . Since

1g ŒG , ( )( ) ( )( )1 2 1 2g g l g g m£ . Therefore, 1 2g g ŒG . (2) The proof is clear. (3) For l ŒF , ( ) ( ) ( ) ( )( )( )1 2 1 2 1 21 1 1 1 1 1 1g g l g g l g g l* = - - = - - - - = -

( )( ) ( )( )1 2 1 21g g l g g l* * *- = . Therefore, ( )1 2 1 2g g g g* * *= .

Theorem 3.39. Let X be a nonempty set, F the family of all intuitionistic fuzzy sets defined on X and 2,i k ŒG . Suppose ( ) ( )ik l k l£ and ( ) ( )ik l kik l£ for every l ŒF . If g is a product of alternating factors i and k , except g ki= , then 2g ŒG . In addition, if ( ) ( )i l ki l£ for every l ŒF and g ki= , then 2g ŒG .

Proof. 2,i k ŒG implies that ,i k ŒG and so by Theorem 3.38 (1), g ŒG . Let l ŒF . By hypothesis ( ) ( )ik l k l£ and so ( ) ( ) ( )kik l kk l k l£ = . Therefore, ( )ikik l £

( )ik l . Again by hypothesis ( ) ( )ik l kik l£ and so ( )iik ikik l£ which implies that

( ) ( )ik l ikik l£ . Hence ( ) ( )ikik l ik l= and so ikik ik= . Further, ( )( )iki iki =

( ) ( )ikiki ikik i iki= = . ( )( ) ( )kik kik kikik k ikik kik= = = . ( ) ( )( )ikik ik ik= =

ikik . ( )( ) ( ) ( ) ( )kiki kiki k ikik iki k ik iki ik ikik i kik= = = = . Thus the statement is valid for three or four factors. Since ikik ik= , a alternating product of 5n ≥ factors is equal to another product of 2n - factors and the statement holds for it. Suppose g ki= . Since by hypothesis, ( ) ( )ik l k l£ , we have ( ) ( )iki l ki l£ and so ( ) ( )kiki l ki l£ .

Since ( ) ( )i l ki l£ , by hypothesis, we have ( ) ( )i l iki l£ and so ( ) ( )ki l kiki l£ . Hence it follows that ki kki= .

Corollary 3.40. Let X be a nonempty set, F the family of all intuitionistic fuzzy sets defined on X , 2i -ŒG and 2k +ŒG . If g is a product of factors of i and k then

2g ŒG .

4. On g -intuitionistic fuzzy semiopen sets


In this section, we introduce a new class of fuzzy set called g -intuitionistic fuzzy semioprn sets, which contains the class of all -IFg open sets. We characterize such sets and establish some of its properties.

Definition 4.1. Let X be a nonempty set, F the family of all fuzzy sets defined on X and g ŒG , l ŒF is said to be a g -intuitionistic fuzzy semiopen (briefly, -IFg semioprn) set if there exists a -IFg open set m such that ( )cgm l m£ £ .

We will denote the family of all -IFg semiopen set by ( )s g . The following Theorem 4.2 gives some properties of -IFg semiopen sets.

Theorem 4.2. Let X be a nonempty set, F the family of all intuitionistic fuzzy sets defined on X and g ŒG . Then the following hold.

(1) Every -IFg open set is a -IFg semiopen set.

(2) 0 is a -IFg semiopen set.

(3) Arbitrary union of -IFg semiopen sets is a -IFg semiopen set.

Proof. (1) Suppose l is -IFg open. Then, ( )cgl l l= £ and so l is -IFg semiopen.

(2) Let { }al a ŒD be the family of -IFg semiopen sets. Let al l= � . Since each

al is -IFg semiopen, there exist a -IFg open set am such that ( )ca a g am l m£ £

and so ( )a a g am l m£ £� � �� . Let am m= � . Then m is a -IFg open set. Also,

( ) ( ) ( ) ( )c c c cg a g a g gm m m m£ £ =� � � � . Thus m is a -IFg open set such that

( )ca gm l m£ £� and so am� is a -IFg semiopen set.

Theorem 4.3. Let X be a nonempty set, l ŒF and g ŒG . Then the following hold.

(1) ( )cs l is the smallest -IFg semiclosed set containing l .

(2) l is -IFg semiclosed if and only if ( )csl l= .

(3) 012cs +ŒG ( 0cs ŒG by Theorem 4.2 (b) and 1cs ŒG by Theorem 4.2 (3)).

(4) ( )si l is the largest -IFg semiopen set contained in l .

(5) l is -IFg semiopen if and only if ( )sl i l= .

(6) 012si -ŒG .

(7) If ( ),x a b is a intuitionistic fuzzy point, then ( ) ( ),x sa b i lŒ if and only if there is a

-IFg semiopen set m containing ( ),x a b such that ( ),x a b m lŒ £ .

The following Theorem 4.4 gives the relation between si and cs .

Theorem 4.4. Let X be a nonempty set and g ŒG . Then the following hold.


(1) cs si* = .

(2) cs si* = .

(3) ( ) ( )1 1 cs si l l- = - for every l ŒF .

(4) ( ) ( )1 1cs sl i l- = - for every l ŒF .

Proof. (1) Let l ŒF . Then ( ) ( ) ( )1 1s si l i l* = - - . Since ( )1si l- is the largest

-IFg semiopen set contained in 1 l- , ( )1 1si l- - is the smallest -IFg semiclosed

set containing l and so ( ) ( )1 1 cs si l l- - = . Hence cs si* = .

(2) By Theorem 3.35 (7) and (1), ( ) ( )( )cs s si i** *= = . This proves (2).

(3) If l ŒF , then ( ) ( ) ( )1 1s si l i l* = - - and so by (2), ( ) ( )1 1cs sl i l= - -

which implies that ( ) ( )1 1 cs si l l- = - for every l ŒF . (4) The proof is similar to the proof of (3).

The following Theorem 4.5 and Theorem 4.6 (1) give characterizations of g -fuzzy semiopen sets in terms of -IFg interior and -IFg closure operators. Theorem 4.6 (2), (3) and (4) give properties of -IFg semiinterior and -IFg semiclosure operators.

Theorem 4.5. Let X be a nonempty set, g ŒG and l ŒF . The following are equivalent

(1) l is -IFg semiopen.

(2) ( )cg gl i l£ .

(3) ( ) ( )c cg g gl i l= .

Proof. (1) fi (2) Suppose l is -IFg semiopen. Then there exists a -IFg open set m such that ( )cgm l m£ £ . Since m is -IFg open, ( )gm i m= and so ( )cg gl i m£ .

Since cg gi ŒG , by Theorem 3.38 (1), and m l£ , it follows that ( )cg gl i l£ , which proves (2).

(2) fi (3) Since cg ŒG and ( )gi l l£ , we have ( ) ( )c cg g gi l l£ . By hypothesis

and Theorem 3.38 (7), ( ) ( ) ( )c c c cg g g g g gl i l i l£ = . Therefore, ( ) ( )c cg g gl i l= .

(3) fi (1) Since ( )gi l is a -IFg open set such that ( ) ( )cg g gi l l i l£ £ , l is -IFg semiopen.

Theorem 4.6. Let X be a nonempty set, g ŒG and l ŒF . Then the following hold.

(1) l is -IFg semiopen if and only if l is -c IFg gi open if and only if ( )cg gil i l= .

(2) cg gs ii i= and cc cg gs i= .

(3) ( ) ( )cs g gi l l i l= Ÿ .


(4) ( ) ( )c cs g gl l i l= ⁄ .

Proof. The proof of (1) follows from Theorem 4.5 (1) and (2). (2) If ( )x si lŒ , then there exists a -IFg semiopen set m such that x m lŒ £ . By (1), m is a -c IFg gi open

set and so ( )cxg gii mŒ . Hence ( ) ( )cg gs ii l i l£ . Similarly, we can prove that ( )cg gi

i l £

( )si l . Therefore, cg gs ii i= . Again, ( )cs si*= , by Theorem 4.4 (1) and so ( )cc

g gs ii*

=

cc g gi= , by Theorem 3.35 (7). (3) Since ( )s s si i l i l= and cs s si l i lÃ for every intuit-

tionistic fuzzy subset l ŒF . Then we have ( )c cg gi g gi l l i l= Ÿ and so by, (2), si l l=

cg gi lŸ . (4) Since ( ) ( )cs g gi l l i l= Ÿ , ( ) ( ) ( )c cs g gl l i l*

= ⁄ = ( ) ( ) ( )cg gl i l* *

⁄ =

( )cg gl i l⁄ , by Theorem 3.38 (3) and Theorem 3.35 (7). By (2), ( ) ( )c cs g gl l i l= ⁄ .

Theorem 4.7. If X is nonempty set, g ŒG , l ŒF , ( )cgl m l£ £ and l is -IFg semiopen, then m is -IFg semiopen. In particular, the g -closure of every -IFg semiopen set is a -IFg semiopen set.

Proof. Since l is -IFg semiopen, by Theorem 4.5 (3), ( ) ( )c cg g gl i l= and so

( ) ( )c cg g gl i m£ . Since ( )cgm l£ , ( )cg gm i m£ and so by Theorem 4.5, m is -IFg semiopen.

The following Theorem 4.8 gives characterizations of -IFg semiclosed sets.

Theorem 4.8. Let X be a nonempty set, g ŒG and g ŒF . Then the following are equivalent.

(1) l is -IFg semiclosed.

(2) ( )cg gi l l£ .

(3) ( ) ( )cg g gi l i l= .

(4) There exists a -IFg closed set v such that ( )v vgi l£ £ .

Proof. (1) fi (2) If l is -IFg semiclosed, then 1 l- is -IFg semiopen and so

( )1 1cg gl i l- £ - , by Theorem 4.5 (2). By Theorem 3.35 (9) and (10), it follows that

( ) ( )1 1c cg g g gi l i l- = - and so cg gi l£ . (2)fi (3) ( )cg gi l l£ implies that ( )cg gi l

l£ . (3)fi (4). If ( )v cg l= , then v is a -IFg closed set such that ( ) ( )v cg g gi i l= =

( ) vgi l l£ £ , which prove (4). (4) fi (1). If there exists a -IFg closed set v such

that ( )v vgi l£ £ , then ( ) ( )1 1 1 1v v c vg gl i- £ - £ - = - . Since 1 v- is -IFg open,

1 l- is -IFg semiopen and so l is -IFg semiclosed.

References


[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [2] D. Coker, An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces, J. Fuzzy math.,

4 (2) (1996), 749-764. [3] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88

(1997), 81-89. [4] D. Coker and M. Demirci, On intuitionistic fuzzy points, Notes on Intuitionistic Fuzzy Sets, 1 (1995),

79-84. [5] H. Gurcay and D. Coker, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math.,

5 (1997), 365-378. [6] I. M. Hanafy, intuitionistic fuzzy g -continuity, Canad. Math. Bull., 52 (4) (2009), 544-554.

[7] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.


__________________________ Received April, 2013


Los Angeles

More on Generalized Intuitionistic Fuzzy Topology

N. Gowrisankar

70/232 6B, Kollupettai Street, M.Chavady, Thanjavur-613001, Tamil-nadu, India E-mail address: [email protected]

N. Rajesh

Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India


V. Vijayabharathi

Department of Mathematics, National Institute of Technology, Tiruchi-Rapalli, Tamilnadu, India E-mail address:[email protected]

Abstract: The purpose of the paper is to investigate some results concering particular monotonic

functions in generalized intuitionistic fuzzy topology.

Key words and phrases: Generalized intuitionistic fuzzy topology

1. Introduction

After the introduction of fuzzy sets by Zadeh [6], there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them. Using the notion of intuitionistic fuzzy sets, Coker [3] introduced the notion of intuitionistic fuzzy topological spaces. The aim of this paper is to present a common approach allowing to obtain familes of intuitionistic fuzzy sets in an intuitionistic fuzzy topological space.

2. Preliminaries


functions :A X Im Æ and :A X Ig Æ denote respectively the degree of membership


(namely ( )A xm ) and the degree of non-membership (namely ( )A xg ) of each element


( ) ( ){ }, , :A AA x x x x Xm g= Œ .

Definition 2.2 [1]. Let X be a nonempty set and let the IFS ’s A andB in the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ ( ) ( ){ }, , :B BB x x x x Xm g= Œ . Let { }:jA j JŒ be

an arbitrary family of IFS ’s in ( ),X t . Then,


(2) ( ) ( ){ }, , :A AA x x x x Xg m= Œ ;

(3) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∩ � � ;

(4) ( ) ( ){ }, , :j jj A AA x x x x Xm g= Œ∪ � � ;

(5) { }1 ,1,0 :X x x X= Œ and { }0 ,0,1 :X x x X= Œ .


(i) 0 ,1X X tŒ ;



In this case, the ordered pair ( ),X t is called an intuitionistic fuzzy topological space ( IFTS , for short) and each IFS in t is known as an intuitionistic fuzzy open set ( IFOS , for short) in X . The complement of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set ( IFCS , for short). The family of all IFOSs (resp. IFCSs ) of ( ),X t is denoted by ( )IFO X (resp. ( )IFC X ).

Definition 2.4 [3]. Let ( ),X t be an IFTS and let ( ) ( ){ }, , :A AA x x x x Xm g= Œ

be an IFS inX . Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A is defined by ( ) { }Int \ is an in and A G G IFOS X G A= Õ∪ and ( ) {Cl \ A G G= ∩

}is an in and IFCS X G A .

Remark 2.5. For any IFS A in ( ),X t , we have, ( ) ( )Cl 1 1 IntA A- = - , ( )Int 1 A-

( )1 Cl A= - .

Let X be a nonempty set and [ ]{ }: 0,1Xl l= ÆF be the family of all

intuitionistic fuzzy sets defined on X . Let :g ÆF F be a function such that l m£

More on Generalized Intuitionistic Fuzzy Topology 37

implies that ( ) ( )g l g m£ for every ,l m ŒF . That is, g is a monotonic function

defined on F by ( )G F or simpily G . We will define the following subclasses of G .

(1) For every [ ]0,1a Œ , define ( ){ }a g g a aG = ŒG = where a is the intuitionistic

fuzzy set defined by ( )xa a= for every x XŒ ,

(2) ( ) ( ){ }22 for every g g l g l lG = ŒG = ŒF ,

(3) ( ){ } for every g l g l l+G = ŒG £ ŒF and

(4) ( ){ } for every g g l l-G = ŒG £ F .

(5) ( )( ){ }2 :g g g l gl-G = ŒG £ .

If Â is a collection of some of the symbols 2, - , + and [ ]0,1a Œ , then

{ } for every ig g iÂG = ŒG ŒG ŒÂ .

3. Main results

Definition 3.1. Let ,i k ŒG and A be an intuitionistic fuzzy subset of X . Then two operations i and k are said to be

(1) ( ),a i k -related for A if A Ai iki£ ;

(2) ( ),s i k -related for A if A Ai ki£ ;

(3) ( ),p i k -related for A if A Ai ik£ ;

(4) ( ),b i k -related for A if A Ai kik£ .

Remark 3.2. (1) Let X be an intuitionistic fuzzy topological space, and let Inti = and Clk = . Then for an intuitionistic fuzzy subset A of X , if A is a -open (resp. preopen), then i and k are ( ),a i k -related (resp. ( ),p i k -related) for A .

(2) Let ( ),X l be a generalized intuitionitsic topological space [4], and let ili = and clk = . Then for an intuitionistic fuzzy subset A , if A is -l a -open (resp. l -

preopen), then i and k are ( ),a i k -related (resp. ( ),p i k -related) for A . (3) In (2), since ,i k ŒG , for the i -open set A of X , if A is iki -open (resp. ki -

open, ik -open, kik -open), then i and k are ( ),a i k -related (resp., ( ),s i k -related,

( ),p i k -related, ( ),b i k -related) for A .

Definition 3.3. Let ,i k ŒG and A be an intuitionistic fuzzy subset of X . Then two operations i and k are said to be ( ),a i k -related (resp., ( ),s i k -related, ( ),p i k -related,

( ),b i k -related) if ( ),a i k -related (resp., ( ),s i k -related, ( ),p i k -related, ( ),b i k -related) for all intuitionistic fuzzy subsets A in X .


Theorem 3.4. For ,i k ŒG , if i and k are ( ),s i k -related and ( ),p i k -related, then

they are ( ),b i k -related.

Proof. For an intuitionistic fuzzy subset A of X , A A A Ai ik ki kik£ fi £ . From the ( ),s i k -relatedness, we have A A Ai ki kik£ £ . Hence i and k are ( ),b i k -related.

Consider the following conditions: For ,i k ŒG , (R1) A Aki k£ for every intuitionistic fuzzy subset A of X . R(2) A Aik k£ for every intuitionistic fuzzy subset A of X .

Theorem 3.5. Let ,i k ŒG . Then

(1) if i and k are ( ),a i k -related and (R1), then they are ( ),p i k -related;

(2) if i and k are ( ),a i k -related and (R2), then they are ( ),s i k -related.

Proof. (1) For an intuitionistic fuzyz subset A of X , from (R1), A A Ai iki ik£ £ . Hence ( ),p i k -relatedness is proved.

(2) For an intuitionistic fuzzy subset A of X , from (R2), A A Ai iki ki£ £ and so they are ( ),s i k -related.

Corollary 3.6. Let ,i k ŒG . Then if i and k are ( ),a i k -related, (R1) and (R2),

then they are ( ),b i k -related.

Proof. It follows from Theorem 3.4 and Theorem 3.5. Remark 3.7. Let ,i k ŒG .

(1) ( )( ) ( ), -relatedness

, -relatedness, -relatedness

s

p

i kb i k

i kÏ ¸Ô ÔÆÌ ˝Ô ÔÓ ˛

.

(2) ( ) ( )( ) ( )

2

1

, -relatedness , -relatedness, -relatedness , -relatedness

R

R

s

p

a i k i ka i k i kÏ ææÆÔÌ ææÆÔÓ

(3) ( ),a i k -related +(R1)+(R2) fi ( ),b i k -relatedness.

Theorem 3.8. Let k +ŒG . Then for every i ŒG , i and k are

(1) ( ),p i k -related;

(2) ( ),s i k -related;

(3) ( ),b i k -related;

(4) if ( ),s i k -related, then ( ),b i k -related;

(5) if ( ),p i k -related, then ( ),b i k -related.

Proof. (1) For every intuitionistic fuzzy subset A of X , easily we have A A A kAk i i£ fi £ .

(2) For every intuitionistic fuzzy subset A of X , A Ai ki£ . (3) For every intuitionistic fuzzy subset A of X , A A Ai ik kik£ £ .


(4), (5) For every intuitionistic fuzzy subset A of X , it follows A A Ai ki kik£ £ and A A Ai ik kik£ £ .

Remark 3.9. For i ŒG , k +ŒG , from Remark 3.7 and Theorem 3.8,

( ) ( ) ( )( ) ( ) ( )

2

1

, -relatedness , -relatedness , -relatedness., -relatedness , -relatedness , -relatedness.

R

R

s

p

a i k i k b i ka i k i k b i kÏ ææÆ ÆÔÌ ææÆ ÆÔÓ

Lemma 3.10. Let i -ŒG . Then for every k ŒG , (R1) and (R2) are fulfilled.

Theorem 3.11. Let i -ŒG . For every k ŒG , if i and k are ( ),a i k -related, then they are



(3) ( ),b i k -related.

Remark 3.12. Let i -ŒG , k ŒG . (1)

( ),a i k -relatedness( )( ) ( ), -relatedness

,, -relatedness

s

p

i kb i k

i kÏ ¸Ô ÔÆ ÆÌ ˝Ô ÔÓ ˛

-relatedness.

(2) If i -ŒG and k +ŒG , then by Theorem 3.8 and Theorem 3.11,

( ) ( ) ( )( ) ( ) ( )

, -relatedness , -relatedness , -relatedness., -relatedness , -relatedness , -relatedness.

s

p

a i k i k b i ka i k i k b i kÏ Æ ÆÔÌ Æ ÆÔÓ

Theorem 3.13. Let 2i ŒG and k ŒG . If i and k are ( ),s i k -related, then they are

( ),a i k -related.

Proof. For every intuitionistic fuzzy subset A of X , since A Aii i= , we have A A A A A Ai ki ii iki i iki£ fi £ fi £ . Hence i and k are ( ),a i k -realted.

Theorem 3.14. Let 2i ŒG and k ŒG . If i and k are ( ),p i k -related, then they are

( ),a i k -related.

Proof. For a intuitionistic fuzzy subset A of X , since 1Ai £ , by hypothesis, ( ) ( )A A Ai i i ik i= £ . Therefore, i and k are ( ),a i k -related.

Theorem 3.15. Let 2i ŒG and k ŒG . Then

(1) if i and k are ( ),s i k -related and (R1), then they are ( ),p i k -related;


(2) if i and k are ( ),p i k -related and (R2), then they are ( ),s i k -realted.

Proof. (1) For an intuitionistic fuzzy subset A of X , A A A A Ai ki i ii iki£ fi = £ £ Aik . Therefore, i and k are ( ),p i k -related.

(2) For an intuitionistic fuzzy subset A of X , since 1Ai £ , by ( ),p i k -relatedness,

A A Aii iki ki£ £ . From A Ai ii= , it follows that i and k are ( ),s i k -related.

Lemma 3.16. Let 2i ŒG and k +ŒG . Then i and k are ( ),a i k -related.

Proof. First, i and k are ( ),s i k -related (or ( ),p i k -realted) from Theorem 3.8.

Thus from Theorem 3.13 (or Theorem 3.14), they are ( ),a i k -related.

Remark 3.17. For 2i ŒG , k ŒG , we have the following:

(1) ( ) ( )( ) ( )

, -relatedness , -relatedness, -relatedness , -relatedness

s

p

i k a i ki k a i k

Ï ÆÔÌ ÆÔÓ

(2) If i and k satisfy (R1), then: ( ),s i k -relatedness ( ),p i k -relatedness ( ),a i k -relatedness.

(3) If i and k satisfy (R2), then: ( ),p i k -relatedness ( ),s i k -relatedness ( ),a i k -relatedness.

Lemma 3.18. Let i ŒG and 2k ŒG . If A Ai k£ , then

(1) they satisfy (R1) and (R2); (2) A Akik k£ .

Proof. Obvious.

Lemma 3.19. Let i ŒG and 2k ŒG . Then if i and k are (R1) (or (R2)), for every intuitionistic fuzzy subset A of X , then A Akik k£ .

Proof. Obvious.

Lemma 3.20. Let i ŒG and 2k ŒG . If i and k are (R1) (resp., (R2)) and ( ),s i k -

related (resp., ( ),p i k -related), then A Ai k£ .

Lemma 3.21. Let 2,i k ŒG . If i and k are (R1) and ( ),s i k -related, then

(1) A Aikik ik= . (2) A Akiki ki= .

Theorem 3.22. Let 2,i k ŒG . If i and k are ( ),b i k -related and (R1) (or (R2)), then they are




(3) ( ),a i k -related.

Proof. (1) For an intuitionistic fuzzy subset A of X , since Ai is an intuitionistic fuzzy subset of X , from the ( ),b i k -relatedness and the condition (R1), it follows

( ) ( ) ( )A A A Ai i kik i kk i ki£ £ £ . Hence by 2i ŒG , i and k are ( ),s i k -related.

(2) For an intuitionistic fuzzy subset A of X , from the ( ),b i k -relatedness and the condition (R1), it follows A A A Ai kik kk k£ £ £ . This implies A A Ai ii ik= £ . Hence i and k are ( ),p i k -related.

(3) It follows from Theorem 3.13.

Corollary 3.23. Let 2,i k ŒG . (1) If i and k satisfy (R1), then:

( ) ( )

( ) ( )

, -relatedness , -relatedness

, -relatedness , -realtednesss p

b i k a i k

i k i kÆ

(2) If i and k satisfy (R2), then:

( ) ( )

( ) ( )

, -relatedness , -relatedness

, -relatedness , -realtednessp s

b i k a i k

i k i kÆ

Proof. (1) It follows from Theorem 3.4, Theorem 3.15 (1), Remark 3.17 and Theorem 3.22.

(2) It follows from Theorem 3.4, Theorem 3.15 (2), Remark 3.17 and Theorem 3.22.

Remark 3.24. Let 2,i k ŒG . Ifi -ŒG , then by Lemma 3.10, ( ),a i k -related ( ),s i k´ -

related ( ),p i k´ -related ( ),b i k´ -realted.

4. ( ),a i k -open, ( ),s i k -open, ( ),p i k -open, ( ),b i k -open

Definition 4.1. Let ,i k ŒG and A be an intuitionistic fuzzy subset of X . Then A is said to be (1) ( ),a i k -open if A Ai iki£ ; (2) ( ),s i k -open if A Ai ki£ ; (3) ( ),p i k -

open if A Ai ik£ ; (4) ( ),b i k -open if A Ai kik£ .

Remark 4.2 (1) Let ( ),X t be an intuitionistic fuzzy topological space, and let Inti = and Clk = . Then for an intuitionistic fuzzy subset A of X , if A is a -open

(resp., preopen), then it is ( ),a i k -oopen (resp. ( ),p i k -open).


(2) Let ( ),X l be a generalized topological space, and let ili = and clk = . Then for an intuitionistic fuzzy subset A of X , if A is -i a -open (resp. i -preopen), then it is ( ),a i k -open (resp. ( ),p i k -open). Furthermore, if A is a l -open set, then A is also

( ),a i k -open ( ),s i k -open, ( ),p i k -open, ( ),b i k -open).

Lemma 4.3. Let ,i k ŒG . Then every ki -open set is ( ),a i k -open.

Theorem 4.4. Let ,i k ŒG and A be an intuitionistic fuzzy subset of X . If 2i -ŒG , then the following hold:

(1) iki -open ( ),a i kfi -open,

(2) ik -open ( ),p i kfi -open.

Theorem 4.5. Let ,i k ŒG and A be an intuitionistic fuzyz subset of X . If i , k satisfy (R2), then the following hold:

(1) iki -open ( ),a i kfi -open,

(2) ki -open ( ),s i kfi -open,

(3) ik -open ( ),p i kfi -open,

(4) kik -open ( ),b i kfi -open.

Corollary 4.6. Let ,i k ŒG and A be an intuitionistic fuzzy subset of X . If i -ŒG , then the following hold:

(1) iki -open ( ),a i kfi -open;

(2) ki -open ( ),s i kfi -open;

(3) ik -open ( ),p i kfi -open;

(4) kik -open ( ),b i kfi -open.

Lemma 4.7. Let ,i k ŒG . For an i -open set A of X , if A is ( ),a i k -open (resp.

( ),s i k -open, ( ),p i k -open, ( ),b i k -open), then it is iki -open (resp. ki -open, ik -open, kik -open).

Remark 4.8. (1) For ,i k ŒG ,( )( ) ( ), -open

, -open, -open

s

p

i kb i k


. In [4], the authors

have shown that if i -ŒG and 2k -ŒG , then: , -open

, -open, -openk i

iki ki ki kÏ ¸

Æ ÆÌ ˝Ó ˛

.

(3) Ifi -ŒG and k ŒG , then: ( ) ( )( ) ( ), -open

, -open , -open, -open

s

p

i ka i k b i k


.


(4) If i -ŒG and k +ŒG , then: i -open ( ) ( ), ,sa i k i kÆ Æ -open ( ),b i kÆ -open.

i -open ( ) ( ), ,pa i k i kÆ Æ -open ( ),b i kÆ -open. Furthermore, since a fuzzy subset A is i -open if and only if A Ai = , we have i -open ikiÆ -open kiÆ -open kikÆ -open. i -open ikiÆ -open ikÆ -open kikÆ -open.

References

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [2] D. Coker, An introduction to fuzzy subspaces in intuitionistic fuzyz topological spaces, J. Fuzzy

Math., 4 (2) (1996), 749-764. [3] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88

(1997), 81-89. [4] N. Gowrisankar, N. Rajesh and V. Vijayabharathi, On generalized intuitionistic fuzyz topology

(submitted). [5] H. Gurcay and D. Coker, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math.,

5 (1997), 365-378. [6] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.


__________________________ Received May, 2013


Los Angeles

Characterization of Connectedness and Surroundness in An Intuitionistic Fuzzy Digital Topology

R. Narmada Devi, E. Roja and M. K. Uma

Department of Mathematics, Sri Sarada College for Women, Salem, Tamil Nadu, India

E-mail: [email protected]

Abstract: Topological relationships among parts of a digital picture, such as connectedness and

surroundedness, play an important role in picture analysis and description. This paper generalizes these concepts to an intuitionistic fuzzy sets and discussed some of their basic properties.

Keywords: Intuitionistic fuzzy strength of a path, intuitionistic fuzzy connectedness, intuitionistic

fuzzy components like intuitionistic fuzzy plateau, intuitionistic fuzzy top, intuitionistic fuzzy bottom and intuitionistic fuzzy surroundness.

1. Introduction

The concept of fuzzy sets was introduced by Zadeh [5] and later Atanassov [1] generalized the idea to intuitionistic fuzzy sets. Geometrical properties and relationships among parts of a digital picture play an important role in picture analysis and description that was studied by A. Rosenfeld and A. Kak [2,3]. Later A. Rosenfeld [4] extend the concepts of digital picture geometry to fuzzy sets. In this paper, the topological concepts of connectedness and surroundness with respect to an intuitionistic fuzzy sets are introduced. Some interesting properties are established.

2. Preliminaries

Let Â be a rectangular array of integer-coordinate points. Thus the point ( ),P x y=

of Â has four horizontal and vertical neighbors, namely ( )1,x y± and ( ), 1x y ± ; and it

also has four diagonal neighbors, namely ( )1, 1x y± ± and ( )1, 1x y± ∓ . We say that former points are 4-adjacent to, or 4-neighbors of P and we say that both types of neighbors are 8-adjacent to, or 8-neighbors of P . Note that if P is on the border B of Â , some of these neighbors may not exist.

46 R. Narmada Devi, E. Roja and M. K. Uma

For all points P , Q of Â , by a path r from P to Q we mean a sequence of points

0 1, , , nP P P P Q= = such that iP is adjacent to 1iP - , 1 i n£ £ . Note that this is two definitions in one (“4-path” and “8-paht”), depending on whether “adjacent” mans “4-adjacent” or “8-adjacent”. The same is true for many of the definitions that follow, but we usually do not mention this. Let S be any subset of Â . We say that the points P , Q of Â are connected in S if there is a path from P to Q consisting entirely of points of S . Readily, “connected” is an equivalence relation: P is connected to P (by a path of length 0); if P is connected to Q then Q is connected to P (the reversal of a path is a path) and if P is connected to Q and Q to R , then P is connected to R (the concatenation of two paths is a path). This relation partitions S into equivalence classes, which are maximal subsets iS of S such that every P , Q belonging to a given iS are connected. These classes are called the (connected) components of S .

Let S S= Â- be the complement of S . We assume, for simplicity, that the border points of Â are all in S . Thus one component of S always contains the border B of Â . The other components, if any, are called holes in S . [it turns out that opposite types of connectedness (4-and 8-, or 8- and 4-) should be sued for S and for S , in order for various algorithms to work properly; see Rosenfeld and Kak [2,3]. If S has no holes, it is called simply connected. Let S and T be disjoint subsets of Â . We say that S surrounds T if any path from T to the border of Â must meet S . Readily, “surrounds” is a strict order relation: If S surrounds T , then T does not surround S ; and if S surrounds T and T surrounds W , then S surrounds W . [More generally, for any subsets U , V , W of Â , we say that V separates U from W if any path from U to W must meet V . Thus S surrounds T if it separates T from the border B of Â ].

Definition 2.1 [1]. Let X be a nonempty fixed set and I be the closed interval [0,1]. An intuitionistic fuzzy set ( IFS ) A is an object of the following form ( ){ , ,AA x xm=

( ) }:A x x Xg Œ , where the mappings :A X Im Æ and :A X Ig Æ denote the degree

of membership (namely ( )A xm ) and the degree of nonmember-ship (namely ( )A xg ) for

each element x XŒ to the set A , respectively, and ( ) ( )0 1A Ax xm g£ + £ for each x XŒ . Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form, ( ) ( ){ }, ,1 :A AA x x x x Xm m= - Œ . For the sake of simplicity, we shall use the

symbol , ,A AA x m g= for the intuitionistic fuzzy set ( ) ( ){ }, , :A AA x x x x Xm g= Œ .

Definition 2.2 [1]. Let X be a nonempty set and the IFSsA and B in the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ , ( ) ( ){ }, , :B BB x x x x Xm g= Œ . Then

(i) A BÕ iff ( ) ( )A Bx xm m£ and ( ) ( )A Bx xg g≥ for all x XŒ ;

(ii) ( ) ( ){ }, , :A AA x x x x Xg m= Œ ;

(iii) ( ) ( ) ( ) ( ){ }, , :A B A BA B x x x x x x Xm m g g« = Ÿ ⁄ Œ ;

Characterization of Connectedness and Surroundness in an Intuitionistic Fuzzy Digital Topology 47

(iv) ( ) ( ) ( ) ( ){ }, , :A B A BA B x x x x x x Xm m g g» = ⁄ Ÿ Œ .

Definition 2.3 [1]. The ~0IFSs and ~1 are defined by { }~0 ,0,1 :x x X= Œ and

{ }~1 ,1,0 :x x X= Œ .

3. Properties of an intuitionistic fuzzy connectedness in an intuitionistic fuzzy digital topology

Definition 3.1. Let Â be a rectangular array of integer-coordinate points. Let , ,A AA P m g= be an intuitionistic fuzzy set in Â . Let 0 1: , , , nP P P P Qr = = be

any path between two points of Â . Then an intuitionistic fuzzy strength of a path with respect to an intuitionistic fuzzy set A in Â is defined and denoted by As with

( ) ( )0min

As A ii n

Pm r m£ £

= and ( ) ( )0max

As A ii n

Pg r g£ £

=

such that ( ) ( ) 1A As sm r g r+ £ .

Definition 3.2. Let Â be a rectangular array of integer-coordinate points. Let , ,A AA P m g= be an intuitionistic fuzzy set in Â . Let 0 1: , , , nP P P P Qr = = be

any path between two points of Â . Then an intuitionistic fuzzy connectedness of the points P and Q of a path with respect to an intuitionistic fuzzy set A in Â is defined and denoted by AC with ( ) ( ), max

A AsP Q

rm m r=C and ( ) ( ), min

A AsP Q

rg g r=C such

that 1A A

m g+ £C C . Clearly, AC is an intuitionistic fuzzy set in Â¥Â .

Proposition 3.1. Let Â be a rectangular array of integer-coordinate points. Let , ,A AA P m g= be an intuitionistic fuzzy set in Â . Then for all P , Q in Â ,

(i) ( ) ( ),A AP P Pm m=C and ( ) ( ),

A AP P Pg g=C .

(ii) ( ) ( ), ,A AP Q Q Pm m=C C and ( ) ( ), ,

A AP Q Q Pg g=C C .

Proof. (i) For any path 0 1: , , , nP P P P Qr = = , the degree of membership of an in-tuitionistic fuzyz strength of a path with respect to an intuitionistic fuzzy set A in Â is always less than or equal to the degree of membership of an intuitionistic fuzzy set A . That is, ( ) ( )

As A Pm r m£ and the degree of nonmembership of an intuitionistic fuzzy

strength of a path with respect to an intuitionistic fuzzy set A inÂ is always greater than or equal to the degree of nonmembership of an intuitionistic fuzzy setA . That is, ( )

Asg r

( )A Pg≥ Now, let ir be any path from P to P itself. Thus ( )iA

AsPrm m= and

iAsrg =

( )A Pg , for P ŒÂ and 1, 2,3, ,i n= . This implies that ( ) ( )maxA

i

A s iPr

m m r= =

( ),AP PmC and ( ) ( ) ( )min ,

A Ai

A s iP P Pr

g g r g= = C

(ii) it follows from the fact that the reversal of a path is a path and reverse path preserves path strength.


Remark 3.1. Let Â be a rectangular array of integer-coordinate points. Let ,A P=

,A Am g be an intuitionistic fuzzy set in Â . Let ( ){ 1AP PmW = ŒÂ = ( ) }and 0A Pg = .

Let 0 1: , , , nP P P P Qr = = be any path between two points of Â . Let As and AC be an intuitionistic fuzyz strength of a path and intuitionistic fuzyz connectedness of the points P and Q of a path with respect to an intuitionistic fuzzy set A in Â . Then

(i) ( ) 1As

m r = and ( ) 0As

g r = if and only if a path r contains entirely of points of W .

(ii) ( ), 1AP Qm =C and ( ), 0

AP Qg =C if and only if P and Q are intuitionistic fuzzy

connected in W .

Proposition 3.2. Let Â be a rectangular array of integer-coordinate points. Let , ,A AA P m g= be an intuitionistic fuzzy set in Â . Then for all ,P QŒÂ , we have

( ) ( ) ( )( ), min ,A A AP Q P Qm m m£C and ( ) ( ) ( )( ), max ,

A A AP Q P Qg g g≥C .

Proof. Let 0 1: , , , nP P P P Qr = = be any path. Then the degree of membership of an intuitionistic fuzyz strength of a path r with respect to an intuitionistic fuzzy set A in Â is given by

( ) ( ) ( ) ( )( ) ( ) ( )( )00min min , min ,

As A i A A n A Ai n

P P P P Qm r m m m m m£ £

= £ = (3.1)

and the degree of nonmembership of an intuitionistic fuzzy strength of a path r with respect to an intuitionistic fuzzy set A in Â is given by

( ) ( ) ( ) ( )( ) ( ) ( )( )00max max , max ,

As A i A A n A Ai n

P P P P Qg r g g g g g£ £

= ≥ = (3.2)

This implies that AC is an intuitionistic fuzzy connected of P and Q with respect to

an intuitionistic fuzzy set A whose ( ) ( ) ( ) ( )( ), max min ,A As A AP Q P Q

rm m r m m= £C

and ( ) ( ) ( ) ( )( ), min max ,A As A AP Q P Q

rg g r g g= ≥C

Definition 3.3. Let Â be a rectangular array of integer-coordinate points. Let A =

, ,A AP m g be an intuitionistic fuzzy set in Â . Let ( ){ ( )1and A AP P Pm gW = ŒÂ =

}0= . Let 0 1: , , , nP P P P Qr = = be any path between two points of Â . Let G Õ Â be any subset of Â . Then intuitionistic fuzzy connectedness of G with respect to an intuitionistic fuzzy set A in Â is defined and denote by ( )A G� with ( ) ,

minA P Q

m G ŒG=�

( ),AP QmC and ( ) ( )

,max ,

AA P QP Qg gG ŒG

= C� such that ( ) ( ) 1A A

m gG G+ £� � .

Corollary 3.1. Let Â be a rectangular array of integer-coordinate points. Let

, ,A AA P m g= be an intuitionistic fuzzy set in Â . Let ( ){ 1 andAP PmW = ŒÂ =

( ) }0A Pg = . Let 0 1: , , , nP P P P Qr = = be any path between two points of Â . Let

G Õ Â be any subset of Â . Then ( ) ( )minA

AP

Pm mG ŒG£� and ( ) ( )min

AA

PPg gG ŒG

≥�


Proof. The proof is obvious.

Definition 3.4. Let Â be a rectangular array of integer-coordinate points. Let

0 1: , , , nP P P P Qr = = be any path between two points of Â . Let A be an intuitionistic fuzzy set in Â . Then P and Q are intuitionistic fuzzy connected with respect to an intuitionistic fuzzy set A if

( ) ( ) ( )( ), min ,A A AP Q P Qm m m=C and ( ) ( ) ( )( ), max ,

A A AP Q P Qg g g=C

Proposition 3.3. Let Â be a rectangular array of integer-coordinate points. Let A = , ,A AP m g be an intuitionisitc fuzzy set in Â . Let 0 1: , , , nP P P P Qr = = be any

path between two points of Â . Then P and Q is an intuitionistic fuzzy connected in an intuitionistic fuzzy set A if and only if there exists a path 0 1: , , , nP P P P Qr = =¢ such

that ( ) ( ) ( )( )min ,A i A AP P Qm m m≥ and ( ) ( ) ( )( )max ,A i A AP P Qg g g£ ,for 0 i n£ £ .

Proof. Suppose that if exists such a path 0 1: , , , nP P P P Qr = =¢ , we have ( ),AP QmC

( ) ( ) ( ) ( ) ( )( )max min min ,A As s A i A A

iP P Q

rm r m r m m m= ≥ = ≥¢ and ( ), min

AP Q

rg =C

( ) ( ) ( ) ( ) ( )( )max max ,A As s A i A A

iP P Qg r g r g g g£ = £¢ By Proposition 3.2, ( ),

AP Qm =C

( ) ( )( )min ,A AP Qm m and ( ) ( ) ( )( ), max ,A A AP Q P Qg g g=C . This implies that P and Q

are intuitionistic fuzzy connected in an intuitionistic fuzzy set A . Conversely, if P andQ are intuitionistic fuzzy connected in an intuitionistic fuzzy set

A . Let 0 1: , , , nP P P P Qr = =¢ be a path, for which ( ) ( ) (max ,A A As s P

rm r m r m= =¢ C

) ( ) ( )( )min ,A AQ P Qm m= and ( ) ( ) ( ) ( ) ( )( )min , max ,A A As s A AP Q P Q

rg r g r g g g= = =¢ C .

Then ( ) ( ) ( )minAA i A i s

iP Pm m m r≥ = ¢ and ( ) ( ) ( )max

AA i A i si

P Pg g g r£ = ¢ for all iP

on a path r ¢ . Hence the proof is completed. Clearly, if intuitionistic fuzzy set A in Â with ( ) ( )1A AP Qm m= = and ( ) 0A Pg =

( )A Qg= , then P and Q are intuitionistic fuzzy connected if and if only there exists a

path r from P to Q such that for any point P ¢ in r , ( ) 1A Pm =¢ and ( ) 0A Pg =¢ . Then two points P and Q of W are intuitionistic fuzzy connected with respect to an intuitionistic fuzzy set A if and only if there are intuitionistic fuzzy connected in W . If

( ) 0A Pm = and ( ) 1A Pg = of an intuitioistic fuzzy set A , then an intuitionistic fuzzy connectedness of P and Q of a path with respect to an intuitioistic fuzzy set A whose

( ), 0AP Qm =C and ( ), 1

AP Qg =C .

Proposition 3.4. Let Â be a rectangular array of integer-coordinate points. Let

0 1: , , , nP P P P Qr = = be any path between two points of Â . Let A be an intuit-

tionistic fuzzy set in Â . Let ( ){ , and are intuitionistic fuzzycon-A P Q P Q= �

}nected in A . Then A � is reflexive and symmetric but not necessarily transitive with respect to an intuitionistic fuzzy set A .


Proof. Reflexive. For all P ŒÂ , ( ) ( ) ( ) ( )( ), min ,A A A AP P P P Pm m m m= =C and

( ) ( ) ( ) ( )( ), max ,A A A AP P P P Pg g g g= =C . This implies that A � is reflexive with to an

intuitionistic fuzzy set A .

Symmetric. For all ,P QŒÂ , ( ) ( ) ( )( ) ( ( ), min , min ,A A A AP Q P Q Qm m m m= =C

( )) ( ),AA P Q Pm m= C and ( ) ( ) ( )( ) ( ) ( )( ), max , max ,

A A A A AP Q P Q Q Pg g g g g= = =C

( ),AQ Pg C . This implies that A� is symmetric with to an intuitionistic fuzzy set A .

Not Transitive. Let Â be the 1-by-3 array P , Q , R . Let ( ),P Q and ( ),Q R be

intuitionistic fuzzy connected in A respectively and let ( ) ( )1A AP Rm m= = , ( ) 1A Qm <

and ( ) ( )0A AP Rg g= = , ( ) 1A Qg > . This implies that ( ),P R is not an intuitionistic

fuzzy connected in A . This implies that A� is not transitive with respect to an intuitionistic fuzzy set A .

4. Intuitionistic fuzzy components related to an intuitionistic fuzzy digitial otpology

4.1. Plateau, Tops and bottom with respect to an intuitionistic fuzzy set

Definition 4.1. Let � be any nonempty subset of Â . Then � is connected with respect to an intuitionistic fuzzy set A in Â if all ,P QŒ� are intuitionistic fuzzy connected in A .

Definition 4.2. Let P be any nonempty subset of Â and let A be an intuitionistic fuzzy set in Â . Then P is said to be an intuitionistic fuzzy plateau in A if it satisfies the following conditions:

(i) P is intuitionistic fuzzy connected in A (ii) ( ) ( )A AP Qm m= and ( ) ( )A AP Qg gπ , for all ,P QŒP

(iii) ( ) ( )A AP Qm mπ and ( ) ( )A AP Qg g= , for all pairs of neighboring points P ŒP , QœP .

Clearly, any P ŒÂ , belongs to exactly one intuitionistic fuzzy plateau in A .

Definition 4.3. Let P be any nonempty subset of Â and let A be an intuitionistic fuzzy set in Â . Then an intuitionistic fuzzy plateau P in A is said to be an intuitionistic fuzzy top in A if ( ) ( )A AP Qm m> and ( ) ( )A AP Qg g£ , for all pairs of neighboring points P ŒP , QœP . Otherwise it is said to be an intuitionistic fuzzy bottom in A .

Definition 4.4. Let P be any intuitionistic fuzzy top in A . Now, some sets associated with an intuitionistic fuzzy top P in A are defined as follows:

{ 0 1there exists a path : , , , such thatnP P P P P QrP = ŒÂ = = ŒP�

( ) ( ) ( ) ( ) }1 1 and ,1A i A i A i A iP P P P i nm m g g- -£ > £ £ .



( ) ( ) ( ) ( ) ( ) ( ) } and ,1A A i A A A i AP P Q P P Q i nm m m g g g£ £ > > £ £ .


( ) ( ) ( ) ( ) } and ,1A A i A A iP P P P i nm m g g£ > £ £ .

Remark 4.1. For any intuitionistic fuzzy top in A , P P PP Õ Õ Õ� � � .

Proof. It is obvious.

Note 4.1. Let P be any intuitionistic fuzzy top in A . If P PŒ� and ( ) ( )A APm m≥ P

and ( ) ( )A APg g< P . Then P ŒP .

4.2. Connectedness in an intuitionistic fuzzy tops P

Proposition 4.1. Let Â be a rectangular array of integer-coordinate points. Let A be any intuitionistic fuzzy set in Â and let P be an intuitionistic fuzzy top. If P PŒ� and P œP . Then ( ) ( )A APm m< P and ( ) ( )A APg g≥ P .

Proof. Assume that if P PŒ� and P œP such that ( ) ( )A APm m≥ P and ( )A Pg <

( )Ag P . This implies that there exists a path r from P to P such that ( ) ( )A i AP Pm m≥

( )Am≥ P and ( ) ( ) ( )A i A AP Pg g g< < P , for all iP on r . But if P œP , then r must

pass through a point Q that is adjacent to P but not in P . Hence, ( ) ( )A AQm m< P

and ( ) ( )A AQg g≥ P , for any such Q with is a contradiction. Therefore, ( )A Pm <

( )Am P and ( ) ( )A APg g≥ P .

Proposition 4.2. Let Â be a rectangular array of integer-coordinate points. Let A be any intuitionistic fuzzy set in Â and let P be an intuitionistic fuzzy top. Then P� is the set of points of Â that are intuitionistic fuzzy connected to points in P .

Proof. Let P be an intuitionistic fuzzy connected to QŒP with respect to an intuitionistic fuzzy set A . Then by Proposition 3.3, there exists a path r from P to Q

such that ( ) ( ) ( )( )min ,A i A AP P Qm m m≥ and ( ) ( ) ( )( )max ,A i A AP P Qg g g£ , for

0 i n£ £ and for all iP on r . If ( ) ( )A AP Qm m> and ( ) ( )A AP Qg g£ and P œP

implies that ( ) ( )A i AP Qm m≥ and ( ) ( )A i AP Qg g£ , for all iP on r . But by Proposition 4.1, it is impossible, since r must pass through a point Q¢ adjacent to P but not in P and for such a point, ( ) ( )A AQm m< P¢ and ( ) ( )A AQg g≥ P¢ . Therefore,

( ) ( )A AP Qm m£ and ( ) ( )A i AP Pm m≥ and ( ) ( )A AP Qg g≥ and ( ) ( )A i AP Pg g≥ , for all iP on r . This implies that P PŒ� .

Conversely, let P PŒ� . Then ( ) ( )A APm m£ P and ( ) ( )A APg g≥ P . Then by Note 4.1, P œP . Again by Proposition 4.1., there exists a path r from P to a point


Q of P such that ( ) ( ) ( ) ( )( )min ,A i A A AP P P Qm m m m≥ = and ( ) ( ) maxA i AP Pg g£ =

( ) ( )( ),A AP Qg g , for all iP on r . By Proposition 3.3., P is intuitionistic fuzzy

connected to Q .

Proposition 4.3. For any P in Â , there exists an intuitionistic fuzzy top such that P PŒ� .

Proof. Let P be in intuitionistic fuzzy plateau 0P . Suppose that if 0P is an intuit-tionistic fuzzy top. Then by Remark 4.1.,

00P PŒP Õ � . If not, let 1P be a neighbor of

0 0P ŒP such that ( ) ( )1 0A AP Pm m> and ( ) ( )1 0A AP Pg g£ . Thus a monotonically non-decreasing path r from 0P to 1P (going through 0P up to a neighbor of 1P . Repeat this argument with 1P replacing P and continue in this way to obtain 2 3, ,P P . This must terminate at nP , since Â is finite. Then nP is an intuitionistic fuzzy top. This implies that for a monotonic nondecresing path r from P to nP . By Remark 4.1,

nP PŒ� .

Proposition 4.4. Let P and P¢ be any two distinct intuitionistic fuzzy tops in Â . Then 0P /P « =¢ � .

Proof. Suppose that P PŒP «¢ � . Then there exists a path r from P to P such that ( ) ( )A A iP Pm m£ and ( ) ( )A A iP Pg g> , for all iP on r . But for a point iP

adjacent to P¢ but not in P , ( ) ( ) ( )A i A AP Pm m m< P =¢ and ( ) ( ) ( )A i A AP Pg g g≥ P =¢ which is a contradiction. Therefore, 0P /P « =¢ � .

Proposition 4.5. Let P and Q be any two points in Â . Then P is intuitionistic fuzzy connected to Q if and only if there exists an intuitionistic fuzzy top P such that P and Q are both in P� .

Proof. Suppose that if there exists an intuitionistic fuzzy top P such that P and Q are in P� . Then there are two paths 1r , 2r from P and Q respectively to P such that ( ) ( )A i AP Pm m≥ and ( ) ( )A i AP Pg g£ , for all iP on 1r and similarly, ( )A iQm ≥

( )A Qm and ( ) ( )A i AQ Qg g£ , for all iQ on 2r . This implies that there is a path 11 2r r-

from P to Q such that for all points R on this path, ( ) ( ) ( )( )min ,A A AR P Qm m m≥

and ( ) ( ) ( )( )max ,A A AR P Qg g g£ By Proposition 3.3, P is intuitionistic fuzzy

connected to Q . Conversely, let P are intuitionistic fuzzy connected to Q and let ( ) ( )A AP Qm m£

and ( ) ( )A AP Qg g≥ . Let r ¢ be a monotonic nondecreasing path from Q to some intuitionsitic fuzzy top P . By Proposition 4.3, Q P PŒ Õ� � . On the other hand, by Proposition 3.3, there is a path r from P to Q such that ( ) ( ( )min ,A i AP Pm m≥

( )) ( )A AQ Pm m= and ( ) ( ) ( )( ) ( )max ,A i A A AP P Q Pg g g g£ = for all iP on r . This


implies that for all iQ on r ¢ , ( ) ( ) ( )A i A AQ Q Pm m m≥ ≥ and ( ) ( )A i AP Qg g£ £

( )A Pg Thus there is a path rr ¢ form P to P which guarantees that P PŒ� .

Corollary 4.1. Let Â be any rectangular-interior points and let A be an intuitionistic fuzzy set in Â . Then Â is intuitionistic fuzzy connected with respect to an intuitionistic fuzzy set A if and only if there exists a unique intuitionistic fuzzy top in A .

Proof. The proof follows from the Proposition 4.4 and 4.5.

Definition 4.5. Let A be any intuitionistic fuzzy set in Â and let P be any intuitionistic fuzzy top in A . Then intuitionistic fuzzy set AP of Â whose

( ) ( ) ( ) , if ,0 otherwise.

A

A AP PP

m mm PP

Ï P Œ= ÌÓ

� and ( ) ( ) ( ) , if ,

1 otherwise.A

A AP PP

g gg PP

Ï P Œ= ÌÓ

�

such that 1A Am gP P+ £ .

Note 4.2. Clearly, by Proposition 4.1, any intuitionistic fuzzy set AP of Â whose ( ) 1A PmP = and ( ) 0A Pg P = if and only if P ŒP .

5. Intuitionistic fuzzy surroundness

Definition 5.1. Let A ,B and C be any three intuitionistic fuzzy sets in Â . Then B separates A from C if for all points ,P R ŒÂ and all paths r from P to R , there

exists a pointQ on r such that ( ) ( ) ( )( )min ,B A CQ P Rm m m≥ and ( ) ( ( )max ,B AQ Pg g£

( ))C Rg .

In particular, B surrounds A if it separates A from the border B of Â . That is, for all P ŒÂ and all paths r from P to B , there exists a point Q on r such that

( ) ( )B AQ Pm m≥ and ( ) ( )B AQ Pg g£ .

Note 5.1. Clearly, if R œB , then ( ) 0C Pm = and ( ) 1C Pg = . Similarly, if R ŒB ,

then ( ) 1C Pm = and ( ) 0C Pg = . This implies that

( ) ( ) ( )( )min , , if for ,

0 otherwise.A C

A

P R RP

m mm

Ï ŒÔ= ÌÔÓ

B

and

( ) ( ) ( )( )max , , if for ,

1 otherwise.A C

A

P R RP

g gg

Ï ŒÔ= ÌÔÓ

B

Proposition 5.1. Let A , B and C be any three intuitionistic fuzzy sets in Â . Then (i) Reflexivity: A surrounds A (ii) Transitivity: If A surrounds B and B surrounds C , then A surrounds C . (iii) Antisymmetry: If A surrounds B and B surrounds A , then A B«

surrounds both of them.


Proof. (i) It is obvious if we take P Q= . (ii) Suppose that A surrounds B and B surrounds C . Then for any P ŒÂ and

any path r from P to B , there is a point Q on r such that ( ) ( )B CQ Pm m≥ and

( ) ( )B CQ Pg g£ . Moreover, on the part of path r between Q and B , there is a point

R such that ( ) ( )A BR Qm m≥ and ( ) ( )A BR Qg g£ . This implies that, for any P ŒÂ

and any path r from P to B , there is a point R such that ( ) ( )A CR Pm m≥ and

( ) ( )A CR Pg g£ . Therefore, A surrounds C . (iii) Let r be any path from P to B and let Q be the last point on r such that

( ) ( )B AQ Pm m≥ and ( ) ( )B AQ Pg g£ . Since A surrounds B , there must a point Q¢

on r beyond (or possibly itself) Q such that ( ) ( )A BQ Qm m≥¢ and ( ) ( )A BQ Qg g£¢ . Since B surrounds A , there must a point Q¢¢ on r beyond (or equal to) Q such that

( ) ( ) ( )B A AQ Q Pm m m≥ ≥¢¢ ¢ and ( ) ( ) ( )B A AQ Q Qg g g£ £¢¢ ¢ . Suppose that Q Q= =¢

Q¢¢ , then ( ) ( ) ( )A A AQ Q Pm m m⁄ ≥ and ( ) ( ) ( )A A AQ Q Pg g gŸ £ . Since P is any

arbitrary point. If ( ) ( )A BQ Qm m= and ( ) ( )A BQ Qg g= , then the two intuitionistic fuzzy sets A and B are equal. Otherwise it is not equal. This implies that A B« surrounds A and B respectively. Respectively.

Note 5.2. Intuitionistic fuzzy surrounds is an weak partial order relation with respects to an intuitionistic fuzzy sets.

References

[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [2] A. Rosenfeld, Fuzzy graphs, in “Fuzzy sets and their applications to cognitive and decision processes”

(L. A. Zadeh et al., Eds.,), Academic Press, New York, (1975), 75-95. [3] A. Rosenfeld and A. C. Kak, Digital picture processing, Academic Press, New York, (1976). [4] A. Rosenfeld, Fuzzy digital topology, Information and Control, 40 (1979), 76-87. [5] L. A. Zadeh, Fuzzy sets, Infor. And Control, 9 (1965), 338-353.


__________________________ Received May, 2013


Los Angeles

Generalization of An Intuitionistic Fuzzy str� Open Sets in

An Intuitionistic Fuzzy Grill Structure Spaces

R. Narmada Devi, E. Roja and M. K. Uma

Department of Mathematics, Sri Sarada College for Women, Salem, Tamil Nadu, India, Email: [email protected]

Abstract: The purpose of this paper is to introduce the concepts of an intuitionistic fuzzy grill,

intuitionistic fuzzy � strucute space, intuitionistic fuzzy str

d� set and intuitionistic fuzzy

stra� open set. The concepts of an intuitionistic fuzzy

str

Ea� continuous function, intuit-

tionistic fuzzy str

a� - iT space, 0,1, 2i = and intuitionistic fuzzy str

a� -co-closed graphs are defined. Some interesting properties are established.

Keywords: Intuitionistic fuzzy grill, intuitionistic fuzzy � structure space, intuitionistic fuzzy

strd� set and intuitionistic fuzzy

stra� (resp.

stra� ,

strsemi� ,

strpre� ,

strregular� and

strb� ) open set, intuitioistic fuzzy

stra� exterior, intuitionistic fuzzy

str

Ea� continuous

function, intuitionistic fuzzy str

a� - iT space, 0,1,2i = and intuitionistic fuzzy str

a� -co-closed graphs.

1. Introduction

The concept of fuzzy sets was introduced by Zadeh [7] and later Atanassov [1] generalized the idea to intuitionistic fuzzy sets. On the otherhand, Coker [2] introduced the notions of an intuitionistic fuzzy topological spaces, intuitionistic fuzzy continuity and some other related concepts. The concept of an intuitionistic fuzzy a -closed set was introduced by H. Gurcay and D. Coker [5]. The concept of fuzzy grill was introduced by Sumita Das, M. N. Mukherjee [6]. Erdal Ekici [4] studied slightly precontinuous functions, separation axioms and pre-co-clsoed graphs in fuzzy topological space. In this paper, the concepts of an intuitionistic fuzzy grill, intuitionistic fuzzy � structure space, intuitionistic fuzzy

strd� set and intuitionistic fuzzy

stra� open

set are introduced. Some interesting properties of separation axioms in intuitionistic


fuzyz grill structure space with intuitionistic fuzzy strEa� continuous function are

established.

2. Preliminaries

Definition 2.1 [1]. Let X be a nonempty fixed set and I be the closed interval [0,1]. An intuitionistic fuzzy set ( )IFS A is an object of the following form { ( ), ,AA x xm=

( ) }:A x x Xg Œ , where the mappings :A X Im Æ and :A X Ig Æ denote the degree of

membership (namely ( )A xm ) and the degree of nonmembership (namely ( )A xg ) for each

element x XŒ to the set A , resepctivley, and ( ) ( )0 1A Ax xm g£ + £ for each x XŒ . Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form,

( ) ( ){ }, ,1 :A AA x x x x Xm m= - Œ . For the sake of simplicity, we shall use the symbol

, ,A AA x m g= for the intuitionistic fuzzy set ( ) ( ){ }, , :A AA x x x x Xm g= Œ .

Definition 2.2 [1]. Let X be a nonempty set and the IFSs A and B in the form

( ) ( ){ }, , :A AA x x x x Xm g= Œ , ( ) ( ){ }, , :B BB x x x x Xm g= Œ . Then

(i) A BÕ iff ( ) ( )A Bx xm m£ and ( ) ( )A Bx xg g≥ for all x XŒ ;

(ii) ( ) ( ){ }, , :A AA x x x x Xg m= Œ ;

(iii) ( ) ( ) ( ) ( ){ }, , :A B A BA B x x x x x x Xm m g g« = Ÿ ⁄ Œ ;

(iv) ( ) ( ) ( ) ( ){ }, , :A B A BA B x x x x x x Xm m g g» = ⁄ Ÿ Œ .

Definition 2.3 [1]. The IFSs ~0 and ~1 are defined by { }~0 ,0,1 :x x X= Œ and

{ }~1 ,1,0 :x x X= Œ .

Definition 2.4 [2]. An intuitionistic fuzzy topology ( )IFT in Coker’s sense on a nonempty set X is a family t of IFSs in X satisfying the following axioms:

(i) ~ ~0 ,1 tŒ ;

(ii) 1 2G G t« Œ for any 1 2,G G tŒ ;

(iii) iG tŒ∪ for arbitrary family { }iG i I tŒ Õ .

In this case the ordered pair ( ),X t is called an intuitionistic fuzzy topological space

( )IFTS on X and each IFS in t is called an intuitionistic fuzzy open set ( )IFOS .

The complement A of an IFOS A in X is called an intuitionistic fuzzy closed set ( )IFCS in X .

Generalization of an Intuitionistic Fuzzy str

� Open Sets in an 57 Intuitionistic Fuzzy Grill Structure Spaces

Definition 2.5 [2]. Let A be an IFS in IFTS X . Then ( ) {int is an A G G= ∪

} in and IFOS X G AÕ is called an intuitionistic fuzzy interior of A ; { isclA G G= ∩

} an in and IFCS X G A is called an intuitionistic fuzzy closure of A .

Proposition 2.1 [1]. For any IFS A in ( ),X t we have

(i) ( ) ( )intcl A A=

(ii) ( ) ( )int A cl A=

Corollary 2.1 [2]. Let A , iA ( )i JŒ be IFSs in X , B , jB ( )j KŒ IFSs in Y and :f X YÆ a function. Then

(i) ( )( )1A f f A-Õ (If f is injective, then ( )( )1A f f A-= ),

(ii) ( )( )1f f B B- Õ (If f is surjective, then ( )( )1f f B B- = ),

(iii) ( ) ( )1 1j jf B f B- -=∪ ∪ ,

(iv) ( ) ( )1 1j jf B f B- -=∩ ∩ ,

(v) ( )1~ ~1 1f - = ,

(vi) ( )1~ ~0 0f - = ,

(vii) ( ) ( )1 1f B f B- -= .

Definition 2.6 [3]. Let X be a nonempty set and x XŒ a fixed element in X . If 0r IŒ , 1s IŒ are fixed real numbers such that 1r s+ £ , then the IFS , , ,r s rx x x=

11 sx -- is called an intuitionistic fuzzy point ( )IFP in X , where r denotes the degree of membership of ,r sx , s denotes the degree of nonmembership of ,r sx and x XŒ the

support of ,r sx . The ,r sIFPx is contained in the IFS ( ),r sA x AŒ if and only if

( )Ar xm< , ( )As xg> .

Definition 2.7 [4]. An IFSs A and B are said to be quasi coincident with the IFSA , denoted by AqB if and only if there exists an element x XŒ such that

( ) ( )A Bx xm g> or ( ) ( )A Bx xg m< . If A is not quasi coincident with B , denoted AqB .

Definition 2.8 [5]. Let A be an IFS of an IFTS X . Then A is called an intuit-tionistic fuzzy a -open set ( IF OSa ) if ( )( )( )int intA cl AÕ . The complement of an

intuitionistic fuzzy a -open set is called an intuitionistic fuzzy a -closed set ( IF CSa ).


3. Intuitionistic fuzzy operators with respect to an intuitionistic fuzzy grills

Definition 3.1. Let Xz be the collection of all intuitionistic fuzzy sets in X . A

collection XzÕ� is said to be an intuitionistic fuzzy stack on X if A BÕ and AŒ� then B Œ� .

Definition 3.2. Let Xz be the collection of all intuitionistic fuzzy sets in X . An intuitionistic fuzzy grill � on X is an intuitionistic fuzzy stack on X if � satisfies the following conditions:

(i) ~0 œ� ,

(ii) If , XA B zŒ and A B» Œ� , then AŒ� (or) B Œ� .

Definition 3.3. Let ( ),X T be an intuitionistic fuzzy topological space and let Xz be the collection of all intuitionistic fuzzy sets in X . Let � be an intuitionistic fuzzy grill on X . A function : X Xz zF Æ� is defined by

( ) ( ){ }int , ,A IF A U AqU A U U TF = « « Œ Œ∪� �

for each XA zŒ . The function F� is an intuitionistic fuzzy operator associated with an intuitionistic fuzzy grill � and an intuitionistic fuzzy topology T .

Remark 3.1. Let ( ),X T be an intuitionistic fuzzy topological space. Let F� be an intuitionistic fuzzy operator associated with an intuitionistic fuzzy grill � and an intuitionistic fuzzy topology T . Then

(i) ( ) ( )~ ~ ~0 0 1F = = F� �

(ii) If , XA B zŒ and A BÕ , then ( ) ( )A BF Õ F� �

Proof. The proof of (i) and (ii) are follows from the definition of F� .

Definition 3.4. Let ( ),X T be an intuitionistic fuzzy topological space and let � be an intuitionistic fuzzy grill on X . Let F� be an intuitionistic fuzzy operator associated with an intuitionistic fuzzy grill � and an intuitionistic fuzzy topology T . A function

: XT zY Æ� is defined by ( ) ( )intA A IFY = » F� � for each A TŒ . The function

Y� is an intuitionistic fuzzy operator associated with F� .

Definition 3.5. Let ( ),X T be an intuitionistic fuzzy topological space and let � be an intuitionistic fuzzy grill on X . Let Y� be an intuitionistic fuzzy operator associated

with F� . A collection ( ){ } { }~1str A A A= Y = »�� is said to be an intuitionistic

fuzzy � structure on X . Then ( ), strX � is said to be an intuitionistic fuzzy �



structure space. Every member of str� is an intuitionistic fuzzy str� open set (in short,

strIF OS� ) and the complement of an intuitionistic fuzzy str� open set is an intuitionistic fuzzy str� closed set (in short, strIF CS� ).

Definition 3.6. Let ( ), strX � be an intuitionistic fuzzy � structure space and let XA zŒ . Then the

(i) intuitionistic fuzzy str� closure of A is denoted and defined by

( ) { } and str

XstrIF cl A B B A Bz= Œ Œ� �∩

(ii) intuitionistic fuzzy str� interior of A is denoted and defined by

( ) { }int and str

XstrIF A B B A Bz= Œ Õ Œ∪� �

Remark 3.2. Let ( ), strX � be an intuitionistic fuzzy � structure space. For any

, XA B zŒ ,

(i) ( )str

IF cl A A=� if and only if A is an intuitionistic fuzzy str� closed set.

(ii) ( )intstr

IF A A=� if and only if A is an intuitionistic fuzzy str� open set.

(iii) ( ) ( )intstr str

IF A A IF cl AÕ Õ� � .

(iv) ( ) ( )~ ~ ~int 1 1 1str str

IF IF cl= =� � and ( ) ( )~ ~ ~int 0 0 0str str

IF IF cl= =� � .

(v) ( ) ( )intstr str

IF A IF cl A=� � and ( ) ( )intstr str

IF cl A IF A=� � .

4. Properties of an intuitionistic fuzzy str

d� set and str

a� open set in an intuitionistic fuzzy � structure spaces

Definition 4.1. Let ( ), strX � be an intuitionistic fuzzy � structure space and let XA zŒ . Then A is said to be an intuitionistic fuzzy

strd� set (in short,

strIF Sd� ) if

( )( ) ( )( )int intstr str str str

IF IF cl A IF cl IF AÕ� � � � .

Proposition 4.1. Let ( ), strX � be an intuitionistic fuzzy � structure space. Then

(i) The complement of an intuitionistic fuzzy str

d� set is an intuitionistic fuzzy str

d� set.

(ii) Finite union of an intuitionistic fuzzy str

d� sets is an intuitionistic fuzzy str

d� set.

(iii) Finite intersection of an intuitionistic fuzzy str

d� sets is an intuitionistic fuzzy

strd� set.

(iv) Every intuitionistic fuzzy str

d� sets is an intuitionistic fuzzy str� open set.


Proof. (i) Let A be an intuitionistic fuzzy str

d� set. Then ( )( )intstr str

IF IF cl A Õ� �

( )( )intstr str

IF cl IF A� � . Taking complement on both sides, we have ( )( )intstr str

IF IF cl A� �

( )( )intstr str

IF cl IF A � � ( )( ) ( )( )int intstr str str str

IF cl IF cl A IF IF A � � � � (str str

IF cl IF� �

( )) ( )( )int intstr str

cl A IF IF A � � Hence A is an intuitionistic fuzzy str

d� set.

(ii) Let A and B be any two intuitionistic fuzzy str

d� sets. Then


IF IF cl A IF cl IF AÕ� � � � (4.1)


IF IF cl B IF cl IF BÕ� � � � (4.2)

Now, taking union of 3.1 and 3.2, we have ( )( ) (int intstr str str str

IF IF cl A IF IF»� � � �

( ))cl B Õ ( )( ) ( )( )int intstr str str str

IF cl IF A IF cl IF B»� � � � ( ( )intstr str str

IF IF cl A IF»� � �

( )) ( ) ( )( )int intstr str str

cl B IF cl IF A IF BÕ »� � � ( )( ) (intstr str str str

IF IF cl A B IF cl IF» Õ� � � �

( ))int A B» . Hence A B» is an intuitionistic fuzzy str

d� set. (iii) The proof is obvious by taking complement of (ii). (iv) It is obvious.

Note 4.1. In general, arbitrary union of an intuitionistic fuzzy str

d� set need note be intuitionistic fuzzy

strd� set.

Definition 4.2. Let ( ), strX � be an intuitionistic fuzzy � structure space and let XA zŒ . Then A is said to be an

(i) intuitionistic fuzzy str

semi� open set (in short, str

IFS OS� ) if (str str

A IF cl IFÕ � �

( ))int A . The complement of an intuitionistic fuzzy str

semi� open set is said to be an

intuitionistic fuzzy str

semi� closed set (in short, str

IFS CS� ).

(ii) intuitionistic fuzzy str

pre� open set (in short, str

IFP OS� ) if (intstr str

A IF IFÕ � �

( ))cl A . The complement of an intuitionistic fuzzy str

pre� open set is said to be an


pre� closed set (in short, str

IFP CS� ).

(iii) intuitionistic fuzzy str

a� open set (in short, str

IF OSa� ) if

( )( )( )int intstr str str

A IF IF cl IF AÕ � � � .

The complement of an intuitionistic fuzzy str

a� open set is said to be an intuitionistic fuzzy

stra� closed set (in short,

strIF CSa� ).

(iv) intuitionistic fuzzy str

b� open set (in short, str

IF OSb� ) if



( )( )( )intstr str str

A IF cl IF IF cl AÕ � � � .


b� open set is said to be an intuitionistic fuzzy

strb� closed set (in short,

strIF CSb� ).

(v) intuitionistic fuzzy str

regular� open set (in short, str

IFR OS� ) if

( )( )intstr str

A IF IF cl A= � � .


regular� open set is said to be an intuitionistic fuzzy

strregular� closed set (in short,

strIFR CS� ).

Note. 4.2. The family of all intuitionistic fuzzy str

semi� (resp. str

pre� , str

a� , str

b�

and str

regualr� ) open sets are denoted by ( )str

IFS O X� (resp. ( )str

IFP O X� , str

IFa�

( )O X , ( )str

IF O Xb� and ( )str

IFR O X� ).

Definition 4.3. Let ( ), strX � be an intuitionistic fuzzy � structure space and let XA zŒ . Then the


a� closure of A is denoted and defined by

( ) ( ){ } and strstr

XIF cl A B B A B IF O Xa z a= Œ Œ∩� �

(ii) intuitionistic fuzzy str

a� interior of A is denoted and defined by

( ) ( ){ }int and strstr

XIF A B B A B IF O Xa z a= Œ Õ Œ∪� �

Remark 4.1. Let ( ), strX � be an intuitionistic fuzzy � structure space. For any

, XA B zŒ ,

(i) ( )str

IF cl A Aa =�

if and only if A is an intuitionistic fuzzy str

a� closed set.

(ii) ( )intstr

IF A Aa =�

if and only if A is an intuitionistic fuzzy str

a� open set.

(iii) ( ) ( )intstr str

IF A A IF cl Aa aÕ Õ� �

.

(iv) ( ) ( )~ ~ ~int 1 1 1str str

IF IF cla a= =� �

and ( ) ( )~ ~ ~int 0 0 0str str

IF IF cla a= =� �

.

(v) ( ) ( )intstr str

IF A IF cl Aa a=� �

and ( ) ( )intstr str

IF cl A IF Aa a=� �

.

Proposition 4.2. Let ( ), strX � be an intuitionistic fuzzy � structure space. Then

(i) Every intuitionistic fuzzy str

regular� open set is an intuitionistic fuzzy str� open set.


(ii) Every intuitionistic fuzzy intuitionistic fuzzy str� open set is an intuitionistic fuzzy

strsemi� (resp.

strpre� ,

stra� and

strb� ) open set.


Remark 4.2. The converse of the Proposition 4.2 need not be true as shown in Example 4.1.

Example 4.1. Let { },X a b= be a nonempty set. Let ( ) (, 0.3, 0.5 , 0.2,A x a b a=

)0.3b , ( ) ( ), 0.3, 0.4 , 0.2, 0.4B x a b a b= , ( ), 0.2, 0.2 ,C x a b= ( )0.7 , 0.5a b

and ( ) ( ), 0.7 , 0.8 , 0.2, 0.2E x a b a b= be intuitionistic fuzzy sets on X . The family

{ }~ ~0 ,1 , , , ,T A B C D= is an intuitionistic fuzzy topology on X and the family =�

( ) ( ){ }0.2 1 and 0 0.8XG GG x xz m gŒ £ £ £ £ is an intuitionistic fuzzy grill on X .

Then the family { }~ ~0 ,1 ,str A=� is an intuitionistic fuzzy � structure on X . There-

fore, ( ), strX � is an intuitionistic fuzzy � structure space. Now,

(i) ( ) ( ), 0.3, 0.5 , 0.2, 0.3F x a b a b= is an intuitionistic fuzzy str� open set but need not be an intuitionistic fuzzy

strregular� open set in X .

(ii) ( ) ( ), 0.3, 0.6 , 0.2, 0.3H x a b a b= is an intuitionistic fuzzy str

a� (resp.

strsemi� ) open set but need not be an intuitionistic fuzzy str� open set in X .

(iii) ( ) ( ), 0.3, 0.7 , 0.2, 0.3K x a b a b= is an intuitionistic fuzzy str

pre� (resp. str

b� ) open set but need not be an intuitionistic fuzzy str� open set in X .

Proposition 4.3. Every intuitionistic fuzzy str

regular� open set is an intuitionistic fuzzy

strd� set.

Proof. Let A be an intuitionistic fuzzy str

regular� open set. Then (intstr str

IF IF� �

( )) ( )str

cl A A IF cl A= Õ � . Since every intuitionistic fuzzy str

regular� open set is an

intuitionistic fuzzy str� open set, ( )( )intstr str

IF IF cl A Õ� � ( )( )intstr str

IF cl IF A� � .

Hence A is an intuitionistic fuzzy str

d� set.


Example 4.2. Let { },X a b= be a nonempty set. Let ( ) (, 0.3, 0.5 , 0.2,A x a b a=

)0.3b , ( ) ( ), 0.3, 0.4 , 0.2, 0.4B x a b a b= , ( ), 0.2, 0.2 ,C x a b= ( )0.7 , 0.5a b

and ( ) ( ), 0.7 , 0.8 , 0.2, 0.2E x a b a b= be intuitionistic fuzzy sets on X . The family

{ }~ ~0 ,1 , , , ,T A B C D= is an intuitionistic fuzzy topology on X and the family =�

( ) ( ){ }0.2 1 and 0 0.8XG GG x xz m gŒ £ £ £ £ is an intuitionistic fuzzy grill on X .



Then the family { }~ ~0 ,1 ,str A=� is an intuitionistic fuzzy � structure on X . There-

fore, ( ), strX � is an intuitionistic fuzzy � structure space. Now, ( ), 0.3, 0.5 ,F x a b=

( )0.2, 0.2a b is an intuitionistic fuzzy str

d� set but need not be an intuitionistic fuzzy

strregular� open set in X .


a� open set is an intuitionistic fuzzy

strsemi� open set.


a� open set. Then

( )( )( ) ( )( )int int intstr str str str str

A IF IF cl IF A IF cl IF AÕ Õ� � � � �


semi� open set.



pre� open set is an intuitionistic fuzzy

strb� open set.


pre� set. Then

( )( ) ( )( )( )int intstr str str str str

A IF IF cl A IF cl IF IF cl AÕ Õ� � � � �


b� open set.


Example 4.3. Let { },X a b= be a nonempty set. Let

( ) ( )1 , 0.2, 0.3 , 0.7 , 0.7A x a b a b= , ( ) ( )2 , 0.4, 0.6 , 0.2, 0.4A x a b a b= ,

( ) ( )3 , 0.7 , 0.6 , 0.3, 0.4A x a b a b= , ( ) ( )4 , 0.1, 0.8 , 0.9, 0.2A x a b a b= ,

( ) ( )5 , 0.1, 0.3 , 0.9, 0.7A x a b a b= , ( ) ( )6 , 0.7 , 0.8 , 0.3, 0.2A x a b a b= ,

( ) ( )7 , 0.7 , 0.6 , 0.2, 0.4A x a b a b= , ( ) ( )8 , 0.7 , 0.8 , 0.2, 0.2A x a b a b= ,

( ) ( )9 , 0.4, 0.8 , 0.2, 0.2A x a b a b= , ( ) ( )10 , 0.2, 0.8 , 0.7 , 0.2A x a b a b= ,

( ) ( )11 , 0.1, 0.6 , 0.9, 0.4A x a b a b= , ( ) ( )12 , 0.4, 0.8 , 0.3, 0.2A x a b a b=

( ) ( )13 , 0.4, 0.6 , 0.3, 0.4A x a b a b=

be intuitionistic fuzzy sets on X . The family { }~ ~0 ,1 , , 1, 2, ,13iT A i= = is an intuitionistic fuzzy topology on X and

the family ( ) ( ){ }0.1 1 and 0 0.9XG GG x xz m g= Œ £ £ £ £� is an intuitionistic fuzzy


grill on X . Then the family { }~ ~ 1 2 3 70 ,1 , , , ,str A A A A=� is an intuitionsitic fuzzy �

structure on X . Therefore, ( ), strX � is an intuitionistic fuzzy � structure space. Now,

(i) ( ) ( ), 0.7 , 0.7 , 0.3, 0.3B x a b a b= is an intuitionistic fuzzy str

semi� open set but need not be an intuitionistic fuzzy

stra� open set in X .

(ii) ( ) ( ), 0.2, 0.3 , 0.4, 0.6C x a b a b= is an intuitionistic fuzzy str

b� open set but need not be an intuitionistic fuzzy

strpre� open set in X .

Remark 4.6. From the following diagram, the following implications holds:

strIF OSa�

strIFS OS�

strIF OSb�

strIFR OS�

strIF OS�

strIF OSd�

strIFP OS�

5. Separation axioms in an intuitionistic fuzzy � structure space

Definition 5.1. Let ( ), strX � be an intuitionistic fuzzy � structure space. Then an intuitionistic fuzzy set A is said to be an intuitionistic fuzzy str� (resp.

stra� ) clopen set

if and only if it is both intuitionistic fuzzy str� (resp. str

a� ) open and intuitionistic fuzzy

str� (resp. str

a� ) closed.

Definition 5.2. Let ( ), strX � be an intuitionistic fuzzy � structure space. Then an intuitionistic fuzzy set A is said to be an intuitionisitc fuzzy

stra� exterior of A if

( ) ( )intstr str

IFExt A IF Aa a=� �

.

Remark 5.1. Let ( ), strX � be an intuitionistic fuzzy � structure space. Then

(i) If A is an intuitionistic fuzzy str

a� clopen set, then ( )str

IFExt A A Aa = =�

.

(ii) If A BÕ then ( ) ( )str str

IFExt A IFExt Ba a � �

(iii) ( )~ ~1 0str

IFExta =�

and ( )~ ~0 1str

IFExta =�

.

Definition 5.3. Let ( )1, strX � and ( )2, strY � be any two intuitionistic fuzzy �

structure space. Let ( ) ( )1 2: , ,str strf X YÆ� � be an intuitionistic fuzzy function. Then




a� continuous function if for each intuitionistic fuzzy point

,r sx in X and each intuitionistic fuzzy 2str� open set B in Y containing ( ),r sf x ,

there exists an intuitionistic fuzzy 1str

a� open set A in X containing ,r sx such that

( )f A BÕ (ii) intuitionistic fuzzy

strEa�

continuous function if for each intuitionistic fuzzy point

,r sx in X and each intuitionistic fuzzy 2str� clopen set B in Y containing ( ),r sf x ,

there exists an intuitionistic fuzzy 1str

a� clopen set A in X containing ,r sx such that

( )( )1str

f IFExt A Ba Õ�

.

(iii) intuitionistic fuzzy str

a� open function if for each intuitionistic fuzzy 1str� open

set A in X , ( )1f A- is an intuitionistic fuzzy 2str

a� open set A in Y .

Proposition 5.1. Let ( )1, strX � and ( )2, strY � be any two intuitionistic fuzzy �

structure spaces. Let ( ) ( )1 2: , ,str strf X YÆ� � be an intuitionistic fuzzy function. Then the following are equivalent

(i) f is intuitionistic fuzzy str

Ea� continuous function.

(ii) ( )1f A- is an intuitionistic fuzzy 1str

a� open set in X , for each intuitionistic fuzzy 2str� clopen set A in Y .

(iii) ( )1f A- is an intuitionistic fuzzy 1str

a� closed set in X , for each intuitionistic fuzzy 2str� clopen set A in Y .

(iv) ( )1f A- is an intuitionistic fuzzy 1str

a� clopen set in X , for each intuitionistic fuzzy 2str� clopen set A in Y .

Proof. (i) fi (ii) Let B be an intuitionistic fuzzy 2str� clopen set in Y and let ,r sx

( )1f B-Œ . Since ( ),r sf x BŒ , by (i), there exists an intuitionistic fuzzy 1str

a� clopen set

A in X containing ,r sx such that ( )( ) ( )( )1 1

intstr str

f IFExt A f IF A Ba a= Õ� �

, 1str

IFa�

( ) ( )1int A f B-Õ . This implies that ( )( )

( )1

, 1

1

intint

str

r sstr

x IF A

f B IF Aa

a-

Œ= ∪

�

�

. Thus ( )1f B- is

an intuitionistic fuzzy 1str

a� open set in X .

(ii) fi (iii) Let B be an intuitionistic fuzzy 2str� clopen set in Y . Then B is an

intuitionistic fuzzy 2str� clopen set in Y . Thus, ( ) ( )1 1f B f B- -= . Since ( )1f B- is an

intuitionistic fuzzy 1str

a� open set in X , ( )1f B- is an intuitionistic fuzzy 1str

a� closed set in X .

(iii) fi (iv) The proof is easily.


(iv) fi (v) Let B be an intuitionistic fuzzy 2str� clopen set in Y containing

( ),r sf x . By (iv), ( )1f B- is an intuitionistic fuzzy 1str

a� clopen set in X . If we take

( )1A f B-= , then ( )f A BÕ . By Remark 5.1, ( )( )1str

f IFExt A Ba Õ�

.

Remark 5.2. Let ( )1, strX � and ( )2, strY � be any two intuitionistic fuzzy �

structure spaces. If ( ) ( )1 2: , ,str strf X YÆ� � is an intuitionistic fuzzy str

a� exterior set connected function, then f is an intuitionistic fuzzy

stra� continuous function.

Definition 5.4. An intuitionistic fuzzy � structure space ( ), strX � is said to be an

(i) intuitionistic fuzzy 0-strclo T� if for each pair of distinct intuitionistic fuzzy points ,r sx and ,m ny in X , there exists an intuitionistic fuzzy str� clopen set A of X containing one intuitionistic fuzzy point ,r sx but not ,m ny .

(ii) intuitionistic fuzzy 1-strclo T� if for each pair of distinct intuitionistic fuzzy points ,r sx and ,m ny in X , there exist an intuitionistic fuzzy str� clopen sets A and B containing ,r sx and ,m ny respectively such that ,m ny Aœ and ,r sx Bœ .

(iii) intuitionistic fuzzy 2-strclo T� if for each pair of distinct intuitionistic fuzzy points ,r sx and ,m ny in X , there exist an intuitionistic fuzzy str� clopen sets A and B containing ,r sx and ,m ny respectively such that ~0A B« = .

(iv) intuitionistic fuzzy strcloa� -regular if for each intuitionistic fuzzy str� clopen

set A and an intuitionistic fuzzy point ,r sx Aœ , there exist disjoint intuitionistic fuzzy

stra� open sets B and C such that A BÕ and ,r sx CŒ .

(v) intuitionistic fuzzy strcloa� -normal if for each pair of disjoint intuitionistic fuzzy

str� clopen sets A and B in X , there exist disjoint intuitionistic fuzzy str

a� open sets C and D such that A CÕ and B DÕ .

Definition 5.5. An intuitionistic fuzzy � structure space ( ), strX � is said to be an

(i) intuitionistic fuzzy 0-strTa� if for each pair of distinct intuitionistic fuzzy points

,r sx and ,m ny in X , there exists an intuitionistic fuzzy str

a� open set A of X containing one intuitionistic fuzzy point ,r sx but not ,m ny .

(ii) intuitionistic fuzzy 1-strTa� if for each pair of distinct intuitionistic fuzzy points

,r sx and ,m ny in X , there exist an intuitionistic fuzzy str

a� open sets A and B containing ,r sx and ,m ny respectively such that ,m ny Aœ and ,r sx Bœ .

(iii) intuitionistic fuzzy 2-strTa� if for each pair of distinct intuitionistic fuzzy points

,r sx and ,m ny in X , there exist an intuitionistic fuzzy str

a� open sets A and B containing ,r sx and ,m ny respectively such that ~0A B« = .



(iv) intuitionistic fuzzy strongly str

a� -regular if for each intuitionistic fuzzy str

a� closed set A and an intuitionistic fuzzy point ,r sx Aœ , there exist disjoint intuitionistic fuzzy str� open sets B and C such that A BÕ and ,r sx CŒ .

(v) intuitionistic fuzzy strongly str

a� -normal if for each pair of disjoint intuitionistic fuzzy

stra� closed sets A and B in X , there exist disjoint intuitionistic fuzzy str�

open sets C and D such that A CÕ and B DÕ .


structure spaces. Let ( ) ( )1 2: , ,str strf X YÆ� � be an injective, intuitionistic fuzzy

strEa�

continuous function.

(i) If ( )2, strY � is intuitionistic fuzzy 2 0-strclo T� , then ( )1, strX � is intuitionistic fuzzy

1 0-strTa� .

(ii) If ( )2, strY � is intuitionistic fuzzy 2 1-strclo T� , then ( )1, strX � is intuitionistic fuzzy

1 1-strTa� .

(iii) If ( )2, strY � is intuitionistic fuzzy 2 2-strclo T� , then ( )1, strX � is intuitionistic fuzzy

1 2-strTa� .

Proof. (i) Suppose that ( )2, strY � is an intuitionistic fuzzy 2 0-strclo T� space. For any distinct intuitionistic fuzzy points ,r sx and ,m ny in X , there exists an intuitionistic

fuzzy 2str� clopen set A in Y such that ( ),r sf x AŒ and ( ),r sf y Aœ . Since f is an


Ea� continuous function, ( )1f A- is an intuitionistic fuzzy

1stra�

open set in X such that ( )1,r sx f A-Œ . This implies that ( )1, strX � is an intuitionistic

fuzzy 1 0-strTa� space.

(ii) Suppose that ( )2, strY � is an intuitionistic fuzzy 2 1-strclo T� space. For any dis-tinct intuitionistic fuzzy points ,r sx and ,m ny in X , there exist an intuitionistic fuzzy

2str� clopen sets A and B in Y such that ( ),r sf x AŒ , ( ),r sf x Bœ , ( ),r sf y Aœ and

( ),r sf y BŒ . Since f is an intuitioistic fuzzy str

Ea� continuous function, ( )1f A- and

( )1f B- are intuitionistic fuzzy 1str

a� open sets in X respectively such that ( )1,r sx f A-Œ ,

( )1,r sx f B-œ , ( )1

,r sy f A-œ and ( )1,r sy f B-Œ . This implies that ( )1, strX � is an

intuitionistic fuzzy 1-strTa� space.

(iii) Suppose that ( )2, strY � is an intuitionistic fuzzy 2 2-strclo T� space. For any dis-tinct intuitionistic fuzzy points ,r sx and ,m ny inX ,there exist an intuitionistic fuzzy 2str�

clopen sets A and B in Y such that ( ),r sf x AŒ , ( ),r sf x Bœ , ( ),r sf y Aœ , ( ),r sf y BŒ

and ~0A B« = . Since f is an intuitionistic fuzzy str

Ea� continuous function, ( )1f A-


and ( )1f B- are intuitionistic fuzzy 1str

a� open sets in X containing ,r sx and ,r sy res-

pectively such that ( ) ( ) ( ) ( )1 1 1 1~ ~0 0f A f B f A B f- - - -« = « = = This implies that

( )1, strX � is an intuitionistic fuzzy 1 2-strTa� space.


structure spaces. Let ( ) ( )1 2: , ,str strf X YÆ� � be an intuitionistic fuzzy function.

(i) If f is an injective, intuitionistic fuzzy str

a� open function and intuitionistic fuzzy

strEa�

continuous function from intuitionistic fuzzy strongly 1str

a� -regular space

( )1, strX � onto an intuitionistic fuzzy � structure space ( )2, strY � , then ( )2, strY � is an intuitionistic fuzzy

2strcloa� -regular space.

(ii) If f is an injective, intuitionistic fuzzy str

a� open function and intuitionistic fuzzy

strEa�

continuous function from intuitionistic fuzzy strongly 1str

a� -normal space

( )1, strX � onto an intuitionistic fuzzy � structure space ( )2, strY � , then ( )2, strY � is an intuitionistic fuzzy

2strcloa� -normal space.

Proof. (i) Let A be an intuitionistic fuzzy 2str� clopen set in Y such that ,m ny Aœ .

Take ( ), ,m n r sy f x= . Since f is an intuitionistic fuzzy str

Ea� continuous function, B =

( )1f A- is an intuitionistic fuzzy 1str

a� closed sets in X such that ,r sx Bœ . Since ( ,X

)1str� is an intuitionistic fuzzy strongly 1str

a� -regular space, there exist disjoint intuit-tionistic fuzzy 1str� open sets C and D such that B CÕ and ,r sx DŒ . Since f is an


a� open function, we have ( ) ( )A f B f C= Õ and ( ), ,m n r sy f x= Œ

( )f D such that ( )f C and ( )f D are disjoint intuitionistic fuzzy 2str

a� open sets in Y .

This implies that ( )2, strY � is an intuitionistic fuzzy 2strcloa� -regular space.

(ii) Let 1A and 2A be disjoint intuitionistic fuzzy 2str� clopen sets in Y . Since f is

an intuitionistic fuzzy str

Ea� continuous function, ( )1

1f A- and ( )12f A- are

intuitionistic fuzzy 1str

a� closed sets in X . Take ( )11B f A-= and ( )1

2C f A-= . This

implies that ~0B C« = . Since ( )1, strX � is an intuitionistic fuzzy strongly 1str

a� -normal space, there exist disjoint intuitionistic fuzzy 1str� open sets 1D and 2D such that 1B DÕ and 2C DÕ . Since f is an intuitionistic fuzzy

stra� open function, we

have ( )1A f B= Õ ( )1f D and ( ) ( )2 2A f C f D= Õ such that ( )1f D and ( )2f D are

disjoint intuitionistic fuzzy 2str

a� open sets in Y . This implies that ( )2, strY � is an intuitionistic fuzzy

2strcloa� -normal space.

Definition 5.6. Let ( )1, strX � and ( )2, strY � be any two intuitionistic fuzzy � structure spaces. Let :f X YÆ be an intuitionistic fuzzy function. An intuitionsitic



fuzzy graph ( ) ( )({ }, , ,, : X X Yr s r s r sG f x f x x z z z= Œ Õ ¥ of an intuitionistic fuzzy

function f is an intuitionistic fuzzy str

a� -co-closed graph if and only if for each

( ) ( ), ,, \X Yr s m nx y G fz zŒ ¥ , there exist an intuitionsitic fuzzy

1stra� open set A in X

containing ,r sx and an intuitionistic fuzzy 2str� clopen set B in Y containing ,m ny

such that ( ) ~0f A B« = .

Proposition 5.4. Let ( )1, strX � , ( )2, strY � and ( ), strX Y¥ � be any three

intuitionistic fuzzy � structure spaces. Let ( ) ( )1 2: , ,str strf X YÆ� � be an intuit-ionsitic fuzzy function.

(i) If f is an intuitionistic fuzzy str

Ea� continuous function and ( )2, strY � is an

intuitionsitic fuzzy intuitionistic fuzzy 2 2-strTa� space, then ( )G f is an intuitionsitic

fuzzy str

a� -co-closed graph in Yz z¥ ¥ .

(ii) If f is an intuitionistic fuzzy str

a� continuous function and ( )2, strY � is an

intuitionistic fuzzy intuitionistic fuzzy 2 1-strclo T� space, then ( )G f is an intuitionistic

fuzzy str

a� -co-closed graph in X Yz z¥ .

(iii) If f is an intuitionistic fuzzy injective function has an intuitionistic fuzzy str

a� -

co-closed graph in X Yz z¥ . Then ( )1, strX � is an intuitionistic fuzzy 1 1-strTa� space.

Proof. (i) Let ( ) ( ), ,, \X Yr s m nx y G fz zŒ ¥ and ( ), ,r s m nf x yœ . Since ( )2, strY � is

an intuitionistic fuzzy 2 2-strTa� space, there exist an intuitionistic fuzzy 2str� clopen

sets A and B in Y containing ( ),r sf x and ,m ny respectively such that ~0A B« = .

Since f is an intuitionistic fuzzy str

Ea� continuous function, there exists an intuitionistic

fuzzy 1str

a� open set C in X containing ,r sx such that ( )( ) ( )str

f Ext C f C Aa = Õ�

and ( ) ~0f C A« = . This implies that ( )G f is an intuitionistic fuzzy str

a� -co-closed

graph in X Yz z¥ .

(ii) Let ( ) ( ), ,, \X Yr s m nx y G fz zŒ ¥ and ( ), ,r s m nf x yœ . Since ( )2, strY � is an intuit-

tionistic fuzzy 2 2-strTa� space, there exist an intuitionistic fuzzy 2str� clopen set A in

Y such that ( ),r sf x AŒ and ( ),r sf x Aœ . Since f is an intuitionistic fuzzy str

a� con-

tinuous function, there exists an intuitionistic fuzzy 1str

a� open set B in X containing

,r sx such that ( )f B AÕ . Therefore, ( ) ~0f B A« = and A is an intuitionistic fuzzy

2str� clopen in Y containing ,m ny . This implies that ( )G f is an intuitionistic fuzzy

stra� -co-closed set in X Yz z¥ .


(iii) Let ,r sx and ,m ny be any two intuitionistic fuzzy points X . Then ( ( ), ,,r s r sx f x Œ

( )\X Y G fz z¥ . Since ( )G f is an intuitionistic fuzzystr

a� -co-closed graph in X Yz z¥ , there exist an intuitionistic fuzzy

1stra� open set A in X and an intuitionistic fuzzy 2str�

clopen set B in Y such that ,r sx AŒ , ( ),m nf y BŒ and ( ) ~0f A B« = . Since f is an

intuitionistic fuzzy injective function, ( )1~0A f B-« = and ( )1f B- is an intuitionistic

fuzzy 2str� clopen set in Y containing ,r sy . And ,m ny Aœ . Thus ( )1, strX � is an intuitionistic fuzzy

1 1-strTa� space.

References

[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [2] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88

(1997), no.1, 81-89. [3] D. Coker and M. Demirci, On intuitionistic fuzzy points, Notes IFS, 1 (1995), no.2, 79-84. [4] Erdal Ekici, Generalization of some fuzzy functions, Bulletin of the Institute of Mathematics, Vol.33,

No.3 (2005), 277-289. [5] H. Gurcay and D. Coker, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math.,

5 (1997), 365-378. [6] Sumita Das, M. N. Mukherjee, Fuzzy generalized closed sets via fuzzy grills, Filomat, 26, No.3,

(2012), 563-571. [7] L. A. Zadeh, Fuzzy Sets, Infor. And Control, 9 (1965), 338-353.


__________________________ Received May, 2013 1 address for communication


Los Angeles

A Note on Fuzzy Continuum on Mixed Fuzzy Topological Spaces

N. R. Das

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India

Brojen Das 1

Department of Mathematics, M. C. College, Barpeta, Assam, India


Abstract: This paper deals with the study of H -continuum and N -continuum in Mixed Fuzzy

Topological Spaces. We have already given the H -continuum and N -continuum structure to Fuzzy topological spaces in our papers “A note on H -continuum” and “Some Aspects of Fuzzy N -continuum”,and Prof. N. R. Das and P. C. Baishya has given the Mixed Topological structure in Fuzzy setting. We have tried to study the continuum structure in Mixed Fuzzy Topology. Hence in this paper we also see that the presence of H -closed and N -closed sets with Mixed Fuzzy topology give arise some new properties which are also be studied. Our perspective is to explore more results in this direction.

Keywords: q -connectedness, d -connectedness, H -closed, N -closed, H -continuum, N -

continuum, q -continuous, d -continuous.

1. Introduction

The study of mixed topology originated from the work of Alexiewicz and Semadini, N. R. Das and P. C. Baishya have constructed a fuzzy topology called Mixed Fuzzy Topology with the help of two given fuzzy topologies on a set X . S. Ganguly and S. Jana introduced H -continuum and N -continuum in general topological spaces. We in our paper have given these two structure to fuzzy topological spaces. H -continuum is a q -connected H -set and N -continuum is a d -connected N -closed set.

In this paper we have combined our continuum structure with the mixed fuzzy topological space. If a fuzzy set is a H -set under one of the topology then it becomes

72 N. R. Das and Brojen Das

H -set under the mixed topology also. Under some conditions if it an H -set under the other topology then also it an H -set under the mixed topology. These results will justify that our study to explore more results in this direction is fruitful.

2. Preliminaries

Most of the concepts, notations and definitions which we have used in this paper are standard by now. But for the sake of completeness, we recall some definitions and results used in the sequel. The remaining definitions and notations which are not explained can be referred to [6,7,8].

Definition 2.1. A fuzzy set in X is called a fuzzy point denoted by xl if it takes value l at x and 0 at else.

Definition 2.2. A fuzzy point xl is said to be q -coincident with a fuzzy set A in X if ( ) 1A xl + > .

Definition 2.3. A fuzzy set A is said to be quasi-coincident with another fuzzy set B in X if there exists x XŒ such that ( ) ( ) 1A x B x+ > .

Definition 2.4. A fuzzy set A in ( ),X t is called a q -nbd of xl if and only if there exists a B tŒ such that x qB Al £ .

Definition 2.5. A fuzzy point xl is said to be a fuzzy q -cluster point of a fuzzy set A in ( ),X t if the closure of every q -nbd of xl is quasi-coincident with A .

The union of all q -cluster points of A is called the q -closure of A and is denoted by Aq .

Definition 2.6. A fuzzy point xl is said to be a fuzzy d -cluster point of a fuzzy set A in ( ),X t if every fuzzy regular open q -nbd U of xl is quasi-coincident with A .

The collection of d -cluster point if A is called d -closure of A and is denoted by Ad .

Definition 2.7. A triplet ( )1 2, ,X t t of a non-empty set X together with tow fuzzy topologies 1t , 2t is called a fuzzy bitopological space.

Definition 2.8. Let ( )1,X t and ( )2,X t be two fuzzy topological sapces and

( ) {1 2 2 1for each , there exists a - -nbd of such that -closureXA I x qA q A xa a at t t t= Œ

}of A Aa £ . Then 1t ( 2t ) is a topology on X called a mixed fuzzy topology.

A Note on Fuzzy Continuum on Mixed Fuzzy Topological Spaces 73

Theorem 2.9 [1]. If ( )1,x t and ( )2,X t be two fuzzy topological spaces, then the

mixed fuzzy topology ( )1 2t t is coarser than 2t i.e. ( )1 2 2t t tÕ .

Theorem 2.10 [1]. If ( )1,x t and ( )2,X t be two fuzzy topological spaces, with 1t

fuzzy regular and 1 2t tÃ , then ( )1 1 2t t tÕ .

Definition 2.11. A pair of fuzzy sets ( ),P Q are said to be q -separation ( d -

separation) of A with respect to X , if P Q A⁄ = and 0P Q P Qq qŸ = Ÿ =

( )0P Q P Qd dŸ = Ÿ = .

Notation. Let A be a fuzzy subset in a fuzzy bitopological space ( )1 2, ,X t t . Then the q -closure of A with respect to 1t (respectively d -closure of A with respect to 1t )

is denoted by ( )1Aq t (resp. ( )1A

d t ). If there is only one topology we simply denoted it by Aq (resp. Ad ).

3. H -continuum in mixed fuzzy topological spaces

Theorem 3.1. Let ( )1,X t and ( )2,X t be two fuzzy topological spaces and A be a fuzzy subset of X . Then

( ) ( )( )1 22A Aq t tq t £

That is the q -closure of A w.r.t. 2t is a fuzzy subset of the q -closure of A w.r.t.

( )1 2t t .

Proof. Let ( )2x Aq t

aŒ . Then xa is a 2 -t q -cluster point of A i.e. 2t -closure of every q -nbd N of xa is q -coincident with A . Now if xN a

is a ( )1 2t t q -nbd of xa then

there exists a fuzzy set ( )1 2V t tŒ such that xx qV Naa £ . Since ( )1 2 2t t tÕ [Theorem

2.9], therefore 2V tŒ . Thus xN a is a q -nbd of xa in 2t also. Since, xa is a 2 -t q -

cluster point of A , therefore ( )2xN qAa

t i.e. ( ) ( ) ( )2 1xN x A xa

t + > . Again ( )1 2 2t t tÕ

implies ( ) ( )2 1 2x xN Na a

t t t£ . Hence ( ) ( ) ( )1 2 1xN x A xa

t t + > , that is ( )1 2t t -closure of xN a is

q -coincident with A , implies that ( )( )1 2x Aq t t

a Œ . Hence the proof.

Theorem 3.2. If ( )1,X t and ( )2,X t be two fts and A be a fuzzy subset of X then

( ) ( )( )1 22A Ad t td t £

Proof. Similar with the proof of Theorem 3.1


Theorem 3.3. Let ( )1,X t and ( )2,X t be two fts such that 1t is regular and 1 2t tÃ .

Then ( )( ) ( )1 2 1A Aq t t q t£ and ( )( ) ( )1 2 1A A

d t t d t£ .

Proof. Follows from the fact that ( )1 1 2t t tÃ by Theorem 2.10.

Theorem 3.4. If ( ),P Q is a q -separation (resp. d -separation) in ( )1 2t t , then

( ),P Q is a q -separation (resp. d -separation) in 2t .

Proof. If possible, suppose ( ),P Q is not a q -separation in 2t . Then ( )2 0P Qq t Ÿ π

or ( )2 0P Qq tŸ π . Without any loss of generality, suppose ( )2 0P Q

q t Ÿ π . Since ( )2Pq t

( )( )1 2Pq t t£ , therefore, ( )( )1 2 0P Q

q t t Ÿ π . It is a contradiction to the fact that ( ),P Q is

q -connected in ( )1 2t t . Hence the result.

Theorem 3.5. If A is an H -subcontinuum (resp. N -subcontinuum) in 2t , then A is H -subcontinuum (resp. N -subcontinuum) in ( )1 2t t .

Proof. Let A be an H -subcontinuum in ( )2,X t . Then A is an H -set in ( )2,X t .

Let { }:Ua a ŒL be an open cover for A in ( )1 2t t . Since ( )1 2 2t t tÕ , therefore

{ }:Ua a ŒL is a 2t -open cover for A also. As A is an H -set in 2t so there exists a

subfamily { }: 1, 2, ,i

U i na = such that ( )21 i

niA U

ta=£ � . But ( ) ( )( )1 22

i iU U

t tta a£ , for each

ia . This implies that A is an H -set in ( )1 2t t . Again from the Theorem 3.4 we have

that A is q -connected in 2t . Thus A is q -connected in ( )1 2t t . Hence A is an H -

subcontinuum in ( )( )1 2,X t t .

Theorem 3.6. If 1t is regular and 1 2t tÕ , then if A is an H -subcontinuum in

( )1 2t t then it is H -subcontinuum in 1t .

Proof. Similar to the proof of Theorem 3.5, as in this case ( )1 1 2t t tÕ .

4. q -continuous

Definition 4.1. A function :f X YÆ from an fts ( )1,X t to another fts ( )2,Y t is called q -continuous if for each fuzzy point xl in X ad for each open q -nbd V of

( )f x l , there exists an open q -nbd U of Xl such that ( )f U V£ .


Theorem 4.2. Let ( )1 2, ,X t t and ( )1 2, ,Y t t* * be two fuzzy bitopological spaces. If

:f X YÆ is 1 1t t *- and 2 2t t *- q -continuous then f is ( ) ( )1 2 1 2t t t t* *- q -

continuous.

Proof. Let xl be a fuzzy point in X and V be a ( )1 2t t* * open q -nbd of ( )f x l .

Since ( )1 2 2t t t* * *Õ , therefore V is 2t* -open q -nbd of ( )f x l . Again f is 2 2t t *- q -

continuous therefore there exists a 2t -open q -nbd U of xl such that ( )( ) ( )22f U Vtt *

£ .

Again since f is 1 1t t *- q -continuous, therefore for any 1t* -open q -nbd 1V of ( )f x

there exists a 1t -open q -nbd 1U of xl such that ( )( ) ( )111 1f U V

tt *

£ .

Now let ( )1U f V-= and xl be quasi-coincident with U . Then ( ) 1U xl + > which

implies ( )( ) 1f U xl + > . I.e. ( )f x l is quasi-coincident V . Since V is ( )1 2t t* * open,

so there exists a 2t* q -nbd Vl

* of ( )f x l such that ( )1

V Vt

l

*

* £ . f is 1 1t t *- -continuous

implies that there exists a 1t -open q -nbd 1U of xl such that ( ) ( )1

1 1f U V Vt *

*£ £ implies

1U U£ .

Also for each fuzzy point xl in X and each 2t* q -nbd Va

* of ( )f x l , there exists a

2 -qt -nbd 2U of xl such that ( )2

2f U Vt

a*£ implies ( )2 1

2U f Vt

a- *£ . Thus ( )1

2U f Va*-£ .

Since 2U is 2 -qt -nbd of xa therefore we have a 2t -open set Aa such that

2x qA Ua a £ . Therefore ( )12A U f Va a

*-£ £ . Hence ( )1 2U t tŒ . Moreover

( )1U f V-= implies ( )( )( ) ( )( )1 2 1 2f U Vt t t t£ . Hence the theorem.

Proposition 4.3. Let ( )1 2, ,X t t and ( )1 2, ,Y t t* * be two fuzzy bitopological spaces. If

:f X YÆ is 1 1t t *- and 2 2t t *- ,d -continuous then f is ( ) ( )1 2 1 2t t t t* *- d -continuous.

Proof. Similar with the proof of Theorem 4.2.

Proposition 4.4. Let ( )1 2, ,X t t and ( )1 2, ,Y t t* * be two fuzzy bitopological spaces

where 1 2t t* *Õ and 1t* is fuzzy regular. If :f X YÆ is ( ) ( )1 2 1 2t t t t* *- q -continuous

then f is 2 1-t t * q -continuous.

Proof. Let B* be an 1t* -open q -nbd of ( )f x l . Since 1 2t t* *Õ and 1t

* is fuzzy

regular, therefore by Theorem 2.10, ( )1 1 2t t t* * *Õ . Therefore, B* is a ( )1 2t t* * -open q -

nbd of ( )f x l . Now f is ( ) ( )1 2 1 2t t t t* *- q -continuous implies that there exists a

( )1 2t t -open q -nbd B of xl such that ( )( ) ( )1 21 2f B B

t tt t* *

*£ . Again, ( )1 2 2t t tÕ


implies that B is 2t -open and ( ) ( )( ) ( )1 2 11 22f B f B B B

t t tt tt* * *

* *£ £ £ . Hence f is 2 1-t t *

q -continuous.


:f X YÆ is ( ) ( )1 2 1 2t t t t* *- q -continuous then f is ( )2 1 2t t t* *- q -continuous.

Proof. Let B* be a ( )1 2t t* * open q -nbd of ( )f x l in Y . Then there exists a ( )1 2t t

open q -nbd B of xl such that ( )( ) ( )( )1 21 2f B f Bt tt t * **£ . Now by Theorem 2.9, we have

( )1 2 2t t tÕ . So, B is 2t -open of xl and ( ) ( )( ) ( )( )1 21 22f B f B f Bt tt tt* **£ £ . So, f is

( )1 1 2t t t* * *- q -continuous.


:f X YÆ is 1 1t t *- q -continuous, and 1t is fuzzy regular with 1 2t tÕ , then f is

( )1 2 1t t t *- q -continuous.

Proof. Let B* be a 1t* open q -nbd of ( )f x l in Y . Then there exists a 1t open q -

nbd B of xl such that ( )( 1 1f B f Bt t **£ . Now by Theorem 2.10, we have ( )1 1 2t t tÕ .

So, B is ( )1 2t t -open and ( )( ) ( )1 2 1 1f B f B Bt t t t **£ £ . Hence, f is ( )1 2 1t t t *- q -

continuous.

Proposition 4.7. If ( )1 2, ,X t t* * , ( )1 2, ,Y t t* * are fuzzy bitopological space and

:f X YÆ is ( )1 2 1t t t q*- -continuous, then f is 2 1t t *- q -continuous.

Proof. If B* is a 1t* open q -nbd of ( )f x l in Y , then there exists a ( )1 2t t open q -

nbd B of xl such that ( )( )1 2 2f B Bt t t **£ . Now, Now by Theorem 2.9, we have

( )1 2 2t t tÕ which implies B is 2t -open and ( ) ( )( )1 22 1f B f B Bt tt t **£ £ . Hence the

result.


:f X YÆ is ( )1 2 2t t t q*- -continuous, then f is 2 2t t *- q -continuous.

Proof. Let B* be a 2t* open q -nbd of ( )f x l in Y . Then there exists a ( )1 2t t -open

q -nbd B of xl in X such that ( )( )1 2 2f B Bt t t **£ . But according to Theorem 2.9,

( )1 2 2t t tÕ . So, 2B tŒ and ( ) ( )( )1 22 2f B f B Bt tt t **£ £ . This proves the result.


5. q -open mappings

Definition 5.1. A function :f X YÆ where ( )1,X t and ( )2,X t are two fuzzy

topological spaces, is called a q -open map if for each q -open set A of X , ( )f A is q -open in Y .

Theorem 5.2. Let ( ),X t and ( ),Y t * be two fuzzy topological spaces. Then if

:f X YÆ is an onto fuzzy open map, then it is q -open.

Proof. Let A be a q -open subset in X and ( ) ( )f x f Al Œ . This implies x Al Œ and A is q -open implies there exists a q -nbd xV of xl such that

xV V A£ £ (1)

xV is a q -nbd of xl implies there exists an open subset U of ( ),X t such that

xx qU Vl £ . Now, x qUl implies ( ) ( )f x qf Ul and ( ) ( )xf U f V£ . Since, f is open

such that ( )f U is a open fuzzy set in Y and so ( )xf V is a q -nbd of ( )f x l . Again,

( ) ( )x xf V f V£ . Therefore, (1) implies ( ) ( )xf V f A£ . So, ( )f A is q -open in Y .

References

[1] N. R. Das and P. C. Baishya, Mixed fuzzy topological spaces, The Journal of fuzzy math, 3 (4) (1995). [2] S. Ganguly and S. Jana, A note on H -continuoum, Bull. Cal. Math. Soc., 95 (6) (2003), 483-496. [3] S. Ganguly and S. Jana, A note on N -continuum, Tamkang Journal of Mathematics, Vol-35, No.4,

Winter, (2004), 371-382. [4] M. N. Mukharjee and S. P. Sinha, Fuzzy q -closure operator on fuzzy topological spaces,

International J. Math. And Math. Sci., 14 (2) (1991), 309-314. [5] S. Ganguly and S. Saha, A note on d -continuity and d -connected sets in fuzzy set theory, Simon

Stevin, 62 (1988), 127-141. [6] Wu Congxin and Wu Jianrong, Fuzzy quasi uniformities and fuzzy bitopolgoical spaces, Fuzzy Sets

and Systems, 46 (1992), 133-137. [7] J. C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc., 3 (13) (1936), 71-89. [8] Pu Pao Ming and Liu Ying Ming, Fuzzy topology I, Neighbourhood structures of a fuzzy point and

moore-smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.


__________________________ Received September, 2013


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Fuzzy Reliability Optimization Based on Fuzzy Geometric Programming Method Using Different Operators

A. K. Shaw and T. K. Roy *

Bengal Engineering and Science University, Shibpur, P. O. –B. Garden, Howrah-711103, India

* Correspondence E-mail: [email protected]

Abstract: In this paper, we summarize the fundamentals of fuzzy GP and present the problem

of optimal reliability for a series system subject to a cost constraint. In real life, it is necessary to improve the reliability of the system under limited available cost of reliability component. Practically some desired system reliability level subject to constraint on cost has always been imprecise and vague in nature. It may be formulated as a fuzzy geometric programming problem. Numerical examples are given to illustrate the model through fuzzy geometric programming by max-min, max-additive and max-product operators.

Key words: Fuzzy GP , reliability, posynomial, signomial.

1. Introduction

To achieve an aim to find out the best way to increase the systems reliability, reliability optimization provides immense help to the reliability engineer. The reliability optimization is an important research topic in engineering and operation research. In practical field, the problem of series system reliability may be formed as a typical non-linear programming problem with non-linear cost-functions in fuzzy environment. Some researchers applied the fuzzy set theory to reliability analysis. Park [12] used fuzzy in the reliability apportionment problem for a two-component series system subject to a single constraint and solved it by FNLP technique. GP method is rarely used to solve the reliability optimization problem. Federowiez and Mazumder [1] first used GP on reliability optimization problem. Govil [11] used GP for a 3-stage series reliability system. Govi [10] also applied GP method on an optimal maintainability problem of a series system with cost constraint. Mahapatra and Roy [8] discussed optimal fuzzy reliability for a series system with cost constraint using fuzzy geometric programming.

80 A. K. Shaw and T. K. Roy

Again Mahapatra and Roy [7] used geometric programming to analyze system reliability. But fuzzy reliability optimization model through FGP is very rare in literature.

The Reliability optimization assumes that systems have redundancy components in series parallel or parallel systems and that alternative designs achieve the goal of optimal system reliability by optimal allocation of redundancy components. Reliability of a multi-stage system can be improved by adding similar components as redundancy to each subsystem, may be some different components that can be considered as design alternatives in a subsystem. Thus the problem is to improve system’s reliability associated with a system design under the limited availability resources. Tillman et al. [5], Singh and Misra [9] and Kuo et al [18] illustrated allocation of redundant component in a system to enhance the system reliability, which is important in reliability engineering. In many practical problems, most of the parameters of an optimization model are not known exactly. Due to this imperfect and unreliable input information, fuzzy numbers become an important aspect in the reliability design of the engineering systems. The problem is to find out the optimum number of redundancies of similar components, which maximize the system reliability subject to the available system cost and weight, regards as the problem of geometric programming in the circumstance of fuzzy number presentation as relatibility, cost and weight of components.

Ceometric Programming Geometric programming ( )GP can be considered to be an innovative modus operandi

to solve a nonlinear problem in comparison with other nonlinear techniques. It was originally developed to design engineering problems. It has become a very popular technique since its inception in solving non-linear problems. The concept of geometric programming ( )GP was introduced by Duffin et al. [15] in their famous book Geometric Programming-Theory and Application. This publication is a landmark in the development of GP . It studied all the theoretical developments up to date providing important examples to illustrate the technique. In addition to elegant proofs, it provided several constructive transformations and approximation for expressing optimization problems in suitable form in order to solve by GP .

Many real-world problems comprise of positive as well as negative coefficients for the cost terms. However the study of GP by Duffin et al. [15] deals with the problem involving only a positive coefficient for the component cost terms. Passy and Wilde [17] made a significant methodological development of GP to deal with this type of problem. They extended the concept of the GP technique to generalized polynomials free from a restrictive environment. Now GP is capable of dealing with any problems involving signomials in both objective and constraint functions. It is important to note that any nonlinear algebraic problem can be transformed into an equivalent posynomial/signomial. For a detailed discussion, one may consult with the book Applied Geometric Programming written by Beightle and Phillips [6].

The advantages of GP are as follows: ∑ This technique provides us with a systematic approach for solving a class of

nonlinear optimization problems by finding the optimal value of the objective function and then the optimal values of the design variable are derived.

∑ This method often reduces a complex nonlinear optimization problem to a set of simultaneous equations.

∑ This approach is more amenable to the digital computers.

Fuzzy Reliability Optimization Based on Fuzzy Geometric Programming Method Using Different Operators 81

∑ This method allows an easy sensitivity analysis, which can be performed in the optimal solution. GP inherits some drawbacks. The main disadvantages of GP lie in the fact that it

requires the objective functions and constraints in the form of posynomials/signomials. Note. Someone guesses that the name GP comes from the many geometrical

problems. There is a difference between GP and geometric optimization ( )GOP . GP is an optimization based on the arithmetic-geometric mean inequality ( ). . .AM GM≥ . However, GOP is an optimization problem involving geometry. GP is an optimization problem of the form

Minimize ( )0g t (1.1)

Subject to ( ) 1jg t £ , 1, 2, ,j m= , ( ) 1kh t = , 1, 2, ,k p= , 0it > , 1, 2, ,i n=

where ( )jg t ( )0,1,2, ,j m= are posynomial or signomial functions, ( )kh t ( 1, 2,k =

), p are monomials and t is the decision variable vector of n components it ( 1,i =

)2, ,n . The problem (1.1) may be written as:

Minimize ( )0g t (1.2)

Subject to ( ) 1jg t £ 1, 2, ,j m= , 0t > , [since ( ) 1jg t £ , ( ) ( )1 1j jh t g t= fi £

where ( ) ( ) ( )( )j j kg t g t h t= be a posynomial ( 1,2, ,j m= ; 1, 2, ,k p= )].

2. Posynomial geometric programming problem

2.1 Primal problem Minimize ( )0g t (2.1)

Subject to ( ) 1jg t £ ( 1, 2, ,j m= ) and 0it > ,( 1,2, ,i n= ) where ( )11

jN njki

j jk iik

g t c ta

===Â ’

Where ( )0jkc > and jkia ( 1, 2, ,i n= ; 1, 2, , jk N= , 0,1,2, ,j m= ) are real

numbers. ( )1 2, , , T

nt t t t∫ .

It is a constrained posynomial primal geometric problem ( )PGP . The number of inequality constraints in the problem (2.1) is m . The number of terms in each posynomial constraint function varies, and tit is denoted by jN for each 0,1,2, ,j m= .

The degree of difficulty ( )DD of a GP is defined as number of terms in a PGP -(number of variables in 1PGP + ).

2.2 Dual problem The dual programming of (2.1) is as follows:


Maximize ( ) 0

0 1

jkj

wNm

jk j

j k jk

c wd w

w= =

Ê ˆ= Á ˜Ë ¯’’ (2.2)

Subject to 0

01

1N

kk

w=

=Â (normality condition) 1 1

0jNm

jki jkj k

wa= =

=ÂÂ , ( 1,2, ,i n= ) (orthogo-

nality condition) 01

0jN

j jkk

w w=

= ≥Â , 0jkw ≥ , ( 1, 2, ,i n= ; 1, 2, , jk N= ), 00 1w = .

There are 1n + independent dual constraint equalities and 1m

jjN N== Â independent

dual variables for each term of the primal problem. In this case ( )1DD N n= - + .

3. Signomial geometric programming problem

3.1 Primal problem Minimize ( )0g t (3.1)

Subject to ( )j jg t d£ , ( 1, 2, ,j m= ) and 0it > , ( 1,2, ,i n= ) where ( )1

jN

j jk jkk

g t cd=

=Â

( )1

0,1, 2, ,n

jkii

i

t j ma

==’ 1jd = ± ( 2, ,j m= ), 1jkd = ± ( 0,1,2, ,j m= ; 1,2,k =

, jN ), ( )1 2, , , T

nt t t t∫ .

3.2 Dual problem

The dual problem of (5) is as follows:

Maximize ( ) ( )0

0 00 1

jjk jk

Nm w

jk j jkj k

d w c w w

da

d= =

È ˘= Í ˙

Î ˚’’ (3.2)

Subject to 0

0 0 01

N

k kk

wd d=

=Â (normality condition) 0 1

0jNm

jk jki jkj k

wd a= =

=Â Â , ( )1,2, ,i n=

(orthogonality condition) where 1jd = ± ( 2, ,j m= ), 1jkd = ± ( 1,2, ,j m= ; 1,k =

2, , jN ), and 00 1w = . 0 1, 1d = + - and non-negativity conditions, 0 1jN

j j jk jkkw wd d=∫ Â

0≥ , 0jkd ≥ , ( 1,2, ,j m= ; 1, 2, , jk N= ) and 00 1w = .

4. Fuzzy geometric programming ( )FGP

( )0Minimizeg t (4.1)

Subject to ( )j jg t b£ ( 1,2, ,j m= ), 0t ≥

Here, the symbol “Minimize ” denotes a relaxed or fuzzy version of “Minimize”. Similarly, the symbol “ £ ” denotes a fuzzy version of “ £ ”. There fuzzy requirements


may be quantified by eliciting membership functions ( )( )j jg tm ( 0,1,2, ,j m= ) from

the decision maker for all functions ( )jg t ( 0,1,2, ,j m= ). By taking account of the rate of increased membership satisfaction, the decision maker must determine the sub-jective membership function ( )( )j jg tm . It is, in general, a strictly monotone decreasing

linear or nonlinear function ( )( )j ju g t with respect to ( )jg t ( 0,1,2, ,j m= ). Here for simplicity, linear membership functions are considered. The linear membership functions can be represented as follows:

( )( )( )

( )( ) ( ) ( )( )

0

0 0

1, if

if

0 if

j j

j j j j j j j j j

j j

g t g

g t g g t g g g g t g

g t g

m

Ï £ÔÔ= - - £ £¢ ¢ ¢ÌÔ ≥ ¢ÔÓ

for 0,1,2, ,j m= .

( )( )j jg tm

a ( ),jg a

( ) 0j jg t g£

1

0 0jg ( )0

j j jg g t g£ £ ¢ jg ¢ ( )j jg t g≥ ¢( )jg t

Figure 1. Membership function

As shown in Figure 1. 0jg ∫ the value of ( )jg t such that the grade of membership function ( )( )j jg tm is 1.

jg ∫¢ the value of ( )jg t such that the grade of membership function ( )( )j jg tm is 0.

jg ∫ the intermediate value of ( )jg t between 0jg and jg ¢ (i.e. ( )0 ,j j jg g gŒ ¢ ) such that

the grade of membership function ( )0,1a Œ .

The Problem (4.1) reduces to FGP when ( )0g t and ( )jg t are signomial and osynomial functions.

Based on fuzzy decision making of Bellman and Zadeh [14], we may write (i) ( ) ( )( )( )( )max min max -min operatorD j jt g tm m* = (4.2)

subject to ( )( ) ( )( ) ( )0j j j j j jg t g g t g gm = - -¢ ¢ ( 0,1,2, ,j m= ), 0t >

(ii)

( ) ( )( ) ( )0

max max -additive operatorm

D j j jj

t g tm l m*

=

Ê ˆ= Á ˜Ë ¯Â (4.3)


(iii)


( ) ( )( )( ) ( )0

max max -product operatorjm

D j jj

t g tl

m m*

=

Ê ˆ= Á ˜Ë ¯’ (4.4)


Here, for jl ( 0,1,2, ,j m= ) are numerical weights considered by decision makers.

For normalized weights0

1m

jj

l=

=Â and [ ]0,1jl Œ .

For equal importance of objective and constraint goals, 1jl = . Unlike GP , FGP in general has not been widely circulated in the oliterature. In

1990, Vermal [16] studied a new concept to use the GP technique for multi-objective fuzzy decision-making problems. He projected the very importance on the product operator, which reduces the DD with a considerable amount. Biswal [13] applied the fuzzy programming technique to solve a multi-objective GP problem as a vector minimization problem. A vector maximization problem can be transformed into a vector minimization problem. Cao [3-4] discussed the properties of a kind of posynomizal GP with an -L R fuzzy coefficient in objective and constraints. In the sequel, Cao [2] published an important book on FGP , which was the most recent book until now.

If ( )jg t ( 0,12, ,j m= )be posynomial function as ( ) 1 1jkijN n

j jk ik ig t c t

a= ==Â Â ( ( )0jkc ≥

and jkia ( 1,2, ,i n= ; 1, 2, , jk N= ; 0,1,2, ,j m= )) then (i) max-min operator (4.2) reduces to

Maximize ( )( ) ( )( )0j j j j jg g t g gl - -¢ ¢ (4.5)

Subject to, ( )( ) ( ) ( )( ) ( )0 0r r r r r j j j j jg g t g g g g t g gl l- - ≥ - -¢ ¢ ¢ ¢ , ( 0,1,2, ,r m= and

r jπ ), 0t > .

So ( )M jV t l* * = ( )( ) ( )0j j jg g g V t* *-¢ ¢ Where t* is obtained by solving the

following signomial GP :

Minimize ( ) ( )( )0

11

j

jki

N n

j j j jk iik

V t g g c tal

=== - -¢ Â ’ (4.6)

Subject to, ( )( ) ( )( )

( )( ) ( )( )0 0

1 11 10 0

j j

jkirki

N Nn n

r r r rk i j j j jk ii ik k

r r r r j j j j

g g c t g g c t

g g g g g g

aal l

l l= == =

- - -¢ ¢

- - -¢ ¢ ¢ ¢

Â Â’ ’,( 0,1, 2r m= and

r jπ ), 0t > (ii) max-additive operator (4.3) reduces to

Maximize ( ) ( )0

10 1

j

jki

Nm n

A j j jk i j jij k

V t g c t g gal

== =

Ê ˆÊ ˆ= - -¢ ¢Á ˜Á ˜Ë ¯Ë ¯Â Â ’ (4.7)

Subject to 0t >


So the optimal decision variable t* with the optimal objective value ( )V t* * can be

obtained by ( ) ( )00

m

j j j jjV t g g g U tl* * * *

== - -¢ ¢Â where t* is the optimal decision variable of the unconstrained geometric programming problem

Minimize ( ) ( )( )0

10 1

j

jki

Nm n

j j j jk iij k

U t g g C tal

== == -¢Â Â ’ (4.8)

Subject to 0t > .

(iii) Similarly, Eq. (4.4) can be solved by GP based on a suitable transformation.

4.1. Numerical example

The series system of reliability has three subsystems with reliability ( )1,2,3iR i = for

the thi sub system. For the series system the system reliability ( )1 2 3 1 2 3, ,sR R R R RR R= .

Now we have to find the maximization of ( )1 2 3, ,sR R R R subject to the limited available cost. Therefore the problem in fuzzy environment becomes

( )1 2 3 1 2 3Max , , 0.5504467sR R R R RR R= ≥ (with tolerance 0.3102683) (4.9)

s.t. 0.94 0.91 0.891 2 2130 140 150 401R R R+ + + £ (with tolerance 51)

Here, the membership functions for the fuzzy objective and constraint goals are Rm =

( )( )1 2 3 0.8607150 0.3102683 0.3102683RR R - - ( )1 2 3 0.5504467 0.3102683RR R= -

( )( )0.94 0.91 0.891 2 3401 130 ,140 ,150 51C R R Rm = -

(i) Based on max-min operator (4.2), FGP (4.9) reduces to Max Rm

Max ( )( )1 2 3Max 0.7758517 0.0705803RR R - (4.10)

s.t. ( )( )0.96 0.895 0.871 2 3385 125 ,135 ,145 10R R R- ≥ ( )1 2 3 0.7758517 0.0705803RR R - , 10 ,R<

2 3, 1R R £ i.e. 1 2 3Max 14.16826. 10.99247RR R - (4.11)

s.t. ( )0.96 0.895 0.871 2 3 1 2 33.85 12.5 ,13.5 ,14.5 14.16826 10.99247R R R RR R- ≥ - i.e. 49.49247

( )0.96 0.895 0.871 2 3 1 2 314.16826 12.5 ,13.5 ,14.5RR R R R R≥ + i.e. ( 0.96

1 2 3 114.16826 12.5 ,RR R R

)0.895 0.872 313.5 ,14.5 49.49247R R £ i.e. ( ) ( )1 2 314.16826 49.49247 , 12.5 49.49247RR R

0.961 ,R ( ) ( )0.895 0.87

2 313.5 49.49247 , 14.5 49.49247 1R R £

To solve equation (4.11), we are to solve the following crisp GP : ( )Min 1 14.16826 1 1 1

1 2 3R R R- - - s.t. 0.96 0.895 0.871 2 3 1 2 30.2525637 ,0.2727688 ,0.2929739 0.286271 1R R R RR R+ £ .

1 2 30 , , 1R R R< £ . For this problem ( )5 3 1 1DD = - + = . The dual problem (DD ) of

this GP is ( ) ( ) ( ) ( )01 11 12

01 11 12Max 1 14.16826 0.2525637 0.2727688W W Wd W W W W= ◊ ◊ ◊


( ) ( )13 14

13 140.2929739 0.286271W WW W ll◊ . Where, 11 13 13 14W W W Wl = + + + s.t. 01W

1= , 01 11 140.96 0W W W- + + = , 01W- + 12 140.895 0W W+ = , 01 13 140.87W W W- + + 0= . So, 11 141 0.96W W= - , 12 1W = - 14 0.895W , 13 141 0.87W W= - .

( ) ( )( )( ) ( )( )141 0.96

14Max 0.0705803 0.2525637 1 0.96W

d W W-

= - ( ((0.2727688 1-

) )) ( )( )141 0.89514 0.895

WW

-= ( )( )( ) ( )( ) ( )( )14 14

1 0.87

14 140.2929739 1 0.87 0.286271W W

W W-

-

( ){ } 141 0.96,1 0.895,1 0.87 1 0.96,1 0.895,1 0.87 1 WÈ ˘- -Î ˚ For optimality of ( )d W , we

have ( )( ) 14log 0d W W∂ ∂ = , That is ( ) ( )142.30841 log 0.286271 1 0.96 logW+ =

( )( )140.2424612 1 W- , ( ) ( )( )141 0.895 log 0.2441281 1 W- , ( ) ( (1 0.87 log 0.2548873 1

))14W- .

Table –1. Optimal solution of (4.9) by max-min operator (4.2)

d* 0.02990114 1R* 0.9245300

11W * 0.1768750 2R* 0.9743700

12W * 0.1897207 3R* 0.9465400

13W * 0.1951724 sR* 0.8526757

14W * 0.8302027 -- --

It is clear from the above table that the system reliability of series system with

objective and cost constraints in fuzzy environment by max-min operator is 0.8526757 which shows that the FGP technique is effective in achieving a superior compromise. (ii) Based on the max-additive operator (4.3), FGP (4.9) reduces to Max R Cm m+ , s.t.

0 1iR£ £ , ( ) ( )1 2 3 1 2 3Max , , 0.5332254 0.3274896A R R R RR R= - , ( ( 0.941401 130 ,R-

))0.91 0.892 3140 ,150 41R R s.t. 0 1iR£ £ . Here, ( ) ( )1 2 3 1 2 3, , 1 0.3274896A R R R RR R=

( )( )0.94 0.91 0.891 2 31 41 130 ,140 ,150R R R- ( ) ( )( ) ( 1 2401 41 0.5332254 0.3274896 , ,A R R+ - -

)3R ( ) ( ) ( )0.94 0.91 0.891 2 3130 41 , 140 41 , 150 41R R R= (1 2 3 1 13.053532 8.152267 ,RR R A R- - =

)2 3, 8.152267R R - At first we take the problem ( ) ( )0.94 0.911 2Min 130 41 , 140 41 ,R R

( ) 0.893 1 2 3150 41 3.053532R RR R- s.t. 1 2 30 , , 1R R R£ £ . For this problem 4DD = -

( )3 1 0+ = . The dual problem ( )DP of this .GP is ( ) ( )( ) 1

1Max 130 41W

d W W=

( )( ) ( )( ) ( )32 4

2 3 4140 41 150 41 3.053532WW W

W W W- s.t. 1 2 3 4 1W W W W+ + - = , 10.94W

4 0W- = , 2 40.91 0W W- = , 3 40.89 0W W- =

Table –2. Optimal solution of (4.9) by max-additive operator (4.3)

( )d W * 7.184581 1R* 0.95321000


1W* 0.4653009 2R

* 0.9356400

2W* 0.4806405 3R

* 0.9608557

3W* 0.4914414

sR* 0.8569501

4W* 0.4373829 -- --


objective and cost constraints in fuzzy environment by max-additive operator is 0.8569501 which shows that the FGP technique is effective in achieving a superior compromise.

(iii) Based on the max-product operator (4.4), FGP reduces to Max R Cm m

( )( )1 2 3Max 0.7758517 0.0705803RR R - , ( )( )0.96 0.895 0.871 2 3385 125 ,135 ,145 10R R R- ,

1 2 30 , , 1R R R£ £ , ( ) 1.961 2 3 1 2 3 1 2 3Max , , 545.4780 177.1032f R R R RR R R R R= - 191.2715-

1.895 1.87 0.961 2 3 1 2 3 1205.4398 137.4059RR R RR R R- + 0.895 0.87

2 3148.3983 159.3908R R+ + - 423.21 Let ( ) ( )1 1 2 3 1 2 3, , , , 423.21f R R R f R R R= - - ( )1 1 2 3Min , , 545.4780f R R R =-

1.961 2 3 1 2 3177.1032RR R R R R+ 1.895 1.87

1 2 3 1 2 3191.2715 205.4398 137.4059RR R RR R+ + - 0.961R

0.895 0.872 3148.3983 159.3908R R- - The dual problem ( )DP of this GP is ( )Maxd W

( ) 1

1545.4780 WWx -È= Î ( )( ) 2

2177.1032W

W ( )( ) 3

3191.2715W

W ( )( ) 4

4203.4398W

W

( ) ( ) ( )5 6 7

5 6 7137.4059 148.3983 159.3908W W WW W W

x- - - ˘◊ ◊ ˚ , 1x = or 1- , Let 1x = - s.t.

1 2 3 4 5 6 7 1 2 3 4 51.96 0.96 0W W W W W W W W W W W Wx- + + + - - - = - + + + - = 1W-

2 3 4 61.895 0.895 0W W W W+ + + - = , 1 2 3 4 71.87 0.87 0W W W W W- + + + - = Again,

1 2 3 1545.4780RR R W d= ◊ , 1.961 2 3 2177.1032R R R W d= - ◊ , 1.895

1 2191.2715RR = 3W d- ◊ , 1.87

1 2 3 4205.4398RR R W d= - ◊ , 0.961 5137.4059R W d= - ◊ , 0.895

2148.3983R = 6W d- ◊ , 0.873 7159.3908R W d= -

Table –3. Optimal solution of (4.9) by max-product operator (4.4)

d* 0.280157 6W* 0.4295830

1W* 0.9999992 7W

* 0.2828283

2W* 0.2875887 1R

* 0.8813088

3W* 0.4295827 2R

* 0.9932752

4W* 0.2828280 3R

* 0.9694974

5W* 0.2875890

sR* 0.8486808


objective and cost constraints in fuzzy environment by max-product operator is 0.8486808 which shows that the FGP technique is effective in achieving a superior compromise.


5. Conclusion

Here we have dealt with series system reliability optimization problem with objective and cost constraints in fuzzy environment. Numerical examples are solved by different operators like max-min operator, max-additive operator and max-product operators to show that the FGP technique is effective in achieving a superior compromise between mutually conflicting objective and constraints when the problem parameters, prescribed goal and design constraint are not known precisely. It is hoped that this technique can be used in many practical engineering model on reliability optimization.

References

[1] A. J. Federowicz and M. Majumder, Use of geometric programming to maximize reliability achieved by redundancy, Operations Research, 16 (5) (1968), 948-954.

[2] B. Y. Cao, Fuzzy geometric programming, Kluwer academic publishers, The Netherlands (2002). [3] B. Y. Cao, Posynomial geometric programming with -L R . Fuzzy coefficients, Fuzzy Sets and

Systems, 64 (1994), 267-276. [4] B. Y. Cao, Fuzzy geometric programming (I), Fuzzy Sets and Systems, 53 (1993), 135-153. [5] F. A. Tillman, C. L. Hwang and W. Kuo, Optimization techniques for systems reliability with

redundancy-A review, IEEE Transactions on Reliability, 26 (1977), 148-155. [6] G. S. Beightler and D. T. Phillips, Applied geometric programming, Wiley, New York, (1976). [7] G. S. Mahapatra and T. K. Roy, Generalized piece-wise linear fuzzy number with its application in

redundancy for optimum reliability of series-parallel system, The Journal of Fuzzy Mathematics, 18 (3) (2010), 541-552.

[8] G. S. Mahapatra and T. K. Roy, Optimal fuzzy reliability for a series system with cost constraint using fuzzy geometric programming, Tamsui Oxford Journal of Management Sciences, 22 (2) (2006), 53-63.

[9] H. Singh and N. Misra, On redundancy allocation in systems, Journal of Applied Probability, 31 (1994), 1004-1014.

[10] K. K. Govil, Optimal maintainability allocation using the geometric programming method, Microelectronic Reliability, 32 (1/2) (1992), 265-266.

[11] K. K. Govil, Geometric programming method for optimal reliability allocation for a series system subject to cost constraint, Microelectronic Reliability, 23 (5) (1983), 783-784.

[12] K. S. Park, Fuzzy apportionment of system reliability, IEEE Transactions on Reliability, 36 (1987), 129-132.

[13] M. P. Biswal, Fuzzy programming technique to solve multi-objective geometric programming problems, Fuzzy Sets and Systems, 51 (1992), 67-71.

[14] R. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment, Management Sciences, 17 (1972), 141-164.

[15] R. J. Duffin, E. L. Peterson and C. Zener, Geometric programming theory and applications, Wiley, New York, (1967).

[16] R. K. Verma, Fuzzy geometric programming with several objective functions, Fuzzy Sets and Systems, 35 (1990), 115-120.

[17] U. Passy and D. J. Wilde, Generalized polynomial optimization, SLAM Journal on Applied Mathematics, 15 (5) (1967), 1344-1356.

[18] W. Kuo, V. R. Prasad, F. A. Tillman and C. Hwang, Optimal reliability design-fundamentals and applications, Cambridge universeity press, Cambridge United Kingdom (2001)


__________________________ Received November, 2013


Los Angeles

Concept Analysis with Interior and Closure Operators

Subrata Bhowmik

Department of Mathematics, Tripura University Suryamaninagar, Tripura, INDIA-799022 E-mail: [email protected]

Abstract: The study of concept analysis is done within the lattice structure. In this paper we are

interested to study concept analysis in some different way. Here we will introduce some algebraic operators and study some algebra. Also we will study different types of Information-System and the rules of decision making using this new type of concept analysis.

Key words: Formal Concept Analysis, Semi-Lattices, Fuzzy Set, Information System.

1. Introduction

Formal concept analysis or context analysis and study of information system is almost similar thing. In this paper we are interested to study concept analysis or context analysis as a different study from the study of the information system.

In your real life if we take any concept (say about certain disease), then there are some attribute associated with this concept. An association of the concepts with the attributes we say a context.

In this paper we will introduce two operators-interior and closure operators and will study some algebra.

Lastly we will study different types of Information Systems (IS).

2. Context

In our study all the sets are assumed to be finite.

Definition 2.1. A context C is a triplet ( ), ,L V v , where L is a Lattice-called

concept lattice, V is the set of attributes and ( ):v L VÆ√ called valuation mapping satisfying the following conditions: for any , La b Œ

90 Subrata Bhowmik

(i) ( )v a π ∆

(ii) ( ) ( )v va b a b£ fi Õ

(iii) ( ) ( ) ( )v v va b a b⁄ = »

(iv) ( ) ( ) ( )v v va b a bŸ = « . If L is a join (or meet) semi-lattice and with conditions (i), (ii) and (iii) (or (iv))

( ), ,L V v is called join (resp. meet) semi-context.

Theorem 2.2. Let be ( ), ,L V v a context, for any X VÕ we define [ ]: 0,1XF LÆ as

( ) ( ) ( )XF v X va a a= « .

XF is a fuzzy subset of L .

Definition 2.3. Let ( ), ,L V v be a context, for any A L∆ π Õ we define fuzzy relation

( ) ( ) [ ]: 0,1A V VD √ ¥√ Æ on ( )V√ defined as ( ),A AX Y aŒD = Â ( ) ( )X YF F Aa a-

called the distance measure between X and Y in ( )V√ with respect to A .

The function ( ) ( ) [ ]: 0,1AS V V√ ¥√ Æ defined as ( ) ( ), 1 ,A AS X Y X Y= - D called

the similarity measure between X and Y in ( )V√ with respect to A .

Theorem 2.4. AD is a pseudo-metric on ( )V√ . Also for any ,X Y VÕ ,

( ) ( ), ,A AS X Y S Y X= .

Proof. It is not very difficult to see that for any , ,X Y Z VÕ ,

(i) ( ), 0A X XD =

(ii) ( ) ( ), ,A AX Y Y XD = D

(iii) ( ) ( ) ( ), , ,A A AX Z X Y Y ZD £ D + D .

Thus AD is a pseudo-metric on ( )V√ . Other part is straight forward.

Definition 2.5. For any A LÕ we define a relation AD

ª on ( )V√ defined as:A

X YDª

iff ( ), 0A X YD = or equivalently ( ), 1AS X Y = . Now we have the following easy consequence:

Theorem 2.6. AD

ª is an equivalence relation on ( )V√ .

Definition 2.7. Let ( ), ,L V v be a context, we define ( ) ( ):I V L√ Æ√ and ( ):C V√

( )LÆ√ called respectively interior operator and closure operator defined as:

Concept Analysis with Interior and Closure Operators 91

( ) ( ){ }:I X L v Xa a= Œ Õ , ( ) ( ){ }:C X L v Xa a= Œ « π ∆ .

( ) ( )X C X I X∂ = - is called the boundary region of X and ( ) ( )E X L C X= - is called the exterior region of X .

Theorem 2.8. Let ( ), ,L V v be a context, for any X VÕ

(i) ( ) ( ){ }: 1XI X L Fa a= Œ =

(ii) ( ) ( ){ }: 0XC X L Fa a= Œ >

(iii) ( ) ( ){ }: 0 1XX L Fa a∂ = Œ < <

(iv) ( ) ( ){ }: 0XE X L Fa a= Œ = .

Definition 2.9. Let ( ), ,L V v be a context. We define three relations Iª , Cª and ª

on ( )V√ as: IX Yª iff ( ) ( )I X I Y= , CX Yª iff ( ) ( )C X C Y= and X Yª iff

( ) ( )I X I Y= and ( ) ( )C X C Y= . Now we have the following easy consequences:

Theorem 2.10. Iª , Cª and ª are all equivalence relation on ( )V√ . Let for any

X VÕ , [ ]I

X ª , [ ]C

X ª and [ ]X ª be the equivalence classes of X for the equivalence

relations Iª , Cª and ª respectively. Let ( )VI

√ ª , ( )VC

√ ª and ( )V√ ª be the collec-tion of equivalence classes for the equivalence relations Iª , Cª and ª respectively.

Theorem 2.11. For a context ( ), ,L V v for any ,X Y VÕ

(i) ( ) ( )X Y I X I YÕ fi Õ and ( ) ( )C X C YÕ ,

(ii) ( ) ( ) ( )I X Y I X I Y« = « ,

(iii) ( ) ( ) ( )C X Y C X C Y» = » ,

(iv) ( ) ( )I V X L C X- = - ,

(v) ( ) ( )C V X L I X- = - ,

(vi) ( ) ( )I C∆ = ∆ = ∆ , ( ) ( )I V C V L= = .

Proof. (i) and (vi) directly follows from the definitions. (ii) X Y X« Õ and X Y Y« Õ , so ( ) ( )I X Y I X« Õ and ( ) ( )I X Y I Y« Õ (by

(i), thus ( ) ( ) ( )I X Y I X I Y« Õ « . To prove the reverse inequality, let ( ) ( )I X I Ya Œ « ,

i.e. ( )I Xa Œ and ( )I Ya Œ , so ( )v Xa Õ and ( )v Ya Ã , thus ( )v X Ya Õ « and

hence ( )I X Ya Œ « , so ( ) ( ) ( )I X I Y I X Y« Õ « . Hence ( ) ( ) ( )I X Y I X I Y« = « .

92 Subrata Bhowmik

(iii) Again using (i) we have ( ) ( ) ( )C X C Y C X Y» Õ » . To prove the reverse

inequality let ( )C X Ya Œ » ,so ( ) ( )v X Ya « » π ∆ , so ( )v Xa « π ∆ or ( )v Ya «

π ∆ , i.e. ( )C Xa Œ or ( )C Ya Œ , so ( ) ( )C X C Ya Œ » . Hence ( ) ( )C X Y C X» Õ

( )C Y» . Thus we have ( ) ( ) ( )C X Y C X C Y» = » .

(iv) Let ( )I V Xa Œ - ( ) ( )v V Xa¤ Õ - , ( )v Xa¤ « = ∆ ( )C Xa¤ œ a¤

( )( )L C XŒ - . Thus ( ) ( )I V X L C X- = - .

(v) Proof is imilar to proof of (iv).

Using the above Theorem 2.11 we can define the following well defined operators:

Definition 2.12. We define the following operators: (i) u on ( ) IV√ ª as [ ] [ ] [ ]

I I I

X Y X Yª ª ª= «u ,

(ii) t on ( ) CV√ ª as [ ] [ ] [ ]C C C

X Y X Yª ª ª= »t ,

(iii) ÿ on ( )V√ ª as [ ] [ ]X V Xª ªÿ = - .

Again using the above Theorem 2.11 we can define the following well defined inequlaties:

Definition 2.13. We define the following operators: (i)

Iª on ( ) IV√ ª as [ ] [ ]

II I

X Yªª ª iff ( ) ( )I X I YÕ .

(ii) Cª

on ( ) CV√ ª as [ ] [ ]CC C

X Yªª ª iff ( ) ( )C X C YÕ .

(iii) ª on ( )V√ ª as [ ] [ ]X Yªª ª iff ( ) ( )I X I YÕ and ( ) ( )C X C YÕ .

Theorem 2.14. (i) ( )( ),IIV ª√ ª is a partial order set with least element [ ]

Iª∆

and greatest element [ ]I

V ª and ( )( ), ,IIV ª√ ª u is a meet-semilattice.

(ii) ( )( ),CCV ª√ ª is a partial order set with least element [ ]

Cª∆ and greatest

element [ ]C

V ª and ( )( ), ,CCV ª√ ª t is a join-semilattice.

(iii) ( )( ),V ª√ ª is a partial order set with least element [ ]ª∆ and greatest

element [ ]V ª and ( )( ), ,IV ª√ ª ÿ is a complemented partial order set.

Proof. We are going to prove this theorem for (i) only, proof of others are similar. (i) A relation is a partial order relation if it is reflexive, transitive and antisymmetric.

It is not much difficult to see that the relation Iª

is a partial order relation on

( ) IV√ ª . Also it is easy to see that the partial order set ( )( ),IIV ª√ ª possess the

least and greatest elements which are none other than [ ]Iª

∆ and [ ]I

V ª respectively. A

partial order set is meet-semilatice if it is closed under finite meet, i.e. any two element have their infimum with respect to the partial order relation.


A lattice is simultaneously a meet-semilattice and join-semilattice both. Now we have the following consequences:

Theorem 2.15. The mappings:

(i) ( ) ( )( ): , ,II If L V ª£ Æ √ ª defined as ( ) ( )

IIf va a

ªÈ ˘= Î ˚ is a meet-semi lattice

homomorphism.

(ii) ( ) ( )( ): , ,CC Cf L V ª£ Æ √ ª defined as ( ) ( )

CCf va a

ªÈ ˘= Î ˚ is a join-semi

lattice homomorphism.

(iii) ( ) ( )( ): , ,f L V ª£ Æ √ ª defined as ( ) ( )f va aª

È ˘= Î ˚ is a partial order set

homomorphism.

Proof. We are going to prove this theorem for (i) only, the proof of others are similar. (i) For , La b Œ , ( ) ( ) ( )( ) ( )( ) ( )

II

v v I v I v va b a b a b a ªªÈ ˘£ fi Õ fi Õ fi Î ˚

( )I

v bª

È ˘Î ˚ , thus If is a partial order homomorphism.

Again ( ) ( ) ( ) ( ) ( ) ( )I I I I

If v v v v va b a b a b a bª ª ª ª

È ˘ È ˘ È ˘ È ˘Ÿ = Ÿ = « =Î ˚ Î ˚ Î ˚ Î ˚u , thus If

is an meet-semilattice homomorphism.

3. Information system

In this section we will study different types of Information-Systems and some methods of decision making in such information-system. In this section we consider all the sets unless otherwise stated are assumed to be finite.

Definition 3.1. An information system (IS) I is a pair ( ),OC , where ( ), ,L V v=C is a context and O is a set of objects. For each a VŒ , we associate a mapping

{ }: 0,1a OÆ . If ( ) 1a x = we say the object x is associated with the attribute a , else we say x is not associated with the attribute a .

Let U OÕ , we define ( ) ( ){ }: 1,I U a V a x x U= Œ = " Œ , ( ) ( ){ : 1,C U a V a x= Œ =

}x U$ Œ and ( ) ( ) ( )U C U I U∂ = - . It is easy to see that ( ) ( )I U C UÕ . We say a concept a is

(i) Strongly associate with U if ( ) ( )v I Ua Õ , i.e ( ) ( ) 1I UF a = . We write this

situation as sU a .

(ii) Weakly associate with U if ( ) ( )v C Ua Õ , i.e. ( ) ( ) 1C UF a = . We write this

situation as wU a .

(iii) Associate with U at strong e -level if ( ) ( )I UF a e≥ , where 0 1e< £ . We

write this situation as sU e a

94 Subrata Bhowmik

(iv) Associate with U at weak e -level if ( ) ( )C UF a e≥ , where 0 1e< £ . We write

this situation as wU e a .

Now we have the following easy consequences:

Theorem 3.2. Let ( ),O=I C , be an information system, let 1 2, ,U U U OÕ , , La b Œ :

(a) s wU Ua afi , s wU Ue ea afi

(b) For ,s w= :

(i) U a , Ub a b£ fi

(ii) 1U a , 2 1 2U U Ub a bfi » ⁄

(c) For ,s we e= , 0 1e< £ :

(i) 1U a , 1 2 2U U U aÕ fi .

Example 3.3. We consider the IS ( ),O=I C with ( ), ,L V v=C as: { }1 2 3, ,L a a a= ,

{ }1 2 3 4, , ,V a a a a= , ( ) { }1 1 2,v a aa = , ( ) { }2 2 3,v a aa = , ( ) { }3 1 4,v a aa = and { 1 ,O x=

}2 3 4 5, , ,x x x x with:

1 2 3 4

1

2

3

4

5

0 1 1 10 0 1 11 0 1 11 1 0 01 1 0 0

a a a a

x

x

x

x

x

{ }2 3,U x x= , ( ) { }3 4,I U a a= , ( ) { }1 3 4, ,C U a a a= and ( ) { }1U a∂ = . So, 3 0.4s Ua ,

3w Ua .

Let us now generalize IS in fuzzy setting as:

Definition 3.4. An information system with fuzzy values (ISFV) I is a pair ( ),OC ,

where ( ), ,L V v=C is a context and O is a set of objects, for each a VŒ , we associate

a mapping [ ]: 0,1a OÆ called object gradation associated with a , which is a fuzzy subset of O , denoted by the same symbol a .

For U OÕ and 0 1e< £ we define ( ) ( ){ }: ,I U a V x U a xe e= Œ " Œ ≥ , ( )C Ue =

( ){ }: ,a V x U a x eŒ $ Œ ≥ and ( ) ( ) ( )U C U I Ue e e∂ = - . Obviously ( ) ( )I U C Ue eÕ . If

U is a singleton set { }x , then ( ) ( ) ( )I U C U IC xe e e= = (say). For 0 1e< £ , U OÕ , we say La Œ is associated with U at

(i) Strong e -level if ( ) ( )v I Uea Õ , we write s Uea


(ii) Weak e -level if ( ) ( )v C Uea Õ , we write w Uea .

Theorem 3.5. For a ( ),ISFV O=I C with ( ), ,L V v=C , for x OŒ and La Œ , if

we define level of a for x as ( ) ( ) ( ){ }max 0 :l x v IC xa ee a= > Õ or ( ) 0l xa = other-

wise; then la is a fuzzy subset of O and ( ) ( ) ( ){ }min :l x a x a va a= Œ .

Theorem 3.6. For an ( ),ISFV O=I C with ( ), ,L V v=C , for U OÕ ,

(i) s wU Ue ea afi

(ii) a b£ , U Ub afi , for ,s we e=

(iii) 1 2U U OÕ Õ , 2 1s sU Ue ea afi and 1 2

w wU Ue ea afi .

Example 3.7. We consider the ( ),ISFV O=I C with ( ), ,L V v=C as: { }1 2 3, ,L a a a= ,

{ }1 2 3 4, , ,V a a a a= , ( ) { }1 1 2,v a aa = , ( ) { }2 2 3,v a aa = , ( ) { }3 1 4,v a aa = and { 1,O x=

}2 3 4 5, , ,x x x x with: Object gradation:

1 2 3 4

1

2

3

4

5

0.1 0.3 0.5 10 0.2 0.5 0.6

0.9 0.3 0.4 0.71 0.2 0.5 0.2

0.1 0.3 0.9 0.9

a a a a

x

x

x

x

x

{ }2 3,U x x= , ( ) { }0.5 4I U a= , ( ) { }0.5 1 3 4, ,C U a a a= and ( ) { }0.5 1 3,U a a∂ = . So,

3 0.5w Ua .

Definition 3.8. An information system with fuzzy attributes ( )ISFA I is a pair

( ),OC , where ( ), ,L V v=C is a context and O is a set of objects, for each La Œ , we

associate a mapping [ ]: 0,1Va Æ called value gradation associated with a , which is a fuzzy subset of V , denoted by the same symbol a and for a b£ , a is a fuzzy subset of b ; for each a VŒ , we associate a mapping { }: 0,1a OÆ .

We denote the e -cut, where 0 1e< £ of a as ( ) ( ){ }:a V aes a a e= Œ ≥ . For U OÕ , we say La Œ associated with U at (i) Strong e -level if ( ) ( )I Ues a Õ , we write s Uea

(ii) Weak e -level if ( ) ( )C Ues a Õ , we write w Uea .

Theorem 3.9. For an ( ),ISFA O=I C , with ( ), ,L V v=C , for U OÕ ,

(i) s wU Ue ea afi

96 Subrata Bhowmik

(ii) a b£ , U Ub afi , for ,s we e=

(iii) 1 2U U OÕ Õ , 1 2U Ua afi , for ,s we e= .

Example 3.10. We consider the ( ),ISFA O=I C with ( ), ,L V v=C as: { 1L a=

}2 3a a< < , { }1 2 3 4, , ,V a a a a= with: Value gradation:

1 2 3

1

2

3

4

0.6 0.7 10.7 0.8 0.90.2 0.4 0.50.3 0.4 0.5

a

a

a

a

a a a

Object-attribute:

1 2 3 4

1

2

3

4

5

0 1 1 10 0 1 11 1 0 11 1 0 01 1 0 0

a a a a

x

x

x

x

x

.

( ) { }0.5 1 1 2,a as a = . { }3 4,U x x= , ( ) { }1 2,I U a a= , ( ) { }1 2 4, ,C U a a a= and

( ) { }4U a∂ = . So, 1 0.5s Ua .

Definition 3.11. An information system with fuzzy values and fuzzy attributes ( )ISFVFA I is a pair ( ),OC , where ( ), ,L V v=C is a context and O is a set of objects,

for each La Œ , we associate a mapping [ ]: 0,1Va Æ , which is a fuzzy subset of V and for a b£ , a is a fuzzy subset of b ; for each a VŒ , we associate a mapping

[ ]: 0,1a OÆ , which is a fuzzy subset of O . Let 0 e< , 1d £ . For U OÕ , we say La Œ associated with U at (i) Strong ( ),e d -level if ( ) ( )I Ue ds a Õ , we write ,

sU e d a ,

(ii) Weak ( ),e d -level if ( ) ( )C Ue ds a Õ , we write ,wU e d a .

Theorem 3.12. For an ( ),ISFVFA O=I C , with ( ), ,L V v=C , for U OÕ ,

(i) , ,s wU Ue d e da afi

(ii) a b£ , U Ub afi , for , ,,s we d e d=

(iii) 1 2U U OÕ Õ , , 2 , 1s sU Ue d e da afi and , 1 , 2

w wU Ue d e da afi .

Example 3.13. We consider the ( ),ISFVFA O=I C with ( ), ,L V v=C as:

{ }1 2 3L a a a= < < , { }1 2 3 4, , ,V a a a a= with: Object gradation:


1 2 3

1

2

3

4

0.6 0.7 10.7 0.8 0.90.2 0.4 0.50.3 0.4 0.5

a

a

a

a

a a a

Object gradation:

1 2 3 4

1

2

3

4

5

0.1 0.3 0.5 10 0.7 0.5 0.6

0.9 0.3 0.4 0.71 0.8 0.5 0.2

0.1 0.3 0.9 0.9

a a a a

x

x

x

x

x

.

( ) { }0.6 3 1 2,a as a = . { }2 4,U x x= , ( )0.5I U { }2 3,a a= , ( ) { }0.5 1 2 3 4, , ,C U a a a a= and

( ) { }0.5 1 4,U a a∂ = . So, 3 0.6,0.5w Ua .

Acknowledgement:

Financial support from the UGC, Govt. of India (F. No.41-1381/2012(SR)) is gratefully acknowledged.

References

[1] S. Bhowmik, IC-topological spaces and applications in soft computing, In: S. O. Kuznetsov et al. (Eds.): PreMI 2011, LNCS, Springer-Verlag Berlin Heidelberg, (2011), 311-317.

[2] B. A. Davy and H. A. Pristley, Introduction to lattices and order, Cambridge university press, 2e, (2002).

[3] P. Pagliani and M. K. Chakraborty, A geometric approximation, Rough set theory: Logic, Algebra and topology of onceptual patterns, Springer, Heidelberg, (2008).

[4] Z. Pei, D. Pei and L. Zheng, Topology vs generalized rough sets, International Journal of Approximate Reasoning, 52 (2011), 231-239.

[5] S. Vickers, Topology via logic, Cambridge university press, (1989). [6] L. A. Zadeh, Fuzzy sets, Inform. And Control, 8 (1965), 338-353.


__________________________ Received November, 2013


Los Angeles

On Somewhat Fuzzy Nearly Continuous Functions

G. Thangaraj

Department of Mathematics Thiruvalluvar University Vellore-632115, Tamilnadu, India.

S. Anjalmose

Research Scholar, Department of Mathematics Thiruvalluvar University Vellore-632115, Tamilnadu, India.

Abstract: In this paper the concept of somewhat fuzzy nearly continuous functions, somewhat

fuzzy nearly open functions are introduced and studied. Besides giving characterizations of these functions, several interesting properties of these functions are also given. Several examples are given to illustrate the concepts introduced in this paper.

Key Words: Somewhat fuzzy continuous, somewhat fuzzy nearly continuous, somewhat fuzzy

open, somewhat fuzzy nearly open, fuzzy irresolute, fuzzy pre semi-open, fuzzy dense, fuzzy nowhere dense, fuzzy Baire space.

1. Introduction

The concept of fuzzy sets and fuzzy set operations were first introduced by L. A. Zadeh in his classical paper [12] in the year 1965. Thereafter the paper of C. L. Chang [3] in 1968 paved the way for the subsequent tremendous growth of the numerous fuzzy topological concepts. Since then much attention has been paid to generalize the basic concepts of General Topology in fuzzy setting and thus a modern theory of fuzzy topology has been developed. The notion of continuity is of fundamental importance essentially in almost all branches of Mathematics. Hence it is of considerable significance from applications view point, to formulate and study new variants of fuzzy continuity.

In classical topology, the class of somewhat continuous functions was introduced and studied by Karil. R. Gentry and Hughes B. Hoyle, III [6]. Later, the concept of “somewhat” in classical topology has been extended to fuzzy topological spaces. Somewhat fuzzy continuous functions, somewhat fuzzy open functions on fuzzy topological spaces were introduced and studied by G. Thangaraj and G. Balasubramanian in [9]. The concept of somewhat nearly continuous functions were introduced and

100 G. Thangaraj and S. Anjalmose

studied in classical topology by Z. Piotrowski [8]. The concept of nearly open functions were introduced and studied extensively in classical topology by D. S. Jankovic and C. Konstadilaki Savvopoulou [5]. In this paper we introduce the concepts of somewhat fuzzy nearly continuous functions, somewhat fuzzy nearly open functions. Besides giving characterizations of these functions, several interesting properties of these functions are also given. Several examples are given to illustrate the concepts introduced in this paper.

2. Preliminaries

Now we introduce some basic notions and results that are used in the sequel. In this work by a fuzzy topological space we shall mean a non-empty set X together with a fuzzy topology T (in the sense of Chang) and denote it by ( ),X T . The interior, closure

and the complement of a fuzzy set l will be denoted by ( )int l , ( )cl l and 1 l- respectively.

Definition 2.1. Let ( ),X T be a fuzzy topological space and l be a fuzzy set in

( ),X T . We define ( ) { }cl ,1 Tl m l m m= £ - Œ� and ( ) { }int , Tl m m l m= £ Œ� .

For any fuzzy set in a fuzzy topological space ( ),X T , it is easy to see that

( ) ( )1 cl int 1l l- = - and ( ) ( )1 int cl 1l l- = - [1].

Definition 2.2. Let ( ),X T and ( ),Y S be any two fuzzy topological spaces. Let f

be a function from the fuzzy topological space ( ),X T to the fuzzy topological space

( ),Y S . Let l be a fuzzy set in ( ),Y S . The inverse image of l under f written as

( )1f l- is the fuzzy set in ( ),X T defined by ( )( ) ( )( )1f x f xl l- = for all x XŒ . Also

the image of l in ( ),X T under f written as ( )f l is the fuzzy set in ( ),Y S defined by

( )( )( )( )

( )1

1sup if is non-empty;

0 otherwise.x f y

x f yf y

ll -

-

Œ

ÏÔ= ÌÔÓ

for each y YŒ .

Lemma 2.1 [3]. Let ( ) ( ): , ,f X T Y SÆ be a mapping. For fuzzy sets l and m of

( ),X T and ( ),Y S respectively, the following statements hold:

(1) ( )1ff m m- £ ;

(2) ( )1f f l l- ≥ ;

(3) ( ) ( )1 1f fl l- ≥ - ;

(4) ( ) ( )1 11 1f fm m- -- = - ;

On Somewhat Fuzzy Nearly Continuous Functions 101

(5) If f is one-to-one, then ( )1f f l l- = ;

(6) If f is onto, then ( )1ff m m- = ;

(7) If f is one-to-one and onto, then ( ) ( )1 1f fl l- = - .

Lemma 2.2 [1]. Let ( ) ( ): , ,f X T Y SÆ be a mapping and { }al be a family of fuzzy sets of Y . Then

(a) ( ) ( )1 1j jf fa al l- -=∪ ∪ .

(b) ( ) ( )1 1j jf fa al l- -=∩ ∩ .

Lemma 2.3 [4]. Let ( ) ( ): , ,f X T Y SÆ be a mapping and { }jA , j JŒ be a family

of fuzzy sets in X . Then

(a) ( ) ( )j J j j J jf A f AŒ Œ=∪ ∪ .

(b) ( ) ( )j J j j J jf A f AŒ Œ£∩ ∩ .

Definition 2.3 [9]. A fuzzy set l in a fuzzy topological space ( ),X T is called fuzzy

dense if there exists no fuzzy closed set m in ( ),X T such that 1l m< < .


nowhere dense if there exists no non-zero fuzzy open set m in ( ),X T such that

( )clm l< . That is, ( )int cl 0l = .


first category if ( )1i il l•== � , where il ’s are fuzzy nowhere dense sets in ( ),X T . Any

other fuzzy set in ( ),X T is said to be of second category.

Definition 2.6. A fuzzy set l in a fuzzy topological space ( ),X T is called:

(a) Fuzzy semi-open [1] if ( )clintl l£ ,

(b) Fuzzy pre-open [2] if ( )int cll l£ .

Definition 2.7. A fuzzy set l in a fuzzy topological space ( ),X T is called:

(a) Fuzzy semi-closed [1] if 1 l- is fuzzy semi-open. (b) Fuzzy pre-closed [2] if 1 l- is fuzzy pre-open.


Definition 2.8 [7]. A function ( ) ( ): , ,f X T Y SÆ from a fuzzy topological space

( ),X T into another fuzzy topological space ( ),Y S is called fuzzy irresolute if ( )1f J-

is a fuzzy semi-open set in ( ),X T for any fuzzy semi-open set J in ( ),Y S .


( ),X T into another fuzzy topological space ( ),Y S is called fuzzy pre-semi open if

( )f J is a fuzzy semi-open set in ( ),Y S for any fuzzy semi-open set J in ( ),X T .


( ),X T into another fuzzy topological space ( ),Y S is called fuzzy pre-continuous if

( )1f m- is a fuzzy pre-open set in ( ),X T for any fuzzy open set m in ( ),Y S .


( ),X T into another fuzzy topological space ( ),Y S is called fuzzy `semi-continuous if

( )1f d- is a fuzzy semi-open set in ( ),X T for any fuzzy open set d in ( ),Y S .


( ),X T into another fuzzy topological space ( ),Y S is called somewhat fuzzy

continuous if Sl Œ and ( )1 0f l- π implies that there exist a fuzzy open set d in

( ),X T such that 0d π and ( )1fd l-£ . That is, ( )1int 0f l-È ˘ πÎ ˚ .


( ),X T into another fuzzy topological space ( ),Y S is called somewhat fuzzy open if

Tl Œ and 0l π implies that there exists a fuzzy open set h in ( ),Y S such that 0h π

and ( )fh l£ . That is, ( )int 0f lÈ ˘ πÎ ˚ .

3. Somewhat fuzzy nearly continuous functions

Motivated by the classical concept introduced in [8] we shall now define:

Definition 3.1. Let ( ),X T and ( ),Y S be any two fuzzy topological spaces. A

function ( ) ( ): , ,f X T Y SÆ is called a somewhat fuzzy nearly continuous function if

Sl Œ and ( )1 0f l- π , there exists a non-zero fuzzy open set m of ( ),X T such that

( )1cl fm l-È ˘£ Î ˚ . That is, ( ) ( ): , ,f X T Y SÆ is a somewhat fuzzy nearly continuous

function if ( )1int cl 0f l- π , for any non-zero fuzzy open set l in ( ),Y S .


Example 3.1. Let { }, ,X a b c= . The fuzzy sets l , m , a and b are defined on X as follows:

[ ]: 0,1Xl Æ is defined as ( ) 0.7al = ; ( ) 0.8bl = ; ( ) 0.6cl = .

[ ]: 0,1Xm Æ is defined as ` ( ) 0.9am = ; ( ) 0.6bm = ; ( ) 0.5cm = .

[ ]: 0,1Xa Æ is defined as ( ) 0.8aa = ; ( ) 0.7ba = ; ( ) 0.6ca = .

[ ]: 0,1Xb Æ is defined as ( ) 0.6ab = ; ( ) 0.9bb = ; ( ) 0.5cb = .

Let { }0, , , , ,1T l m l m l m= ⁄ Ÿ and { }0, , , , ,1S a b a b a b= ⁄ Ÿ . Then T and S

are clearly fuzzy topologies on X . Define a function ( ) ( ): , ,f X T X SÆ by ( )f a b= ;

( )f b a= ; ( )f c c= . Then, for the non-zero fuzzy open sets a , b , [ ]a b⁄ , [ ]a bŸ

and 1 in ( ),X S , we have ( )1int cl 0f a-È ˘ πÎ ˚ , ( )1int cl 0f b-È ˘ πÎ ˚ , ( )1int cl 0f a b-È ˘⁄ πÎ ˚ ,

( )1int cl 0f a b-È ˘Ÿ πÎ ˚ and ( )1int cl 1 0f -È ˘ πÎ ˚ respectively in ( ),X T . Hence f is a

somewhat fuzzy nearly continuous function from ( ),X T into ( ),X S .



[ ]: 0,1Xm Æ is defined as ( ) 0.6am = ; ( ) 0.5bm = ; ( ) 0.4cm = .




are clearly fuzzy topologies on X . Define a function ( ) ( ): , ,f X T X SÆ by ( )f a a= ;

( )f b b= ; ( )f c c= . Then, for the non-zero fuzzy open sets a , b , [ ]a b⁄ , [ ]a bŸ

and 1 in ( ),X S ,we have ( ) [ ]1int cl 1 0f a l m-È ˘ = - ⁄ πÎ ˚ , ( )1int cl 1f b-È ˘ = -Î ˚ [ ] 0l mŸ π ,

( )1int cl 0f a b-È ˘⁄ =Î ˚ , ( )1int cl 0f a b m-È ˘Ÿ = πÎ ˚ and ( )1int cl 1 0f -È ˘ πÎ ˚ respectively

in ( ),X T . Since ( )1int cl 0f a b-È ˘⁄ =Î ˚ , f is not a somewhat fuzzy nearly continuous

function from ( ),X T into ( ),X S .

Proposition 3.1. If a function ( ) ( ): , ,f X T Y SÆ from a fuzzy topological space

( ),X T into another fuzzy topological space ( ),Y S is a somewhat fuzzy continuous function, then f is a somewhat fuzzy nearly continuous function.

Proof. Let l be a non-zero fuzzy open set in ( ),Y S such that ( )1 0f l- π . Since f is a somewhat fuzzy continuous function, there exists a non-zero fuzzy open set m of

( ),X T such that ( )1fm l-£ . Now ( ) ( )1 1clf fl l- -È ˘£ Î ˚ , implies that ( )1cl fm l-È ˘£ Î ˚ .

That is, ( )1int cl 0f l-È ˘ πÎ ˚ . Hence f is a somewhat fuzzy nearly continuous function.


Remarks. The implications contained in the following diagram are true.

Fuzzy continuity

Somewhat fuzzy continuity Fuzzy pre-continuity

Somewhat fuzzy nearly continuity

However, none of the above implications is reversed as is shown in the following examples:


[ ]: 0,1Xl Æ is defined as ( ) 1al = ; ( ) 0.2bl = ; ( ) 0.7cl = .

[ ]: 0,1Xm Æ is defined as ( ) 0.3am = ; ( ) 1bm = ; ( ) 0.2cm = .

[ ]: 0,1XJ Æ is defined as ( ) 0.7aJ = ; ( ) 0.4bJ = ; ( ) 1cJ = .

[ ]: 0,1Xa Æ is defined as ( ) 0.6aa = ; ( ) 0.5a b = ; ( ) 0.7a c = .


Let [ ] [ ]{0, , , , , , , , , , , ,T l m J l m l J m J l m l J m J l m J m l J= ⁄ ⁄ ⁄ Ÿ Ÿ Ÿ ⁄ Ÿ ⁄ Ÿ

[ ] },1J l mŸ ⁄ and { }0, , , , ,1S a b a b a b= ⁄ Ÿ . Then T and S are clearly fuzzy

topologies on X . Define a function ( ) ( ): , ,f X T X SÆ by ( )f a a= ; ( )f b b= ; ( )f c

c= . Then, for the non-zero fuzzy open sets a , b , [ ]a b⁄ , [ ]a bŸ and 1 in ( ),X S ,

we have ( ) ( ) ( )1 1 1int cl int cl int clf f fa b a b- - -È ˘ È ˘ È ˘= = ⁄ =Î ˚ Î ˚ Î ˚ ( )1int cl f a b J-È ˘Ÿ =Î ˚

[ ] 0l mŸ ⁄ π and ( )1int cl 1 0f -È ˘ πÎ ˚ respectively in ( ),X T . Hence f is a somewhat

fuzzy nearly continuous function from ( ),X T into ( ),X S . But f is not a fuzzy pre-

continuous function from ( ),X T into ( ),X S , since in ( ),X T , we have ( ) inta b⁄ £/

( ) [ ]1cl f a b J l m-È ˘⁄ = Ÿ ⁄Î ˚ . Also f is not a fuzzy continuous function from ( ),X T

into ( ),X S , since ( )1f Ta a-È ˘ = œÎ ˚ .




[ ]: 0,1XJ Æ is defined as ( ) 0.7aJ = ; ( ) 0.5bJ = ; ( ) 0.9cJ = .




[ ]: 0,1Xd Æ is defined as ( ) 0.6ad = ; ( ) 0.9bd = ; ( ) 0.7cd = .

Let [ ] [ ]{0, , , , , , , , , , , ,T l m J l m l J m J l m l J m J l m J m l J= ⁄ ⁄ ⁄ Ÿ Ÿ Ÿ ⁄ Ÿ ⁄ Ÿ

[ ] [ ] [ ] [ ] }, , , , , ,1J l m l m J m l J J l m l m J l m J⁄ Ÿ Ÿ ⁄ Ÿ ⁄ Ÿ ⁄ Ÿ Ÿ ⁄ ⁄ and {0, ,S a=

}, , , , , , , ,1b d a b a d b d a b a d b d⁄ ⁄ ⁄ Ÿ Ÿ Ÿ . Then T and S are clearly fuzzy topo-

logies on X . Define a function ( ) ( ): , ,f X T X SÆ by ( )f a a= ; ( )f b b= ; ( )f c c= .

Then, f is a somewhat fuzzy nearly continuous function from ( ),X T into ( ),X S . But

f is not a somewhat fuzzy continuous function from ( ),X T into ( ),X S . Since

( ) ( )1int int 0f a a-È ˘ = =Î ˚ , f is not a fuzzy continuous function.

Proposition 3.2. Let ( ),X T and ( ),Y S be any two fuzzy topological spaces. Let

( ) ( ): , ,f X T Y SÆ be a non-to one and onto function. Then the following are equivalent:

(1) f is somewhat fuzzy nearly continuous.

(2) If l is a fuzzy open and fuzzy dense set in ( ),X T , then ( )f l is a fuzzy dense

set in ( ),Y S .

(3) If l is a fuzzy nowhere dense in ( ),X T , then ( )int 0f lÈ ˘ =Î ˚ in ( ),Y S .

Proof. (1) fi (2) Let f be a somewhat fuzzy nearly continuous function from a fuzzy topological space ( ),X T into a fuzzy topological space ( ),Y S and suppose that

l is a fuzzy open and fuzzy dense set in ( ),X T . We claim that ( )cl 1f lÈ ˘ =Î ˚ . Assume

the contrary. Then ( )cl 1f lÈ ˘ πÎ ˚ . Then there exists a non zero fuzzy closed set d of

( ),Y S such that ( ) 1f l d< < . Then ( ) ( )1 1f f fl d- -< . Now ( )1f fl l-£ , implies

that l < ( )1f d- . Then [ ] ( )11 1 fl d-È ˘- > -Î ˚ and hence ( )11 1fl d-- > - . Since

( )1d π is a non-zero fuzzy closed set d of ( ),Y S , 1 d- is a non-zero fuzzy open set in

( ),Y S . By hypothesis, ( )1int cl 1 0f d-È ˘- πÎ ˚ . Then ( )1int cl 1 0f d-È ˘- πÎ ˚ , implies that

1 cl int- ( )1 0f d-È ˘ πÎ ˚ . That is, ( )1cl int 1f d-È ˘ πÎ ˚ . Now ( )1fl d-< , implies that

( )cl int l < ( )1cl int f d-È ˘Î ˚ . Then ( )cl int 1l π . Since l is a fuzzy open set in ( ),X T ,

( )int l l= and hence we have ( )cl 1l π , which is a contradiction to l being a fuzzy

dense set in ( ),X T . Hence we must have ( )cl 1f lÈ ˘ =Î ˚ .

(2)fi (3) Suppose that l is a fuzzy nowhere dense in ( ),X T .Then we have [ ]int cl l

0= . Now [ ]1 int cl 1 0 1l- = - = , implies that [ ]cl int 1 1l- = . Hence [ ]int 1 l- is a

fuzzy open and fuzzy dense set in ( ),X T . By hypothesis, [ ]( )int 1f l- is a fuzzy dense

set in ( ),Y S . That is, [ ]( ){ }cl f int 1 1l- = . Now [ ] [ ]int 1 1l l- £ - , implies that

[ ]( ) [ ]( )cl int 1 cl 1f fl l- £ - . Then we have [ ]( )1 cl 1f l£ - . That is, [ ]( )cl 1 1f l- = .


Since the function f is one to one and onto, [ ]( ) ( )1 1f fl l- = - . Then ( )cl 1 1f lÈ ˘- =Î ˚ .

This implies that ( )1 int 1f lÈ ˘- =Î ˚ . Therefore ( )int 0f lÈ ˘ =Î ˚ in ( ),Y S .

(3)fi (1). Suppose that l is a fuzzy open in ( ),Y S . We claim that ( )1int cl 0f l- π .

Assume the contrary. That is, ( )1int cl 0f l- = . This implies that ( )1f l- is a fuzzy

nowhere dense in ( ),X T . Then, by hypothesis, ( )( )1int 0f f l-È ˘ =Î ˚ in ( ),Y S . Since

the function f is one to one and onto, ( )( 1f f l l-È ˘ =Î ˚ . Then we have ( )int 0l = ,

which implies that 0l = (since l is fuzzy open in ( ),Y S , ( )int l l= ). But this is a

contradiction to l being a non-zero fuzzy (open) set in ( ),Y S . Hence we must have

( )1int cl 0f l- π .

Theorem 3.1 [10]. If a non-zero fuzzy set l in a fuzzy topological space ( ),X T , is a

fuzzy nowhere dense set, then l is a fuzzy semi-closed set in ( ),X T .

Proposition 3.3. Let ( ) ( ): , ,f X T Y SÆ be a one-to-one, fuzzy pre semi-open

function from a fuzzy topological space ( ),X T onto the fuzzy topological space ( ),Y S . Then the following are equivalent:

(1) f is somewhat fuzzy nearly continuous.

(2) If l is a fuzzy nowhere dense set in ( ),X T , then ( )f l is a fuzzy nowhere

dense set in ( ),Y S .

Proof. (1) fi (2) Assume (1) suppose that l is a fuzzy nowhere dense set in ( ),X T .

Then by Theorem 3.1, l is a fuzzy semi-closed set in ( ),X T . Hence 1 l- is a fuzzy

semi-open set in ( ),X T . Since the function f is a fuzzy pre semi-open function, ( )1f l-

is a fuzzy semi-open set in ( ),Y S . Then we have ( ) ( )1 clint 1f fl lÈ ˘- £ -Î ˚ . Since f

is one-to-one and onto, ( ) ( )1 1f fl l- = - . Then we have ( ) ( )1 clint 1f fl lÈ ˘- £ -Î ˚ .

This implies that ( ) ( )1 1 int clf fl lÈ ˘- £ - Î ˚ . Then we have ( ) ( )int cl f fl lÈ ˘ £Î ˚ . Now

( )( ) ( )int int cl intf fl lÈ ˘ È ˘£Î ˚ Î ˚ implies that ( )( ) ( )int cl intf fl lÈ ˘ È ˘£Î ˚ Î ˚ (since ( )int cl f lÈ ˘Î ˚

is fuzzy open in ( ),Y S , ( )( ) ( )int int cl int clf fl lÈ ˘ È ˘=Î ˚ Î ˚ . Then, we have ( ) )int cl f lÈ ˘Î ˚

( )int f lÈ ˘£ Î ˚ (A). Since f is a somewhat fuzzy nearly continuous function and l is a

fuzzy nowhere dense set in ( ),X T , by Proposition 3.2, ( )int 0f lÈ ˘ =Î ˚ in ( ),Y S .

Hence from (A), we have, ( ) )int cl 0f lÈ ˘ £Î ˚ . That is, ( ) )int cl 0f lÈ ˘ =Î ˚ in ( ),Y S .

Therefore ( )f l is a fuzzy nowhere dense set in ( ),Y S .

(2)fi (1) Assume (2). Suppose that l is a fuzzy open and fuzzy dense set in ( ),X T .

Then ( )int l l= and ( )cl 1l = ,implies that ( )cl int 1l = . Then we have ( )1 cl int 0l- = .


This implies that ( )int cl 1 0l- = . Hence 1 l- is a fuzzy nowhere dense set in ( ),X T .

By hypothesis, ( )1f l- is a fuzzy nowhere dense set in ( ),Y S . Then, ( )( )int cl 1f lÈ ˘-Î ˚

0= . Since f is one-to-one and onto, ( )( )int cl 1 0f l- = . This implies that 1 cl int-

( )( ) 0f l = . Since ( )( ) ( )( )int f fl l£ , ( )( )1 cl 1f l- £ - ( )( )cl int f l . Then, we have

( )( )1 cl 0f lÈ ˘- £Î ˚ . That is, ( )( )1 cl 0f lÈ ˘- =Î ˚ implies that ( )cl f 1lÈ ˘ =Î ˚ . Hence ( )f l

is a fuzzy dense set in ( ),Y S . Thus, for the fuzzy open and fuzzy dense set l in ( ),X T ,

we have ( ) 1cl f lÈ ˘ =Î ˚ . Therefore, by Proposition 3.2, f is a somewhat fuzzy nearly continuous function.

Proposition 3.4. Let ( ) ( ): , ,f X T Y SÆ b ea one-to-one, fuzzy pre semi-open and

somewhat fuzzy nearly continuous function from a fuzzy topological space ( ),X T onto

the fuzzy topological space ( ),Y S . If l is a fuzzy first category in ( ),X T , then ( )f l

is a fuzzy first category in ( ),Y S .

Proof. Let l be a fuzzy first category in ( ),X T . Then ( )1i il l•== � , where il ’s

are fuzzy nowhere dense sets in ( ),X T . Now ( ) ( )( ) ( )1 1i i i if f fl l l• •= == =� � . Since

f is a one-to-one, fuzzy pre semi-open and somewhat fuzzy nearly continuous function from ( ),X T onto ( ),Y S and il ’s are fuzzy nowhere dense sets in ( ),X T , by Proposi-

tion 3.3, ( )if l ’s are fuzzy nowhere dense sets in ( ),Y S . Hence ( ) ( )1i if fl l•== � ,

where ( )if l ’s are fuzzy nowhere dense sets in ( ),Y S , implies that ( )f l is a fuzzy first

category in ( ),Y S .

Proposition 3.5. If ( ) ( ): , ,f X T Y SÆ is a somewhat fuzzy nearly continuous

function from a fuzzy topological space ( ),X T into a fuzzy topological space ( ),Y S

and ( ) ( ): , ,g Y S Z WÆ is a fuzzy continuous function from ( ),Y S into a fuzzy

topological space ( ),Z W , then ( ) ( ): , ,g f X T Z WÆ is a somewhat fuzzy nearly

continuous function from ( ),X T into ( ),Z W .

Proof. Let l be a non-zero fuzzy open set in ( ),Z W . Since g is a fuzzy continuous

function from ( ),Y S into ( ),Z W , ( )1g l- is a fuzzy open set in ( ),Y S . Since f is a

somewhat fuzzy nearly continuous function from ( ),X T into ( ),Y S , there exists a non-

zero fuzzy open set m of ( ),X T such that ( )( )1 1clf gm l- -£ . That is, ( ) ( )1cl g fm l-£ .

Hence ( ) ( )1int cl 0g f l- π . Therefore g f is a somewhat fuzzy nearly continuous

function from ( ),X T into ( ),Z W .


Proposition 3.6. If ( ) ( ): , ,f X T Y SÆ is a function from a fuzzy topological space

( ),X T into a fuzzy topological space ( ),Y S and the graph function :g X X YÆ ¥ of f is somewhat fuzzy nearly continuous, then f is somewhat fuzzy nearly continuous.

Proof. Let l be a non-zero fuzzy open set in ( ),Y S . Then 1 l¥ is a fuzzy open set

in X Y¥ . Since g is somewhat fuzzy nearly continuous, we have ( )1intcl 1 0g l-È ˘¥ πÎ ˚ .

But ( ) ( ) ( )1 1 11 1f f gl l l- - -= Ÿ = ¥ . This implies that ( )1int cl 0f l-È ˘ πÎ ˚ . Hence f is

a somewhat fuzzy nearly continuous function from ( ),X T into ( ),Y S .

4. Somewhat fuzzy nearly open functions

Motivated by the classical concept introduced in [5] we shall now define:

Definition 4.1. Let ( ),X T and ( ),Y S be any two fuzzy topological spaces. A

function ( ) ( ): , ,f X T Y SÆ is called a somewhat fuzzy nearly open function if for all

Tl Œ and ( ) 0f l π , there exists a non-zero fuzzy open set m of ( ),Y S such that

( )cl fm lÈ ˘£ Î ˚ . That is, a function ( ) ( ): , ,f X T Y SÆ is a somewhat fuzzy nearly open

function if ( )int cl 0f l π , for every non-zero fuzzy open set l of ( ),X T .








( )f b b= ; ( )f c c= . Then, for the non-zero fuzzy open sets l , m , l m⁄ , l mŸ , and 1

in ( ),X T ,we have ( ) ( )int cl int 1f l bÈ ˘ = - =Î ˚ 0l π , ( ) ( )int cl int 1 0f m a mÈ ˘ = - = πÎ ˚ ,

( ) [ ]( ) [ ]int cl int 1f l m a b l mÈ ˘⁄ = - Ÿ = ⁄Î ˚ 0π , ( ) [ ]( )int cl int 1f l m a bÈ ˘Ÿ = - ⁄ =Î ˚

[ ] 0l mŸ π and ( )int cl 1 0fÈ ˘ πÎ ˚ respectively in ( ),X S . Hence f is a somewhat fuzzy

nearly open function from ( ),X T into ( ),X S .

Example 4.2. Let { },X a b= . The fuzzy sets l , m , a and b are defined on X as follows:

[ ]: 0,1Xl Æ is defined as ( ) 0.2al = ; ( ) 0.4bl = .


[ ]: 0,1Xm Æ is defined as ( ) 0.5am = ; ( ) 0.3bm = .

[ ]: 0,1Xa Æ is defined as ( ) 0.5aa = ; ( ) 0.7ba = .

[ ]: 0,1Xb Æ is defined as ( ) 0.8ab = ; ( ) 0.4bb = .



( )f b b= . Then, for the non-zero fuzzy open sets l , m in ( ),X T ,we have ( )int cl f lÈ ˘Î ˚

( )int 1 0b= - = and ( ) ( )int cl int 1 0f m aÈ ˘ = - =Î ˚ in ( ),X S . Hence f is not a some-

what fuzzy nearly open function from ( ),X T into ( ),X S .

Proposition 4.1. If a function ( ) ( ): , ,f X T Y SÆ from a fuzzy topological space

( ),X T into another fuzzy topological space ( ),Y S is a somewhat fuzzy open function, then f is a somewhat fuzzy nearly open function.

Proof. Let l be a non-zero fuzzy open set in ( ),X T such that ( ) 0f l π . Since f

is a somewhat fuzzy open function, there exists a non-zero fuzzy open set m of ( ),X T

such that ( )fm l£ . Now ( ) ( )clf fl lÈ ˘£ Î ˚ , implies that ( )cl fm lÈ ˘£ Î ˚ . That is,

( )int cl 0f lÈ ˘ πÎ ˚ . Hence f is a somewhat fuzzy nearly open function.

Remarks. The implications contained in the following diagram are true.

Fuzzy open

Somewhat fuzzy open Fuzzy pre-open

Somewhat fuzzy nearly open

However, none of the above implications is reversed as is shown in the following examples:









( )f b b= ; ( )f c c= . Then, for the non-zero fuzzy open sets l , m , l m⁄ , l mŸ and 1

in ( ),X T ,we have ( )int cl 0f lÈ ˘ πÎ ˚ , ( )int cl 0f mÈ ˘ πÎ ˚ , ( )int cl 0f l mÈ ˘⁄ πÎ ˚ , (int cl f lÈÎ

) 0m ˘Ÿ π˚ and ( )int cl 1 0fÈ ˘ πÎ ˚ respectively in ( ),X S . Hence f is a somewhat fuzzy

nearly open function from ( ),X T into ( ),X S . But f is not a somewhat fuzzy open

function, since ( )int 0f lÈ ˘ =Î ˚ in ( ),X S .

Also f is not a fuzzy open function, since ( )f Sl l= œ in ( ),X S .

Example 4.4. Let { }, ,X a b c= . The fuzzy sets a , b , J , l , m and d are defined on X as follows:



[ ]: 0,1XJ Æ is defined as ( ) 0.4aJ = ; ( ) 0.7bJ = ; ( ) 0.5cJ = .



[ ]: 0,1Xd Æ is defined as ( ) 0.3ad = ; ( ) 0.4bd = ; ( ) 0.6cd = .

Let [ ] [ ]{0, , , , , , , , , , , ,T a b J a b a J b J a b a J b J a b J J a b= ⁄ ⁄ Ÿ Ÿ Ÿ Ÿ ⁄ Ÿ ⁄ Ÿ

[ ] [ ] }, ,1b a J a b JŸ ⁄ ⁄ ⁄ and {0, , , , , , , , ,S l m d l m l d m d l m l d= ⁄ ⁄ ⁄ Ÿ Ÿ ,m dŸ

[ ] [ ] [ ] [ ] }, , , ,1m l d d l m l m d l m J⁄ Ÿ ⁄ Ÿ Ÿ ⁄ ⁄ ⁄ . Then T and S are clearly fuzzy

topologies on X . Define a function ( ) ( ): , ,f X T X SÆ by ( )f a a= ; ( )f b c= ; ( )f c

b= . Now, for the non-zero fuzzy open set a in ( ),X T , ( ) [ ]int cl int cl 1f a lÈ ˘ = - =Î ˚

[ ]int 1 l l d- = Ÿ and ( ) ( )int clf fa aÈ ˘£/ Î ˚ . Then ( )f a is not a fuzzy pre-open set in

( ),X S and hence f is not a fuzzy pre-open function from ( ),X T into ( ),X S and f is

a somewhat fuzzy nearly open function from ( ),X T into ( ),X S , since ( )int cl 0f h π ,

for every non-zero fuzzy open set h of ( ),X T .

Proposition 4.2. If ( ) ( ): , ,f X T Y SÆ is a fuzzy open function from a fuzzy topolo-

gical space ( ),X T into a fuzzy topological space ( ),Y S and ( ) ( ): , ,g Y S Z WÆ is a

somewhat fuzzy nearly open function from ( ),Y S into a fuzzy topological space ( ),Z W ,

then ( ) ( ): , ,g f X T Z WÆ is a somewhat fuzzy nearly open function from ( ),X T into

( ),Z W .

Proof. Let l be a non-zero fuzzy open set in ( ),X T . Since f is a fuzzy open

function from ( ),X T into ( ),Y S , ( )f l is a fuzzy open set in ( ),Y S . Since g is a

somewhat fuzzy nearly open function from ( ),Y S into ( ),Z W , there exists a non-zero

fuzzy open set m of ( ),Z W such that ( )( )clg fm l£ . Since ( )( ) ( )( )g f g fl l= , we


have ( )( )cl g fm l£ . Hence ( )( )int cl 0g f lÈ ˘ πÎ ˚ . Therefore g f is a somewhat

fuzzy nearly open function from ( ),X T into ( ),Z W .

Proposition 4.3. Let ( ) ( ): , ,f X T Y SÆ be a function from a fuzzy topological

space ( ),X T onto a fuzzy topological space ( ),Y S and ( ) ( ): , ,g Y S Z WÆ be a

function from ( ),Y S into a fuzzy topological space ( ),Z W . If f is a fuzzy continuous

function and g f is a somewhat fuzzy nearly open function from ( ),X T into ( ),Z W ,

then g is a somewhat fuzzy nearly open function from ( ),Y S into ( ),Z W .

Proof. Let l be a non-zero fuzzy open set in ( ),Y S . Since f is a fuzzy continuous

function from ( ),X T into ( ),Y S , ( )1f l- is a fuzzy open set in ( ),X T . Since g f is

a somewhat fuzzy nearly open function, ( ) ( )( )1int cl 0g f f l- π . Then, [ ( 1int cl g f f -ÈÎ

( )) 0l ˘˘ π˚˚ . Since f is onto, ( )( )1f l l- = . Then we have ( )int cl 0g lÈ ˘ πÎ ˚ . Hence g

is a somewhat fuzzy nearly open function from ( ),Y S into ( ),Z W .

Proposition 4.4. Let ( ),X T and ( ),Y S be any two fuzzy topological spaces. Let

( ) ( ): , ,f X T Y SÆ be a function. Then the following conditions are equivalent:

(1) f is somewhat fuzzy nearly open.

(2) If l is a fuzzy open and fuzzy dense set in ( ),Y S , then ( )1f l- is a fuzzy dense

set in ( ),X T .

(3) If l is a fuzzy nowhere dense in ( ),Y S , then ( )1int 0f l-È ˘ =Î ˚ in ( ),X T .

Proof. (1) fi (2) Assume (1). Suppose that l is a fuzzy open and fuzzy dense set in ( ),Y S . We claim that ( )1f l- is a fuzzy dense set in ( ),X T . Assume the contrary. Then

( )1cl 1f l-È ˘ πÎ ˚ . Then ( )11 cl 0f l-È ˘- πÎ ˚ . This implies that ( )( )1int 1 0f l-È ˘- πÎ ˚ . Then

( )( )1int 1 0f l- - π . Since f is a somewhat fuzzy nearly open function and ( )( )1int 1f l- -

is a fuzzy open set in ( ),X T , ( )( ){ }1int cl int 1 0f f l-È ˘- πÎ ˚ . Now ( )( )1int 1f l- - £

( )1 1f l- - implies that ( )( ){ }1int cl int 1f f l-È ˘- £Î ˚ ( )( )1int cl 1f f l-È ˘-Î ˚ . Then, int cl

( ( ))1 1 0f f l-È ˘- π˚Î . Now ( )( ) ( )1 1 1f f l l-È ˘- £ -Î ˚ implies that ( )int cl 1 0lÈ ˘- πÎ ˚ (A).

Since 1 l- is a fuzzy closed set, from (A), we have ( )int 1 0l- π , which implies that

( )1 cl l- 0π . That is, ( )cl 1l π , a contradiction to l being a fuzzy dense set in ( ),Y S .

Hence we must have ( )1cl 1f l-È ˘ =Î ˚ and therefore that ( )1f l- is a fuzzy dense set in

( ),X T .


(2) fi (3) Assume (2). Suppose that l is a fuzzy nowhere dense in ( ),Y S . Then

[ ]int cl 0l = . Now ( )1 int cl 1 0 1l- = - = , implies that [ ]cl int 1 1l- = . Hence [ ]int 1 l-

is a fuzzy open and fuzzy dense set in ( ),Y S . By hypothesis, [ ]( )1 int 1f l- - is a fuzzy

dense set in ( ),Y S . That is, [ ]( ){ }1cl int 1 1f l- - = . Now [ ] [ ]int 1 1l l- £ - , implies

that [ ]( ) [ ]( )1 1cl int 1 cl 1f fl l- -- £ - . Then we have [ ]( )11 cl 1f l-£ - . That is, [(1cl 1f -

]) 1l- = , which implies that ( )11 int 1f l-È ˘- =Î ˚ . Therefore ( )1int 0f l-È ˘ =Î ˚ in ( ),X T .

(3)fi (1). Assume (3). Suppose that l is a non-zero fuzzy open in ( ),X T . We claim

that ( )int cl 0f lÈ ˘ πÎ ˚ . Assume the contrary. Suppose that, ( )int cl 0f lÈ ˘ =Î ˚ .This implies

that ( )f l is a fuzzy nowhere dense in ( ),Y S . Then, by hypothesis, ( )( )1int 0f f l- È ˘ =Î ˚ .

But ( )1f f l l- ≥ implies that ( ) [ ]1int intf f l l-È ˘ ≥Î ˚ in ( ),X T . Then [ ]0 int l≥ . That

is, ( )int 0l = , which implies that 0l = (since l is fuzzy open in ( ),X T , ( )int l l= ).

But this is a contradiction to l being a non-zero fuzzy (open) set in ( ),X T . Hence we

must have ( )int cl 0f lÈ ˘ πÎ ˚ . Therefore f is a somewhat fuzzy nearly open function

from a fuzzy topological space ( ),X T into the fuzzy topological space ( ),Y S .

Proposition 4.5. Let ( ) ( ): , ,f X T Y SÆ be a fuzzy irresolute function from a fuzzy

topological space ( ),X T into a fuzzy topological space ( ),Y S . Then, the following are equivalent:

(1) f is somewhat fuzzy nearly open.

(2) If l is a fuzzy nowhere dense set in ( ),Y S , then ( )1f l- is a fuzzy nowhere

dense set in ( ),X T .

Proof. (1)fi (2) Assume (1). Suppose that l is a fuzzy nowhere dense set in ( ),Y S .

Then by Theorem 3.1, l is a fuzzy semi-closed set in ( ),Y S . Hence 1 l- is a fuzzy

semi-open set in ( ),Y S . Since the function f is a fuzzy irresolute function, ( )1 1f l- -

is a fuzzy semi-open set in ( ),X T . Then we have ( ) ( )1 11 cl int 1f fl l- -È ˘- £ -Î ˚ . Then

we have ( ) ( )1 11 clint 1f fl l- -È ˘- £ -Î ˚ , which implies that ( ) ( )1 11 1 int clf fl l- -È ˘- £ - Î ˚ .

Then, ( ) ( )1 1int cl f fl l- -È ˘ £Î ˚ . Now ( )( ) ( )1 1int int cl intf fl l- -È ˘ È ˘£Î ˚ Î ˚ implies that

( ) ( )1 1int cl intf fl l- -È ˘ È ˘£Î ˚ Î ˚ (since ( )1int cl f l-È ˘Î ˚ is fuzzy open in ( ),X T , (int int cl .

( ) ) ( )1 1int clf fl l- -È ˘ È ˘=Î ˚ Î ˚ Then, we have ( ) ( )1 1int cl intf fl l- -È ˘ È ˘£Î ˚ Î ˚ (B). Since f

is a somewhat fuzzy nearly open function and l is a fuzzy nowhere dense set in ( ),Y S ,

by Proposition 4.4, ( )1int 0f l-È ˘ =Î ˚ in ( ),X T . Hence from (B), we have,


( )1int cl 0f l-È ˘ £Î ˚ . That is, ( )1int cl 0f l-È ˘ =Î ˚ in ( ),X T . Therefore ( )1f l- is a fuzzy

nowhere dense set in ( ),X T .

(2)fi (1). Assume (2). Suppose that l is a fuzzy open and fuzzy dense set in ( ),Y S .

Then ( )int l l= and ( )cl 1l = , implies that ( )cl int 1l = . Then we have ( )1 cl int l- =

0 . This implies that ( )int cl 1 0l- = . Hence1 l- is a fuzzy nowhere dense set in ( ),Y S .

By hypothesis, ( )1 1f l- - is a fuzzy nowhere dense set in ( ),X T . Then, ( (1int cl 1f - ÈÎ

) ) 0l ˘- =˚ , implies that ( )( )11 cl int 0f l-- = . Since ( )( ) ( )( )1 1int f fl l- -£ , ( 11 cl f --

( )) ( )( )11 cl int fl l-È ˘£ - Î ˚ . Then, we have ( )( )11 cl 0f l-È ˘- £Î ˚ . That is, ( )( )11 cl f l-È ˘- Î ˚

0= implies that ( )1cl 1f l-È ˘ =Î ˚ . Hence ( )1f l- is a fuzzy dense set in ( ),X T . Thus,

for the fuzzy open and fuzzy dense set l in ( ),Y S , we have ( )1cl 1f l-È ˘ =Î ˚ in ( ),X T .

Therefore, by Proposition 4.3, f is a somewhat fuzzy nearly open function from ( ),X T

into ( ),Y S .

Proposition 4.6. Let ( ) ( ): , ,f X T Y SÆ be a fuzzy irresolute and somewhat fuzzy

nearly open function from a fuzzy topological space ( ),X T into the fuzzy topological

space ( ),Y S . If l is a fuzzy first category in ( ),Y S , then ( )1f l- is a fuzzy first

category in ( ),X T .

Proof. Let l be a fuzzy first category in ( ),Y S . Then ( )1i il l•== � , where il ’s

are fuzzy nowhere dense sets in ( ),Y S . Now ( ) ( )( ) ( )1 1 11 1i i i if f fl l l- - • • -= == =� � .

Since f is a fuzzy irresolute function and somewhat fuzzy nearly open from ( ),X T into

( ),Y S and il ’s are fuzzy nowhere dense sets in ( ),Y S , by Proposition 4.4, ( )1if l- ’s

are fuzzy nowhere dense sets in ( ),X T . Hence we have ( ) ( )1 11i if fl l- • -== � , where

( )1if l- ’s are fuzzy nowhere dense sets in ( ),Y S . Hence ( )1f l- is a fuzzy first

category in ( ),X T .

5. Baire spaces and functions

Definition 5.1 [10]. Let ( ),X T be a fuzzy topological space. Then ( ),X T is called

a fuzzy Baire space if ( )( )1int 0i il•= =� , where il ’s are fuzzy nowhere dense sets in

( ),X T .

Theorem 5.1 [10]. Let ( ),X T be a fuzzy topological space. Then the following are equivalent:


(1) ( ),X T is a fuzzy baire space.

(2) ( )Int 0l = for every fuzzy first category set l in ( ),X T .

(3) ( )cl 1m = for every fuzzy residual set m in ( ),X T .

Proposition 5.1. Let ( ) ( ): , ,f X T Y SÆ be a fuzzy irresolute and somewhat fuzzy

nearly open function from a fuzzy topological space ( ),X T into a fuzzy topological

space ( ),Y S . If ( ),X T is a fuzzy Baire space, then ( ),Y S is a fuzzy Baire space.

Proof. Let l be a fuzzy first category in ( ),Y S . Now ( )int l is a fuzzy open set in

( ),Y S . Since every fuzzy open set is fuzzy semi-open in a fuzzy topological space,

( )int l is a fuzzy semi-open set in ( ),Y S . Since f is a fuzzy irresolute function from

( ),X T into ( ),Y S , ( )( )1 intf l- is a fuzzy semi-open set in ( ),X T . Then, ( )( )1 intf l-

( )( )1cl int intf l-£ . Then, ( )( ) ( )1 1int cl intf fl l- -È ˘£ Î ˚ (A). Since l is a fuzzy first

category in ( ),Y S , by Proposition 4.4, ( )1f l- is a fuzzy first category in ( ),X T . Also

since ( ),X T is a fuzzy Baire space, by Theorem 5.1, ( )1int 0f l-È ˘ =Î ˚ . Then, 1cl int f -ÈÎ

( ) 0l ˘ =˚ . Hence from (A), we have ( )( )1 int 0f l- = . This implies that ( )int 0l = .

Therefore by Theorem 5.1, ( ),Y S is a fuzzy Baire Space.

Proposition 5.2. Let ( ) ( ): , ,f X T Y SÆ be a one-to-one, fuzzy pre semi-open and

somewhat fuzzy nearly continuous function from a fuzzy topological space ( ),X T onto

the fuzzy topological space ( ),Y S . If ( ),Y S is a fuzzy Baire space, then ( ),X T is a fuzzy Baire space.

Proof. Let l be a fuzzy first category in ( ),X T . Now ( )int l is a fuzzy open set in

( ),X T . Since every fuzzy open set is fuzzy semi-open in a fuzzy topological space, int

( )l is a fuzzy semi-open set in ( ),X T . Since f is a fuzzy pre semi-open function from

( ),X T onto ( ),Y S , ( )intf lÈ ˘Î ˚ is a fuzzy semi-open set in ( ),Y S . Then, ( )intf lÈ ˘ £Î ˚

( )( )cl int intf lÈ ˘Î ˚ . Then, ( ) ( )( )int cl intf fl lÈ ˘ È ˘£Î ˚ Î ˚ (A). Since l is a fuzzy first

category in ( ),X T , by Proposition 3.4, ( )f l is a fuzzy first category in ( ),Y S . Also

since ( ),Y S is a fuzzy Baire space, by Theorem 5.1, ( )int 0f lÈ ˘ =Î ˚ . Then, ( )cl int f lÈ ˘Î ˚

( )cl 0 0= = . Hence from (A), we have ( )int 0f lÈ ˘ =Î ˚ . This implies that ( )int 0l = .

Therefore by Theorem 5.1, ( ),Y S is a fuzzy Baire space.

References


[1] K. K. Azad, On fuzzy semi continuity, Fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl., 82 (1981), 14-32.

[2] A. S. Bin Shahna, On fuzzy strong semi continuity and fuzzy precontinuity, Fuzzy Sets and Systems, 44 (1991), 303-308.

[3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [4] David H Foster, Fuzzy Topological groups, J. Math. Anal. Appl., 67 (1979), 549-564. [5] D. S. Jankovic and C. Konstadilaki-savvopoulou, On a -continuous functions, Math. Bohem., 117

(1992), 259-270. [6] Karl. R. Gentry and Hughes, B. Hoyle III, Somewhat continuous functions, Czech. Math. Jour., 21

(96) (1971), 5-12. [7] M. N. Mukherjee and S. P. Sinha, Irresoute and almost open functions between fuzzy topological

spaces, Fuzzy Sets and Systems, 29 (1989), 381-388. [8] Z. Piotrowski, A survey of results concerning generalized continuity on topological spaces, Acta.

Math. Univ. Comenianae, 52-53, (1987), 91-109. [9] G. Thangaraj and G. Balasubramanian, On somewhat fuzzy continuous functions, J. Fuzzy Math.,

Vol.11, No.2 (2003), 725-736. [10] G. Thangaraj and S. Anjalmose, On fuzzy baire spaces, J. Fuzzy Math., Vol.21, No.3, (2013) 667-676. [11] T. H. Yalvac, Semi-interior and semi-closure of a fuzzy set, J. Math. Anal. Appl., 132 (1988), 356-364. [12] L. A. Zadeh, Fuzzy sets, Information and Control, Vol.8, (1965), 338-358.


__________________________ Received December, 2013


Los Angeles

Some Fixed Point Results in Fuzzy Metric Spaces under Common Limit Range Property

Sunny Chauhan

Near Nehru Training Centre H. No.274, Nai Basti B-14, Bijnor-246701, Uttar Pradesh, India


Mujahid Abbas

Dept. of Mathematics and Applied Mathematics, University of Preteria, Lynvvood Road, Pradesh, InPretoria, 002, South Africa


Ishak Altun

Dept. of Mathematics, Fuculty of Science and Arts, Kirikkale University 71450 Yahsihan, Kirikkale, Turkey.

E-mail:[email protected]

Jelena Vujakovic

University of Pristina, Faculty of Sciences and Mathematics Lole Ribara 29 38 200, Kosovska Mitrovica, Serbia


Abstract: In this paper, some new common fixed point theorems under certain strict contractive

conditions for mappings sharing the ( )STCLR property in fuzzy metric spaces are proved. Examples in support of our results are also given. A fixed point theorem for four finite families of self mappings in fuzzy metric space is also obtained. Our results improve and extend the results of Sedghi and Shobe [Common fixed point theorems under strict contractive conditions in fuzzy metric spaces using property (E.A). Commun. Korean Math. Soc. 27 (2), 399-410 (2012)].

Key words and phrases: Fuzzy metric space; weakly compatible mappings; property (E. A); common property

(E. A); common limit range property; fixed point.

1. Introduction

118 Sunny Chauhan, Mujahid Abbas, Ishak Altun and Jelena Vujakovic

Probability theory provides some tools to measure uncertainty in our real world of activities. In fact, the more general (i.e., not necessarily probabilistic in nature) concept of “uncertainty” is considered a basic ingredient of some basic mathematical structures. The concept of fuzzy sets [1] constitutes an example, where the concept of uncertainty was introduced in the theory of sets, in a non probabilistic manner. Fuzzy set theory has applications in applied sciences such as mathematical programming, modeling theory, engineering sciences, image processing, control theory, communication etc. In 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric space as a generalization of the statistical (probabilistic) metric space. In fact the study of such spaces received an impetus with the pioneering work of Heilpern [3]. Afterwards, Grabiec [4] defined the completeness of the fuzzy metric space and extended the Banach contraction principle to fuzzy metric spaces. Further, Fang [5] established some new fixed point theorems for contractive type mappings in fuzzy metric spaces. Soon after, Mishra et al. [6] also proved several fixed point theorems for asymptotically commuting mappings in such spaces. Since then, the notion of a complete fuzzy metric space presented by George and Veeramani [7] which constitutes a modification of the one due to Kramosil and Michalek [2]. Many authors have contributed to the development of this theory and apply to fixed point theory, for instance [8,9,10,11,12].

Mishra et al. [6] extended the notion of compatible mappings (introduced by Jungck [13] in metric space) to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space and containment of the ranges amongst involved mappings. Further, Singh and Jain [14] (introduced by Jungck and Rhoades [15] in metric space) weakened the notion of compatibility by using the notion of weakly compatible mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible but converse is not true. However, the study of common fixed points of non-compatible mappings (introduced by Pant [16] in metric space) in fuzzy metric space is also of great interest due to Pant and Pant [17]. In 2009, Abbas et al. [18] proved a common fixed point theorem for two pairs of weakly compatible mappings in fuzzy metric space by using the property (E. A) (introduced by Aamri and El-Moutawakil [19] in metric space). They also utilized the notion of common property (E. A) (introduced by Liu et al. [20] in metric space) to prove the same result. It is observed that the notion of common property (E. A) relatively relaxes the required containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates (see [21]). In recent past, several authors proved various fixed point theorems employing relatively more general contractive conditions (e.g., [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]).

Recently, Sintunavarat and Kumam [44] coined the idea of “common limit range property” in fuzzy metric spaces. They showed that common limit range property never requires the closedness of the subspace (also see [45]) for the existence of fixed point. Most recently, Sedghi and Shobe [46] proved a common fixed point theorem for weakly compatible mappings satisfying strict contractive conditions in fuzzy metric spaces by using the property (E. A).

The aim of this paper is to prove a common fixed point theorem for two pairs of weakly compatible mappings in fuzzy metric space by using common limit range property. Some examples are proved to validate our results. We extend our main result to four finite families of mappings by using the notion of pairwise commuting property of two finite families of mappings due to Imdad et al. [47].

Some Fixed Point Results in Fuzzy Metric Spaces under Common Limit Range Property 119

2. Preliminaries

Definition 2.1 [1]. Let X be any set. A fuzzy set A in X is a function with domain X and values in [ ]0,1 .

Let F be the collection of all fuzzy sets on [ )2 0,X ¥ • , that is, { [ )2: 0,f X= ¥ •F

[ ]}0,1Æ .

Definition 2.2 [46]. Let ,f g ŒF . The algebraic sum f g≈ of f and g is defined by

( ) ( ) ( ) ( ){ }1 2

1 2, , , , sup min , , , , ,t t t

f x y t g x y t f x y t g x y t+ =

≈ =¢ ¢

Remark 2.1 [46]. For every ,x y XŒ and every 0t > , we have

(1) ( ) ( ) ( ) ( ){ } ( ), , 2 , , 2 min , , , , , , ,f x y t f x y t f x y t f x y t f x y t≈ ≥ = .

(2) ( ) ( ) ( ){ } ( ), , 1 min , , , , , , ,f x y t f x y t f x x f x y te e e≈ ≥ - = - .

Letting 0e Æ , we get ( ) ( ), , 1 , ,f x y t f x y t≈ ≥ . Throughout this paper F denotes a family of mappings such that for each f ŒF ,

[ ] [ ]3: 0,1 0,1f Æ is containous and increasing in each co-ordinate variable. Also

( ) ( ), ,t t t t tg f= > for every [ ]0,1tŒ .

Example 2.1 [46]. Let [ ] [ ]3: 0,1 0,1f Æ be defined by

( ) { }, , min , ,x y z x y zf = .

Definition 2.3 [48]. A mapping [ ] [ ] [ ]: 0,1 0,1 0,1* ¥ Æ is called a continuous t -norm

if [ ]( )0,1 ,* is an abelian topological monoid with unit 1 such that a b c d* £ * , for a c£ , b d£ .

Three typical examples of t -norms are { }min ,a b a b* = (minimum t -norm),

a b ab* = (product t -norm), and { }max 1,0a b a b* = + - (Lukasiewicz t -norm).

Definition 2.4 [2]. The 3-tuple ( ), ,X M * is called a fuzzy metric space if X is an

arbitrary set, * is a continuous t -norm and M is a fuzzy set in [ )2 0,X ¥ • satisfying the following conditions:

(a) ( ), , 0M x y t > ,

(b) ( ), , 1M x y t = for all 0t > if and only if x y= ,

(c) ( ) ( ), , , ,M x y t M y x t= ,

(d) ( ) ( ) ( ), , , , , ,M x y t M y z s M x z t s* £ + ,

(e) ( ) ( ) [ ], , : 0, 0,1M x y ◊ • Æ is continuous, for each , ,x y z XŒ and , 0t s > .


Definition 2.5. Then M is called a fuzzy metric on X . Note that, ( ), ,M x y t can be thought of as the definition of nearness between x and y with respect to t . It is known that ( ), ,M x y ◊ is nondecreasing for all ,x y XŒ [4]. Throughout this work, X will represent a fuzzy metric space equipped with a continuous t -norm and a fuzzy set M until or unless stated otherwise.

From the following example, it is assured that every metric induces a fuzzy metric.

Example 2.2 [7]. Let ( ),X d be a metric space. Denote a b ab* = (or { }min ,a b a b* = )

for all [ ], 0,1a bŒ and let dM be fuzzy sets on ( )2 0,X ¥ • defined as follows: ( ,dM x

) ( )( ), ,y t t t d x y= + . Then ( ), ,dX M * is a fuzzy metric space and the fuzzy metric M induced by the metric d is often referred to as the standard fuzzy metric.

Definition 2.6 [7]. A sequence { }nx in X is said to be: (1) convergent to some x in X if for each 0e > and 0t > , there exists 0n Œ such

that ( ), , 1nM x x t e> - for all 0n n≥ ; i.e., ( ), , 1nM x x t Æ as nÆ• for all 0t > . (2) Cauchy if for each 0e > and 0t > , there exists 0n Œ such that

( ), , 1n mM x x t e> - for all 0,n m n≥ ; i.e., ( ), , 1n mM x x t Æ as ,n mÆ• for all 0t > . A fuzzy metric space X in which every Cauchy sequence is convergent is said to be

complete.

Definition 2.7 [4]. A fuzzy set M is said to be continuous on ( )2 0,X ¥ • if

( ) ( )lim , , , ,n n nn

M x y t M x y tÆ•

=

whenever ( ), ,n n nx y t is a sequence in ( )2 0,X ¥ • converges to a point

( ) ( )2, , 0,x y t XŒ ¥ • ; i.e.,

( ) ( )lim , , lim , , 1n nn n

M x x t M y y tÆ• Æ•

= = , and ( ) ( )lim , , , ,nn

M x y t M x y tÆ•

= .

Definition 2.8 [18]. Two self mappings A and S on X are said to be: (i) compatible if and only if ( ), , 1n nM ASx SAx t Æ for all 0t > , whenever { }nx

is a sequence in X such that ,n nAx Sx zÆ for some z XŒ as nÆ• [6]. (ii) non-compatible, if there exists a sequence { }nx inX such that limn nAxÆ• =

limn nSx zÆ• = for some z XŒ , but for some 0t > , ( )lim , ,n n nM ASx SAx tÆ• is either less than 1 or nonexistent.

(iii) weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Az Bz= some z XŒ , then ABz BAz= [15].

It is known that a pair ( ),A S of compatible mappings is weakly compatible but converse is not true in general.


Definition 2.9. A pair of self mappings ( ),A S defined on a fuzzy metric space X is said to satisfy:

(iv) the property (E. A) if there exists a sequence { }nx in X such that limn nAxÆ• = limn nSx zÆ• = , for some z XŒ [18].

(v) the common limit range property with respect to mapping S (briefly, ( )SCLR

property) if there exists a sequence { }nx in X such that lim limn nn n

Ax Sx zÆ• Æ•

= = , where

( )z S XŒ [44]. It is straight forward to notice that every pair of non-compatible self mappings of a

fuzzy metric space X satisfies the property (E. A) but not conversely (see [5, Example 1]). Here, it can be pointed out that weakly compatibility and the property (E. A) are independent to each other (see [49, Example 2.2]). It is evident that a pair ( ),A S

satisfying the property (E. A) along with closedness of the subspace ( )S X always

enjoys the ( )SCLR property.

With a view to extend the ( )SCLR property to two pair of self mappings, we defined

the ( )STCLR property with respect to mappings S and T in the following definition.

Definition 2.10. Two pairs of self mappings ( ),A S and ( ),B T of a fuzzy metric space X are said to satisfy

(vi) the common property (E. A), if there exist two sequences { }nx , { }ny in X for some z in X such that lim lim lim limn n n n n n n nAx Sx By Ty zÆ• Æ• Æ• Æ•= = = = [18].

(vii) The common limit range property with respect to mappings S and T (briefly, ( )STCLR property) if there exist two sequences { }nx , { }ny in X such that

lim lim lim limn n n nn n n n

Ax Sx By Ty zÆ• Æ• Æ• Æ•

= = = = ,

where ( ) ( )z S X T XŒ « .

Example 2.3. Let ( ), ,X M * be a fuzzy metric space, where [ ]3, 20X = , with t -

norm * is defined by { }min ,a b a b* = for all [ ], 0,1a b Œ and ( ), ,M x y t =

( ) , if 0;0, if 0,t t x y t

t

Ï + - >ÔÌ

=ÔÓ for all ,x y XŒ and 0t > . Define the self mappings A , B ,

S and T by ( )( )

7, if 3;5, if 3 14;

1 5, if 14.

x

A x x

x x

Ï =Ô= < £ÌÔ + >Ó

( ) ( )4, if 3;4 3 5, if 3 14;

13, if 14.

x

B x x x

x

=ÏÔ= + < £ÌÔ >Ó

( )( )

5, if 3;15, if 3 14;2 1 9, if 14.

x

S x x

x x

Ï =Ô= < £ÌÔ - >Ó

( ) ( )6, if 3;

3 2, if 3 14;17, if 14.

x

T x x x

x

=ÏÔ= + < £ÌÔ >Ó

If we choose two

sequence as { } { }14 1n nx n Œ= + and { } { }3 1n n

y n Œ= + , then the pairs ( ),A S and


( ),B T enjoy the common property (E. A) for all 0t > : lim lim limn n nn n n

Ax Sx ByÆ• Æ• Æ•

= =

lim 3nn

Ty XÆ•

= = Œ . Here it is noticed that ( ) ( )3 S X T Xœ « . Therefore, the pairs

( ),A S and ( ),B T do not satisfy the common limit range property with respect to mappings S and T .

In view of Example 2.3, the following proposition is predictable.

Proposition 2.1. If the pairs ( ),A S and ( ),B T enjoy the common property (E. A)

and ( )S X as well as ( )T X are closed subsets of X , then the pairs also enjoy the

( )STCLR property. The following definition is essentially contained in Imdad et al. [47].

Definition 2.11. Two families of self mappings { } 1m

i iA = and { } 1

n

k kS = are said to be

pairwise commuting if (1) i j j iA A A A= for all { }, 1, 2, ,i j mŒ ,

(2) k l l kS S S S= for all { }, 1, 2, ,k l nŒ ,

(3) i k k iA S S A= for all { }1, 2, ,i mŒ and { }1,2, ,k nŒ .

3. Results

In this section, it is assumed that M is a continuous fuzzy metric on ( )2 0,X ¥ • such

that ( )lim , , 1tM x y t

Æ•= , for all ,x y XŒ .

Sedghi and Shobe [46] proved the following fixed point theorem for two pairs of weakly compatible mappings satisfying strict contractive condition in fuzzy metric spaces using (E. A) property.

Theorem 3.1 [46, Theorem 2.4]. Let A , B , S and T be self mappings on a fuzzy metric space X satisfying the following conditions:

(1) ( ) ( )A X T XÕ , ( ) ( )B X S XÕ , (2)

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4M Sx Ty t k M Ax Sx t k M By Ty t k

M Ax By tM Ax Ty t k M Sx By t k

fÊ ˆ≈

≥ Á ˜≈Ë ¯, (3.1)

for all ,x y XŒ , 0t > , f ŒF and 0 2k£ < . Suppose that one of the pairs ( ),A S and

( ),B T satisfies the (E. A) property, ( ),A S and ( ),B T are weakly compatible and one

of ( )A X , ( )B X , ( )S X and ( )T X is a complete subspace of X . Then A , B , S and T have a unique common fixed point in X .

Lemma 3.1. Let A ,B ,S and T be self mappings of a fuzzy metric space X satisfy-ing inequality (3.1) of Theorem 3.1. Suppose that


(1) The pair ( ),A S satisfies the ( )SCLR property (or the pair ( ),B T satisfies the

( )TCLR property),

(2) ( ) ( )A X T XÃ (or ( ) ( )B X S XÃ ),

(3) ( )T X (or ( )S X ) is a closed subset of X ,

(4) ( )nB y converges for every sequence { }ny in X whenever ( )nT y converges

(or ( )nA x converges for every sequence { }nx in X whenever ( )nS x converges).

Then the pairs ( ),A S and ( ),B T share the ( )STCLR property.

Proof. If pair ( ),A S satisfies the ( )SCLR property, then there exists a sequence

{ }nx in X such that lim limn n

n nAx Sx z

Æ• Æ•= = , (3.2)

where ( )z S XŒ . As ( ) ( )A X T XÃ , for a sequence { }nx XÃ there corresponds a

sequence { }ny XÃ such that n nAx Ty= . Therefore,

lim limn nn n

Ty Ax zÆ• Æ•

= = , (3.3)

where ( ) ( )z S X T XŒ « . Thus, we have nAx zÆ , nSx zÆ and nTy zÆ . Now we claim that nBy zÆ . Using (3.1) with nx x= , ny y= , we get

( ) ( ) ( ) ( )( ) ( )

, , 2 , , ,2 , ,2 ,, ,

, , 4 , ,4n n n n n n

n n

n n n n

M Sx Ty t k M Ax Sx t k M By Ty t kM Ax By t

M Ax Ty t k M Sx By t kfÊ ˆ≈

≥ Á ˜≈Ë ¯ (3.4)

Since ( ) ( ){ }lim inf , , 2 , , 2n n n nn

M Ax Sx t k M By Ty t kÆ•

≈ ({lim inf min , ,n nn

M Ax SxÆ•

≥

) ( )}, , , 2n nM By Ty t ke e- ( )lim inf , , 2n nn

M By Ty t k eÆ•

= - . Letting 0e Æ , in above

inequality, we obtain ( ) ( ){ } (liminf , , 2 , , 2 liminf ,n n n n nn n

M Ax Sx t k M By Ty t k M ByÆ• Æ•

≈ ≥

), 2nTy t k . Also, ( ) ( ){ }lim inf , , 4 , , 4n n n nn

M Ax Ty t k M Sx By t kÆ•

≈ lim inf minnÆ•

≥

( ) ( ){ }, , , , , 4n n n nM Ax Ty M Sx By t ke e- ( )lim inf , , 4n nn

M Sx By t k eÆ•

= - Letting

0e Æ , we get ( ) ( ){ }lim inf , , 4 , , 4n n n nn

M Ax Ty t k M Sx By t kÆ•

≈ (lim inf ,nn

M S xÆ•

≥

), 4nBy t k ( )lim inf , ,3n nn

M Sx By t kÆ•

≥ passing to limit as nÆ• in inequality (3.4),

we obtain ( )lim inf , ,n nn

M Ax By tÆ•

≥

)( ))( ) )( ){ })( ) )( ){ }

lim inf , , , 2 ,

lim inf min , , , 2 , , , , 2 ,

lim inf min , , , 4 , , , , 4

n n n n

n n n n n n n

n n n n n n n

M Sx Ty By t k

M Ax Sx By t k M By Ty By t k

M Ax Ty By t k M Sx By By t k

fÆ•

Æ•

Æ•

Ê ˆÁ ˜Á ˜Á ˜Á ˜Ë ¯


( ) ( )( )

lim inf , ,2 , lim inf , ,2 ,lim inf , ,2

n n n n n n

n n n

M Sx Ty t k M By Ty t k

M Sx By t kf Æ• Æ•

Æ•

Ê ˆ≥ Á ˜Ë ¯

,

and so

( ) ( ) ( )( )

, , 2 , lim inf , , 2 ,, lim inf ,

, lim inf ,2n n

nn

n n

M z z t k M By z t kM z By t

M z By t kf Æ•

Æ•Æ•

Ê ˆ≥ Á ˜Ë ¯

( )( )( )

lim inf , , 2 ,lim inf , , 2 ,

, lim inf ,2

n n

n n

n n

M By z t k

M By z t k

M z By t k

fÆ•

Æ•

Æ•

Ê ˆÁ ˜≥ Á ˜Á ˜Ë ¯

( ), lim inf , 2nn

M z By t kÆ•

≥

( )( ), lim inf , 2 1n

nn

M z By t k tÆ•

≥ Æ .

Similarly, ( ) ( )lim sup , , , lim sup ,n nn n

M z By t M z By tÆ• Æ•

= . Hence, ( )lim , , 1n nM z By tÆ• = .

It implies nBy zÆ asnÆ• . Therefore the pairs ( ),A S and ( ),B T share the ( )STCLR property.

Theorem 3.2. Let A ,B ,S and T be soft mappings of a fuzzy metric space ( ), ,X M *

satisfying inequality (3.1) of Theorem 3.1 If pair ( ),A S and ( ),B T enjoy the ( )STCLR

property, then ( ),A S and ( ),B T have a coincidence point each. Moreover, A , B , S

and T have a unique common fixed point provided that both the pairs ( ),A S and ( ),B T are weakly compatible.

Proof. Since the pairs ( ),A S and ( ),B T share the ( )STCLR property, there exist

two sequences { }nx and { }ny in X such that lim lim lim limn n n nn n n n

Ax Sx Ty ByÆ• Æ• Æ• Æ•

= = =

z= , where ( ) ( )z S X T XŒ « . Since ( )z S XŒ , there exists a point u XŒ such that Su z= . Now we assert that Au Su= . Using inequality (3.1) with x u= , ny y= , we obtain

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4n n n

n

n n

M Su Ty t k M Au Su t k M By Ty t kM Au By t

M Au Ty t k M Su By t kfÊ ˆ≈

≥ Á ˜≈Ë ¯,

that is,

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4n n n

n

n n

M z Ty t k M Au z t k M By Ty t kM Au By t

M Au Ty t k M z By t kfÊ ˆ≈

≥ Á ˜≈Ë ¯ (3.5)

Since ( ) ( ){ }liminf , , 2 , , 2n nn

M Au z t k M By Ty t kÆ•

≈ { ( )lim inf min , , 2 ,n

M Au z t k eÆ•

≥ -

( )}, ,n nM By Ty e , for all ( )0, 2t ke Œ . Letting 0e Æ in above inequality, we get

( ) ( ){ } ( )lim inf , , 2 , , 2 lim inf , , 2n nn n

M Au z t k M By Ty t k M Au z t kÆ• Æ•

≈ ≥ . Also lim infnÆ•

( ) ( ){ }, , 4 , , 4n nM Au Ty t k M z By t k≈ ( )lim inf , , 4 1nn

M Au Ty t kÆ•

= ≈ ( , ,M Au z≥


)4t k ( ), , 2M Au z t k≥ . Now passing to limit as nÆ• in inequality (3.5), we obtain

( ) ( ) ( )( ), , 1, , , 2 , , , 2M Au z t M Au z t k M Au z t kf≥ ( )( (, , 2 , , ,M Au z t k M Au zf≥

) ( ))2 , , , 2t k M Au z t k ( ), , 2M Au z t k≥ ( )( ), , 2 1nM Au z k t≥ Æ . Hence,

( ), , 1M Au z t = , that is, Au Su z= = and hence u is a coincidence point of ( ),A S .

Also ( )z T XŒ , there exists a point v XŒ such that Tv z= . Now we show that Bv Tv= . On using inequality (3.1) with x u= , y v= , we get

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4M Su Tv t k M Au Su t k M Bv Tv t k

M Au Bv tM Au Tv t k M Su Bv t k

fÊ ˆ≈

≥ Á ˜≈Ë ¯

( ) ( ) ( )( ), , 1,1 , , 2 ,1 , , 4M z Bv t M Bv z t k M z Bv t kf≥ ≈ ≈ ( ( ) (1, , , 2 , ,M Bv z t k M zf≥

)), 4Bv t k ( ) ( ) ( )( ), , 2 , , , 2 , , , 2M Bv z t k M Bv z t k M z Bv t kf≥ ( ), , 2M z Bv t k≥

( )( ), , 2 1nM Bv z k t≥ Æ . Hence z Bv= . Therefore Bv Tv z= = and hence v is a

coincidence point of ( ),B T . Since the pair ( ),A S is weakly compatible, therefore Az ASu SAu Sz= = = . Taking x z= and y v= in inequality (3.1), we have

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4M Sz Tv t k M Az Sz t k M Bv Tv t k

M Az Bv tM Az Tv t k M Sz Bv t k

fÊ ˆ≈

≥ Á ˜≈Ë ¯

( ) ( ) ( ) ( )( ), , , , 2 ,1, , , 4 , , 4M Az z t M Az z t k M Az z t k M Az z t kf≥ ≈ (( , ,M Az zf≥

) ( ))2 ,1, , , 4t k M Az z t k ( ) ( ) ( )( ), , 2 , , , 4 , , , 2M Az z t k M Az z t k M Az z t kf≥ M≥

( ), , 2Az z t k ( )( ), , 2 1nM Az z k t≥ Æ , which implies that Az z Sz= = , therefore

z is a common fixed point of the pair ( ),A S . Also the pair ( ),B T is weakly compatible, therefore Bz BTv TBv Tz= = = . On using inequality (3.1) with x u= , y z= , we have

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4M Su Tz t k M Au Su t k M Bz Tz t k

M Au Bz tM Au Tz t k M Su Bz t k

fÊ ˆ≈

≥ Á ˜≈Ë ¯

( ) ( ) ( ) ( )( ), , , , 2 ,1, , , 4 , , 4M z Bz t M z Bz t k M z Bz t k M z Bz t kf≥ ≈ (( , ,M z Bzf≥

) ( ))2 ,1, , , 4t k M z Bz t k ( ) ( ) ( )( ), , 2 , , , 2 , , , 2M z Bz t k M z Bz t k M z Bz t kf≥ ( ,M z≥

( )( ), , 2 1nM z Bz k t≥ Æ , which implies that Bz z= . Therefore, Bz z Tz= = .

Hence z is a common fixed point of both the pairs ( ),A S and ( ),B T . The uniqueness of common fixed point is an easy consequence of inequality (3.1).

Following example illustrates Theorem 3.2.


Example 3.1. Let ( ), ,X M * be a fuzzy metric space, where [ )5,25X = , with t -

norm * is defined by { }min ,a b a b* = for all [ ], 0,1a b Œ and

( ) ( ) , if 0;, ,

0, if 0t t x y t

M x y tt

Ï + - >Ô= Ì=ÔÓ

for all ,x y XŒ and 0t > . Let [ ] [ ]3: 0,1 0,1f Æ be the same as given in Example 2.1.

Define self mappings A , B , S and T by ( ) { } ( )( ]

5, if 5 10, 25 ;22, if 5,10 .

xA x

x

Ï Œ »Ô= Ì ŒÔÓ

( ) { } ( )( ]

5, if 5 10, 25 ;12, if 5,10 .

xB x

x

Ï Œ »Ô= Ì ŒÔÓ ( ) ( ]

( ) ( )

5, if 5;11, if 5,10 ;

5 3, if 10,25 .

x

S x x

x x

Ï =Ô= ŒÌÔ + ŒÓ

( )T x =

( ]( )

5, if 5;14, if 5,10 ;

5, if 10,25 .

x

x

x x

Ï =Ô ŒÌÔ - ŒÓ

Taking { } { }10 1nx n= + , { } { }5ny = or { } { }5nx = , { }ny =

{ }10 1 n+ , it is clear that both the pairs ( ),A S and ( ),B T satisfy the ( )STCLR pro-

perty ( ) ( )lim lim lim lim 5n n n nn n n n

Ax Sx By Ty S X T XÆ• Æ• Æ• Æ•

= = = = Œ « Then ( ) { }5, 22A X =

[ ) ( )5,20 T XÕ =/ and ( ) { } [ ) { } ( )5,12 5,10 11B X S X= Õ » =/ . Thus, all the conditions

of Theorem 3.2 are satisfied for some fixed [ )0, 2kŒ . Moreover 5 is the unique

common fixed point of the pairs ( ),A S and ( ),B T . It is worth noting that in this

example ( )S X and ( )T X are not closed subsets of X . Also, all mappings considered herein are dis-continuous even at their unique common fixed point 5.

Theorem 3.3. Let A , B , S and T be self mappings of a fuzzy metric space ( ), ,X M * , where * is a continuous t -norm satisfying (3.1) of Theorem 3.1. Then A ,

B , S and T have a unique common fixed point provided both the pairs ( ),A S and

( ),B T are weakly compatible and conditions (a)-(d) of Lemma 3.1 hold.

Proof. In view of Lemma 3.1, pairs ( ),A S and ( ),B T share the ( )SRCLR property,

that is, there exist two sequences { }nx and { }ny in X such that lim limn nn n

Ax SxÆ• Æ•

= =

lim limn nn n

Ty By zÆ• Æ•

= = , where ( ) ( )z S X T XŒ « . The rest of the proof follows using

similar arguments to those given in the proof of Theorem 3.2.

Example 3.2. In Example 3.1, replace the self mappings S and T by the following

( ) ( ]( ) ( )

5, if 5;3, if 5,10 ;5 3, if 10,25 .

x

S x x x

x x

Ï =Ô= + ŒÌÔ + ŒÓ

( ) ( ]( )

5, if 5;13 , if 5,10 ;

5, if 10, 25 .

x

T x x x

x x

Ï =Ô= + ŒÌÔ - ŒÓ


Clearly, both the pairs ( ),A S and ( ),B T satisfy the ( )STCLR property lim nn

AxÆ•

=

( ) ( )lim lim lim 5n n nn n n

Sx By Ty S X T XÆ• Æ• Æ•

= = = Œ « . Notice that ( ) { } [ ]5, 22 5, 23A X = Ã

( )T X= and ( ) { } [ ] ( )5,12 5,13B X S X= Ã = . Also the remaining conditions of

Theorem 3.3 can be easily verified for some fixed [ )0, 2k Œ . Moreover 5 is the unique

common fixed point of the pairs ( ),A S and ( ),B T . Note that Theorem 3.2 can not be

used in the context of this example as ( )S X and ( )T X are closed subsets of X . Again mappings involved herein are discontinuous seven at their unique common fixed point.

Remark 3.1. The conclusions of Lemma 3.1, Theorem 3.2 and Theorem 3.3 remain true if we replace inequality (3.1) by the following:

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, , min

, , 4 , , 4M Sx Ty t k M Ax Sx t k M By Ty t k

M Ax By tM Ax Ty t k M Sx By t k

Ï ¸≈Ô Ô≥ Ì ˝≈Ô ÔÓ ˛, (3.6)

for all ,x y XŒ , 0t > and 0 2k£ < .

By choosing A , B , S and T suitably, we can deduce corollaries involving two or three self mappings. As a sample, we obtain the following natural result for a pair of self mappings.

Corollary 3.1. Let A and S be self mappings of a fuzzy metric space ( ), ,X M * . Suppose that

(1) The pair ( ),A S satisfies the ( )SCLR property,

(2) ( )S X is a closed subset of X , (3)

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4M Sx Sy t k M Ax Sx t k M Ay Sy t k

M Ax Ay tM Ax Sy t k M Sx Ay t k

fÊ ˆ≈

≥ Á ˜≈Ë ¯ (3.7)

for all ,x y XŒ , 0t > , f ŒF and 0 2k£ < . Then ( ),A S has a coincidence point.

Moreover if the pair ( ),A S is weakly compatible then it has a unique common fixed point in X .

Now, we utilize Definition 2.11 (which is indeed a natural extension of commutativity condition to two finite families) to prove a common fixed point theorem for four finite families of weakly compatible mappings in fuzzy metric space (as an application of Theorem 3.2).

Theorem 3.4. Let { } 1m

i iA = , { } 1

n

r rB = , { } 1

p

k kS = and { } 1

q

h hT = be four finite families of

self mappings of a fuzzy metric space ( ), ,X M * with 1 2 mA AA A= , 1 2 nB BB B= ,

1 2 pS S S S= and 1 2 qT T T T= satisfying the inequality (3.1) of Theorem 3.1 such


that the pairs ( ),A S and ( ),B T share the ( )STCLR property, then each ( ),A S and

( ),B T have a point of coincidence.

Then { } 1m

i iA = , { } 1

n

r rB = , { } 1

p

k kS = and { } 1

q

h hT = have a unique common fixed point

provided the pairs of families ( { }iA , { }kS ) and { } { }( ),r hB T commute pairwise

wherein { }1, 2, ,i mŒ , { }1, 2, ,k pŒ , { }1,2, ,r nŒ and { }1, 2, ,h qŒ .

Proof. Since the proof of this theorem is similar to the result contained in [21], hence it is omitted.

By setting 1 2 mA A A A= = = = , 1 2 nB B B B= = = = , 1 2 pS S S S= = = = and 1 2 qT T T T= = = = in Theorem 3.4, we deduce the following:

Corollary 3.2. Let A , B , S and T be self mappings of a fuzzy metric space ( ), ,X M * . Suppose that

(1) The pairs ( ),m pA S and ( ),n qB T share the ( ),p qS TCLR property,

(2)

( ) ( ) ( ) ( )( ) ( )

, , 2 , , , 2 , , 2 ,, ,

, , 4 , , 4

p q m p n q

m n

m q p n

M S x T y t k M A x S x t k M B y T y t kM A x B y t

M A x T y t k M S x B y t kfÊ ˆ≈

≥ Á ˜Á ˜≈Ë ¯

,

for all ,x y XŒ , 0t > , f ŒF , 0 2k£ < , m , n , p and q are fixed positive integers. Then A , B , S and T have a unique common fixed point provided AS SA= and BT TB= .

Remark 3.2. The results similar to Theorem 3.4 and Corollary 3.2 can also be outlined in respect of inequality (3.6).

4. Conclusion

Theorem 3.2 is proved for two pairs of single-valued mappings in fuzzy metric spaces employing the ( )STCLR property. It improves the results of Sedghi and Shobe [46, Theorem 2.4] in the sense that the conditions on completeness (or closedness) of the underlying space (or subspaces), containment of ranges amongst involved mappings together with conditions on continuity in respect of any one of the involved mappings are relaxed. A natural result is obtained in the form of a corollary. Theorem 3.4 is presented as an extension of Theorem 3.2 to four finite families of mappings.

Acknowledgment

The authors would like to express their sincere thanks to Professor S. L. Singh for his paper [24].

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__________________________ Received December, 2013


Los Angeles

Some Fixed Point Theorems on Generalized M-fuzzy Metric Space

A. Singadurai # and G. Pushpalakshmi

Department of Mathematics, TDMNS college, T. Kallikulam-627117 Tamilnadu, India

# Email: [email protected]

Abstract: We define compatibility and weak compatibility of a collection of self maps on

generalized M -fuzzy metric space. This concept motivates us to establish a unique fixed point for a collection of self maps. Also we study a fixed point theorem for multivalued maps on generalized M -fuzzy metric space.

Keywords: Generalized M -fuzzy metric space, self map, compatiblemaps and weakly

compatible maps, multivalued map.

0. Introduction

After the introduction of fuzzy sets by Zadeh [25], many researchers concentrated on fuzzy Mathematics. Probabilistic metric [16] was introduced by Menger in 1947.

Subsequently Kramosil and Michalek [15] introduced the concept of fuzzy metric spaces. Recently Sedghi and Shobe [20] introduced M -fuzzy metric space. We introduced generalized M -fuzzy metric space [21]. The main objective of this paper is to study some fixed point theorems on a generalized M -fuzzy metric space. Also we define compatibility and weakly compatibility collection of self maps. A 3n ≥ is arbitrary in a generalized M -fuzzy metric space, we establish a fixed point theorem for an arbitrary collection of self maps. Finally we define multifunction and study a fixed point theorem for some multifunction on a generalized M -fuzzy metric space.

1. Preliminaries

132 A. Singadurai and G. Pushpalakshmi

Definition 1.1. A binary operation [ ] [ ] [ ]: 0,1 0,1 0,1* ¥ Æ is a continuous t -norm if it satisfies the following conditions

(i) * is associative and commutative (ii) 1a a* = for all [ ]0,1aŒ

(iii) a b c d* £ * whenever a c£ and b d£ , for each [ ], , , 0,1a b c d Œ

(iv) * is continuous.

Example 1.2. The following are some usual examples of continuous t -norm

a b ab* = the product of a and b , { }min ,a b a b* = .

Definition 1.3 [20]. A 3-tuple ( ), ,X *M is called a M -fuzzy metric space if X is an arbitrary nonempty set, * is a continuous t -norm, and M is a fuzzy set on

( )3 0,X ¥ • , satisfying the following conditions: for each , , ,x y z a XŒ and , 0t s > ,

(i) ( ), , , 0x y z t >M ,

(ii) ( ), , , 1x y z t =M if and only if x y z= =

(iii) ( ) { }( ), , , , , ,x y z t p x y z t=M M (symmetry) where p is a permutation function

(iv) ( ) ( ) ( ), , , , , , , , ,x y a t a z z s x y z t s* £ +M M M ,

(v) ( ) ( ) [ ], , , : 0, 0,1x y z ◊ • ÆM is continuous

Definition 1.4 [21]. A 3-tuple ( ), ,X *M is called a generalized M -fuzzy metric

space if X is an arbitrary nonempty set, * is a continuous t -norm and ( ): 0,nX ¥ •M

[ ]0,1Æ , 3n ≥ satisfying the following conditions: for each 1 2, , , nx x x , nx XŒ¢ and , 0t s >

(i) ( )1 2, , , , 0nx x x t >M ,

(ii) ( )1 2, , , , 1nx x x t =M , for all 0t > if and only if 1 2 nx x x= = =

(iii) ( ) { }( )1 2 1 2, , , , , , , ,n nx x x t p x x x t=M M where p is a permutation function

(iv) ( ) ( ) ( )1 2 1 2, , , , , , , , , , , ,n n n n nx x x t s x x x t x x x s+ ≥ *¢ ¢M M M

(v) ( ) ( ) [ ]1 2, , , , : 0, 0,1nx x x ◊ • ÆM is continuous

(vi) ( )1 2, , , , 1nx x x t ÆM as tÆ• .

Example 1.5. Let * be defined by a b ab* = or a b a b* = Ÿ . Let ( ), ,X M * be a

KM fuzzy metric space as defined in [15] and ( )1 2, , , ,nx x x t =M ( ) (1 2 2, , ,M x x t M x*

Some Fixed Point Theorem on Generalized M -fuzzy Metric Space 133

) ( )3 1, , ,n nx t M x x t-* * for every 1 2, , , nx x x XŒ . Then ( ), ,X *M is a generalized M -fuzzy metric space.

Remark 1.6. It is easy to observe that in a generalized M -fuzzy metric space ( ), ,X *M , ( ) ( ), , , , , , , , , ,x x x y t y x x x t=M M for all 0t > and ,x y XŒ .

Definition 1.7. Let ( ), ,X *M be a generalized M -fuzzy metric space. A sequence

{ }kx in X is said to converge to a point x XŒ if and only if ( ), , , , , 1kx x x x t ÆM as kÆ• for all 0t > .

Definition 1.8. Let ( ), ,X *M be a generalized M -fuzzy metric space. A sequence

{ }kx in X is said to be Cauchy sequence if ( ), , , , 1k m mx x x t ÆM as ,k mÆ• for all 0t > .

Lemma 1.9. Let ( ), ,X *M be a generalized M -fuzzy metric space. Then M is a

continuous function on ( )0,nX ¥ • .

Definition 1.10. A generalized M -fuzzy metric space ( ), ,X *M is said to be complete, if every Cauchy sequence converges.

2. Fixed points for self maps

Definition 2.1. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let 1T and

2T be self maps on X . If there exists a sequence { }kx in X and x XŒ such that

( )1lim , , , , , 1k kT x x x tÆ• =M and ( )2lim , , , , 1k kT x x x tÆ• =M . Then 1T and 2T are said to have the same equivalent property.

Definition 2.2. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let 1T and

2T be self maps on X . (i) The pair 1T and 2T of self maps on X is said to be weakly compatible, if they

commute at their coinciding point. I.e., 1 2T x T x= implies 1 2 2 1T T x T T x= . (ii) The pair 1T and 2T of self maps on X is said to be compatible, if

( )1 2 2 1 2 1lim , , , , 1k k k kT T x T T x T T x tÆ• =M , for all 0t > whenever { }kx is a sequence in X such that 1 2lim limk k k kT x T x x XÆ• Æ•= = Œ .

Definition 2.3. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let 1, , kT T be self maps on X . They are said to be weakly compatible, if any pair of 1, , kT T is weakly compatible. They are said to be compatible, if any pair of 1, , kT T is compatible.


They are said to have the same equivalent property if there exists a sequence { }1x inX

and x XŒ such that ( ) ( )1 1 1lim , , , , lim , , , , 1l l kT x x x t T x x x tÆ• Æ•= = =M M .

Theorem 2.4. Any pair 1T and 2T of compatible maps is weakly compatible. Proof. Let 1 2T x T x= for some x XŒ . Consider { }kx , where kx x= for all k .

Then 1 1limk kT x T xÆ• = and 2 2limk kT x T xÆ• = . Hence 1limk kT xÆ• = 2limk kT xÆ•

1 2T x T x= = and so by definition of compatible maps ( 2 2 11lim , , ,k k kT T x T T xÆ• M

)2 1 , 1kT T x t = , for all 0t > . This means that ( )1 2 2 1 2 1, , , , 1T T x T T x T T x t =M for all 0t > . Hence 1 2 2 1T T x T T x= .

Theorem 2.5. Let ( ), ,X *M be a generalized M -fuzzy metric space. Then any compatible collection is weakly compatible.

Theorem 2.6. Let ( ), ,X *M be a generalized M -fuzzy metric space. A collection of weakly compatible maps need not be compatible.

Proof. Consider [ ]0, 2X = and * is given by a b ab* = , the usual product of a and

b . Define ( )1 2 1

1 2, , , ,n nx x x x

t tnx x x t e e

-- - - -

= ¥ ¥M . Then ( ), ,X *M is a generalized M -fuzzy metric space. Consider the maps 1T , 2T , 3T defined as follows

( )1

2 if 0 1if 1 2

x xT x

x x

- £ <Ï= Ì £ £Ó

, ( ) ( ) ( )2

3 2 2 if 0 12 if 1 2

x xT x

x

Ï - £ <= Ì £ £Ó

,

( ) ( ) ( )3

3 4 4 if 0 12 if 1 2

x xT x

x

Ï + £ <= Ì £ £Ó

Then ( ) 2i jT T x = ,if 1 2x£ £ , for all i , j and so 1T , 2T , 3T are weakly compatible

maps. Consider 1 1kx k= - . Then it is clear that ( )1 1lim , , ,1, 1k k kT x T x tÆ• =M ,

( )2 2lim , , ,1, 1k k kT x T x tÆ• =M and ( )3 3lim , , ,1, 1k k kT x T x tÆ• =M . Hence limkÆ•

1 2 3lim lim 1k k k k kT x T x T xÆ• Æ•= = = . Now 1 3 1 1 4kT T x k= + , 3 1 2kT T x = , 1 2 2kT T x = ,

2 1 2kT T x = , 2 3 1 1 8kT T x k= + and 3 2 2kT T x = . Hence ( 1 2 2 1lim , , ,k k kT T x T T xÆ•M

) ( )2 1 , lim 2, 2, 1k kT T x t tÆ•= =M , ( )2 3 3 2 3 2lim , , , ,k k k kT T x T T x T T x tÆ• =M ( )( ) ( )2 1 1 8 1 1 8lim lim limk t k t

k k ke e- - + - -

Æ• Æ• Æ•= = 1 1 8 1 1 8 1limk t t kt tke e e e- - - -Æ•= ◊ = . That

is ( )2 3 3 2 3 2lim , , , , 1k k k kT T x T T x T T x tÆ• πM . Also ( 3 3 11lim , ,k k kT T x T T xÆ• M

)3 1, , 1kT T x t π . This shows that weakly compatible maps need not be compatible.

Theorem 2.7. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let 1 2, , ,A A

nA , B , 1, , nT T , S are self maps on X satisfying the following properties:


(i) 1 1 2 2 n nT y T y T y= = = implies 1 2 ny y y= = =

(ii) B , 1, , nT T have the same equivalent property

(iii) ( ) ( )i iA X T XÃ for 1,2, ,i n= and ( ) ( )B X S XÃ and ( )S X is a complete space.

(iv) The collection of maps 1 2, , , nA A A , S is weakly compatible.

(v) The collection of maps 1, , nT T , B is weakly compatible.

(vi) For all 1 2 1, , , ny y y - and x in X , ( )1 2 1, , , , ,i nA x By By By t- ≥M

( ) ( ){ }1 1 1 2 1 1 1 2 1min , , , , , , , , , , , ,n n n n nSx T y T y T y kt Sx T y T y T y kt- -M M for some

1k > and for 1,2, ,i n= . Then all the maps have the same unique fixed point.

Proof. Since B , 1, , nT T have the same equivalent property, there exists a sequence

{ }kx XŒ , u XŒ such that ( ) ( )lim , , , , lim , , , ,k k k i kBx u u t T x u u tÆ• Æ•=M M , 1, 2, ,i n= . Hence lim limk k k i kBx T x uÆ• Æ•= = for all 1,2, ,i n= . Since BX Ã

SX , there exists ky in X such that k kBx Sy= . Hence lim limk k k i kBx T xÆ• Æ•= = limk kSy uÆ• = , 1, 2, ,i n= . Since ( )S X is complete, there exists a point x XŒ

such that Sx u= . Now to prove iA x u= for any i , by (vi) one can write ( , ,i kA x BxM

) ( ){1 1 1 1, , , min , , , , ,k k k kBx Bx t Sx T x T x kt+ +≥ M ( )2 2 1, , , , , ,k kSx T x T x kt+M

( )}1, , , ,n k n kSx T x T x kt+M . By Lemma 1.9 it is clear that (lim , , ,k i kA x BxÆ• M

) ( ) ( ){1, , , , , , min , , , , ,k k iBx Bx t A x u u t u u u kt+ = ≥M M ( ) (, , , , , ,u u kt uM M

)}, , 1u kt = for all 0t > . This means that iA x u Sx= = , 1, 2, ,i n= . Since the collection 1 2, , , nA A A , S is weakly compatible, i iSA x ASx= , for all 1, 2, ,i n= . Now, ( ) ( ) ( ) ( )i i i i i iA A x A A x A Sx S A x S Sx SSx= = = = = for al 1,2, ,i n= . Since

( ) ( )i iA X T XÃ , 1, 2, ,i n= , we have i i iA x T y= , for some iy XŒ , 1,2, ,i n= . It

is observed that ( )1 1 2 2, , , ,n nT y T y T y t =M ( ), , , , 1u u u t =M , for all 0t > and so

1 1 2 2 n nT y T y T y= = = . By (i) 1 2 ny y y y= = = = (say). Now we prove; iT y By= , 1, 2, ,i n= and By u= . Suppose By uπ . Then ( ) {, , , , miniA x By By t ≥M M

( )1 1, , , , , ,Sx T y T y kt ( )}, , , ,n nSx T y T y ktM . This means that ( , , , ,iA x By ByM

) 1t = , for all 0t > and so ( ), , , , 1u By By t =M , for all 0t > . Hence By u= . This implies that iT y By u= = , 1,2, ,i n= . The following argument leads to iA u u= .

Suppose iA u uπ . Then ( ) ( ) ({ 1, , , , , , , , min , ,i iA u u u t A u By By t Su T y= ≥M M M

) (1, , , , ,T y kt SuM )} ( ), , , , , , ,n nT y T y kt Su u u kt=M . Now, ( )Su S Sx= =

( )i i iA A x A u= . Hence ( ) ( ), , , , , , , ,i iA u u u t A u u u kt≥M M . This means that

( ), , , ,iA u u u t >M ( ), , , ,iA u u u ktM and so ( ), , , , 1iA u u u t =M , for all 0t > . Hence iA u u= , 1, 2, ,i n= . Next to prove Bu u= . Suppose Bu uπ . Then


( ) ( ) ( ) ({ 1 1, , , , , , , , min , , , , , , ,iu Bu Bu t A u Bu Bu t Su T u T u kt Su= ≥M M M M

)}, , ,n nT u T u kt .Now by (v) we have ( ) ( )i i iBu B T y T By T u= = = , 1,2, ,i n= .

Also i i iSu SSx AA x A u= = = . Hence ( ) ( ), , , , , , , ,u Bu Bu t Su Bu Bu kt≥ =M M

( ) ( ), , , , , , , ,iA u Bu Bu kt u Bu Bu kt=M M . Hence ( ), , , ,u Bu Bu t ≥M ( ,uM

), , ,Bu Bu kt . This implies that u Bu= and so iBu T u u= = .

Uniqueness of u : Suppose u vπ , u , v are two distinct fixed points of maps iA , B , iT , S , 1,2, ,i n= . Then ( ) ( ) (, , , , , , , , , ,iu u u v t A v Bu Bu t Sv= ≥M M M

), , ,Bu Bu t ( ) ( ){ }1 1min , , , , , , , , , ,n nSv T u T u kt Sv T u T u kt≥ M M ( ,iA v≥M

) ( ), , , , , , ,Bu Bu kt v u u kt≥M . Hence ( ) ( ), , , , , , , , , ,u u u v t u u u v kt≥M M and so u v= .

3. Fixed points for multivalued maps

Definition 3.1. Let ( ), ,X *M be a generalized M -fuzzy metric space. A multivalued

map or multifunction is a mapping : 2XT XÆ which assign to each x XŒ , a non empty subset Tx of X . Fixed points of multifunction T are those points x XŒ for which x TxŒ .

Definition 3.2. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let

1 2, , , nS S S be subsets of X and 1 2, , , nx x x XŒ . Define the following

(i) ( ) ( ){ }1 2 1 2 1 1, , , , sup , , , , :n nS x x t a x x t a S= ŒM M

(ii) ( ) ( ){ }1 2 3 1 2 3 1 1 2 2, , , , , sup , , , , , : ,n nS S x x t a a x x t a S a S= Œ ŒM M

(iii) ( ) ( ){1 2 1 1 2 1 1 1 2 2 1, , , , , sup , , , , , : , , ,n n n n nS S S x t a a a x t a S a S a- - -= Œ ŒM M

}1nS -Œ and

(iv) ( ) ( ){ } ({1 2 1 2 1 2 1, , , , min inf , , , , , , inf , , , ,n n nS S S t a S S t S S S -È= ÎM M M

)},na t ˘̊

Definition 3.3. Let ( ), ,X *M be a generalized M -fuzzy metric space. Let 1T , 2T be multivalued mappings of X . Then 2 1T TÃ means for any 0x XŒ with 1 1 0x T xŒ we have 2 1 1 0T x T xÃ .

Theorem 3.4. Let ( ), ,X *M be a complete generalized M -fuzzy metric space. Let

( )0,1q Œ . Let 1 2, , , nT T T be multivalued mappings with the following properties:

(i) 1 2 nT T T… … …


(ii) ( ) ( ) ( ){1 1 2 2 1 1 1 1 1 1, , , , min , , , , , , , , , , ,n n nT x T x T x qt x x t x T x x x t≥M M M

( )}, , , , ,n n n n nx T x x x tM

(iii) ( ) ( ) ( ), , , , , , , , , , , , ,x x x t x x y t x x y z t≥ ≥ ≥M M M for all , , ,x y z XŒ Then 1 2, , , nT T T have a same common fixed point.

Proof. Let 0x XŒ and 0e > be given. Then there exists 1 1 0x T xŒ such that ( 01 ,T xM

) ( )0 0 1 0 0, , , 2 , , , ,x x qt x x x qte- <M . Now, since 2 1 1 0T x T xÃ , there exists 2x Œ

2 1T x such that ( ) ( )1 2 1 1 1 1 2 1 1, , , , , 2 , , , ,x T x x x qt x x x x qte- <M M and ( 0 2 11 , ,T x T xM

) ( )1 1 1 2 1 1, , , 2 , , , ,x x qt x x x x qte- <M . Since 3 2 2 1 1 0T x T x T xÃ Ã , it is clear that

there is a 3 3 2x T xŒ such that ( ) (2 2 3 2 2 2 2 2 3 2, , , , , , 2 , , , ,x x T x x x qt x x x xe- <M M

)2, ,x qt , ( ) ( )2 2 1 3 2 2 2 2 2 3 2 2, , , , , , 2 , , , , , ,x T x T x x x qt x x x x x qte- <M M ( 01 ,T xM

) ( )2 1 3 2 2 2 2 2 3 2 2, , , , , 2 , , , , , ,T x T x x x qt x x x x x qte- <M Continuing in this way we

get there is a point 1n n nx T x -Œ such that ( ) (1 1 1 1, , , , 2 ,n n n n nx x T x qt xe- - - -- <M M

)1 1, , , ,n n nx x x qt- - , ( ) (1 1 1 2 1 11, , , , 2 , , ,n n n n n n nnx x T x T x qt x xe- - - - - --- <M M

)1, ,n nx x qt- ( ) ( )1 0 2 1 1 1 1 1, , , , 2 , , , , ,n n n n n nT x T x T x qt x x x x qte- - - -- <M M .

Similarly we will get 1 2, ,n nx x+ ,such that 1 1n nx T x+ Œ and ( ) 21 , , , , 2n n nT x x x qt e-M

( )1, , , ,n n nx x x qt+<M etc. Proceeding in this way we get a sequence { }kx . Claim:

{ }kx is a Cauchy sequence. It is clear that ( ) ( )( ) (1 1 1, , , ,nk nnk n nk nx x x qt T++ - + - ≥M M

)2 1 1, , , , 2knk nk n nk nx T x T x qt e+ + - - ( ) ({ 1 1min , , , , , , , ,nk nk nk n nk nkx x x t x x+ + -≥ M M

)1 , , ,nkT x t ( )( )}1 11, , , , 2knk n n nk nnk nx x T x t e+ - + -+ - -M ({ 1min , , ,nk nkx x +≥ M

( ) ) ( )11 , , , , , , , ,nk nk nknk nx t x x x t++ - M ( )( )}( 1) 1, , , , 2knk n nk nnk n

x x x t e+ - ++ - - ≥M

( )( )1 1, , , , 2knk nk nk nx x x t e+ + - -M . Now, ( )( ) (1 1 1

, , , , ,nk nk nnk n nkx x x t T x+ + - -

>M M

( ) )1 2 1 1 2, , , ,nk nk n nk nT x T x T x t+ - + - ( )( )1 2 1 1 12, , , , ,nk nk n n nknk n

T x T x T x T x t+ - -+ -= ≥M

( )( ){ ( )1 1 12min , , , , , , , , , , , ,nk nk nk nk nk nknk nx x x x t q x x x t q+ - ++ - M ( )( 2 , ,

nk nx + -M

( ) ( ) ) ( )}1 1 12 1, , , , , , , ,nk nk nk nknk n nk nx x t q x x x x t q- - -+ - + - M ( 1 2, , , ,nk nk nk nx x x+ + -≥M

)1,nkx t q- . Hence ( )( ) ( )1 1 1 21, , , , , , , , ,nk nk nk nk nk nk nnk nx x x t x x x x t q+ - + + -+ - ≥M M

( )22 1 3, , , ,nk nk nk nx x x t q- - + -≥M etc. This means that ( )( )1 1, , , , 1nk nk nk n

x x x t+ + - ÆM

as kÆ• . This implies that ( ) ( )( )1 1, , , , 1nk nnk n nk nx x x t++ - + - ÆM as kÆ• . Again,

( )1, , , ,nk nk nkx x x qt+ ≥M ( )1 1 1 1, , , , ,n nk n nk n nk nkT x T x T x T x qt- - - ≥M ( 1 1,n nk nT x T- -M

( ) )2 2 11, , , ,nk nknk nx T x T x qt- - - = ( )( )1 1 2 2 11 1, , , , , minn nk n nk nkk n

T x T x T x T x qt- - - - + ≥M


( )( ) ( ) ( ){ 1 2 1 1 2 2 11 1, , , , , , , , , , , , , , , ,nk nk nk nk nk nk nk nk nkk nx x x x t x x x t x x x t- - - - - - -- +M M M

( ) ( ) ( ) ( )( ) ( )} ({1 11 1 1 1 1 1 1 2, , , , , , , , , , , min ,nk nk nk nk nkk n k n k n k nx x x x t x x x x t x+ -- + - + - + - + =M M M

( ) )2 1, , , , ,nk nknk nx x x t- - - ( )1 1, , , , ,nk nk nkx x x t- -M ( )2 2 1, , , , , ,nk nk nkx x x t- - -M

( ) ( ) ( ) ( )( )1 1 1 2, , , , , ,nk n nk n nk n nk nx x x x t- - - - - - - -M ( ) (1 1, , , , , ,nk nk nk nk nkx x x x t x+ -=M M

( ) )2 1, , , ,nk nknk nx x x t- - - . Hence ( ) (1 1 2, , , , , , ,nk nk nk nk nkx x x qt x x+ - -≥M M

( ) )1 , , 1nknk nx x t- - Æ as kÆ• . This implies that ( )1, , , , , 1k k k kx x x x t+ ÆM as

kÆ• . Now for any k , m with m k> we have ( ) (, , , , ,k m m kx x x t x≥M M

) ( )1 1 1 1, , , , 2 , , , , , 2k k k k m m mx x x t x x x x t*

+ + + +M ( )1 1 1, , , , , 2k k k kx x x x t+ + +≥ M

( )21 2 2 2, , , , , 2k k k kx x x x t+ + + +* *M ( )1, , , , 2m k

m m mx x x t --*M . Hence ( ,kxM

), , , 1m mx x t Æ as k and mÆ• .Since ( ), ,X *M is a generalized M -fuzzy metric space, there is a z in X such that kx zÆ . Next to prove iz T zŒ for all 1,2, ,i n= . It is clear that ( ) ( ( )1 2 3 1 2 1 3 2 1, , , , , , , , , ,nk nk nk n nk nk n nk n

T z x x x qt T z T x T x T x+ + + + + + -≥M M

) ( )( ) ( ){ 1 2 11min , , , , , , , , , , ,nk nk nk nqt z x x x t z z T z t+ + + -≥ M M ( 1 1 1, , , ,nk nk nkx x x+ + +M

) ( ) ( ) ( )( )}2 1 1 1 1, , , , , , , minnk nnk n nk n nk nT x t x x T x t+ + - + - + - ≥M ({ 1 2, , , ,nk nkz x x+ +M

( ) ) ( ) ( )1 1 1 1 21 , , , , , , , , , , , , , ,nk nk nk nknk nx t z z T z t x x x x t+ + + ++ - M ( )( ( )1 1, , ,

nk n nk nx x+ - + -M

)},nk nx t+ . Since M is continuous, by letting kÆ• , we have ( )1 , , , ,T z z z qt ≥M

( )1, , , ,z z T z tM . Hence ( ) (1 1, , , , , , , , ,T z z z t T z z z≥M M ) ( 1 , , , ,t q T z z z≥M

) ( )2 31 , , , , 1t q T z z z t q≥ ≥ =M and so ( )1 , , , , 1T z z z t =M , for all 0t > . This

means that 1z T zŒ . Similarly we can prove iz T zŒ for 2,3, ,i n= .

Acknowledgement:

This research work was supported by the University Grants Commission of India under Minor Research Projects to the first author.

References

[1] K. K. Azad, On fuzzy semi continuity, Fuzzy almost continuity and fuzzy weakly continuity, Jl. Of Math. Analy. Applns, 82 (1981), 14-32.

[2] C. L. Chang, Fuzzy topological spaces, J. Math. Analy. Applns., 24 (1968), 182-190. [3] S. S. Chauhan and NidhiJoshi, Common fixed point theorem in M -fuzzy metric spaces using implicit

relations, International Mathematical Forum, 4 (2009), No.47, 2311-2316. [4] N. R. Das and Pankaja Das, Fuzzy topology generated by fuzzy norm, Fuzzy Sets and Systems, 107

(1999), 349-354.


[5] N. R. Das and Pankaja Das, Fuzzy bitopological spaces induced by fuzzy quasi pseudo norm and its conjugate, Fuzzy Sets and Systems, 119 (2001), 539-542.

[6] S. Elumalai and Vijayaragavan, Best simultaneous approximation in linear 2-normed spaces, General Mathematics, Vol.16, No.1 (2008), 73-81.

[7] M. A. Erceg, Metric spaces in fuzzy set theory, Jl. Of Math. Analy. Applns, 69 (1979), 205-230. [8] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64

(1994), 395-399. [9] U. Hole and S. E. Rodabaugh, eds, Mathematics of fuzzy sets: Logic, Topology, and measure theory,

The handbooks of fuzzy sets and series, Vol.3 (1999), Kluwar AcademicPublishers. [10] Jesus Rodriguez-Lopez, Salvaderromaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets

and Systems, 147 (2004), 273-283. [11] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), 215-229. [12] A. K. Katsaras and D. B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math Analy.

Applns, 58 (1977), 135-146. [13] A. K. Katsaras, Fuzzy topological vector spaces I, Fuzzy Sets and Systems, 6 (1981), 85-95. [14] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143-154. [15] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 336-

344. [16] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 28 (1942), 535-537. [17] PhullanduDas, Fuzzy vector spaces under triangular norms, Fuzzy Sets and Systems, 25 (1988), 73-85. [18] G. Rajendran and G. Balasubramanian, Fuzzy contraction mapping theorem for fuzzy metric spaces,

Bull.cal. Math. Soc., 94 (6) (2002), 453-458. [19] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Maths, 10 (1960), 313-334. [20] S. Sedghi and N. Shobe, Fixed point theorem in M -fuzzy metric spaces with property (E), Advances

in Fuzzy Mathematics, Vol.1 No.1 (2006), 55-65. [21] A. Singadurai and G. Pushpalakshmi, Generalized M -fuzzy metric spaces, The Jl. Of Fuzzy

Mathematics, Vol.21, No.1 (2013), 143-148. [22] T. TamizhChelvam and A. Singadurai, I -topological vector spaces generated by F -norm, The Jl. Of

Fuzzy Mathematics, Vol.14, No.2 (2006), 255-265. [23] T. TamizhChelvam and A. Singadurai, I -bitopological spaces Generated by fuzzy-norm, The Jl. of

Fuzzy Mathematics, Vol.16, No.2 (2008), 483-494. [24] Zike Deng, Fuzzy pseudo-metric spaces, Jl. Math. Analy. Applns, 86 (1982), 74-95. [25] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.


__________________________ Received January, 2014


Los Angeles

Fuzzy Implication Groupoids

Ravi Kumar Bandaru

Department of Engineering Mathematics, GITAM University, Hyderabad Campus, Andhra Pradesh, India-502329


Bijan Davvaz

Department of Mathematics, Yazd University, Yazd, IRAN E-mail: [email protected]

Abstract: In this paper, we fuzzify the concept of implication groupoids and investigate some of

their properties. We give a characterization of fuzzy implication groupoid, and discuss a characterization of fuzzy implication groupoids in terms of level subalgebras of fuzzy implication groupoids.

Keywords: Implication groupoid, distributive implication groupiod, level subalgebra, normal

fuzzy implication groupoid.

1. Introduction

I. Chajda and R. Halas [4] introduced the concept of implication groupoid as a generalization of the implication reduct of intuitionitsic logic, i.e. a Hilbert algebra and studied some connections among ideals, deductive systems and congruence kernels whenever implication groupoid is distributive. In [2], R. K. Bandaru further studied the properties of ideals in implication groupoids. In [6,7], Y. B. Jun et. al introduced the concept of fuzzy ideal, fuzzy deductive systems in Hilbert algebras and discuss the relation between fuzzy ideals and fuzzy deductive systems. In this paper we fuzzify the concept of implication groupoids and investigate some of their properties. We give a characterization of fuzzy implication groupoid, and discuss a characterization of fuzzy implication groupoids in terms of level subalgebras of fuzzy implication groupoids.

2. Preliminaries

142 Ravi Kumar Bandaru and Bijan Davvaz

Definition 2.1 [4]. An algebra ( ), ,1A * of type ( )2,0 is called an implication groupoid if it satisfies the identities:

(1) 1x x* = (2) 1 x x* = for all ,x y AŒ .

Definition 2.2 [4]. An implication groupoid ( ), ,1A * of type ( )2,0 is called a distributive implication groupoid if it satisfies the following identity:

( )LD ( ) ( ) ( )x y z x y x z* * = * * * (left distributivity)

for all , ,x y z AŒ .

Example 2.3. Let { }1, , , ,A a b c d= in which * is defined by

* I a b c d I I a b c d a I I b b I b I a I I d c I a I I d d I I c c I

Then ( ), ,1A * is a distributive implication groupoid. In every implication groupoid, on ecan introduce the so called induced relation £ by

the setting x y£ if and only 1x y* = the induced relation £ is quasiorder.

Lemma 2.4 [4]. Let ( ), ,1A * be a distributive implication groupoid. Then A satisfies the identities

1 1x* = and ( ) 1x y x* * =

3. Fuzzy implication groupoids Through out this section, implication groupoid ( ), ,1X * means distributive implication

groupoid. First we begin with following definition.

Definition 3.1 [7]. Let X be a set. A fuzzy set in X is a function [ ]: 0,1Xm Æ .

Definition 3.2 [7]. Let m be a fuzzy set in a set X . For [ ]0,1a Œ , the set

( ){ }x X xam m a= Œ ≥ is called a level subset of m .

Definition 3.3. A fuzzy set m in X is called a fuzzy implication groupoid of X if it

satisfies, for all ,x y XŒ , ( ) ( ) ( ){ }min ,x y x ym m m* ≥ .

Fuzzy Implication Groupoids 143

Example 3.4. Let { }1, , , ,X a b c d= be the implication groupoid an in Example 2.3

and { }1, ,A a b= . Let [ ]1 2, 0,1t t Œ be such that 1 2t t> . Define a mapping [ ]: 0,1Xm Æ

by ( ) ( ) ( ) 11 a b tm m m= = = and ( ) ( ) 2c d tm m= = . Then m is a fuzzy implication groupoid of X .

Corollary 3.5. If m is a fuzzy implication groupoid of X , then ( ) ( )1 xm m≥ for all x XŒ .

Proposition 3.6. Let m be a fuzzy implication groupoid of X and a XŒ . If m is decreasing, then it is constant.

Proof. We note that ( ) ( )1xm m£ for all x XŒ . Since 1x £ for all x XŒ ,

( ) ( )1xm m≥ because m is decreasing. Hence ( ) ( )1xm m= for all x XŒ . Thus m is constant.

Theorem 3.7. Let m be a fuzzy set in implication groupoid X . Then m is a fuzzy implication groupoid of X if and only if for every [ ]0,1a Œ , the level subset am is a subalgebra of X , when 0am /π .

Proof. Let m be a fuzzy implication groupoid of X and ,x y amŒ for every [ ]0,1a Œ

with 0am /π . Then ( ) ( ) ( ){ }min ,x y x ym m m a* ≥ ≥ , which implies that ( )x ym a* ≥ . Hence x y am* Œ . Thus am is a subalgebra of X . Conversely, assume that am is a subalgebra of X for every [ ]0,1a Œ with 0am /π . Let ,x y XŒ and ( ) 1xm a= and

( ) 2ym a= . Then 1

x amŒ and 2

y amŒ . Without loss of generality, we may assume that

1 2a a£ . Then 2 1a am mÕ and so

1y amŒ . Since

1am is a subalgebra of X , we have

1x y am* Œ . Hence ( ) ( ) ( ){ }1 min ,x y x ym a m m* ≥ = . Therefore m is a fuzzy implica-tion groupoid of X .

Corollary 3.8. If m is a fuzzy implication groupoid of X , then the set {X xm = Œ

( ) ( )}1X xm m= is a subalgebra of X .

Proof. Since ( ) ( )1xm m£ for all x XŒ , we have

( ) ( ) ( ){ } ( ) ( ){ }1 1 1x X x x X x Xmmm m m m m= Œ ≥ = Œ = = .

By Theorem 3.7, we know that Xm is a subalgebra of X .

Definition 3.9. Let m be a fuzzy implication groupoid of X . Each subalgebra am of X , [ ]0,1a Œ , is called a level subalgebra of m , when 0am /π .


Theorem 3.10. Let A be a subalgebra of implication groupoid X and let [ ]: 0,1Xm Æ be a fuzzy set defined by, for all x XŒ ,

( ) 0

1

if if x A

xx A

am

aŒÏ

= Ì œÓ

where [ ]0 1, 0,1a a Œ 0 1a a> . Then m is a fuzzy implication groupoid of X .

Proof. Let ,x y XŒ . If at least one of x and y does not belong to A , then ( )x ym *

( ) ( ){ }1 min ,x ya m m≥ = , since 1a is the minimum value of m . If ,x y AŒ , then x y*

AŒ . Hence ( ) ( ) ( ){ }min ,x y x ym m m* = . Therefore m is a fuzzy implication groupoid of X .

Corollary 3.11. Any proper subalgebra of implication groupoid of X can be realized as a level subalgebra of some fuzzy implication groupoid of X .

Proof. Let A be a proper subalgebra of X and let m be a fuzzy set in X defined by

( ) if 0 if .

x Ax

x A

am

ŒÏ= Ì œÓ

where a is fixed number in ( ]0,1 . Taking 0a a= and 1 0a = in Theorem 3.10, we know that m is a fuzzy implication groupoid of X , and obviously Aam = . This completes the proof.

We can generalize Theorem 3.10 as follows:

Theorem 3.12. Let { } 0n nA

•= be a strictly decreasing sequence of subalgebras of

implication groupoid 0X A= and let { } 0n na •

= be a strictly increasing sequence in ( )0,1 . Then there is a fuzzy implication groupoid m o f X such that

n nAam = for all 0,1,2,n = .

Proof. Define a fuzzy set [ ]: 0,1Xm Æ by

( )1

1

if

lim if .∩n n n

n n nn

x A A

xx A

am

a

+•

Æ•=

Œ -ÏÔ= Ì ŒÔÓ

It can be easily seen that m is a fuzzy implication groupoid of X and that n nAam =

for all 0,1,2,n = .


Theorem 3.13. Let m be a fuzzy implication groupoid of X . Then two level subalgebras

1am , 2am with 1 2a a< of m are equal if and only if there is no x XŒ such

that ( )1 2xa m a£ < .

Proof. Suppose that 1 2a a< and 1 2a am m= . If there exists x XŒ such that 1a £

( ) 2xm a< , then 2am is a proper subset of

1am . This is impossible. Conversely, suppose

that there is no x XŒ such that ( )1 2xa m a£ < . Note that 1 2a a< implies 2 1a am mÕ .

If 1

x amŒ , then ( ) 1xm a≥ , and so ( ) 2xm a≥ because ( ) 2xm a</ . Hence 2

x amŒ , which says that

1 2a am mÕ . Thus 1 2a am m= . This completes the proof.

Remark 3.14. As a consequence of Theorem 3.13, the lvel subalgebras of a fuzzy implication groupoid m of X which has a countable image form a chain. But ( )xm £

( )1m for all x XŒ , and so ( )1mm is the smallest level subalgebra of a fuzzy implication

groupoid, but not always ( ) { }1 1mm = as shown in the following example. Thus we have

a chain 0 1 k

X a a am m m= ,where 0 1 ka a a< < < < and ( )1 limn nm aÆ•= .

Example 3.15. Let A be a proper subalgebra of implication groupoid X and let m be a fuzzy implication groupoid of X in the proof to Corollary 3.11. Then ( )Im m =

{ }0,a , and two level subalgebras of m are 0 Xm = and Aam = . Thus we have ( )1m

a= but { }1Aam = π .

Corollary 3.16. Let m be a fuzzy implication groupoid of X . If ( ) { }1Im , , nm a a= ,

where 1 2 na a a< < < , then the family of subalgebras iam of ( )1,2, ,i nm = con-

stitutes all the level subalgebras of m .

Proof. Let [ ]0,1a Œ and ( )Ima mœ . If 1a a< , then 1a am mÕ . Since

1Xam = , we

have Xam = and 1a am m= . If 1i ia a a +< < ( )1 1i n£ £ - , then there is no x XŒ

such that ( ) 1ixa m a +£ < . Using Theorem 3.13, we obtain 1ia am m+

= . This shows that

for any [ ]0,1a Œ with ( )1a m£ , the level subalgebra am is in { }1i

i nam £ £ .

Theorem 3.17. Let m be a fuzzy implication groupoid ofX with ( ) { }Im i im a= ŒD

and { }iiam= ŒDA where D is an arbitrary index set. Then

(i) There exists a unique 0i ŒD such that 0i ia a> for all iŒD .

(ii) 0

∩i iiXm a am mŒD= = .

(iii) ∪iiX amŒD= .

(iv) The members of A form a chain.


(v) If m attains its infimum on all subalgebras of X , then A contains all level subalgebras of m .

Proof. (i) Since ( ) ( )1 Imm mŒ , there exists a unique 0i ŒD such that

( ) ( )0

1i xa m m= ≥ for all x XŒ so that 0i

a a≥ for all iŒD .

(ii) We know that ( ){ } ( ){ }0 00ii ix X x x X xam m a m a= Œ ≥ = Œ = = { ( )x X xmŒ =

( )}1 Xmm = . Since 0i ia a≥ for all iŒD , therefore clearly

0i ia am mŒ for all iŒD .

Hence 0

∩i iia am mŒDÕ , and so

0∩

i iia am mŒD= , because 0i ŒD .

(iii) It is sufficient to show that ∪iiX amŒDÕ . Let x XŒ . Then ( ) ( )Imxm mŒ and

so there exists ( )i x ŒD such that ( ) ( )i xxm a= . This implies ( ) ∪

iiix aam mŒDŒ Ã . This

proves (iii). (iv) Note that for any ,i j ŒD , either i ja a≥ or i ja a£ , hence

i ja am mÕ or

j ia am mÕ . Therefore the members of A form a chain.

(v) Assume that m attains its infimum on all subalgebras of X . Let am be a level subalgebra of m . If ia a= for some iŒD , then clearly Aam Œ . Assume that ia aπ

for all iŒD . Then there is no x XŒ such that ( )xm a= . Let ( ){ }A x X xm a= Œ > .

Obviously 1 AŒ , and so 0A /π . Let ,x y AŒ . Then ( )xm a> and ( )ym a> . Since

m is a fuzzy implication groupoid of X , it follows that ( ) ( ) ( ){ }min ,x y x ym m m* ≥ >

a so that ( )x ym a* > , i.e., x y A* Œ . Hence A is a subalgebra of X . By hypothesis,

there exists y AŒ such that ( ) ( ){ }infy x x Xm m= Œ . Now ( ) ( )Imym mŒ implies ( )ym

ia= for some iŒD . Obviously ia a≥ , and so by assumption ia a> . Note that there is no z XŒ such that ( ) iza m a£ < . It follows from Theorem 3.13 that

ia am m= . Hence am ŒA . This completes the proof.

4. Normal fuzzy implication groupoids

Definition 4.1. A fuzzy implication groupoid m of X is said to be normal if there exists x XŒ such that ( ) 1xm = .

Let m and s be any two fuzzy subsets of a set X . Then m is said to be contained in s denoted by m sÕ , if ( ) ( )x xm s£ of all x XŒ . If ( ) ( )x xm s= for all x XŒ , m and s are said to be equal and we write m s= . We note that if m is a normal fuzzy implication groupoid of X , then ( )1 1m = . Hence we have the following characteriza-tion.

Theorem 4.2. A fuzzy implication groupoid m of X is normal if and only if

( )1 1m = .


Theorem 4.3. If m is a fuzzy implication groupoid of X , then the fuzzy set m+ of

X defined by ( ) ( ) ( )1 1x xm m m+ = + - for all x XŒ is a normal fuzzy implication groupoid of X containing m .

Proof. Assume that m is a fuzzy implication groupoid of X and let ,x y XŒ . Then

( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( ){ ( )1 1 min , 1 1 min 1 1 ,x y x y x y x ym m m m m m m m m+ * = * + - ≥ + - = + -

( )} ( ) ( ){ }1 1 min ,x ym m m+ ++ - = and ( ) ( ) ( )1 1 1 1 1m m m+ = + - = . Hence m+ is a normal

fuzzy implication groupoid of X , and clearly m m+Ã .

Theorem 4.4. Let m and n be fuzzy implication groupoids of X . If m nÃ and

( ) ( )1 1m n= , then X Xm nÃ .

Proof. Assume that m nÃ and ( ) ( )1 1m n= . If x XmŒ then ( ) ( ) ( ) ( )1 1x xn m m n≥ = = .

Noticing that ( ) ( )1xn n£ for all x XŒ , we have ( ) ( )1xn n= , i.e., x XnŒ .

Corollary 4.5. If m and n are normal fuzzy implication groupoids of X satisfying m nÃ , then X Xm nÃ .

Theorem 4.6. A fuzzy implication groupoid m of X is normal if and only if m m+ = .

Proof. The sufficiency is obvious. Assume that m is a normal fuzzy implication

grou-poid of X and let x XŒ . Then ( ) ( ) ( ) ( )1 1x x xm m m m+ = + - = , and hence

m m+ = .

Theorem 4.7. If m is a fuzzy implication groupoid of X , then ( )m m++ += .

Proof. For any x XŒ , we have ( ) ( ) ( ) ( ) ( )1 1x x xm m m m++ + + += + - = , completing

the proof.

Theorem 4.8. Let m be a fuzzy implication groupoid of X . If there exists a fuzzy

implication groupoid n of X satisfying n m+ Ã , then m is normal.

Proof. Suppose there exists a fuzzy implication groupoid n of X such that n m+ Ã .

Then ( ) ( )1 1 1n m+= £ and hence ( )1 1m = .

Corollary 4.9. Let m be a fuzzy implication groupoid of X . If there exists a fuzzy

implication groupoid n of X satisfying n m+ Ã , then m m+ = .

Theorem 4.10. Let m be a fuzzy implication groupoid of X and let ( ): 0, 1f mÈ ˘ ÆÎ ˚

[ ]0,1 be an increasing function. Define a fuzzy set [ ]: 0,1f Xm Æ by ( ) ( )( )f x f xm m=


for all x XŒ . Then fm is a fuzzy implication groupoid of X . In particular, if ( )( )1f m

1= , then fm is normal, and if ( )f a a≥ for all ( )0, 1a mÈ ˘ŒÎ ˚ , then fm mÃ .

Proof. Let ,x y XŒ . Then ( ) ( )( ) ( ) ( ){ }( )min ,f x y f x y f x ym m m m m* = ≥ = min

( )( ) ( )( ){ } ( ) ( ){ }, min ,f ff x f y x ym m m m= . Hence fm is a fuzzy implication groupoid

of X . If ( )( )1 1f m = then clearly fm is normal. Assume that ( )f a a≥ for all [0,a Œ

( )1m ˘̊ . Then ( ) ( )( ) ( )f x f x xm m m= ≥ for all x XŒ , which proves that fm mÃ .

Theorem 4.11. Let m be a non-constant normal fuzzy implication groupoid of X , which is maximal in the poset of normal fuzzy implication groupoids under set inclusion. Then m takes only the values 0 and 1.

Proof. By Theorem 4.2, ( )1 1m = . Let x XŒ be such that ( ) 1xm π . It sufficient to

show that ( ) 0xm = . Assume that there exists a XŒ such that ( )0 1am< < . Define a

fuzzy set [ ]: 0,1Xn Æ by ( ) ( ) ( ){ }1 2x x an m m= + for all x XŒ . Then clearly n is

well-defined. Let ,x y XŒ . Then ( ) ( ) ( ){ } ( )1 2 1 2 1 2x y x y x yn m m a m* = * + = * +

( ) ( ) ( ){ } ( )1 2min ; 1 2a x y am m m m≥ + ( ) ( )( ) ( ) ( )( ){ }min 1 2 ,1 2x a y am m m m= + + =

( ) ( ){ }min ,x yn n , which proves that n is a fuzzy implication groupoid of X . Now we

have ( ) ( ) ( ) ( ) ( ){ } ( ) ( ){ } ( ){ }1 1 1 2 1 1 2 1 1 2 1x x x a a xn n n m m m m m+ = + - = + + - + = +

and so ( ) ( ){ }1 1 2 1 1 1n m+ = + = . Hence n + is a normal fuzzy implication groupoid of

X . From ( ) ( )a an m+ > it follows that m is not maximal. This is contradiction.

References

[1] J. C. Abbott, Semi-boolean algebras, Mathematicki Vensik, 4 (19) (1976), 177-198. [2] R. K. Bandaru, On ideals of implication groupoids, Advances in decision sciences, (2012), Article ID

652814, 9p. [3] I. Chajda and R. Halas, Congruences and ideals in Hilbert algebras, Kyungpook Math. J., 39 (1999),

429-432. [4] I. Chajda and R. Halas, Distributive implication groupoids, Central European Journal of Mathematics,

5 (3) (2007), 484-492. [5] W. A. Dudek, On ideals in Hilbert algebras, Acta Uni. Palackianae Olomuciensis Fac. Rer. nat ser.

Math., 38 (19999), 31-34. [6] W. A. Dudek, On fuzzy ideals in Hilbert algebras, Novi Sad J. Math., 29 (2) (1999), 193-207. [7] Y. B. Jun and S. M. Hong, On fuzzy deductive systems of Hilbert algebras, Indian Journal of Pure

and Applied Mathematics, 27 (2) (1996), 141-151.


__________________________ Received January, 2014 The present work is partially supported by Special Assistance Programme (SAP) of UGC, New Delhi, India

[Grant No. F. 510/4/DRS/2009 (SAP-I)] 1066-8950/15 $8.50

© 2015 International Fuzzy Mathematics Institute Los Angeles

Some Results on D* -fuzzy Cone Metric Spaces and Fixed Point Theorems in Such Spaces

T. Bag

Department of Mathematics, Visva-Bharati University West Bengal, India


Abstract: In this paper, an idea of D* -fuzzy cone metric space is introduced. Some basic

definitions viz. convergence of sequence, Cauchy sequence, closedness, completeness etc are given and study some related properties. Some fixed point theorems are established in such spaces.

Keywords: D* -fuzzy cone metric space, weakly compatible mapping.

0. Introduction

The concept of fuzzy set is introduced by L. A. Zadeh [14] in 1965. After that, to use this concept in topology and analysis different authors have expansively developed the theory of fuzzy sets and its application in different direction.

On the other hand, there have been a number of generalizations of metric spaces. One such generalization is generalized metric space (D -metric space) introduced by Dhage [5] in 1992. Many other authors viz. Sedghi et al. [13] made a significant contribution in fixed point theory of D* -metric space (which is similar to D -metric space). Recently Sedghi et al. [12]introduced the concept of M -fuzzy metric space which is a generalization of fuzzy metric space due to George & Veeramoni [7].

The present author [3] redefined M -fuzzy metric space and called it D* -fuzzy metric space and established a decomposition theorem and with the help of this theorem some results were established in such spaces. H. Long-Guang et al [9] generalized the notion of general metric spaces, replacing the real numbers by an ordered Banach space and defined cone metric space. They have proved Banach contraction mapping theorem and some other contraction mapping theorem and some other fixed point theorems of contractive type mappings in cone metric spaces. Subsequently, Rezapour and Hamlbarani [11] contributed some fixed point theorems for contractive type mappings in

150 T. Bag

cone metric space. In an earlier paper [4], idea of fuzzy cone metric space has been introduced and some basic definitions are given. Here the range of fuzzy cone metric is considered as ordering fuzzy real numbers defined on a real fuzzy Banach space (Felbin’s type [6]).

The idea of generalized D* -metric space is relatively new and it is introduced by Age & Salunke [1]. They have also established some fixed point theorems in such spaces. Following the idea of generalized D* -metric space introduced by Age & Salunke [1], in this paper, definition of D* -fuzzy cone metric space is given. Some fixed point theorems are established in such space.

The organization of the paper is as follows: In Section 1, comprises some preliminary results which are used in this paper. Definition of D* -fuzzy cone metric space and some basic properties are discussed in

Section 2. In Section 3, some fixed point theorems for contractive mappings are established.

1. Some preliminary results

A fuzzy number is a mapping [ ]: 0,1x RÆ over the set R of all reals.

A fuzzy number x is convex if ( ) ( ) ( )( )min ,x t x s x r≥ where s t r£ £ . If there

exists a 0t RŒ such that ( )0 1x t = , then x is called normal. For 0 1a< £ , a -level set

of an upper semi continuous convex normal fuzzy number (denoted by [ ]ah ) is a closed

interval [ ],a ba a , where aa = -• and ba = +• are admissible. When aa = -• , for

instance, then [ ],a ba a means the interval ( ],ba-• . Similar is the case when ba = +• .

A fuzzy number x is called non-ngeative if ( ) 0x t = , 0t" < . Kaleva (Felbin) denoted the set of all convex, normal, upper semicontinuous fuzzy

real numbers by ( )( )E R I and the set of all non-negative, convex, normal, upper

semicontinuous fuzzy real numbers by ( )( )G R I* .

A partial ordering “≺ ” in E is defined by ≺h d if and only if 1 2a aa a£ and 1 2b ba a£

for all ( ]0,1a Œ where [ ] 1 1,a ba aah È ˘= Î ˚ and [ ] 2 2,a ba aad È ˘= Î ˚ . The strict inequality in E

is defined by ≺h d if and only if 1 2a aa a< and 1 2b ba a< for each ( ]0,1a Œ . According to Mizumoto and Tanaka [10], the arithmetic operations ≈ , , on

E E¥ are defined by

( )( ) ( ) ( ){ }sup min ,s Rx y t x s y t sŒ≈ = - , t RŒ

( )( ) ( ) ( ){ }sup min ,s Rx y t x s y s tŒ= - , t RŒ

( )( ) ( ), 0sup min ,s R s

tx y t x s y

sŒ π

Ï ¸Ê ˆ= Ì ˝Á ˜Ë ¯Ó ˛, t RŒ

Some Results on D* -fuzzy Cone Metric Spaces and Fixed Point Theorems in Such Spaces 151

Proposition 1.1 [10]. Let ( )( ), E R Ih d Œ and [ ] 1 1,a ba aah È ˘= Î ˚ , [ ] 2 2,a ba aad È ˘=Î ˚ , ( ]0,1aŒ .

Then [ ] 1 2 1 2,a a b ba a a aah d È ˘≈ = + +Î ˚ , [ ] 1 2 1 2,a b b aa a a aah d È ˘= - -Î ˚ , [ ] 1 2 1 2,a a b ba a a aah d È ˘= Î ˚

Definition 1.1 [8]. A sequence { }nh in E is said to be convergent and converges to

h denoted by limn nh hÆ• = if lim nn a aa aÆ• = and lim n

n b ba aÆ• = where [ ]n ah =

,n na ba aÈ ˘Î ˚ and [ ] [ ],a ba aah = ( ]0,1a" Œ .

Note 1.1 [8]. If ( )( ), G R Ih d *Œ then ( )( )G R Ih d *≈ Œ .

Note 1.2 [8]. For any scalar t , the fuzzy real number th is defined as ( ) 0t sh = if

0t = otherwise ( ) ( )t s s th h= .

Definition of fuzzy norm on a linear space as introduced by C. Felbin is given below:

Definition 1.2 [6]. Let X be a vector space over R . Let ( ):X R I*Æ and let the mappings [ ] [ ] [ ], : 0,1 0,1 0,1L U ¥ Æ be symmetric,

nondecreasing in both arguments and satisfy ( )0,0 0L = and ( )1,1 1U = .

Write 1 2,x x xa aaÈ ˘È ˘ =Î ˚ Î ˚ for x XŒ , 0 1a< £ and suppose for all x XŒ , 0x π ,

there exists ( ]0 0,1a Œ independent of x such that for all 0a a£ ,

(A) 2x a < •

(B) 1inf 0x a > .

The quadruple ( ), , ,X L U is called a fuzzy normed linear space and is a fuzzy norm if

(i) 0x = if and only if 0x = ;

(ii) rx r x= , x XŒ , r RŒ ;

(iii) for all ,x y XŒ ,

(a) whenever 11s x£ , 1

1t y£ and 11s t x y+ £ + , ( ) ( ) ( )( ),x y s t L x s y t+ + ≥ ,

(b) whenever 11s x≥ , 1

1t y≥ and 11s t x y+ ≥ + , ( ) ( ) ( )( ),x y s t U x s y t+ + £ .

Remark 1.1 [6]. Felbin proved that, if ( )MinL = � and ( )MaxU = � then the

triangle inequality (iii) in the Definition 1.1 is equivalent to ≺x y x y+ ≈ .

Further i

a ; 1, 2i = are crisp norms on X for each ( ]0,1a Œ . When minL = and

maxU = , then we write ( ),X instead of ( ), , ,X L U .

Definition 1.3 [4]. Let ( ),E be a fuzzy real Banach space where ( ):E R I*Æ .

Denote the range of by ( )E I* . Thus ( ) ( )E I R I* *Ã .

152 T. Bag

Definition 1.4 [4]. A member ( )R Ih *Œ is said to be an interior point if 0r$ > such that

( ) ( ){ } ( ), : ≺S r R I r E Ih d h d* *= Œ Ã .

Set of all interior points of ( )R I* is called interior of ( )R I* .

Definition 1.5 [4]. A subset of F of ( )E I* is said to be fuzzy closed if for any

sequence { }nh such that limn h hÆ• • = implies Fh Œ .

Definition 1.6 [4]. A subset P of ( )E I* is called a fuzzy cone if

(i) P is fuzzy closed, nonempty and { }0P π ;

(ii) ,a b RŒ , , 0a b ≥ , , P a b Ph d h dŒ fi ≈ Œ ; (iii) Ph Œ and 0Ph h- Œ fi = .

Given a fuzzy cone ( )P E I*Ã , define a partial ordering £ with respect to P by h d£ iff Pd h Œ and h d< indicates that h d£ but h dπ while h d will stand for IntPd h Œ where IntP denotes the interior of P .

The fuzzy cone P is called normal if there is a number 0K > such that for all ( ), E Ih d *Œ , with 0 h d£ £ implies ≺ Kh d . The lest positive number satisfying

above is called the normal constant of P . The fuzzy cone P is called regular if every increasing sequence which is bounded

from above is convergent. That is if { }nh is a sequence such that 1 2 nh h h£ £ £ £

h£ for some ( )E Ih *Œ , then there is ( )E Id *Œ such that nh dÆ as nÆ• . Equivalently, the fuzzy cone P is regular if every decreasing sequence which is

bounded below is convergent. It is clear that a regular fuzzy cone is a normal fuzzy cone. In the following we always assume that E is a Felbin’s type fuzzy real Banach space,

P is a fuzzy cone in E with IntP fπ and £ is a partial ordering with respect to P .

Definition 1.7 [4]. Let X be a nonempty set. Suppose the mapping :d X X¥ Æ ( )E I* satisfies:

(Fd1) ( )0 ,d x y£ ,x y X" Œ and ( ), 0d x y = iff x y= ;

(Fd2) ( ) ( ), ,d x y d y x= ,x y X" Œ ;

(Fd3) ( ) ( ) ( ), , ,d x y d x z d z y£ ≈ , ,x y z X" Œ .

Then d is called a fuzzy cone metric and ( ),X d is called a fuzzy cone metric space.

Definition 1.8. A point u of a set X is said to be coincidence point of a pair of mappings ( ), :f g X XÆ if there exists x XŒ such that ( ) ( )f x g x u= = .

Definition 1.9. Let f and g be two self mappings defined on a set X . f and g are called weakly compatible if they commute at coincidence points.


Proposition 1.2 [2]. (M. Abbas et al.) Let f and g be weakly compatible self maps of a set X . If f and g have a unique point of coincidence w fx gx= = , then w is the unique common fixed point of f and g .

2. D* -fuzzy cone metric space

In this section an ideal of D* -fuzzy cone metric space is introduced and study some properties.

Definition 2.1. Let X be a nonempty set. Suppose that mapping :D X X X* ¥ ¥ Æ ( )E I* satisfying the following conditions:

( )1FD* ( ), , 0D x y z* ≥ ,x y X" Œ ;

( )2FD* ( ), , 0D x y z* = iff x y z= = ;

( )3FD* ( ) { }( ), , , ,D x y z D p x y z* *= where p stands for permutation;

( )4FD* ( ) ( ) ( ), , , , , ,D x y z D x y a D a z z* * *£ ≈ , , ,x y z a X" Œ .

Then the function D* is called a D* -fuzzy cone metric and the pair ( ),X D* is called

a D* -fuzzy cone metric space.

Example 2.1. Consider the Banach space ,EÊ ˆ¢Ë ¯ where 2E = R and x ¢ =

( )1 2221 iix=Â , ( )1 2,x x x= . Define ( ):X R I*Æ by ( ) 1 if

0 if

t xx t

t x

Ï ¢>Ô= Ì¢Ô £Ó

Then

,x x xa

È ˘¢ ¢È ˘ =Î ˚ Í ˙Î ˚ ( ]0,1a" Œ . It is easy to verify that,

(i) 0x = iff 0x = (ii) rx r x= (iii) ≺x y x y+ ≈ . Thus ( ),E is a Felbin’s

type ( minL = and maxU = ) fuzzy normed linear space. Let ( ){ },n nx y be a Cauchy

sequence in ( ),X . So, ( ) ( ),lim , , 0m n n n m mx y x yÆ• - = . (,lim ,n m nm n

x x yÆ•

fi - -

) ( ){ }0 ,m n ny x y¢ = fi be a Cauchy sequence in ,XÊ ˆ¢Ë ¯ . Since ,XÊ ˆ¢

Ë ¯ is complete,

( ),x y X$ Œ such that ( ) ( )lim , , 0n n nx y x yÆ•¢- = . i.e. ( ) ( )lim , , 0n n

nx y x y

Æ•- = . Thus

( ),E is a real fuzzy Banach space. Define ( ){ }: 0P E Ih h*= Œ (where ( )E I*

( )( )R I*Ã is the range of ).

(i) P is fuzzy closed. For, consider a sequence { }nd in P such that limn nd dÆ• Æ .

i.e. 1 1,limn n a ad dÆ = and 2 2

,limn n a ad dÆ = where [ ] 1 2, ,,n n na aad d dÈ ˘= Î ˚ and [ ] 1 ,aad dÈ= Î

2ad ˘̊ ( ]0,1a" Œ . Now 0n nd " . So, 1

, 0n ad ≥ and 2, 0n ad ≥ ( ]0,1a" Œ . limnÆfi

154 T. Bag

1, 0n ad ≥ and 2

,lim 0n n adÆ ≥ ( ]0,1a" Œ 1 0adfi ≥ and 2 0ad ≥ ( ]0,1a" Œ 0dfi . So Pd Œ . Hence P is fuzzy closed.

(ii) It is obvious that, ,a b RŒ , , 0a b ≥ , P a b Ph d h dŒ fi ≈ Œ . (iii) Let Ph Œ . If Ph- Œ , then for all 0t < we have ( )( ) ( ) ( ) 0t t sh h h- = - = ≥

for ( ) 0s t= - > . If ( ) 0sh = 0s" > then 0h = . Otherwise for some ( ) 0s t= - > ,

( ) 0sh > i.e. for some 0t < , ( )( ) 0th- > . In that case h- does not belong to P .

Hence Ph Œ and Ph- Œ implies 0h = . Thus P is a fuzzy cone in E . Take X = R

and choose the ordering £ as ≺ . Define ( ):D X X X E I* *¥ ¥ Æ by ( )( ), ,D x y z t* =

( ) if 0 if x y y z z x t t x y y z z x

t x y y z z x

Ï - + - + - ≥ - + - + -ÔÌ < - + - + -ÔÓ

a -level sets are given by

( ), , ,D x y z x y y z z xa

*È ˘ = È - + - + -ÎÎ ˚ x y y z z x a- + - + - ˘̊ . Now we show

that D* satisfies the conditions ( ) ( )1 4FD FD* *- . We have, ( )1 , ,D x y z x ya* = - +

y z z x- + - and ( ) ( )2 , ,D x y z x y y z z xa a* = - + - + - ( ]0,1a" Œ . Conditions

( )1FD* , ( )2FD* and ( )3FD* can be easily verified. We have, x y y z- + - +

z x x y y a a z z a a x- = - + - + - + - + - £ x y y a a x a z z a- + - + - + - + -

z z+ - . So, ( ) ( ) ( )1 1 1, , , , , ,D x y z D x y a D a z za a a* * *£ + and ( ) ( )2 2, , , ,D x y z D x y aa a

* *£ +

( )2 , ,D a z za* ( ]0,1a" Œ . Thus ( ) ( ) ( ), , , , , ,≺D x y z D x y z D a z z* * *≈ , , ,x y z a X" Œ .

Hence D* is a D* -fuzzy cone metric and ( ),X D* is a D* -fuzzy cone metric space.

Definition 2.1. Let ( ),X D* be a D* -fuzzy cone metric space. Let { }nx be a

sequence in X and x XŒ . If for every c EŒ with 0 c there is a positive integer

N such that for all n N≥ , ( ), ,n mD x x x c* , then { }nx is said to be convergent and

converges to x and x is called the limit of { }nx . We denote it by limn nx xÆ• = .

Proposition 2.1. Let ( ),X D* be a D* -fuzzy cone metric space. Then for all , ,x y ,

z XŒ , ( ) ( ), , , ,D x x y D x y y* *= .

Proof. We have

( ) ( ) ( ) ( ), , , , , , , ,D x x y D x x x D x y y D x y y* * * *£ ≈ = (i)

Again ( ) ( ) ( ) ( ), , , , , , , ,D y y x D y y y D y x x D y x x* * * *£ ≈ = i.e.

( ) ( ), , , ,D x y y D x x y* *£ (ii)

From (i) and (ii) we get, ( ) ( ), , , ,D x y y D x x y* *= .


Lemma 2.1. Let ( ),X D* be a D* -fuzzy cone metric space and P be a normal fuzzy

cone with normal constant K . Let { }nx be a sequence in X . Then { }nx converges to

x iff ( ), , 0m nD x x x* Æ as nÆ• .

Proof. First we suppose that { }nx converges to x . For every real 0e > , choose c Œ

E with 0 c and ≺K c e . Then $ a natural number N , such that ,m n N" ≥ ,

( ), ,m nD x x x c* . So that when ,m n N≥ , ( ), , ≺ ≺m nD x x x K c e* (since P is

normal). ( )1 , ,m nD x x xa e*fi < , ( )2 , ,m nD x x xa e* < ,m n N" ≥ , ( ]0,1a" Œ ,limm nÆ•

fi

( )1 , , 0m nD x x xa* = and ( )1

,lim , , 0m nm n

D x x xa*

Æ•= ( ]0,1a" Œ fi ( )

,lim , , 0m nm n

D x x x*

Æ•= .

Conversely, suppose that ( ),lim , , 0m n m nD x x x*Æ• = . Let c EŒ with 0 c . Choose

0d > such that ≺x d . This implies that { }≺c c x d . i.e. Intc x PŒ .

For this d , there is a positive integer N such that ,m n N" ≥ , ( ), , ≺m nD x x x d* . Let

( ) ,, ,m n m nD x x x y* = where ,m ny EŒ . So , ≺m ny d ,m n N" > . So, ,m nc y Œ

IntP ,m n N" ≥ ,m ny cfi ,m n N" ≥ ( ), ,m nD x x x c*fi ,m n N" ≥ fi

( ), , 0m nD x x x* Æ as ,m nÆ• (since c EŒ is arbitrary).

Lemma 2.2. If ( ),X D* is a D* -fuzzy cone metric space, then the following results

are equivalent. (i) { }nx converges to x .

(ii) ( ), , 0n nD x x x* Æ as nÆ• .

(iii) ( ), , 0nD x x x* Æ as nÆ• .

Proof. Suppose (i) holds. From Lemma 3.1, we have ( ), , 0m nD x x x* Æ as ,m nÆ• .

It follows that, ( ), , 0n nD x x x* Æ as nÆ• . So (ii) holds.

Assume (ii) holds. I.e. ( ), , 0n nD x x x* Æ as nÆ• . We have

( ) ( ) ( ) ( ), , , , , , , ,D x x y D x x x D x y y D x y y* * * *£ ≈ = (i)

Again ( ) ( ) ( ) ( ), , , , , , , ,D y y x D y y y D y x x D y x x* * * *£ ≈ = i.e.

( ) ( ), , , ,D x y y D x x y* *= (ii)

From (i) and (ii), we get ( ) ( ), , , ,D x y y D x x y* *= .So ( ) ( ), , , ,n n nD x x x D x x x* *= . Thus

( ), , 0n nD x x x* Æ as nÆ• ( ), , 0nD x x x*fi Æ as nÆ• . Hence (ii) fi (iii).

Next suppose (iii) holds and we have to show that (iii) fi (i). Since ( ), , 0nD x x x* Æ

as nÆ• and ( ) ( ), , , ,n n nD x x x D x x x* *= , We have ( ), , 0n nD x x x* Æ as nÆ• .

Then for any c EŒ with 0 c there is a positive integer N such that n N" Œ ,

156 T. Bag

( ), , 2nD x x x c* £ and ( ), , 2n nD x x x c* £ n N" ≥ . Now ,m n N" ≥ we get,

( ) ( ) ( ) ( ), , , , , , , ,m n m n m n nD x x x D x x x D x x x D x x x* * * *= £ ≈ .i.e. ( ), , 2m nD x x x c* £

2c c≈ = . I.e. { }nx is convergent. This completes the proof.

Lemma 2.3. Let ( ),X D* be a D* -fuzzy cone metric space and P be a normal cone

with normal constant K . Let { }nx be a sequence in X . If { }nx is convergent then its limit is unique.

Proof. If possible suppose that limn xÆ•= and limn yÆ•= . For a given 0e > ,choose c EŒ with 0 c and ≺K c e . Then there exists a natural number N such that

n N" ≥ , ( ), , 2n nD x x x c* and ( ), , 2n nD x x y c* . We have ( ), ,D x x y D* *£

( ) ( ), , , ,n nx x x d x y y≈ = ( ) ( ), , , ,n n n nD x x x d x x y c* ≈ . Since X is a normal cone

with normal constant K ,we have, ( ), ,D x x y K c* ≺ . This implies that ( ), , ≺D x x y e* .

i.e. ( )1 , ,D x x ya e* < and ( )2 , ,D x x ya e* < ( ]0,1a" Œ . Since 0e > is arbitrary, we have

( )1 , , 0D x x ya* = and ( )2 , , 0D x x ya

* = ( ]0,1a" Œ . Hence ( ), , 0D x x y = . Thus x y= .

Definition 2.2. Let ( ),X D* be a D* -fuzzy cone metric space. A sequence { }nx in

X is said to be a Cauchy sequence if for any c EŒ with 0 c , there exists a natural

number N such that , ,l m n N" ≥ , ( ), ,l m nD x x x c* .

Definition 2.4. Let ( ),X D* be a D* -fuzzy cone metric space. If every Cauchy

sequence in X is convergent in X , then X is called a complete D* -fuzzy cone metric space.

Lemma 2.4. Let ( ),X D* be a D* -fuzzy cone metric space and { }nx be a sequence

in X . If { }nx is convergent then it is a Cauchy sequence.

Proof. Let{ }nx converges to x . So for any c EŒ with 0 c there exists a natural

number N such that , ,l m n N" > , ( ), , 2m nD x x x c* and ( ), , 2l lD x x x c* .

Thus ( ) ( ) ( ), , , , , ,m n l m n l lD x x x D x x x D x x x c* * *£ ≈ , ,l m n N" > Thus { }nx is a Cauchy sequence.

Lemma 2.5. Let ( ),X D* be a D* -fuzzy cone metric space, P be a normal fuzzy

cone with normal constant K . Let { }nx be a sequence in X . Then { }nx be a Cauchy

sequence iff ( ), , 0m n lD x x x* Æ as , ,l m nÆ• .

Proof. Let { }nx be a Cauchy sequence in X . For 0e > choose c EŒ with 0 c

such that ≺K c e . Then there is a natural number N such that , ,l m n N" ≥ , ( ,mD x*

),n lx x c . i.e. , ,l m n N" ≥ , ( ), ,m n lD x x x c* < . So that when , ,l m n N≥ , ( ,mD x*

),n lx x K c e≺ ≺ (since P is normal). Since 0e > is arbitrary, it follows that ( ,mD x*


), 0n lx x Æ as , ,l m nÆ• . Conversely suppose that ( ), , 0m n lD x x x* Æ as , ,l m nÆ• .

For c EŒ with 0 c , there is 0d > such that ≺x d . I.e. (c c )x d≺

implies Intc x PŒ . For this 0d > , there exists a natural number N such that ,l"

,m n N≥ , ( ), , ≺m n lD x x x d* . i.e. , , ≺m n lz d if we write ( ) , ,, ,m n l m n lD x x x z* =

where , ,m n lz EŒ . , , Intm n lc z Pfi Œ , ,m n l N" ≥ , ,m n lz cfi , ,m n l N" ≥

( ), ,m n lD x x x c*fi , ,m n l N" ≥ { }nxfi is a Cauchy sequence.

Lemma 2.6. Let ( ),X D* be a D* -fuzzy cone metric space, P be a normal fuzzy

cone with normal constant K . Let { }nx , { }ny and { }nz be three sequences in X

such that nx xÆ , ny yÆ , nz zÆ as nÆ• . Then ( ) ( ), , , ,n n nD x y z D x y z* *Æ as nÆ• .

Proof. Let 0e > be arbitrary. Choose c EŒ with 0 c and 3 ≺K c e . Since

nx xÆ , ny yÆ and nz zÆ , there exists a positive integer N such that n N" ≥ ,

( ), ,n nD x x x c* , ( ), ,n nD y y y c* and ( ), ,n nD z z z c* .We have, ( , ,n nD x y*

) ( ) ( ) ( ) ( ), , , , , , , ,n n n n n n n n nz D x y z D z z z D z x y D z z z* * * *£ ≈ = ≈ , i.e. ( ), ,n n nD x y z* £

( ) ( ) ( ), , , , , ,n n n n nD z x y D y y y D z z z* * *≈ ≈ . i.e. ( ) ( ) (, , , , ,n n n n nD x y z D y z x D y* * *= ≈

) ( ), , ,n n ny y D z z z*≈ . i.e. ( ) ( ) ( ) ( ), , , , , , , ,n n n n n n nD x y z D y z x D x x x D y y y* * * *£ ≈ ≈

( ), ,n nD z z z*≈ . i.e. ( ) ( ), , 3 , ,n n nD x y z c D x y z n N* *£ ≈ " ≥ . ( ), ,n n nD x y z*fi

( ), , 3D x y z c* £ n N" ≥ . So, ( ) ( ), , , , 3≺n n nD x y z D x y z K c* * n N" ≥ (since

P is normal). i.e. ( ) ( ), , , , ≺n n nD x y z D x y z e* * n N" ≥ , i.e. ( )1 , ,n n nD x y za* -

( )2 , ,D x y za e* < and ( ) ( )2 1, , , ,n n nD x y z D x y za a e* *- < n N" ≥ ( ]0,1a" Œ . Since

0e > is arbitrary, we have ( ) ( )1 2, , , , 0n n nD x y z D x y za a* *- = as nÆ• , ( ]0,1a" Œ .

( ) ( )2 1, , , , 0n n nD x y z D x y za a* *- = as nÆ• , ( ]0,1a" Œ ( ) ( ), , , ,n n nD x y z D x y z* *fi

0Æ as nÆ• . i.e. ( ) ( ), , , ,n n nD x y z D x y z* *Æ as nÆ• .

3. Fixed point theorems in D* -fuzzy cone metric spaces

In this Section some fixed point theorems of self mappings are established in D* -fuzzy cone metric spaces.

Theorem 3.1. Let ( ),X D* be a D* -fuzzy cone metric space, P be a normal fuzzy

cone with normal constant K . Let , :S T X XÆ be two self mappings satisfying the following conditions:

(i) ( ) ( )T X S XÃ

(ii) ( )T X or ( )S X is complete and

158 T. Bag

(iii) ( ) ( ) ( ) ( ), , , , , , , ,D Tx Ty Tz aD Sx Sy Sz bD Sx Tx Tx cD Sy Ty Ty dD* * * * *£ ≈ ≈ ≈

( ), ,Sz Tz Tz , ,x y z X" Œ where , , , 0a b c d ≥ , 1a b c d+ + + < . Then S and T have a unique point of coincidence in X . Moreover if S and T are weakly compatible, then S and T have a unique common fixed point.

Proof. Let 0x be arbitrary. There exists 1x XŒ such that 0 1Tx Sx= . In this way we

get a sequence { }nSx with 1n nTx Sx- = . We have, ( ) (1 1 1, , ,n n n nD Sx Sx Sx D Tx* *+ + -=

),n nTx Tx . Thus by (iii) we have, ( ) ( )1 1 1, , , ,n n n n n nD Sx Sx Sx aD Sx Sx Sx* *+ + -£ ≈

( )1 1 1, ,n n nbD Sx Tx Tx*- - - ( ) ( ), , , ,n n n n n ncD Sx Tx Tx dD Sx Tx Tx* *≈ ≈ . i.e. ( ,nD Sx*

) ( ) ( )1 1 1 1, , , , ,n n n n n n n nSx Sx aD Sx Sx Sx bD Sx Sx Sx* *+ + - -£ ≈ ( ,ncD Sx*≈ )1 1,n nSx Sx- +

( )1 1, ,n n ndD Sx Sx Sx*+ +≈ . i.e. ( ) ( ) (1 1 1, , ,n n n nD Sx Sx Sx a b D Sx* *

+ + -£ + ),n nSx Sx

( ) ( )1 1, ,n n nc d D Sx Sx Sx*+ +≈ + . This implies that, ( )1 1, ,n n nD Sx Sx Sx*

+ + ( 1,nqD Sx*-£

),n nSx Sx where ( )1q a b c d= + - + and also 0 1q£ < . By repeated application of above inequality we have,

( ) ( )1 1 0 1 1, , , ,nn n nD Sx Sx Sx q D Sx Sx Sx* *

+ + £ (1)

Then for all ,m n NŒ , m n> and by using (IFN4) we have, ( ), ,n m mD Sx Sx Sx* =

( ), ,m m nD Sx Sx Sx* ( ) ( ), , , ,m m m m n nD Sx Sx Sx D Sx Sx Sx* *£ ≈ .i.e.

( ), ,n m mD Sx Sx Sx* ( ), ,n n mD Sx Sx Sx*£ . i.e.

( ) ( ) ( )1 1, , , , , ,n m m n n n n m mD Sx Sx Sx D Sx Sx Sx D Sx Sx Sx* * *+ +£ ≈ (2)

In similar way we get,

( ) ( ) ( )1 1 1 2 2, , , , , ,n m m n n n n m mD Sx Sx Sx D Sx Sx Sx D Sx Sx Sx* * *+ + + + +£ ≈ (3)

( ) ( ) ( )1 1 1, , , , , ,m m m m m m m m mD Sx Sx Sx D Sx Sx Sx D Sx Sx Sx* * *- - -£ ≈ . i.e.

( ) ( )1 1 1, , , ,m m m m m mD Sx Sx Sx D Sx Sx Sx* *- - -£ (4)

From (2), (3) and (4) we have, ( ) ( ) (1 1, , , , ,n m m n n n nD Sx Sx Sx D Sx Sx Sx D Sx* * *+ +£ ≈

)1 2,n nSx Sx+ + ( ) ( )2 2 3 1 1, , , ,n n n m m mD Sx Sx Sx D Sx Sx Sx* *+ + + - -≈ ≈ ≈ i.e. ( ,nD Sx*

) ( ) ( )1 1 1 1 2, , , , ,m m n n n n n nSx Sx D Sx Sx Sx D Sx Sx Sx* *+ + + + +£ ≈ ( )2 2 3, ,n n nD Sx Sx Sx*

+ + +≈

( )1 1, ,m m mD Sx Sx Sx*- -≈ ≈ i.e. ( ) ( ) (1 1

0, , ,n n mn m mD Sx Sx Sx q q q D S x* + - *£ + + + .

)1 1,Sx Sx . i.e. ( ) ( )0 1 1, , 1 , ,nn m mD Sx Sx Sx q qD Sx Sx Sx* *= - . Since P is a normal

cone with normal constant K we have, ( ) ( )0 1 1, , , ,1

≺n

n m m

qD Sx Sx Sx K D Sx Sx Sx

q* *

-.


( ) ( )1 10 1 1, , , ,

1

n

n m m

qD Sx Sx Sx D Sx Sx Sx

qa a* *fi £

- and ( )2 2, ,

1

n

n m m

qD Sx Sx Sx D

qa a* *£

-

( )0 1 1, ,Sx Sx Sx ( ]0,1a" Œ ( )1

,lim , , 0n m mm n

D Sx Sx Sxa*

Æ•fi = and (2

,lim , ,n mm n

D Sx Sxa*

Æ•

) 0mSx = ( ]0,1a" Œ (since 1q < ) (5) Again for , ,l m n NŒ , we have ( , ,n mD Sx Sx*

) ( ) ( ), , , ,l n m m m l lSx D Sx Sx Sx D Sx Sx Sx* *£ ≈ . Since P is a normal cone we have,

( ) ( ) ( ){ }, , , , , ,≺n m l n m m m l lD Sx Sx Sx K D Sx Sx Sx D Sx Sx Sx* * *≈ (1 , ,n mD Sx Sxa*fi

) ( ) ( )1 1, , , ,l n m m m l lSx KD Sx Sx Sx KD Sx Sx Sxa a* *£ + and ( )2 2, ,n m lD Sx Sx Sx KDa a

* *£

( ) ( )2, , , ,n m m m l lSx Sx Sx KD Sx Sx Sxa*+ ( ]0,1a" Œ ( )1

, ,lim , , 0n m l

l m nD Sx Sx Sxa

*

Æ•fi =

and ( )1

, ,lim , , 0n m l

l m nD Sx Sx Sxa

*

Æ•= ( ]0,1a" Œ by (5) ( )

, ,lim , , 0n m l

l m nD Sx Sx Sx*

Æ•fi = .

Thus { }nSx is a Cauchy sequence in ( ),X D* . Since ( )S X is complete, ( )u S X$ Œ

such that { }nSx uÆ as nÆ• and there exists p XŒ such that Sp u= . If ( )T X is

complete, then there exists ( )u T XŒ such that nSx uÆ . As ( ) ( )T X S XÃ , we have

( )u S XŒ . Then there exists p XŒ such that Sp u= . We claim that Tp u= . Now,

( ) ( ) ( ) ( ), , , , , , , ,n nD Tp u u D Tp Tp u D Tp Tp Tx D Tx u u* * * *= £ ≈ i.e. ( ), ,D Tp u u* £

( ) ( ) ( ), , , , , ,naD Sp Sp Sx bD Sp Tp Tp cD Sp Tp Tp* * *≈ ≈ ( ), ,n n ndD Sx Tx Tx D* *≈ ≈

( )1, ,nSx u u+ i.e. ( ) ( ) ( ) ( ), , , , , , , ,nD Tp u u aD u u Sx bD u Tp Tp cD u Tp Tp* * * *£ ≈ ≈ ≈

( ) ( )1 1 1, , , ,n n n ndD Sx Sx Sx D Sx u u* *+ + +≈ . i.e. ( ) ( )( )( ), , 1 1D Tp Tp u b c* £ - + {aD*

( ) ( )1 1, , , ,n n n nu u Sx dD Sx Sx Sx*+ +≈ ( )}1, ,nD Sx u u*

+≈ . Since P is a normal cone with

normal constant K we have, ( ) ( )( )( ){ ( ), , 1 1 , , nD Tp Tp u K b c aD u u S x dD* * *- + ≈≺

( )1 1, ,n n nSx Sx Sx+ + ( )}1, ,nD S u u*+≈ . Since { }nSx is a Cauchy sequence, thus right

hand side of the above expression tends to 0 as nÆ• . Hence ( ), ,D Tp Tp u* 0= . i.e. Tp u= . So Tp Sp u= = . Thus p is a point of coincidence point of S and T . Now we show that S and T have a unique point of coincidence. Assume that there exists a point q on X such that Sq Tq= . Now, ( ) ( ), , , ,D Tp Tp Tq aD Sp Sp Sq* *£ ≈ ( , ,bD Sp Tp*

)Tp ( ) ( ), , , ,cD Sp Tp Tp dD Sq Tq Tq* *≈ ≈ . i.e. ( ) (, , ,D Tp Tp Tq aD Tp* *£ ),Tp Tq ≈

( ) ( ) ( ), , , , , ,bD Tp Tp Tp cD Tp Tp Tp dD Tq Tq Tq* * *≈ ≈ . i.e. ( ), ,D Tp Tp Tq* ( ,aD Tp*£

),Tp Tq ( ) ( )1 , ,a D Tp Tp Tq P*fi - Œ .Since 1 0a - < ,it implies that ( ,D Tp* ),Tp Tq =

0 , i.e. Tp Tq= . Thus p is a unique point of coincidence of S and T . By Proposition 1.2, it follows that S and T have a unique common fixed point.

Theorem 3.2. Let ( ),X D* be a complete D* -fuzzy cone metric space, P be a

normal fuzzy cone with normal constant K and let :T X XÆ be a mapping satisfies the following conditions:

160 T. Bag

( ) ( ) ( ) ( ) ( ), , , , , , , , , ,D Tx Ty Tz aD x y z bD x Tx Tx cD y Ty Ty dD z Tz Tz* * * * *£ ≈ ≈ ≈ , , ,x y z X" Œ ; , , , 0a b c d ≥ , 1a b c d+ + + < .

Then T has a unique fixed point in X .

Proof. Proof follows from the Theorem 4.1 by replacing S by identity mapping.

Conclusion

In this paper, an idea of D* -fuzzy cone metric space is introduced which is a generalization of D* -fuzzy metric space. In fuzzy cone metric space, range of fuzzy metric is considered as ordering fuzzy real numbers defined on a real fuzzy Banach space. It is seen that Kaleva et al. type ( )max, min fuzzy metric space is a particular case of fuzzy cone metric space. I think that there is a large scope of developing more results of fuzzy functional analysis in this context.

References

[1] C. T. Age and J. N. Salunke, Some fixed point theorems in generalized D* -metric spaces, Applied Sciences, 12 (2010), 1-13.

[2] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal., 341 (2008), 416-420.

[3] T. Bag, Some results on D* -fuzzy metric spaces, International Journal of Mathematics and Scientific Computing, 2, No.1 (2012), 29-33.

[4] T. Bag, Fuzzy cone metric spaces and fixed point theorems of contractive mappings, Annals of Fuzzy Mathematics and Informatics, 2, No.1 (2012), 29-33.

[5] B. C. Dhage, Generalized metric spaces and mappings with fixed point, Bull. Cal. Math. Soc., 84 (1992), 329-336.

[6] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems, 48 (1992), 239-248. [7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64

(1994), 395-399. [8] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), 215-229. [9] H. Long-Guang and Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings, J.

Math. Anal. Appl., 332 (2007), 1468-1476. [10] M. Mizumoto and J. Tanaka, Some properties of fuzzy numbers in: M. M. Gupta et al. Editors,

Advances in Fuzzy Set Theory and Applications (North-Holland, New-York), (1979), 153-164. [11] Sh. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point

theorems of contractive mappings”, J. Math. Anal. Appl., 345 (2008), 719-724. [12] S. Sedghi and N. Shobe, Fixed point theorem in M -fuzzy metric spaces with property (E), Advances

in Fuzzy Mathematics, 1 (1), (2006), 55-65.

[13] S. Sedghi, N. Shobe and H. Zhou, A common fixed point theorem in D* -metric spaces, Fixed Point Theory and Applications, 2007, article ID. 27906, (2007), 1-13.

[14] L. A. Zadeh, Fuzzy sets, Inform. and Control, Vol.8 (1965), 338-353.


__________________________ Received January, 2014


Los Angeles

New Distance and Similarity Measures for Soft Sets

Wang Ling, Qin Keyun and Liu Yaya

College of Mathematics, Southwest Jiaotong University, Sichuan, Chengdu, 610031, China E-mail: [email protected], [email protected], [email protected]

Abstract: In this paper, we will show that some results presented in [3] may be unreasonable and

proper results will be introduced. Meanwhile new distance measure and similarity measures for oft sets will be proposed. Furthermore, some application examples about soft sets are also given.

Keywords: Soft set, Distance measure, Similarity measure

1. Introduction

In order to handle the uncertain problem that exist in engineering, management, economic and medical sciences, researchers have proposed probability theory, fuzzy set theory [5], rough set theory [6], intuitionistic fuzzy set theory [7] etc. They are convenient to describe uncertainties and help us to handle lots of practical problems in our life. However, these approaches have their limitations [1]. In 1999, Molodtsov [1] introduced the soft set theory, it is a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. And it is free from the inadequacy arising form the theories mentioned above.

Currently, research work on soft sets has made great progress. Maji et al. [9] introduced the application of soft sets to a decision making problem. Chen et al. [13] proposed a new definition of parameterization reduction of soft sets and compared it with the concept of attributes reduction in rough sets theory. Maji et al. [2] proposed some operations on soft sets and made a theoretical study of the soft set theory in more detail. M. Irfan et al. [12] also gave some new operations such as the restricted intersection, the restricted union, and the extended intersection of two soft sets. At the same time, they improved the notion of complement of soft set. Aktas and Cagman [15] compared soft set to the related concepts of fuzzy sets and rough sets. They also gave a definition of soft groups and derived their basic properties. Jun [16] introduced the notion of soft

/BCK BCI -algebras and soft subalgebras, and derived their basic properties. Babitha et al. [14] discussed the transitive closures, and orderings on a soft set is defined and also

162 Wang Ling, Qin Keyun and Liu Yaya

obtain some set theoretical results. Qin et al. [10] presented the concept of soft equality and introduced some related properties. Feng et al. [11] introduced the notion of soft rough sets and soft rough fuzzy set.

Majumdar and Samanta initiated the study of uncertainty measures for soft sets. They proposesd some distance measures for soft sets with the same parameter sets in [4], and also introduced similarity measure for soft sets. Kharal introduced a general approach to study the distance measure and similarity measures of soft sets in [3]. Yang has also proposed new similarity measure in [8].

In this paper, we will present some counterexamples to show that some propositions in [3] may be unreasonable and proper results will be introduced. Furthermore, in Section 4, new distance measures and similarity measures are proposed for soft sets. Meanwhile some propositions and example of its application will be introduced.

2. Preliminaries

In this section, we review some basic notions in soft set theory. Let U be an initial universe set and E be the set of all possible parameters under consideration with respect to U . The set of all subsets of U is denoted by ( )P U . Molodtsov [1] defined the notion of a soft set in the following way.

Definition 1 [1]. A pair ( ),F A is called a soft set over U , where A EÕ , and F is

a mapping given by ( ):F A P UÆ . In other words, the soft set is a parameterized family of subsets of the set U . For any

parameter Ae Œ , ( )F Ae Õ may be considered as the set of e -approximate elements

of the soft set ( ),F A .

Definition 2 [3]. Let U be a universe and E be a set of parameters. Then the pair ( ),U E , called a soft space, is the collection of all soft sets on U with parameters from E .

Definition 3 [2]. For two soft sets ( ),F A and ( ),G B over U , we say that ( ),F A is

a soft subset of ( ),G B , if

(i) A BÕ ;

(ii) Ae" Œ , ( ) ( )F Ge eÕ .

We write ( ) ( ), ,F A G BÕ . ( ),F A is said to be a soft super set of ( ),G B , if ( ),G B

is a soft subset of ( ),F A .

Definition 4 [12]. The complement of a soft set ( ),F A is denoted by ( ), cF A , and is

defined by ( ) ( ), ,c cF A F A= , where ( ):cF A P UÆ is a mapping given by ( )cF e =

( )( )cF e , Ae" Œ .

New Distance and Similarity Measures for Soft Sets 163

Definition 5 [3]. A mapping ( ) ( ) [ ]: , , 0,1S U E U E¥ Æ is said to be similarity

measure if its values ( ) ( )( ), , ,S F A G B , for arbitrary soft sets ( ),F A and ( ),G B in the

soft space ( ),U E , satisfies following axioms:

(S1) ( ) ( )( )0 , , , 1S F A G B£ £ ;

(S2) ( ) ( )( ), , , 1S F A F A = ;

(S3) ( ) ( )( ) ( ) ( )( ), , , , , ,S F A G B S G B F A= ;

(S4) If ( ) ( ) ( ), , ,F A G B H CÕ Õ , then ( ) ( )( ) ( ) ( )( ), , , , , ,S F A H C S F A G B£ and

( ) ( )( ) ( ) ( )( ), , , , , ,S F A H C S G B H C£ .

Definition 6 [3]. Let ( ),F A , ( ),G B and ( ),H C be soft sets in a soft space ( ),U E

and :d X X R+¥ Æ a mapping. Then (1) d is said to be quasi-metric if it satisfies ( )1M ( ) ( )( ), , , 0d F A G B ≥ .

( )2M ( ) ( )( ) ( ) ( )( ), , , , , ,d F A G B d G B F A= . (2) A quasi-metric d is said to be semi-metric if ( )3M ( ) ( )( ) ( ) ( )( ) ( ) ( )( ), , , , , , , , ,d F A G B d G B H C d F A H C+ ≥ .

(3) A semi-metric d is said to be pseudo metric if ( )4M ( ) ( ) ( ) ( )( ), , , , , 0F A G B d F A G B= fi = .

(4) A pseudo metric d is said to be metric if ( )5M ( ) ( )( ) ( ) ( ), , , 0 , ,d F A G B F A G B= fi = .

Definition 7 [2]. Denote the absolute null and absolute whole soft sets in a soft space ( ),U E as ( ),F E∆ , ( ),UF E resepctivley, they have been defined as ( )F e∆ = ∆ ,

( )UF Ue = for each Ee Œ .

3. Counterexamples

In this section, we will show that some Lemma and propositions presented in [3] are not reasonable. And we will modify them.

Symmetric difference between two sets A and B is denoted and defined as:

( ) ( )\A B A B A BD = » « .

Definition 8 [3]. For two soft sets ( ),F A and ( ),G B in a soft space ( ),U E , where A B» π ∆ , we define Euclidean distance as:


( ) ( )( ) ( ) ( ) 2, , ,

A B

e F A G B A B F Ge

e eŒ «

= D + DÂ

Normalized Euclidean distance as:

( ) ( )( ) ( ), , ,A B

A Bq F A G B

A B ec e

Œ «

D= +

»Â

where ( ) ( ) ( ) ( ) ( )2F G F Gc e e e e e= D » if ( ) ( )F Ge e» π ∆ ; ( ) 0c e = otherwise.

Lemma 9 [3]. For the soft sets ( ),F E∆ , ( ),UF E and an arbitrary soft set ( ),F A in

a soft space ( ),U E , we have:

(1) ( ) ( )( ), , , 2ce F A F A A= .

(2) ( ) ( )( ), , , 2cq F A F A A= .

(3) ( ) ( )( ), , ,Ue F E F E E U∆ = .

(4) ( ) ( )( ), , ,Uq F E F E E∆ = .

Example 1. Let ( ),U E be a soft space with { }, , ,U a b c d= and { }1 2 3 4, , ,E e e e e= ,

suppose ( ),F A is a soft set defined by: { }1 2 3, ,A e e e= , ( ) { }1 ,F e a d= , ( ) { }2 ,F e b c= ,

( ) { }3 , ,F e a b c= . Then ( ) { }1 ,cF e b c= , ( ) { }2 ,cF e a d= , ( ) { }3cF e d= . Hence

( ) ( )( ) ( ) ( ) 2, , , 48 2 8c c

A B

e F A F A A A F F Ae

e eŒ «

= D + D = π =Â .

( ) ( )( ) ( ), , , 12 2 8c

A B

A Aq F A F A A

A A ec e

Œ «

D= + = π =

»Â .

By this Example, (3) and (4) in Lemma 9 are not suitable. In fact, we have the following theorem.

Theorem 10. For the soft sets ( ),F E∆ , ( ),UF E and an arbitrary soft set ( ),F A in

a soft space ( ),U E , we have:

(1’) ( ) ( )( ), , , ce F A F A X A= .

(2’) ( ) ( )( ), , , cq F A F A X A= .

(3’) ( ) ( )( ), , ,Ue F E F E U E∆ = .

(4’) ( ) ( )( ), , ,Uq F E F E U E∆ = .


Proof. (1’) For each e AŒ , we have ( ) ( ) ( ) ( )( ) ( )(\c cF F F F Fe e e e eD = » «

( ))cF Ue = . And consequently ( ) ( )( ) ( ) ( ) 2, , , c c

A

e F A F A A A F Fe

e eŒ

= D + D =Â

U A .

(4’) For any Ee Œ , by ( ) ( )XF F Xe e∆ » = π ∆ , we have ( ) ( ) ( )( ) ( )

2U

U

F F

F F

e ec e

e e∆

∆

D=

»

2U

UU

= = . And consequently

( ) ( )( ) ( ) ( )( ) ( )

2

, , , U

UA A U

F FE Eq F E F E E U

F FE E e

e ee e

∆∆

Œ « ∆

DD= + = ◊

»»Â .

(2’), (3’) can be proven in a similar way as (1’), (4’).

Definition 11 [3]. For two soft sets ( ),F A and ( ),G B in a soft space ( ),U E , we define a set theoretic matching function similarity measure as:

( ) ( )( ) ( )( ) ( )( ) ( )( ), , ,

max , max ,A B

A B

F GA B

S F A G BA B F G

e

e

e e

e eŒ «

Œ «

««= +

Â

Â.

Proposition 12 [3]. For an arbitrary soft set ( ),F A in a soft space ( ),U E , we have

( ) ( )( ), , , 0cS F A F A = .

By the Definition 8, we can get the Proposition 12 is not correct, it should be modified as ( ) ( )( ), , , 1cS F A F A = .

4. New distance measure and new similarity measure

4.1 New distance measure Majumdar [4] have proposed distance measure for soft sets, but those distance

measures for soft sets with the same parameter set E . The measure is a partial measure. Kharal thought that distance measures and similarity measures for soft sets should be related between both the values of parameters and the mapping on it. Hence we define distance measures for soft sets in the following way:

Definition 13. Let ( ),F A and ( ),G B be soft sets in a soft space ( ),U E ,

( ) ( ): , ,d U E U E R¥ Æ be a mapping d is called a distance measure if it satisfied the following conditions:

(1) ( ) ( )( ), , , 0d F A G B ≥ ;


(2) ( ) ( )( ) ( ) ( )( ), , , , , ,d F A G B d G B F A= ;

(3) ( ) ( )( ), , , 0d F A G B = if and only if ( ) ( ), ,F A G B= ;

(4) If ( ) ( ) ( ), , ,F A G B H CÕ Õ , then ( ) ( )( ) ( ) ( )( ), , , , , ,d F A G B d F A H C£ and

( ) ( )( ) ( ) ( )( ), , , , , ,d G B H C d F A H C£ .

Theorem 14. ( ) ( )( ), , ,d F A G B is a distance measure, where

( ) ( )( ) ( ), , ,A B

d F A G B A Be

c eŒ «

= D + Â

( ) ( ) ( )( ) ( )F G

F G

e ec e

e eD

=»

if ( ) ( ) 0F Ge e» π and ( ) 0c e = otherwise.

Proof. (1), (2) and (3) are trivial. (4) When ( ) ( ) 0F Ge e» π , then we have ( ) ( ) ( ) ( ) ( )F G F Gc e e e e e= D » .

Let ( )F me = , ( )G ne = , ( )H pe = . ( )m n p£ £ . Since

( ) ( ) ( ), , ,F A G B H CÕ Õ , Ae" Œ , then

( ) ( )( ) ( )F G n m

nF G

e ee e

D -=

»,

( ) ( )( ) ( )F H p m

pF H

e ee e

D -=

».

By ( ) 0p m p n m n m p n pn- - - = - ≥ , and A C A BD ≥ D we have

( ) ( )( ) ( )

( ) ( )( ) ( )A B A C

F G F HA B A C

F G F He e

e e e ee e e eŒ « Œ «

D DD + £ D +

» »Â Â .

As a result, we get ( ) ( )( ) ( ) ( )( ), , , , , ,d F A G B d F A H C£ . Similarly we can get

( ) ( )( ) ( ) ( )( ), , , , , ,d G B H C d F A H C£ .

When ( ) ( ) 0F Ge e» = , the proof is trivial. This completes the proof of Theorem 14. The normalized distance can be defined as:

( ) ( )( ) ( ( ) ( ), , ,A B

d F A G B A B A Be

c eŒ «

= D + »¢ Â .

Example 2. Let ( ),U E be a soft space with { }1 2 3 4 5, , , ,U x x x x x= and { 1 2, ,E e e=

}3 4,e e . Suppose ( ),F A and ( ),G B are two soft sets in soft space ( ),U E given by:


{ }1 3,A e e= , ( ) { }1 1 3,F e x x= , ( ) { }3 1 2 4, ,F e x x x= ; { }1 2 4, ,B e e e= , ( ) { }1 1 3 5, ,G e x x x= ,

( ) { }2 2 3 4, ,G e x x x= , ( ) { }4 1 4 5, ,G e x x x= ; { }1 2 3 4, , ,C e e e e= , ( ) { }1 2 3 4, ,H e x x x= ,

( ) { }2 1 2 5, ,H e x x x= , ( ) { }3 2 3,H e x x= , ( ) { }4 1 3 4 5, , ,H e x x x x= . Then ( )( (, , ,d F A G

)) 31 3B = , ( ) ( )( ), , , 57 20d G B H C = , ( ) ( )( ), , , 31 2d F A H C = .

4.2 New similarity measure It is obvious that a soft set is determined by both parameter set and a mapping on it.

So similarity measures for soft sets reflect not only the values of the same parameters, but also related the mapping about the parameters. Based upon this idea, we proposed new similarity measure in the following way:

Theorem 15. 1S is a similarity measure, where

( ) ( )( )1 , , ,A B

S F A G B tA B

«= ¥

»,

( ) ( )( ) ( )

A B

A B

F G

tF G

e

e

e e

e eŒ «

Œ «

«=

»

Â

Â if ( ) ( ) 0

A B

F Ge

e eŒ «

» πÂ , and 1t = otherwise.

Proof. (1) A Be" Œ « , since ( ) ( ) ( ) ( )F G F Ge e e e« £ » , hence

( ) ( )( )( ) ( )( ) ( )10 , , , 1 1 1A B

A B

F e G eA B

S F A G BA B F e G e

e

e

Œ «

Œ «

««£ = ¥ £ ¥ =

» »

Â

Â.

(2) If ( ) ( ), ,F A G B= , then ( ) ( ) ( ) ( )F G F Ge e e e« = » , hence

( ) ( )( )( ) ( )( ) ( )1 , , , 1 1 1A A

A A

F e F eA A

S F A F AA A F e F e

e

e

Œ «

Œ «

««= ¥ = ¥ =

» »

Â

Â.

(3) It is obvious that

( ) ( )( )( ) ( )( ) ( ) ( ) ( )( )1 1, , , , , ,A B

A B

F e G eA B

S F A G B S G B F AA B F e G e

e

e

Œ «

Œ «

««= ¥ =

» »

Â

Â.

(4) Assume that ( ) ( ) ( ), , ,F A G B H CÕ Õ . Thus A B CÕ Õ and for every

Ae Œ ¸ ( ) ( ) ( )F e G e H eÕ Õ . This implies


( ) ( )( )( ) ( )( ) ( )

( )( )1 , , , A C A

A C A

F e H e F eA C A

S F A H CA C CF e H e H e

e e

e e

Œ « Œ

Œ « Œ

««= ¥ = ¥

» »

Â Â

Â Â

( )( )

( ) ( )( ) ( ) ( ) ( )( )1 , , ,A A B

A A B

F e F e G eA A B

S F A G BB A BG e F e G e

e e

e e

Œ Œ «

Œ Œ «

««£ ¥ = ¥ =

» »

Â Â

Â Â.

Similarly, we can get ( ) ( )( ) ( ) ( )( )1 1, , , , , ,S F A H C S G B H C£ . This completes the proof of Theorem 15. Similarly, we have the following theorem.

Theorem 16. 2S is a similarity measure, where

( ) ( )( )2 , , ,A B A B

S F A G B tA B

« «= ¥ ¥ ,

( ) ( )( ) ( )

A B

A B

F G

tF G

e

e

e e

e eŒ «

Œ «

«=

»

Â

Â if ( ) ( ) 0

A B

F Ge

e eŒ «

» πÂ , and 1t = otherwise.

Example 3 [3]. Profile 1. The firm ABC maintains a bearish future outlook as well as same behaviour in trading of its share prices. During last fiscal year the profit-earning ratio continued to rise. Inflation is increasing continuously ABC has a low amount of paid-up capital and a similar situation is seen in foreign direct investment flowing into ABC .

Profile 2. The firm XYZ showed a fluctuating share price and hence a varying future outlook. Like ABC profit-earning ratio remained bearish. As both firms are in the same economy, inflation is also rising for XYZ and may be consider even high in view of XYZ . Competition in the business area of XYZ is increasing. Debit level went high but the paid-up capital lowered.

For this, we first construct a model soft for liquidity-problem and the soft sets for the firm profile. Next we find the similarity measure of these soft sets. For the sake of ease in mathematical manipulation we denote the indication and labels by symbols as follows:

inflationi = , profit earning ratiop = , share prices = 1 fluctuatinge = , paid-up c = capital 2 lowe = , competitionm = 3 risinge = , business diversificationd = 4 highe = ,

future outlooko = 5 bearishe = , debt levell = , foreign direct investmentf = , x = fixed income .

Thus we have soft space ( ),U E , { }, , , , , , , ,U i b c m d o l f x= and { }1 2 3 4 5, , , ,E e e e e e= . The soft sets of firm profile are as follows:

{ }2 3 5, ,A e e e= , ( ) { }2 ,F e s f= , ( ) { }3 ,F e p i= , ( ) { }5 ,F e o s= .

{ }1 2 3 4 5, , , ,B e e e e e= , ( ) { }1 ,G e o s= , ( ) { }2G e c= , ( ) { }3 ,G e i m=


( ) { }4 ,G e i l= , ( ) { }5 ,G e p f= .

A standard soft set for a firm suffering from liquidity problem will be given. We can take it to be as follows:

{ }1 2 4 5, , ,C e e e e= , ( ) { }1 ,H e o s= , ( ) { }2H e c= , ( ) { }4 ,H e i l= , ( ) { }5 ,H e p f= .

Therefore:

( ) ( )( )1 , , , 0.4 0 0S F A H C = ¥ = , ( ) ( )( )1 , , , 0.8 1 0.8S G B H C = ¥ = .

Hence we conclude that the firm XYZ is suffering from a liquidity problem as its soft set is signification similar to the standard liquidity problem, whereas the firm XYZ is likely to be suffering the same problem.

5. Conclusion

Majumdar and Samanta proposed some distance measures for soft sets with the same parameter sets. But the measure is a partial measure. Kharal introduced new distance measures and similarity measures for soft sets. In this paper, we have pointed out some unreasonable propositions in [3] and introduced proper results. Meanwhile, we proposed some distance and similarity measures for soft sets. And application example about soft sets is also given. We believe these approaches can help us to handle uncertain problems.

Acknowledgements

This work has been supported by the National Natural Science Foundation of China (Grant No.61175044, 61175055) and the Fundamental Research Funds for the Central Universities of China (Grant No. SWJTU11ZT29).

References

[1] D. Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications, 37 (1999), 19-31.

[2] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Computers and Mathematics with Applications, 45 (2003), 555-562.

[3] A. Kharal, Distance and similarity measures for soft sets, New Mathematics and Natural Computation, 6 (2010), 321-334.

[4] P. Majumdar and S. K. Samanta, Similarity measure of soft sets, New Mathematics and Natural Computation, 4 (2008), 1-12.

[5] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. [6] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982), 341-

356. [7] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.


[8] Wei Yang, New similarity measure for soft sets and their application, Fuzzy Information and Engineering, 1 (2013), 19-25.

[9] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Computers and Mathematics with Applications, 44 (2002), 1077-1083.

[10] K. Qin and Z. Hong, On soft equality, Journal of Computational and Applied Mathematics, 234 (2010), 1347-1355.

[11] F. Feng, C. X. Li, B. Davvaz and M. I. Ali, Soft sets combined with fuzzy sets and rough sets, A Tentative Approach, Soft Computing, 14 (2010), 899-911.

[12] M. Irfan Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications, 57 (2009), 1547-1553.

[13] D. G. Chen, E. C. C. Tsang, D. S. Yeung and X. Z. Wang, The parameterization reduction of soft sets and its applications, Computers and Mathematics with Applications, 49 (2005), 757-763.

[14] K. V. Babitha and Jacob John Sunil, Transitive closures and orderings on soft sets, Computers and Mathematics with Applications, 62 (2011), 2235-2239.

[15] H. Aktas and N. Cagman, Soft sets and soft groups, Information Science, 177 (2007), 2726-2735.

[16] Y. B. Jun, Soft BCK BCI -algebras, Computers and Mathematics with Applications, 56 (2008), 1408-1413.


__________________________ Received February, 2014


Los Angeles

On Fuzzy Upper and Lower b -irresolute Multifunctions

S. E. Abbas

Department of Mathematics, Faculty of Science, Jazan University, Saudi Arabia.

M. A. Hebeshi and I. M. Taha

Department of Mathematics, Faculty of Science, Sohag University, Egypt E-mail: [email protected]

Abstract: The main purpose of this paper is to introduce and study fuzzy upper and fuzzy lower

b -irresolute, b -continuous and strongly semi b -irresolute multifunctions. Also, several characterizations and properties of these multifunctions along with their mutual relationships are established in fuzzy topological spaces.

Key words and phrases: Fuzzy topology, fuzzy multifunction, graph multifunction, upper and lower b -

continuous, b -irresolute, strongly semi b -irresolute, composition, union and compactness.

1. Introduction and preliminaries

Kubiak [15] and Sostak [24] introduced the fundamental concept of a fuzzy topological structure, as an extension of both crisp topology and fuzzy topology [6], in the sense that not only the objects are fuzzified, but also the axiomatics. In [25,26], Sostak gave some rules and showed how such an extension can be realized. Chat-topadhyay et al., [7] have redefined the same concept under the name gradation of openness. A general approach to the study of topological type structures on fuzzy power sets was developed in [10-16].

Berge [5] introduced the concept multimapping :F X YÆ where X and Y are topological spaces and Popa [20,21] introduced the notion of irresolute multimapping. After Chang introduced the concept of fuzzy topology [6], continuity of multifunctions in fuzzy topological spaces have been defined and studied by many authors from different view points (e.g. see [3,4,17-19]). Later, Tsiporkova et al., [27, 28] introduced the Continuity of fuzzy multivalued mappings in the Chang’s fuzzy topology [6].

172 S. E. Abbas, M. A. Hebeshi and I. M. Taha

Throughout this paper, nonempty set swill be denoted by X , Y etc., [ ]0,1L = and

( ]0,1L = . The family of all fuzzy sets in X is denoted by XL . For La Œ , ( )xa a=

for all x XŒ . A fuzzy point tx for 0t LŒ is an element of XL such that

( ) , if ,0, if .t

t y xx y

y x

=Ï= Ì πÓ

The family of all fuzzy points in X is denoted by ( )Pt X . A fuzzy point tx lŒ iff

( )t xl< . All other notations are standard notations of fuzzy set theory.

Definition 1.1 [2]. Let :F X Y . Then F is called a fuzzy multifunction (FM , for short) iff ( ) YF x LŒ for each x XŒ . The degree of membership of y in ( )F x is

denoted by ( )( ) ( ),FF x y G x y= for any ( ),x y X YŒ ¥ .

The domain of F , denoted by ( )dom F and the range of F , denoted by ( )rng F ,

for any x XŒ , ( )( ) ( ): ,y Y Fy Y dom F x G x yŒŒ = � and ( )( ) ( ),x X Frng F y G x yŒ= � .

Definition 1.2 [2]. Let :F X Y be a FM . Then F is called: (1) Normalized iff for each x XŒ , there exists 0y YŒ such that ( )0, 1FG x y = .

(2) A crisp iff ( ), 1FG x y = for each x XŒ and y YŒ .

Definition 1.3 [2]. Let :F X Y be a FM . Then, (1) The image of XLl Œ is a fuzzy set ( ) YF Ll Œ and defined by:

( )( ) ( ) ( ),x X FF y G x y xl lŒ È ˘= ŸÎ ˚� .

(2) The lower inverse of YLm Œ is a fuzzy set ( )l XF Lm Œ and defined by:

( )( ) ( ) ( ),ly Y FF x G x y ym mŒ È ˘= ŸÎ ˚� .

(3) The upper inverse of YLm Œ is a fuzzy set ( )u XF Lm Œ and defined by:

( )( ) ( ) ( ),u cy Y FF x G x y ym mŒ È ˘= ⁄Î ˚� .

Theorem 1.4 [2]. Let :F X Y be a FM . Then, (1) ( ) ( )1 2F Fl l£ if 1 2l l£ .

(2) ( ) ( )1 2l lF Fm m£ and ( ) ( )1 2

u uF Fm m£ if 1 2m m£ .

(3) ( ) ( )1 1l uF Fm m- = - .

(4) ( ) ( )1 1u lF Fm m- = - .

(5) ( )( )uF F m m£ if F is a crisp.

(6) ( )( )uF F l l≥ if F is a crisp.

On Fuzzy Upper and Lower b -irresolute Multifunctions 173

Definition 1.5 [2]. Let :F X Y and :H Y Z be two FM ’s. Then the composition H F is defined by: ( )( )( )( ) ( ) ( ), ,y Y F HH F x z G x y G y zŒ È ˘= ŸÎ ˚� .

Theorem 1.6 [2]. Let :F X Y and :H Y Z be two FM ’s. Then we have the following:

(1) ( ) ( )H F F H= .

(2) ( ) ( )u u uH F F H= .

(3) ( ) ( )l l lH F F H= .

Theorem 1.7 [2]. Let :iF X Y be a FM . Then, (1) ( )( ) ( )i i i iF Fl lŒG ŒG=∪ � .

(2) ( ) ( ) ( )l li i i iF Fm mŒG ŒG=∪ � .

(3) ( ) ( ) ( )u ui i i iF Fm mŒG ŒG=∪ � .

Definition 1.8 [22,24]. A fuzzy topological space (fts, for short) is pair ( ),X t , where

X is a nonempty set and : XL Lt Æ is a mapping satisfying the following properties: (O1) ( ) ( )0 1 1t t= = ,

(O2) ( ) ( ) ( )1 2 1 2t l l t l t lŸ ≥ Ÿ , for any 1 2, XLl l Œ .

(O3) ( ) ( )i i i it l t lŒG ŒG≥� � , for any { } Xi i

Ll ŒG Ã . Then t is called a fuzzy topology on X .

Theorem 1.9 [8,23]. Let ( ),X t be a fts. Then for each XLl Œ , r LŒ we define

operators Ct and : X XI L L Lt ¥ Æ as follows:

( ) ( ){ }, : , 1XC r L rt l m l m t m= Œ £ - ≥� .

( ) ( ){ }, : ,XI r L rt l m m l t m= Œ £ ≥� .

For , XLl m Œ and ,r s LŒ the operator Ct satisfies the following statements: (C1) ( )0, 0C rt = .

(C2) ( ),C rtl l£ .

(C3) ( ) ( ) ( ), , ,C r C r C rt t tl m l m⁄ = ⁄ .

(C4) ( )( ) ( ), , ,C C r r C rt t tl l= .

(C5) ( ),C rt l l= iff ( )1 rt l- ≥ .

(C6) ( ) ( )1 , 1 ,C r I rt tl l- = - and ( ) ( )1 , 1 ,I r C rt tl l- = - .

Definition 1.10 [13,23]. Let ( ),X t be a fts. Then for each , XLl m Œ and r LŒ . Then l is called:


(1) r -fuzzy semi-open ( -fsor , for short) iff ( )( ), ,C I r rt tl l£ .

(2) r -fuzzy a -open ( -r f oa , for short) iff ( )( )( ), , ,I C I r r rt t tl l£ .

(3) r -fuzzy b -open ( -r f ob , for short) iff ( )( )( ), , ,C I C r r rt t tl l£ .

(4) r -fuzzy semi-closed ( -r fsc , for short) iff ( )( ), ,I C r rt t l l£ .

(5) r -fuzzy a -closed ( -r f ca , for short) iff ( )( )( ), , ,C I C r r rt t t l l£ .

(6) r -fuzzy b -closed ( -r f cb , for short) iff ( )( )( ), , ,I C I r r rt t t l l£ .

Theorem 1.11 [13]. Let ( ),X t be a fts. Then for each XLl Œ , r LŒ we define

operators SCt and : X XSI L L Lt ¥ Æ as follows:

( ) { }, : , is -XSC r L r fsct l m l m m= Œ £�

( ) { }, : , is -XSI r L r fsot l m m l m= Œ £� .

Theorem 1.12 [1]. Let ( ),X t be a fts. Then for each XLl Œ , r LŒ we define

operators Ctb and : X XI L L Ltb ¥ Æ as follows:

( ) { }, : , is -XC r L r f ctb l m l m m b= Œ £� .

( ) { }, : , is -XI r L r f otb l m m l m b= Œ £� .

For , XLl m Œ and ,r s LŒ the operator Ctb satisfies the following statements: (1) ( )0, 0C rtb = .

(2) ( ),C rtl b l£ .

(3) ( ) ( ) ( ), , ,C r C r C rt t tb l b m b l m⁄ £ ⁄ .

(4) ( )( ) ( ), , ,C C r r C rt t tb b l b l= .

(5) l is -r f cb iff ( ),C rtb l l= .

(6) ( ) ( )1 , 1 ,C r I rt tb l b l- = - and ( ) ( )1 , 1 ,I r C rt tb l b l- = - .

Definition 1.13 [9]. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and r LŒ . Then F is called:

(1) FUa -continuous at ( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ and

( ) rh m ≥ there exists XLl Œ , l is -r f oa and tx lŒ such that ( ) ( )udom F Fl mŸ £ .

(2) FLa -continuous at ( )tx dom FŒ iff ( )ltx F mŒ for each YLm Œ and

( ) rh m ≥ there exists XLl Œ , l is -r f oa and tx lŒ such that ( )lFl m£ . (3) FUa -continuous (resp. FLa -continuous) iff it is FUa -continuous (resp.

FLa -continuous) at every ( )tx dom FŒ .


Theorem 1.14 [2,9]. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h

and YLm Œ . Then we have the following:

(1) F is FLS -continuous iff ( )( ) ( )lFt m h m≥ .

(2) If F is normalized, then F is FUS -continuous iff ( )( ) ( )uFt m h m≥ .

(3) F is FLa -continuous iff ( )lF m is -r f oa , ( ) rh m" ≥ .

(4) If F is normalized, then F is FUa -continuous iff ( )uF m is -r f oa ,

( ) rh m" ≥ .

Definition 1.15 [14]. A fuzzy set l in a fts ( ),X t is called r -fuzzy compact iff

every family in ( ){ }: , and Xr L r Lm t m m> Œ Œ covering l has a finite subcover.

Definition 1.16. A fuzzy set l in a fts ( ),X t is called r -fuzzy b -compact iff

every family in { }: is - , and Xr f o L r Lm m b m Œ Œ covering l has a finite subcover.

Definition 1.17 [2]. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and

r LŒ . Then F is called compact-valued iff ( )tF x is r -fuzzy compact for each

( )tx dom FŒ .

Definition 1.18. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and

r LŒ . Then F is called b -compact-valued iff ( )tF x is r -fuzzy b -compact for

each ( )tx dom FŒ .

2. Fuzzy upper and lower b -irresolute multifunctions

Definition 2.1. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and r LŒ . Then F is called:

(1) Fuzzy upper b -irresolute ( FUb -irresolute, for short) at a fuzzy point

( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ and m is -r f ob there exists XLl Œ , l

is -r f ob and tx lŒ such that ( ) ( )udom F Fl mŸ £ . (2) Fuzzy lower b -irresolute ( FLb -irresolute, for short) at a fuzzy point

( )tx dom FŒ iff ( )ltx F mŒ for each YLm Œ and m is -r f ob there exists XLl Œ , l

is -r f ob and tx lŒ such that ( )lFl m£ . (3) FUb -irresolute (resp. FLb -irresolute iff it is FUb -irresolute (resp. FLb -

irresolute) at every ( )tx dom FŒ .



(1) Fuzzy upper b -continuous ( FUb -continuous, for short) at a fuzzy point

( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ and ( ) rh m ≥ there exists XLl Œ , l is

-r f ob and tx lŒ such that ( ) ( )udom F Fl mŸ £ . (2) Fuzzy lower b -continuous ( FLb -continuous, for short) at a fuzzy point

( )tx dom FŒ iff ( )ltx F mŒ for each YLm Œ and ( ) rh m ≥ there exists XLl Œ , l is

-r f ob and tx lŒ such that ( )lFl m£ . (3) FUb -continuous (resp. FLb -continuous) iff it is FUb -continuous (resp.

FLb -continuous) at every ( )tx dom FŒ .

Proposition 2.3. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and r LŒ . If F is normalized, then F is:

(1) FUb -irresolute at a fuzzy point ( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ

and m is -r f ob there exists XLl Œ , l is -r f ob and tx lŒ such that ( )uFl m£ .

(2) FUb -continuous at a fuzzy point ( )tx dom FŒ iff ( )utx F mŒ for each

YLm Œ and ( ) rh m ≥ there exists XLl Œ , l is -r f ob and tx lŒ such that

( )uFl m£ .

Theorem 2.4. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h and YLm Œ the following are equivalent:

(1) F is FLb -irresolute.

(2) ( )lF m is -r f ob , for any m is -r f ob .

(3) ( )uF m is -r f cb , for any m is -r f cb .

(4) ( )( ) ( )( ), ,u uC F r F C rt hb m b m£ , for any YLm Œ .

(5) ( )( )( )( ) ( )( ), , , ,u uI C I F r r r F C rt t t hm b m£ , for any YLm Œ .

Proof. (1) fi (2) Let ( )tx dom FŒ , m is -r f ob and ( )ltx F mŒ then, there exists

XLl Œ , l is -r f ob and tx lŒ such that ( )lFl m£ thus, ( ( ( ( ) ), ,l

tx C I C F rt t t mŒ

) ),r r . Then, we obtain ( ) ( )( )( )( ), , ,l lF C I C F r r rt t tm m£ . Hence, ( )lF m is -r f ob .

(2) fi (3) Let m is -r f cb hence by (2), ( ) ( )1 1l uF Fm m- = - is -r f ob . Then,

( )uF m is -r f cb .

(3) fi (4) Let YLm Œ hence by (3), ( )( ),uF C rhb m is -r f cb . Then, we obtain

( )( ) ( )( ), ,u uC F r F C rt hb m b m£ .


(4) fi (5) Let YLm Œ hence by (4), we obtain ( )( )( )( ), , ,uI C I F r r r Ct t t tm b£

( ( ) ) ( )( ), ,u uF r F C rhm b m£

(5)fi (2) Let m is -r f ob hence by (5), we have ( ) ( )1 1l uF Fm m- = - (I Ct t≥

( ( )( ) ) )1 , , ,uI F r r rt m- ( )( )( )( )1 , , ,lI C I F r r rt t t m= - ( ( ( ( )1 ,lC I C Ft t t m= -

) ) ), ,r r r . Then, we obtain ( ) ( )( )( )( ), , ,l lF C I C F r r rt t tm m£ and hence ( )lF m is

-r f ob .

(2)fi (1) Let ( )tx dom FŒ , m is -r f ob and ( )ltx F mŒ we have by (2), ( )lF m l=

(say) is -r f ob then, there exists l is -r f ob and tx lŒ such that ( )lFl m£ . Thus F is FLb -irresolute.


(1) F is FLb -continuous.

(2) ( )lF m is -r f ob , for any ( ) rh m ≥ .

(3) ( )uF m is -r f cb , for any ( )1 rh m- ≥ .

(4) ( )( ) ( )( ), ,u uC F r F C rt hb m m£ , for any YLm Œ .

(5) ( )( )( )( ) ( )( ), , , ,u uI C I F r r r F C rt t t hm m£ , for any YLm Œ .

Proof. (1) fi (2) Let ( )tx dom FŒ , YLm Œ , ( ) rh m ≥ and ( )ltx F mŒ then, there

exists XLl Œ , l is -r f ob and tx lŒ such that ( )lFl m£ and hence ( (tx C I Ct t tŒ

( )( ) ) ), , ,lF r r rm . Thus, we obtain ( ) ( )( )( )( ), , ,l lF C I C F r r rt t tm m£ . Then,

( )lF m is -r f ob .

(2) fi (3) Let YLm Œ and ( )1 rh m- ≥ hence by (2), ( ) ( )1 1l uF Fm m- = - is

-r f ob . Then, ( )uF m is -r f cb .

(3) fi (4) Let YLm Œ hence by (3), ( )( ),uF C rh m is -r f cb . Then, we obtain

( )( ) ( )( ), ,u uC F r F C rt hb m m£ .

(4) fi (5) Let YLm Œ hence by (4), we obtain ( )( )( )( ), , ,uI C I F r r r Ct t t tm b£

( )( ) ( )( ), ,u uF r F C rhm m£ .

(5) fi (2) Let YLm Œ , ( ) rh m ≥ hence by (5), we have ( ) ( )1 1l uF Fm m- = - ≥

( )( )( )( )1 , , ,uI C I F r r rt t t m- ( )( )( )( )1 , , ,lI C I F r r rt t t m= - ( ( (1 lC I C Ft t t= -

( ) ) ) ), , ,r r rm . Then, we obtain ( ) ( )( )( )( ), , ,l lF C I C F r r rt t tm m£ and hence ( )lF m

is -r f ob .


(2)fi (1) Let ( )tx dom FŒ , YLm Œ , ( ) rh m ≥ and ( )ltx F mŒ we have by (2), ( )lF m

l= (say) is -r f ob then, there exists l is -r f ob and tx lŒ such that ( )lFl m£ . Thus F is FLb -continuous.

Theorem 2.6. Let :F X Y be a FM and normalized between two fts’s ( ),X t ,

( ),Y h and YLm Œ the following are equivalent: (1) F is FUb -irresolute.

(2) ( )uF m is -r f ob , for any m is -r f ob set.

(3) ( )lF m is -r f cb , for any m is -r f cb set.

(4) ( )( ) ( )( ), ,l lC F r F C rt hb m b m£ , for any YLm Œ .

(5) ( )( )( )( ) ( )( ), , , ,l lI C I F r r r F C rt t t hm b m£ , for any YLm Œ .

Proof. This can be proved in a similar way as Theorem 2.4.


( ),Y h and YLm Œ the following are equivalent: (1) F is FUb -continuous.

(2) ( )uF m is -r f ob , for any ( ) rh m ≥ .

(3) ( )lF m is -r f cb , for any ( )1 rh m- ≥ .

(4) ( )( ) ( )( ), ,l lC F r F C rt hb m m£ , for any YLm Œ .

(5) ( )( )( )( ) ( )( ), , , ,l lI C I F r r r F C rt t t hm m£ , for any YLm Œ .


Corollary 2.8. Let :F X Y be a FM between two fts’s ( ),X t , ( ),Y h . Then (1) If F is normalized, then F is FUb -irresolute (resp. FUb -continuous) at tx iff

( )( )( )( ), , ,utx C I C F r r rt t t mŒ for each m is -r f ob (resp. ( ) rh m ≥ ) and ( )u

tx F mŒ .

(2) F is FLb -irresolute (resp. FLb -continuous) at a fuzzy point tx iff (tx C It tŒ

( )( )( ) ), , ,lC F r r rt m for each m is -r f ob (resp. ( ) rh m ≥ ) and ( )ltx F mŒ .

The following implications hold: (1) FUa -continuous FUbfi -continuous FUb‹ -irresolute. (2) FLa -continuous FLbfi -continuous FLb‹ -irresolute. In general the converses are not true.


Example 2.9. Let { }1 2,X x x= , { }1 2 3, ,Y y y y= and :F X Y be a FM defined

by ( )1 1, 0.2FG x y = , ( )1 2, 1FG x y = , ( )1 3, 0.3FG x y = , ( )2 1, 0.5FG x y = , ( )2 2, 0.3FG x y =

and ( )2 3, 1FG x y = . Define fuzzy topologies : XL Lt Æ and : YL Lh Æ as follows:

( ){ }1, if 0, 1 ,

1 2, if 0.5,0, otherwise,

lt l l

Ï ŒÔ= =ÌÔÓ

( )

{ }1, if 0, 1 ,1 2, if 0.5,1 3, if 0.4,0, otherwise.

mm

h mm

Ï ŒÔ =Ô= Ì =ÔÔÓ

(1) F is FUb -continuous but not FUa -continuous because ( )0.4 1 3h = in ( ),Y h

and ( )0.4 0.4uF = is not 1 3-f oa .

(2) F is FLb -continuous but not FLa -continuous because ( )0.4 1 3h = in ( ),Y h

and ( )0.4 0.4lF = is not 1 3-f oa .


by ( )1 1, 0.2FG x y = , ( )1 2, 1FG x y = , ( )1 3, 0.3FG x y = , ( )2 1, 0.5FG x y = , ( )2 2,FG x y =

0.3 and ( )2 3, 1FG x y = . Define fuzzy topologies : XL Lt Æ and : YL Lh Æ as follows:

( )

{ }1, if 0, 1 ,1 2, if 0.5,1 3, if 0.4,0, otherwise,

ll

t lm

Ï ŒÔ =Ô= Ì =ÔÔÓ

( ){ }1, if 0, 1 ,

1 2, if 0.5,0, otherwise.

mh m m

Ï ŒÔ= =ÌÔÓ

(1) F is FUb -continuous but not FUb -irresolute because 0.6 is 1 3-f ob in

( ),Y h and ( )0.6 0.6uF = is not 1 3-f ob . (2) F is FLb -continuous but not FLb -irresolute because 0.6 is 1 3-f ob in

( ),Y h and ( )0.6 0.6lF = is not 1 3-f ob .

Theorem 2.11. Let { }i iF ŒG be a family of FLb -irresolute between two fts’s ( ),X t

and ( ),Y h . Then i iFŒG∪ is FLb -irresolute.

Proof. Let YLm Œ ,then ( ) ( ) ( )l li i i iF Fm mŒG ŒG=∪ � by Theorem 1.7 (2). Since{ }i i

F ŒG

is a family of FLb -irresolute between two fts’s ( ),X t and ( ),Y h , then ( )liF m is

-r f ob for any m is -r f ob . Then we have ( ) ( ) ( )l li i i iF Fm mŒG ŒG=∪ � is -r f ob for

any m is -r f ob . Hence i iFŒG∪ is FLb -irresolute.

Theorem 2.12. Let { }i iF ŒG be a family of FLb -continuous between two fts’s ( ),X t

and ( ),Y h . Then i iFŒG∪ is FLb -continuous.



Theorem 2.13. Let :F X Y be a crisp FUb -irresolute and b -compact-valued between two fts’s ( ),X t , ( ),Y h . If l is r -fuzzy b -compact, then ( )F l is r -fuzzy b -compact.

Proof. Let l is r -fuzzy b -compact set in X and { }: is - ,i i r f o ig g b ŒG be a family

covering of ( )F l i.e., ( ) i iF l gŒG£ � . Since tx txll Œ=� ,we have ( )

t

tx

F F xl

lŒ

Ê ˆ= Á ˜Ë ¯�

( )t

t ix i

F xl

gŒ ŒG

= £� � . It follows that for each tx lŒ , ( )t i iF x gŒG£ � . Since F is b -

compact-valued, then there exists finite subset tx

G of G such that ( )xt

t n nF x gŒG£ =�

txg . By Theorem 1.4 (6), we have ( )( ) ( )

t

u ut t xx F F x F g£ £ and

t t

ut

x x

x Fl l

lŒ Œ

= £� �

( )tx

g . Since tx

g is -r f ob then from Theorem 2.6 (2), we have ( )t

uxF g is -r f ob .

Hence ( ) ( ){ }: is - ,t t

u ux x tF F r f o xg g b lŒ is the family covering the set l . Since l

is r -fuzzy b -compact, then there exists finite index set N such that ( )n

un N xtFl gŒ£� .

From Theorem 1.4 (5), we have ( ) ( ) ( )( )n n n

u uxt xt xt

n N n N n N

F F F F Fl g g gŒ Œ Œ

Ê ˆ£ = £Á ˜Ë ¯� � � .

Then, ( )F l is r -fuzzy b -compact.

Theorem 2.14. Let :F X Y be a crisp FUb -continuous and compact-valued between two fts’s ( ),X t , ( ),Y h . If l is r -fuzzy b -compact, then ( )F l is r -fuzzy compact.


Theorem 2.15. Let :F X Y and :H Y Z be two FM ’s and let ( ),X t ,

( ),Y h and ( ),Z d be three fts’s. If F is FLb -irresolute and H is FLb -irresolute, then H F is FLb -irresolute.

Proof. Let F is FLb -irresolute, H is FLb -irresolute and ZLg Œ , g is -r f ob .

Then from Theorem 2.4 (2) we have ( ) ( ) ( )( )l l lH F F Hg g= is -r f ob with ( )lH g is

-r f ob . Thus H F is FLb -irresolute.


( ),Y h and ( ),Z d be three fts’s. If F and H are normalized, F is FUb -irresolute and H is FUb -irresolute, then H F is FUb -irresolute.




( ),Y h and ( ),Z d be three fts’s. If F is FLb -irresolute and H is FLb -continuous, then H F is FLb -continuous.

Proof. Let F is FLb -irresolute, H is FLb -continuous and ZLg Œ , ( ) rd g ≥ .

Then from Theorem 2.4 (2) and Theorem 2.5 (2) we have ( ) ( ) ( )( )l l lH F F Hg g= is

-r f ob with ( )lH g is -r f ob . Thus H F is FLb -continuous.

Theorem 2.18. Let :F X Y and :H Y Z be twoFM ’s and let ( ),X t , ( ),Y h

and ( ),Z d be three fts’s. If F and H are normalized, F is FUb -irresolute and H is FUb -continuous, then H F is FUb -continuous.


Theorem 2.19. Let :F X Y and :H Y Z be twoFM ’s and let ( ),X t , ( ),Y h

and ( ),Z d be three fts’s. If F is FLb -continuous and H is FLS -continuous, then H F is FLb -continuous.

Proof. Let F is FLb -continuous. H is FLS -continuous and ZLg Œ , ( ) rd g ≥ .


-r f ob with ( )( ) ( )lH rh g d g≥ ≥ . Thus H F is FLb -continuous.

Theorem 2.20. Let :F X Y and :H Y Z be two FM ’s and let ( ),X t , ( ),Y h

and ( ),Z d be three fts’s. If F and H are normalized, F is FUb -continuous and H is FUS -continuous, then H F is FUb -continuous.


Remark 2.21 [4,29]. Let ( ),X t and ( ),Y h be a fts’s. The fuzzy sets of the form

l m¥ with ( ) rt l ≥ and ( ) rh m ≥ form a basis for the product fuzzy topology t h¥

on X Y¥ , where for any ( ),x y X YŒ ¥ , ( )( ) ( ) ( ){ }, min ,x y x yl m l m¥ = .

Definition 2.22 [4,18]. Let :F X Y be a FM between two fts’s ( ),X t and ( ),Y h .

The graph fuzzy multifunction :fG X X Y¥ of F is defined as ( ) ( )1fG x x F x= ¥ for every x XŒ .

Theorem 2.23. Let :F X Y be a FM between two fts’s ( ),X t and ( ),Y h . If

fG is FLb -continuous, then F is FLb -continuous.

Proof. For the fuzzy sets XLr Œ , ( ) rt r ≥ , YLn Œ and ( ) rh n ≥ we take ( )r n¥


( ) ( )0, if ,

,, if .

xx y

y x

rn r

œÏ= Ì ŒÓ

Let ( )tx dom FŒ , YLm Œ and ( ) rh m ≥ with ( )ltx F mŒ , then we have (lt fx G XŒ ¥

)m and ( )X rh m¥ ≥ . Since fG is FLb -continuous, it follows that there exists XLl Œ is -r f ob and tx lŒ such that ( )l

fG Xl m£ ¥ . From here, we obtain that

( )lFl m£ . Thus F is FLb -continuous.

Theorem 2.24. Let :F X Y be a FM between two fts’s ( ),X t and ( ),Y h . If

fG is FUb -continuous, then F is FUb -continuous.


Theorem 2.25. Let ( ),X t and ( ),i iX t be fts’s ( )i IŒ . If a : i I iFMF X XŒ’ is an FLb -continuous (where i I iXŒ’ is the product space), then iP F is an FLb -continuous for each i IŒ , where :i i I i iP X XŒ’ is the projection multifunction

which is defined by ( )( ) { }i i iP x x= for each i IŒ .

Proof. Let 0

iXi Lm Œ and ( )0i i rt m ≥ . Then ( ) ( ) ( )( )0 0 0 0

ll l

i i i iP F F Pm m= = ( 0

liF m

)0i i iXπ¥’ . Since F is FLb -continuous and ( )0 0i i i i iX rt m π¥’ ≥ , it follows that

( )0 0

li i i iF Xm π¥’ is -r f ob . Then iP F is an FLb -continuous.

Theorem 2.26. Let ( ),X t and ( ),i iX t be fts’s ( )i IŒ . If a : i I iFMF X XŒ’ is an FUb -continuous (where i I iXŒ’ is the product space), then iP F is an FUb -continuous for each i IŒ , where :i i I i iP X XŒ’ is the projection multifunction

which is defined by ( )( ) { }i i iP x x= for each i IŒ .


Theorem 2.27. Let ( ),i iX t , ( ),i iY h be fts’s and :i i iF X Y be a FM for each

i IŒ . Suppose that : i I i i I iF X YŒ Œ’ ’ is defined by ( )( ) ( )i i I i iF x F xŒ= ’ . If F

is FLb -continuous, then iF is FLb -continuous for each i IŒ .

Proof. Let iYi Lm Œ and ( )i i rh m ≥ . Then ( )i i i j jY rh m π¥’ ≥ . Since F is FLb -

continuous, it follows that ( ) ( )l li i j j i i j jF Y F Xm mπ π¥’ = ¥’ is -r f ob . Consequently,

we obtain that ( )liF m is -r f ob for each i IŒ . Thus, iF is FLb -continuous.


Theorem 2.28. Let ( ),i iX t , ( ),i iY h be fts’s and :i i iF X Y be a FM for each

i IŒ . Suppose that : i I i i I iF X YŒ Œ’ ’ is defined by ( )( ) ( )i i I i iF x F xŒ= ’ . If F

is FUb -continuous, then iF is FUb -continuous for each i IŒ .


3. Fuzzy upper and lower strongly semi b -irresolute multifunctions


(1) Fuzzy upper strongly semi b -irresolute ( FUSSb -irresolute, for short) at a

fuzzy point ( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ and m is -r f ob there exists

XLl Œ , l is -r fso and tx lŒ such that ( ) ( )udom F Fl mŸ £ . (2) Fuzzy lower strongly semi b -irresolute ( FLSSb -irresolute, for short) at a

fuzzy point ( )tx dom FŒ iff ( )ltx F mŒ for each YLm Œ and m is -r f ob there exists

XLl Œ , l is -r fso and tx lŒ such that ( )lFl m£ . (3) FUSSb -irresolute (resp. FLSSb -irresolute (resp. FLSSb -irresolute) iff it is

FUSSb -irresolute (resp. FLSSb -irresolute) at every ( )tx dom FŒ .

Proposition 3.2. If F is normalized, then F is FUSSb -irresolute at a fuzzy point

( )tx dom FŒ iff ( )utx F mŒ for each YLm Œ and m is -r f ob there exists XLl Œ , l

is -r fso and tx lŒ such that ( )uFl m£ .


(1) F is FLSSb -irresolute.

(2) ( )lF m is -r fso , for any m is -r f ob set.

(3) ( )uF m is -r fsc , for any m is -r f cb set.

(4) ( )( ) ( )( ), ,u uSC F r F C rt hm b m£ , for any YLm Œ .

(5) ( )( )( ) ( )( ), , ,u uI C F r r F C rt t hm b m£ , for any YLm Œ .

Proof. (1) fi (2) Let ( )tx dom FŒ , m is -r f ob and ( )ltx F mŒ then, there exists

XLl Œ , l is -r fso and tx lŒ such that ( )lFl m£ and hence ( ( )( ), ,ltx C I F rt t mŒ

)r . Thus, we obtain ( ) ( )( )( ), ,l lF C I F r rt tm m£ . Then, ( )lF m is -r fso .

(2) fi (3) Let m is -r f cb hence by (2), ( ) ( )1 1l uF Fm m- = - is -r fso . Then,

( )uF m is -r fsc .


(3) fi (4) Let YLm Œ hence by (3), ( )( ),uF C rhb m is -r fsc . Then, we obtain

( )( ) ( )( ), ,u uSC F r F C rt hm b m£ .

(4) fi (5) Let YLm Œ hence by (4), we obtain ( )( )( ) ( )( ), , ,u uI C F r r SC F rt t tm m£

( )( ),uF C rhb m£ .

(5) fi (2) Let m is -r f ob hence by (5), we have ( ) ( )1 1l uF Fm m- = - (I Ct t≥

( )( ) )1 , ,uF r rm- ( )( )( )1 , ,lI C F r rt t m= - ( )( )( )1 , ,lC I F r rt t m= - . Then, we

obtain ( ) ( )( )( ), ,l lF C I F r rt tm m£ and hence ( )lF m is -r fso .

(2)fi (1) Let ( )tx dom FŒ , m is -r f ob and ( )ltx F mŒ we have by (2), ( )lF m l=

(say) is -r fso then, there exists l is -r fso and tx lŒ such that ( )lFl m£ . Thus F is FLSSb -irresolute.


( ),Y h and YLm Œ the following are equivalent: (1) F is FUSSb -irresolute.

(2) ( )uF m is -r fso , for any m is -r f ob set.

(3) ( )lF m is -r fsc , for any m is -r f cb set.

(4) ( )( ) ( )( ), ,l lSC F r F C rt hm b m£ , for any YLm Œ .

(5) ( )( )( ) ( )( ), , ,l lI C F r r F C rt t hm b m£ , for any YLm Œ .


The following implications hold: (1) FUSSb -irresolute FUbfi -irresolute. (2) FLSSb -irresolute FLbfi -irresolute. In general the converse are not true.


by ( )1 1, 0.2FG x y = , ( )1 2, 1FG x y = , ( )1 3, 0.3FG x y = , ( )2 1, 0.5FG x y = , ( )2 2, 0.3FG x y =

and ( )2 3, 1FG x y = . Define fuzzy topologies : XL Lt Æ and : YL Lh Æ as follows:

( ){ }1, if 0, 1 ,


lt l l

Ï ŒÔ= =ÌÔÓ

( ){ }1, if 0, 1 ,


mh m m

Ï ŒÔ= =ÌÔÓ

We can obtain the followings:


( )0, if 0, ,

, , if 0 1,0 1 2,1, otherwise,

r L

C r rt

lb l l l

= ŒÏÔ= < < < £ÌÔÓ

( ),C rhb l = 0, if 0, ,

, if 0 1,0 1 2,1, otherwise.

r L

r

ll l

= ŒÏÔ < < < £ÌÔÓ

(1) F is FUb -irresolute but not FUSSb -irresolute because 0.3 is 1 2-f cb in

( ),Y h and ( )0.3 0.3lF = is not 1 2-fsc . (2) F is FLb -irresolute but not FLSSb -irresolute because 0.3 is 1 2-f cb in

( ),Y h and ( )0.3 0.3uF = is not 1 2-fsc .

Theorem 3.6. Let { }i iF ŒG be a family of FLSSb -irresolute between two fts’s

( ),X t and ( ),Y h . Then i iFŒG∪ is FLSSb -irresolute.

Proof. Let YLm Œ , then ( ) ( ) ( )l li i i iF Fm mŒG ŒG=∪ � by Theorem 1.7 (2). Since

{ }i iF ŒG is a family of FLSSb -irresolute between two fts’s ( ),X t and ( ),Y h , then

( )liF m is -r fso for any m is -r f ob . Then we have ( ) ( ) ( )l l

i i i iF Fm mŒG ŒG=∪ � is -r fso for any m is -r f ob . Hence i iFŒG∪ is FLSSb -irresolute.


and ( ),Z d be three fts’s. If F is FLSSb -irresolute and H is FLb -irresolute, then H F is FLSSb -irresolute.

Proof. Let F is FLSSb -irresolute, H is FLb -irresolute and ZLg Œ , g is -r f ob .


-r fso with ( )lH g is -r f ob . Thus H F is FLSSb -irresolute.


and ( ),Z d be three fts’s. If F and H are normalized, F is FUSSb -irresolute and H is FUb -irresolute, then H F is FUSSb -irresolute.



and ( ),Z d be three fts’s. If F is FLb -irresolute and H is FLSSb -irresolute, then H F is FLb -irresolute.


Proof. Let F is FLb -irresolute, H is FLSSb -irresolute and ZLg Œ , g is -r f ob .


-r f ob with ( )lH g is -r fso . Thus H F is FLb -irresolute.


( ),Y h and ( ),Z d be three fts’s. If F and H are normalized, F is FUb -irresolute and H is FUSSb -irresolute, then H F is FUb -irresolute.


4. Conclusion

It is well known that fuzzy set theory has been regarded as a generalization of classical set theory in one way. Furthermore, this is an important mathematical tool to deal with uncertainty. One of the main contributions of this paper is introduce the concepts of fuzzy upper and lower b -continuous, b -irresolute, strongly semi b -irresolute multifunctions on fuzzy topological spaces in Sostak’s sense and investigate some of their properties. Also, the relationship between these multifucntions are investigated. Later, some applications are introduced and studied.

References

[1] S. E. Abbas, Fuzzy b -irresolute functions, Applied Mathematics and Computation, 157 (2004), 369-380.

[2] S. E. Abbas, M. A. Hebeshi and I. M. Taha, On fuzzy upper and lower semi-continuous multifunctions, J. Fuzzy Math. Vol.4, No. 4 (2014) 951-962.

[3] K. M. A. Al-hamadi and S. B. Nimse, On fuzzy a -continuous multifunctions, Miskolc Mathematical Notes, 11 (2) (2010), 105-112.

[4] M. Alimohammady, E. Ekici, S. Jafari and M. Roohi, On fuzzy upper and lower contra-continuous multifunctions, Iranian Journal of Fuzzy Systems, 8 (3) (2011), 149-158.

[5] C. Berge, Topological spaces, Including a treatment of multi-vlaued functions, Vector spaces and convexity, Oliver, Boyd London (1963).

[6] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [7] K. C. Chattopadhyay, R. N. Hazra and S. K. Samanta, Gradation of opensess, fuzzy topology, Fuzzy

Sets and Systems, 49 (1992), 237-242. [8] K. C. Chattopadhyay and S. K. Samanta, Fuzzy topology, Fuzzy closure operator, Fuzzy compactness

and fuzzy connectedness, Fuzzy Sets and Systems, 54 (2) (1993), 207-212. [9] M. A. Hebeshi and I. M. Taha, On upper and lower a -continuous Fuzzy multifunctions, Journal of

Intelligent & fuzzy Systems (Accepted). [10] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78 (1980), 659-673. [11] U. Hohle and A. P. Sostak, A general theory of fuzzy topological spaces, Fuzzy Sets and Systems, 73

(1995), 131-149. [12] U. Hohle and A. P. Sostak, Axiomatic foundations of fixed-basis fuzzy topology, The hand-books of

fuzzy sets series, Volume 3, Kluwer academic publishers, Dordrecht (Chapter 3), (1999). [13] Y. C. Kim, A. A. Ramadan and S. E. Abbas, Weaker forms of continuity in sostaks fuzzy topology,

Indian J. Pure Appl. Math., 34 (2) (2003), 311-333.


[14] Y. C. Kim and S. E. Abbas, On several types of R -fuzzy compactness, J. Fuzzy Math., 12 (4) (2004), 827-844.

[15] T. Kubiak, On fuzzy topologies, Ph. D. Thesis, A. Mickiewicz, Poznan (1985). [16] T. Kubiak and A. P. Sostak, Lower set-valued fuzzy topologies, Quaestions Math., 20 (3) (1997),

423-429. [17] R. A. Mahmoud, An application of continuous fuzzy multifucntions, Chaos, Solitons and Fractals, 17

(2003), 833-841. [18] M. N. Mukherjee and S. Malakar, On almost continuous and weakly continuous fuzzy multifunctions,

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425. [20] V. Popa, On characterizations of irresolute multimapping, J. Univ. Kuwait (Sci), 15 (1988), 21-25. [21] V. Popa, Irresolute multifunctions, Internat. J. Math. And Math. Sci., 13 (2) (1990), 275-280. [22] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, 48 (1992), 371-375. [23] A. A. Ramadan, S. E. Abbas and Y. C. Kim, Fuzzy irresolute mappings in smooth fuzzy topological

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521.




Los Angeles

Rough Prime Bi-ideals in G -semigroups

N. Thillaigovindan

Veltech Ranga Sanku Arts College, Avadi , Chennai-600 062, India E-mail: [email protected]

V. S. Subha

Department of Mathematics Annamalai University Annamalainagar-608002, India


Abstract: In this paper we introduce the notions rough prime bi-ideals, strongly rough prime bi-

ideals, rough semiprime bi-ideals, strongly rough irreducible and rough irreducible bi-ideals of G -semigroups. We have shown that the lower and upper approximation of a prime bi-ideal are also prime bi-ideals.

Keywords: Prime bi-ideals, Strongly rough prime bi-ideals, Irreducible bi-ideals, Rough

semiprime bi-ideals, Strongly rough irreducible bi-ideals, Rough irreducible bi-ideals.

1. Introduction

The notion of a rough set was originally proposed by Pawalk [10-13] as a formal tool for modeling and processing incomplete data in information systems. In rough set theory any subset of the universal set which has uncertainty or incomplete in nature is represented by two ordinary subsets of the universe known as the lower and upper approximations of the given set. The equivalence classes are used as the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all equivalence classes which are subsets of the set and the upper approximation is union of all equivalence classes which have a nonempty intersection with the set. Some authors have studied the algebraic properties of rough sets. Aslam et.al [1], Biswas and Nanda [2], Chinram [3,4], Davvaz [5], Jun [6], Kuroki and Mordeson [7], Kuroki [8] and Thillaigovindan et. al [18,19], Xiao et. al [20] are some notable works where roughness has been applied in different algebraic structures.

The fundamental concept of G -semigroup was introduced by Sen in [16, 17]. The G -semigroup is a generalization of semigroup. Many researchers have worked on G -semigroup and its sub structures. Many classical notions of semi-groups have been

190 N. Thillaigovindan and V. S. Subha

extended to G -semigroup and its sub structures. Many classical notions of semi-groups have been extended to G -semigroups by Sha and Sen in [14], [15] and [17]. On the other hand the concept of prime bi-ideal, strongly prime bi-ideal, semiprime bi-ideal, strongly irreducible bi-ideal and irreducible bi-ideal of G -semigroups are studied in [9].

In this paper we introduce the concept of rough prime bi-ideals, strongly rough prime bi-ideals, rough semiprime bi-ideals, strongly rough irreducible and rough irreducible bi-ideals of G -semigroups.

2. Preliminaries

In this section we reproduce some basic concepts which are needed in the sequel. Let U be a universal set. For an equivalence relation q on U , the set of elements of

U that are related to x UŒ , is called the equivalence class of x and is denoted by [ ]x q . Let U q denote the family of all equivalence classes induced by q on U . U q is a partition of U such that each element of U is contained in exactly one equivalence class.

Definition 2.1. A pair ( ),U q where U fπ and q is an equivalence relation on U , is called an approximation space.

Definition 2.2. For an approximation ( ),U q by a rough approximation in ( ),U q we

mean a mapping ( ) ( ) ( ): U U Ur Æ ¥� � � defined as ( ) ( ) ( )( ),X X Xr r r= for

X UÕ , where ( ) [ ]{ }X x U x Xqr = Œ Õ , ( ) [ ]{ }X x U x Xqr f= Œ « π . ( )Xr is

called a lower rough approximation of X in ( ),U q where as ( )Xr is called upper

approximation of X in ( ),U q .

Definition 2.3. Given an approximation space ( ),U q , a pair ( ) ( ) ( ),A B U UŒ ¥� �

is called a rough set in ( ),U q if and only if ( ) ( ),A B Xr= for some X UÕ .

Definition 2.4. Let M be a G -semigroup. A sub G -semigroup B of M is called a bi-ideal of M if B M B BG G Õ .

Definition 2.5. Let M be a G -semigroup. An ideal P of M is said to be prime ideal of M if for any two ideals A and B of M . A B PG Õ implies either A PÕ or B PÕ .

Definition 2.6. Let M be a G -semigroup. An ideal P of M is said to be semiprime ideal of M if for any ideal A of M , A A PG Õ implies that A PÕ .

3. Rough prime bi-ideals in G -semigroups

Rough Prime Bi-ideals in G -semigroups 191

In this section we introduce rough prime bi-ideals, strongly rough prime bi-ideals, rough semiprime bi-idelas, strongly rough irreducible and rough irreducible bi-ideals of G -semigroups. Through out his paper M denotes a G -semigroup unless otherwise specified.

Let q be an equivalence relation on M . q is called a congruence relation on M , whenever ( ),a b qŒ implies ( ),a x b xg g qŒ and ( ),x a x bg g qŒ for all x MŒ and

g ŒG . We denote by [ ]a q the equivalence class containing the element a MŒ .

A congruence relation q on M is said to be complete if [ ] [ ] [ ]a b a bq q qG = G for all ,a b MŒ .

Definition 3.1. Let A be a nonempty subset of M and q be a congruence relation on M . Then the q -lower and q -upper approximations of A are defined by ( )Aq =

[ ]{ }x M x AqŒ Õ , ( ) [ ]{ }A x M x Aqq f= Œ « π .

A nonempty subset A of M is called q -upper (resp. q -lower) rough sub G -semigroup of M if ( )Aq (resp. ( )Aq ) is a sub G -semigroup of M .

Definition 3.2. A nonempty subset A of M is called a q -upper (resp. q -lower) rough bi-ideal of M if ( )Aq (resp. ( )Aq ) is a bi-ideal of M , where q is a con-gruence relation on M .

Theorem 3.3 [18]. Let q and Y be congruence relations on M and let A and B be nonempty subsets of M . Then the following are true:

(i) ( ) ( )A A Aq qÕ Õ ,

(ii) ( ) ( )q f f q f= = ,

(iii) ( ) ( )M M Mq q= = and ( ) ( )q qG = G = G ,

(iv) ( ) ( ) ( )A B A Bq q q» = » ,

(v) ( ) ( ) ( )A B A Bq q q« = « ,

(vi) A BÕ implies ( ) ( )A Bq qÕ and ( ) ( )A Bq qÕ ,

(vii) ( ) ( ) ( )A B A Bq q q» « ,

(viii) ( ) ( ) ( )A B A Bq q q« Õ « ,

(ix) q Õ Y implies ( ) ( )A AqY Õ and ( ) ( )A Aq Õ Y ,

(x) ( ) ( ) ( )A B A Bq q qG Õ G ,

(xi) If q is complete, then ( ) ( ) ( )A B A Bq q qG Õ G ,

(xii) ( )( ) ( ) ( )A A Aq q«Y Õ «Y ,

(xiii) ( )( ) ( ) ( )A A Aq q«Y «Y ,

(xiv) ( )( ) ( )A Aq q q= ,

(xv) ( )( ) ( )A Aq q q= ,


(xvi) ( )( ) ( )A Aq q q= ,

(xvii) ( )( ) ( )A Aq q q= .

Theorem 3.4 [18]. Let q be a congruence relation on M and let A be a bi-ideal of M . Then

(i) ( )Aq is a bi-ideal of M

(ii) If q is complete then ( )Aq is a bi-ideal of M .

Theorem 3.5 [18]. Let q be a congruence relation on M and let A be a left (right) ideal of M and ( )Aq be nonempty, then

(i) ( )Aq is both a bi-ideal and quasi-ideal of M and

(ii) ( )Aq is both a bi-ideal and quasi-ideal of M .

Definition 3.6. A bi-ideal B of a G -semigroup M is called prime bi-ideal (strongly prime bi-ideal) of M if 1 2B B BG Õ ( )1 2 2 1B B B B BG « G Õ implies that either 1B BÕ or 2B BÕ for any bi-ideals 1B and 2B of M .

Definition 3.7. A bi-ideal B of a G -semigroup M is called semiprime bi-ideal of M if A A BG Õ implies that A BÕ for any bi-ideal A of M .

Definition 3.8. Let q be a congruence relation on M . A bi-ideal B of M is called rough prime bi-ideal (strongly rough prime bi-ideal) of M if ( )Bq and ( )Bq are prime bi-ideals (strongly prime bi-ideals) of M .

Definition 3.9. Let q be a congruence relation on M . A bi-ideal B of M is called rough semiprime bi-ideal of M if ( )Bq and ( )Bq are semiprime bi-ideals of M .

Definition 3.10. A bi-ideal B of a G -semigroup is called irreducible (strongly ire-ducible) bi-ideal of M if ( )1 2 1 2B B B B B B« = « Õ implies that either (1 1B B B= Õ

)B or ( )2 2B B B B= Õ for all bi-ideals 1B , 2B of M .

Definition 3.11. Let q be a congruence relation on M . A bi-ideal B of M is called rough irreducible (strongly irreducible) bi-ideal of M if ( )Bq and ( )Bq are irreducible (strongly irreducible) bi-ideals of M .

Theorem 3.12. Let q be a congruence relation on M . If A is a prime bi-ideal of M , then ( )Aq is a prime bi-ideal of M .

Proof. Let A be a prime bi-ideal of M . Then 1 2A A AG Õ implies that either 1A AÕ or 2A AÕ for any bi-ideals 1A and 2A of M . Since A be a bi-ideal of M . By Theorem 3.4 ( )Aq is a bi-ideal of M . Assume that ( ) ( ) ( )1 2A A Aq q qG Õ , ( )1Aq Õ/


( )Aq and ( ) ( )2A Aq qÕ/ . Since A is a prime bi-ideal of M , A is a semiprime bi-

ideal of M . Therefore 1A AÕ or 2A AÕ . These implies that ( ) ( )1A Aq qÕ or

( ) ( )2A Aq qÕ which is a contradiction to our assumption. Hence ( )Aq is a prime bi-ideal of M .

Theorem 3.13. Let q be a congruence relation on M . If A is a prime bi-ideal of M , then ( )Aq is a prime bi-ideal of M .

Proof. The proof is similar to Theorem 3.12.

Theorem 3.14. Let q be a complete congruence relation on M and A be a prime bi-ideal of M . If ( )Aq is nonempty then ( )Aq is rough prime bi-ideal of M .

Proof. By Theorem 3.12 ( )Aq is a prime bi-ideal of M and by Theorem 3.13 ( )Aq

is a prime bi-ideal of M . Hence ( )Aq is a rough prime bi-ideal of M .

Theorem 3.15. The intersection of a family of prime bi-ideals of M is a rough semiprime bi-ideal of M .

Proof. The intersection of family of prime bi-ideals of M is a semiprime bi-ideal of M . Therefore it is a rough semipriem bi-ideal of M .

Theorem 3.16. Every strongly irreducible semiprime bi-ideal of M is a strongly rough prime bi-ideal of M .

Proof. Let B be strongly irreducible semiprime bi-ideal of M . Then ( )Bq is strong-ly irreducible semiprime bi-ideal of M . Let 1B and 2B be any two bi-ideals ofM ,then

( )1Bq and ( )2Bq are bi-ideals of M such that ( ) ( )( ) ( ) ( )( )1 2 2 1B B B Bq q q qG « G Õ ( )Bq .

Since ( ) ( )( ) ( ) ( )( ) ( ) ( )1 2 2 1 1 2B B B B B Bq q q q q q« G « Õ G and ( ) ( )( )1 2B Bq q«

( ) ( )( ) ( ) ( )2 1 2 1B B B Bq q q qG « Õ G it follows that ( ) ( )( ) ( )( ( ))1 2 2 1B B B Bq q q q« G «

( ) ( )( ) ( ) ( )( ) ( )1 2 2 1B B B B Bq q q q qÕ G « G Õ . Since ( )Bq is semiprime bi-ideal of

M , ( ) ( ) ( )1 2B B Bq q q« Õ . Also since ( )Bq is strongly irreducible bi-ideal of M ,

we have either ( ) ( )1B Bq qÕ or ( ) ( )2B Bq qÕ . Thus ( )Bq is a strongly prime bi-

ideal of M . Similarly ( )Bq is also a strongly prime bi-ideal of M . Hence ( )Bq is a strongly rough prime bi-ideals of M .

Lemma 3.17. Let B be a bi-ideal of M and a MŒ such that a Bœ . Then there exists a rough irreducible bi-ideal ( )Iq of M such that ( ) ( )B Iq qÕ and ( )a Iqœ .

Proof. Let B be a bi-ideal of M . Then ( )Bq is also a bi-ideal of M . Let � be

the collection of all rough bi-ideals of M which contain ( )Bq and do not contain a .

Since ( )Bq Œ� , � is nonempty. The collection � is a partially ordered set under set


inclusion. If b is any totally ordered subset of � , then the union of all the subset in b is a bi-ideal of M containing ( )Bq and not containing a .

Hence by Zorn’s lemma there exists a maximal element ( )Iq in � . We show that

( )Iq is an irreducible bi-ideal of M . Let ( )Cq and ( )Dq be two bi-ideals if M

such that ( ) ( ) ( )I C Dq q q= « . If both ( )Cq and ( )Dq properly contain ( )Iq , then

( )a CqŒ and ( )a DqŒ . Thus ( ) ( ) ( )a C D Iq q qŒ « = . This contradicts the fact

that ( )a Iqœ . Thus either ( ) ( )I Cq q= or ( ) ( )I Dq q= . Hence ( )Iq is irreducible

bi-ideal. Similary ( )Iq is irreducible bi-ideal. Thus ( )Iq is a rough irreducible bi-idea.

Definition 3.18. An element a MŒ is called regular in a G -semigroup M if a a M aŒ G G where ( ){ }: and ,a M a axb ya b M x yG G = Œ ŒG . M is called regular if every element of M is regular.

Definition 3.19. An element a MŒ is called intra-regular in a G -semigroup M if a M a a MŒ G G G . M is called intra-regular if every element of M is intra-regular.

Theorem 3.20. For the G -semigroup M the following conditions are equivalent (i) M is intra-regular. (ii) ( ) ( ) ( ) ( )R L L Rq q q q« = G for every right ideal R and left ideal L of M .

Proof. (i)fi (ii) Suppose M is intra-regular. Let ( ) ( )x R Lq qŒ « . Then ( )x RqŒ

and ( )x LqŒ . Since M is intra-regular for any x MŒ , we have x M x x MŒ G G G =

( ) ( )M x x MG G G . This implies that ( ) ( )x L Rq qŒ G . Thus ( ) ( ) ( )R L Lq q q« Õ G

( )Rq . A similar proof holds for ( ) ( ) ( ) ( )R L L Rq q q q« Õ G . Thus ( ) ( )R Lq q« Õ

( ) ( )L Rq qG .

(ii) fi (i) Suppose ( ) ( ) ( ) ( )R L L Rq q q q« Õ G . Since ( )Lq is a left ideal and

( )Rq is a right ideal of M , by hypothesis we have ( )( ) ( )( )R M M Lq qG « G Õ

( )( ) ( )( ) ( ) ( )( )M L R M M L R Mq q q qG G G Õ G G G . Let ( )( ) ( )( )y R M M Lq qŒ G « G

then ( )( )y R MqŒ G and ( )( )y M LqŒ G . These imply that ( )y RqŒ and ( )y LqŒ .

Therefore y M y y MŒ G G G for all y MŒ . Hence M is intra-regular.

Theorem 3.21. For a G -semigroup M , the following conditions are equivalent. (i) M is both regular and intra regular. (ii) ( ) ( ) ( )B B Bq q qG = for every ideal B of M .

(iii) ( ) ( ) ( ) ( )( ) ( ) ( )( )1 2 1 2 2 1B B B B B Bq q q q q q« = G « G , for all bi-ideals ( )1Bq

and ( )2Bq of M . (iv) Each bi-ideal of M is semiprime. (v) Each proper bi-ideal of M is the intersection of irreducible semiprime bi-ideals

of M which contain it.


Proof. (i) fi (ii) Let M be both regular and intra-regular and let B be a bi-ideal of M . By Theorem 3.4 ( )Bq is a bi-ideal of M .

( ) ( ) ( ) ( )B B B B Bq q q qG Õ G Õ (1)

Let ( )x BqŒ . Since M is regular there exists ( )x M MqŒ = such that x x M xŒ G G

( )x M x M x= G G G G . Since M is intra-regular, for x MŒ , x M x x MŒ G G G . Thus

( ) ( )( ) ( )( )x x M M x x M M x x M M x x M M xŒ G G G G G G G = G G G G G G G . These imply that

( ) ( ) ( )( ) ( ) ( ) ( )( )x B M B B M Bq q q q q qŒ G G G G G

( ) ( ) ( ) ( )B M B B M B B Bq q q qÕ G G G G G Õ G (2)

Thus ( ) ( ) ( )B B Bq q qG = [by (1) and (2)]. Similarly ( ) ( ) ( )B B Bq q qG = . Hence

( ) ( ) ( )B B Bq q qG =

(ii) fi (iii) Let 1B and 2B are two bi-ideals of M . Then ( )1Bq and ( )2Bq are bi-

ideals of M . By hypothesis ( ) ( ) ( ) ( )( ) ( ) ( )( )1 2 1 2 1 2B B B B B Bq q q q q q« = « G « Õ

( ) ( )1 2B Bq qG . Similarly ( ) ( ) ( ) ( )1 2 2 1B B B Bq q q q« Õ G . Therefore ( ) ( )1B Bq qG 2

and ( ) ( )B Bq qG2 1 are bi-ideals, and so ( ) ( )( ) ( ) ( )( )1 2 2 1B B B Bq q q qG G G is a bi-

ideal of M . Thus ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )(1 2 2 1 1 1 2B B B B B B Bq q q q q q qG « G = G «

( ))1BqG G ( ) ( )( ) ( ) ( )( )1 2 2 1B B B Bq q q qG « G ( ) ( ) ( ) ( )1 2 2 1B B B Bq q q qÕ G G G .

Therefore ( ) ( )( ) ( ) ( )( )1 2 2 1B B B Bq q q qG « G ( ) ( ) ( ) ( )1 1 1B M B Bq q q qÕ G G Õ .

Similarly ( ) ( )( ) ( ) ( )( ) ( )1 2 2 1 2B B B B Bq q q q qG « G Õ . Thus ( ) ( )( )1 2B Bq qG «

( ) ( )( ) ( ) ( )2 1 1 2B B B Bq q q qG Õ « . Hence ( ) ( )( ) ( ) ( )( )1 2 2 1B B B Bq q q qG « G =

( ) ( )1 2B Bq q« . It is also true for the bi-ideals ( )1Bq and ( )2Bq of M . Therefore

( ) ( ) ( )1 2 1 2 2 1B B B B B Bq q q« = G « G .

(iii) fi (iv) Let 1B and B be any two bi-ideals of M . Then ( )1Bq and ( )Bq are

also bi-ideals of M such that ( ) ( ) ( )1 1B B Bq q qG Õ . By hypothesis ( ) ( )1 1B Bq q=

( )1Bq« ( ) ( )( ) ( ) ( )( )1 1 1 1B B B Bq q q q= G « G ( ) ( ) ( )1 1B B Bq q q= G Õ .So ( )1Bq Õ

( )Bq . Hence ( )Bq is a semiprime bi-ideal of M . Similarly ( )Bq is also a

semiprime bi-ideal of M . Therefore ( )Bq is a rough semiprime bi-ideal of M .

(iv) fi (v) Let B be a proper bi-ideal of M , then ( )Bq is also a proper bi-ideal of

M and ( )Bq is contained in the intersection of all rough irreducible bi-ideals of M

which contain ( )Bq . Lemma 3.17 guarantees the existence of such irreducible bi-ideals.

If ( )a Bqœ then there exists an irreducible bi-ideal of M which contains ( )Bq but

does not contain a . Hence ( )Bq is the intersection of all q -lower rough irreducible

semiprime bi-ideals of M which contain ( )Bq . By hypothesis, every q -lower rough


bi-ideal is semiprime and so each q -lower rough bi-ideal is the intersection of q -lower rough irreducible semiprime bi-ideals of M containing q -lower rough bi-ideal. Similarly every q -upper rough bi-ideal is semiprime and so each q -upper rough bi-ideal is the intersection of q -upper rough irreducible semiprime abi-ideals of M containing q -upper rough bi-ideal. Hence each proper bi-ideal of M is the intersection of irreducible rough semiprime bi-ideals of M which containing it.

(v)fi (ii) Let B be a bi-ideal of M . Then ( )Bq is a bi-ideal of M . If ( ) ( )B Bq qG

( )M Mq= = , then clearly ( )Bq is idempotent. If ( ) ( ) ( )B B Mq q qG π then ( )Bq

( )BqG is a proper bi-ideal of M containing ( ) ( )B Bq qG . Thus by hypothesis ( )Bq

( ) ( ) ( ){ :B B Ba a aq q qG = ∩ is an irreducible semiprime bi-ideal of M containing

( ) ( )}B Bq qG . Since each ( )Baq is a semiprime bi-ideal of M , and ( ) ( )B Baq qÕ

for all a implies that ( ) ( ) ( ) ( )B B B Ba aq q q qÕ = G∩ . But ( ) ( ) ( )B B Bq q qG Õ

(because ( )Bq is a sub G -semigroup). Thus each q -upper rough bi-ideal of M is

idem-potent. A similar proof holds for the bi-ideal ( )Bq of M . Hence each rough bi-ideal of M is idempotent.

(ii) fi (i) Suppose that ( ) ( )B Bq qG for all bi-ideals ( )Bq of M . By Theorem 3.5

( ) ( )L Rq q« is a bi-ideal of M for all right ideals R and left ideals L of M . By

hypothesis ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )L R L R L R L Rq q q q q q q q« = « G « Õ G . Thus by

Theorem 3.21 M is intra-regular. Also ( ) ( ) ( ) ( )( ) ( )(L R L R Lq q q q q« = « G «

( )) ( ) ( )R R Lq q qÕ G . But ( ) ( ) ( ) ( )R L L Rq q q qG Õ « always holds. Thus ( )Rq

( ) ( ) ( )L L Rq q qG = . Therefore ( ) ( ) ( ) ( ) ( ) ( )L R R L R M M Lq q q q q q« = G = G G G

( ) ( )R M Lq qÕ G G . Let ( ) ( )x L Rq qŒ « , then ( )x LqŒ and ( )x RqŒ . Thus xŒ x M xG G for all x MŒ . Hence M is regular.

Theorem 3.22. Let M be regular and intra-regular. Then the following are equivalent for the bi-ideal B of M

(i) ( )Bq is strongly rough irreducible.

(ii) ( )Bq is strongly rough prime Proof. (i) fi (ii) Let M be regular and intra-regular and B be a bi-ideal of M .

Then ( )Bq is bi-ideal of M . By Theorem 3.21, ( )Bq is semiprime. Since ( )Bq is

strongly irreducible, by Lemma 3.17, ( )Bq is strongly prime bi-ideal of M . The proof

of ( )Bq is strongly prime bi-ideal of M is similar. Thus ( )Bq is strongly rough prime bi-ideal of M .

(ii) fi (i) Let B be strongly prime bi-ideal of M and let 1B and 2B be any two bi-ideals of M . Then ( )1Bq and ( )2Bq are also bi-ideals of M such that ( ) ( )1 2B Bq q«

( )BqÕ . Since M is regular and intra-regular, by Theorem 3.21 ( ) ( )( )1 2B Bq qG «

( ) ( )( ) ( ) ( ) ( )2 1 1 2B B B B Bq q q q qG = « Õ . Thus by hypothesis either ( ) ( )1B Bq qÕ


or ( ) ( )2B Bq qÕ . Hence ( )Bq is strongly irreducible. Similarly ( )Bq is strongly

irreducible. Therefore ( )Bq is strongly rough irreducible.

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Mathematics, 5 (1) (1997), 183-191. [8] N. Kuroki, Rough ideals in semigroups, Information Sciences, 100 (1997), 139-163. [9] Muhammad Shabir and Shujaat Ali, Prime bi-ideals of G -semigroups, Journal of Advanced Research

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Los Angeles

Compatible Mappings and Common Fixed Points in Fuzzy Metric Spaces

Nanda Ram Das

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India.

Email: [email protected]

Mintu Lal Saha

Department of Mathematics, Handique Girls’College, Guwahati 781001, Assam, India. Email: [email protected]

Abstract: In this paper, we state and prove some common fixed point Theorems in fuzzy metric

spaces in the sense of Kramosil and Michalek, using the notions of compatibility and reciprocal continuity of maps. Our work contains proper generalizations of many important Theorems mainly due to Murthy, Jungck and Cho. We deduce some Corollaries to our Theorems and also illustrate them with suitable examples.

Keywords: Fuzzy metric spaces, compatible maps, reciprocal continuous maps, common fixed

points.

1. Introduction

The concept of fuzzy metric spaces was introduced in various ways by several authors such as Deng [5], Erceg [6], Kaleva and Seikkala [17], Kramosil and Michalek [18]. In particular, Kramosil and Michalek introduced the notion of fuzzy metric spaces in the year 1975 by generalzing the concept of probabilistic metric spaces introduced by Menger, to fuzzy setting. George and Veeramani [7] modified the notion of fuzzy metric spaces introduced by Kramosil and Michalek with the help of continuous t -norm and obtained a Hausdorff topology for this kind of fuzzy metric spaces and proved that every metric d on X induces a fuzzy metric dM on X . Many authors think that the George and Veeramani’s definition is an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained mainly from the classical theorems.

200 Nanda Ram Das and Mintu Lal Saha

In recent years, the study of common fixed point theorems satisfying some contractive-type condition has been at the center stage of some intense research activity and a significant number of interesting results have been obtained by various authors such as [1, 2, 11, 22, 29], etc. Jungck [13] introduced the notion of compatible maps by generalizing the concept of commutating maps and established some important common fixed point theorems in a series of his paper [12, 13, 14, 15] satisfying various compatibility conditions. Recently, Singh et al. [27, 28] extended the concept of compatibility and semi-compatibility of maps to fuzzy metric spaces and proved some common fixed point theorems.

Vasuki [30] has generalized the common fixed point theorem of Pant [22] to fuzzy metric spaces. Cho and Jung [2] established some common fixed point theorems for four weakly compatible maps of an e -chainable fuzzy metric space. Mishra et al. [20] used reciprocal continuity while proving some common fixed point theorems in fuzzy metric spaces and showed that the notion of reciprocal continuity can widen the scope of many interesting fixed point theorems in fuzzy metric spaces. Recent literatures on fixed point theory in fuzzy metric spaces can also be viewed in [3, 4, 9, 10, 19]. For basic analysis, we refer to [24].

Our present work is in the direction of extending the classical common fixed point theorems mainly due to Murthy [21] and Jungck [15] to fuzzy metric spaces in the sense of Kramosil and Michalek. The structure of the paper is as follows. After the preliminaries, we prove the fuzzy analogues of Murthy and Jungck’s common fixed point Theorem. We use a weaker notion of ‘reciprocal continuity’ in place of ‘continuity’ in our Theorems to obtain proper generalizations. We have deduced some Corollaries including the fuzzy Banach contraction Theorem due to Grabiec [8] and have incorporated some examples to support our results.

2. Preliminaries

In this section, we recall some definitions and important results which are already in the literature.

Definition 2.1 [31]. A fuzzy set A in X is a mapping [ ]: 0,1A XÆ . For x XŒ ,

( )A x is called the grade of membership of x .

Definition 2.2 [26]. A binary operation [ ] [ ] [ ]: 0,1 0,1 0,1* ¥ Æ is called a continuous

t -norm, if [ ]( )0,1 ,◊ is an abelian topological monoid with unity 1 such that a b c d* £ * ,

whenever a c£ , b d£ , for all [ ], , , 0,1a b c d Œ .

Example 2.3. { }min ,a b a b* = , a b ab* = , [ ], 0,1a bŒ are continuous t -norms.

Definition 2.4 [18]. The 3-tuple ( ), ,X M * is called a fuzzy metric space, if X is an

arbitrary set, * is a continuous t -norm and M is a fuzzy set in [ [2 0,X ¥ • , satisfying the following conditions:

For all , ,x y z XŒ and , 0s t > ,

Compatible Mappings and Common Fixed Points in Fuzzy Metric Spaces 201

( ), ,0 0M x y = (2.4.1)

( ), , 1M x y t = , for all 0t > , if and only if x y= , (2.4.2)

( ) ( ), , , ,M x y t M y x t= , (2.4.3)

( ) ( ) ( ), , , , , ,M x z s t M x y s M y z t+ ≥ * (2.4.4)

( ) [ [ [ ], , : 0, 0,1M x y ◊ • Æ is left continuous, and (2.4.5)

( )lim , , 1t M x y tÆ• = (2.4.6)

Example 2.5 [7]. Let ( ),X d be a metric space and { }min ,a b a b* = or ab for every

[ ], 0,1a b Œ . Let dM be a fuzzy set in [ [2 0,X ¥ • given by ( ) ( ), , ,dM x y t t t d x y= +

if 0t > and ( ), ,0 0dM x y = . Then ( ), ,dX M * is a fuzzy metric space and dM is called the standard fuzzy metric induced by d . Thus every metric d induces a fuzzy metric

dM on X . For further examples and elementary properties, we refer to [25].

Definition 2.6 [8]. A sequence { }nx in a fuzzy metric space ( ), ,X M * is called

(a) a Cauchy Sequence, if ( )lim , , 1n n n pM x x tÆ• + = , for all 0t > , 0p > .

(b) convergent to x (in symbols, limn nx xÆ• = or nx xÆ ), if (lim , ,n nM x xÆ•

) 1t = , for all 0t > .

Definition 2.7 [8]. A fuzzy metric space ( ), ,X M * is said to be complete, if every Cauchy sequence in X is convergent.

Definition 2.8. A self map T of a fuzzy metric space ( ), ,X M * is said to be continuous, if for every x XŒ , nx xÆ implies nTx TxÆ .

Definition 2.9. A pair [ ],A B of self maps of a fuzzy metric space ( ), ,X M * is said to be

(a) compatible [28], if ( )lim , , 1n n nM ABx BAx tÆ• = , for all 0t > , whenever

{ }nx is a sequence in X such that lim limn n n nAx Bx xÆ• Æ•= = , for some x XŒ .

(b) semi-compatible [27], if limn nABx BxÆ• = , whenever { }nx is a sequence in X such that lim limn n n nAx Bx xÆ• Æ•= = , for some x XŒ .

(c) weak compatible or coincidentally commuting [16], if they commute at their points of coincidence, i.e., for every x XŒ , Ax Bx= implies ABx BAx= .

(d) reciprocal continuous [23], if limn nABx AxÆ• = , limn nBAx BxÆ• = , whenever { }nx is a sequence in X such that lim limn n n nAx Bx xÆ• Æ•= = , for some x XŒ .


Lemma 2.10. If [ ],A B is a pair of compatible self maps of a fuzzy metric space

( ), ,X M * , then they are weak compatible. But the converse is not true. For example

[2], let [ ]0, 2X = and M be the standard fuzzy metric on X induced by the metric

( ),d x y x y= - . Let A , B be self maps of X given by

2 , if 0 12, if 1 2x x

Axx

- £ £Ï= Ì < £Ó

and if 0 1

2, if 1 2x x

Bxx

£ £Ï= Ì < £Ó

For 1 1nx n X= - Œ , we get, 1nAx Æ , 1nBx Æ , but ( )lim , ,n n nM ABx BAx tÆ• π 1 .

Therefore [ ],A B is not compatible. It is easy to see that [ ],A B is weak compatible.

Lemma 2.11. Let ( ), ,X M * be a fuzzy metric space. If there exists 0 1k< < such

that ( ) ( ), , , ,M x y kt M x y t≥ , for all 0t > , then x y= .

Lemma 2.12. If { }ny is a sequence in a fuzzy metric space ( ), ,X M * such that there

exists 0 1k< < satisfying ( ) ( )1 1, , , ,n n n nM y y kt M y y t+ -≥ for all 0t > , 0n > , then

{ }ny is a Cauchy sequence.

Lemma 2.13 [1]. Let [ ],A B be a compatible pair of self maps of a fuzzy metric

space ( ), ,X M * such that A is continuous. If ,n nAx Bx zÆ , for some z XŒ , then

nBAx AzÆ .

Lemma 2.14. If [ ],A B is a pair of continuous self maps of a fuzzy metric space

( ), ,X M * , then [ ],A B is reciprocal continuous. But the converse is not true. For

example [20], let [ ]2,20X = and M be the standard fuzzy metric on X induced by the

metric ( ),d x y x y= - . Let [ ],A B be a pair of self maps of X given by

2, if 23, if 2

xAx

x

=Ï= Ì >Ó

and 2, if 26, if 2

xBx

x

=Ï= Ì >Ó

.

Then [ ],A B is reciprocal continuous but not continuous.

Lemma 2.15 [20]. Let [ ],A B be a pair of reciprocal continuous self maps of a fuzzy

metric space ( ), ,X M * . Then [ ],A B is semi-compatible if and only if [ ],A B is compatible.


Theorem 2.16 (Murthy [21]). Let ( ),X d be a complete metric space and f , g be

self mappings of X such that the ordered pair ( ),f g is weak compatible and satisfy the condition:

For ,x y XŒ , ( ) ( ) ( ) ( )( )( ), max , , , , ,d fx fy d gx gy d fx gy d fx gxl j£ ◊ where [ )0,1l Œ

and :j + +Æ is a continuous mapping such that ( )t tj < for every 0t > . If the range of g contains the range of f and if f is continuous, then f and g have a unique common fixed point.

Theorem 2.17 (Jungck [15]). Let A ,B ,S and T be self maps of a complete metric space ( ),X d . Suppose that S and T are continuous, the pairs [ ],A S and [ ],B T are

compatible and AX TXÕ , BX SXÕ . If there exists ( )0,1r Œ such that ( ),d Ax By

max xyr M£ , for x , y in X , where ( ) ( ){ ( ), , , , , ,1 2xyM d Ax Sx d By Ty d Sx Ty=

( ) ( )( )}, ,d Ax Ty d By Sx+ ,then there is a unique point z in X such that z Az Bz= =

Sz Tz= = . In what follows,we assume that ( ), ,X M * is a fuzzy metric space with the condition:

nx xÆ , ny yÆ in X implies ( ) ( ), , , ,n nM x y t M x y tÆ (2.17.1)

3. Main results

In this section, we state and prove the main results of our paper. We extend the Theorems (2.16) and (2.17) to fuzzy metric spaces and deduce some Corollaries. In paricular, we deduce the fuzzy Banach contraction Theorem due to Grabiec [8] as a Corollary. We further construct examples to illustrate our results.

Theorem 3.1. Let A , B be self maps of a complete fuzzy metric space ( ), ,X M * satisfying the following conditions:

AX BXÕ , (3.1.1)

[ ],A B is compatible, (3.1.2)

A or B is continuous, and (3.1.3)

( ) ( ) ( ) ( ){ }, , min , , , , , , , ,M Ax Ay kt M Bx By t M Ax By t M Ax Bx t≥ , (3.1.4)

for all ,x y XŒ , 0t > and some 0 1k< < . Then A and B have a unique common fixed point.

Proof. Let 0x XŒ be arbitrary. Then 0 1Ax Bx= for some 1x XŒ by (3.1.1) and so on. We thus obtain a sequence { }ny in X such that 1n n ny Ax Bx-= = , nŒ . We get

( ) ( )1 1, , , ,n n n nM y y kt M Ax Ax kt+ -= ( ){ ( )1 1min , , , , , ,n n n nM Bx Bx t M Ax Bx t- -≥


( )}1 1, ,n nM Ax Bx t- - by (3.1.4). ( ) ( ) ( ){ }1 1min , , , , , , , ,n n n n n nM y y t M y y t M y y t- -= =

( )1, ,n nM y y t- , for every 0t > . Therefore, by Lemma (2.12), { }ny is a Cauchy sequence in X . As X is complete, ny yÆ for some y XŒ , i.e., 1n n ny Ax Bx y-= = Æ .

(a) Let A be continuous. As [ ],A B is compatible, we have by Lemma (2.13), nBAx

AyÆ . Now ( ) { ( ) ( ) (, , min , , , , , , ,n n n n n n nM Ax AAx kt M Bx BAx t M Ax BAx t M Ax≥

)},nBx t , for every 0t > . Letting nÆ• , we get, ( ) ( ){, , min , , ,M y Ay kt M y Ay t≥

( ) ( )} ( ), , , , , , ,M y Ay t M y y t M y Ay t= , for every 0t > . Therefore, by Lemma (2.11),

we get Ay y= . By (3.1.1), we get y Ay By= = ¢ for some y XŒ¢ . Also, ( ,nM AAx

) ( ) ( ) ( ){ }, min , , , , , , , ,n n n nAy t M BAx By t k M AAx By t k M AAx BAx t k≥¢ ¢ ¢ . On

letting nÆ• , we get ( ) { ( ) ( ), , min , , , , , ,M Ay Ay t M Ay By t k M Ay By t k≥¢ ¢ ¢

( )}, , 1M Ay Ay t k = . Therefore, Ay Ay= ¢ and thus y Ay Ay By= = =¢ ¢ . Now we

get, ( ) ( ), , , ,M Ay By t M Ay BAy t= ¢ ( ), ,M Ay ABy t= ¢ , by (3.1.2) and Lemma (2.10).

( ), , 1M Ay Ay t= = . Therefore, Ay By= and thus Ay By y= = , a common fixed point of A and B .

(b) Let B be continuous. As [ ],A B is compatible, we have Lemma (2.13), nABx Æ

By . Now ( )1, ,n nM Ax ABx kt- ≥ { ( ) ( ) (1 1min , , , , , ,n n n nM Bx BBx t M Ax BBx t M A- -

)}1 1, ,n nx Bx t- - , which in the limit as nÆ• , gives ( ) { ( ), , min , , ,M y By kt M y By t≥

( ) ( )} ( ), , , , , , ,M y By t M y y t M y By t= , for every 0t > . Therefore, by Lemma (2.11),

we get, By y= . Now we get, ( ), ,nM ABx Ay t ≥ { ( ) (min , , , ,n nM BBx By t k M ABx

) ( )}, , , ,n nBy t k M ABx BBx t k . Letting nÆ• ,we get ( ) { (, , min ,M By Ay t M By≥

) ( ) ( )}, , , , , , , 1By t k M By By t k M By By t k = , for every 0t > . Therefore, By Ay= and thus Ay By y= = , a common fixed point of A and B .

(c) Uniqueness. Let w XŒ be such that Aw Bw w= = . Now, ( ) (, , ,M y w kt M Ay=

) ( ) ( ) ( ){ }, min , , , , , , , ,Aw kt M By Bw t M Ay Bw t M Ay By t≥ { ( ) (min , , , ,M y w t M y=

) ( )}, , , ,w t M y y t ( ), ,M y w t= , for every 0t > . Therefore, y w= and so the common fixed point is unique.

Considering B I= , the identity map of X , we get the following Corollary.

Corollary 3.2. Let A be a self map of a complete fuzzy metric space ( ), ,X M * such

that ( ) ( ) ( ) ( ){ }, , min , , , , , , , ,M Ax Ay kt M x y t M Ax y t M Ax x t≥ , forall ,x y XŒ , 0t > and some 0 1k< < . Then A has a unique fixed point.

The following result is the fuzzy Banach contraction Theorem due to Grabiec [8].



that ( ) ( ), , , ,M Ax Ay kt M x y t≥ for all ,x y XŒ , 0t > and some 0 1k< < . Then A has a unique fixed point.

Proof. It follows immediately from the Corollary 3.2.

Example 3.4. Let [ ]0,1X = and M be the standard fuzzy metric on X induced by

( ),d x y x y= - . Now ( ), ,X M * is a complete fuzzy metric space. Let A , B be self maps of X given by 1Ax = ; 1Bx = , if x is rational and 0Bx = , if x is irrational. Then A , B satisfy all the conditions of the Theorem (3.1) and 1 XŒ is the unique common fixed point A and B .

Theorem 3.5. Let A , B , S and T be self maps of a complete fuzzy metric space ( ), ,X M * satisfying the following conditions:

AX TXÕ , BX SXÕ , (3.5.1)

The pairs [ ],A S and [ ],B T are semi-compatible, (3.5.2)

The pairs [ ],A S and [ ],B T are reciprocal continuous, (3.5.3)

( ) ( ) ( ) ( ){ }, , min , , , , , , , ,M Ax By kt M Ax Sx t M By Ty t M Sx Ty t≥ , (3.5.4)

for every ,x y XŒ , 0t > and some 0 1k< < . Then A , B , S and T have a unique common fixed point.

Proof. Let 0x XŒ be arbitrary. Then there exists 1 2,x x XŒ such that 0 1Ax Tx= ,

1 2Bx Sx= by (3.5.1), and so on. We thus obtain a sequence { }ny in X such that

2 1 2 2 2 1n n ny Ax Tx- - -= = , 2 2 1 2n n ny Bx Sx-= = , nŒ . We get ( )2 2 1, ,n nM y y kt+ =

( ) ( )2 1 2 2 2 1, , , ,n n n nM Bx Ax kt M Ax Bx kt- -= ( ){ (2 2 2 1min , , , ,n n nM Ax Sx t M Bx -≥

) ( )}2 1 2 2 1, , , ,n n nTx t M Sx Tx t- - ( ) ( ) ( ){ }2 1 2 2 2 1 2 2 1min , , , , , , , ,n n n n n nM y y t M y y t M y y t+ - -=

( ) ( ){ }2 2 1 2 1 2min , , , , ,n n n nM y y t M y y t+ -= , for every 0t > . Therefore, we get ( 2 ,n

M y

) ( )2 1 2 1 2, , ,n n ny kt M y y t+ -≥ , for every 0t > , 0n > .

Similarly, ( ) ( )2 1 2 2 2 2 1, , , ,n n n nM y y kt M y y t- - -≥ , for every 0t > , 0n > . These give,

( ) ( )1 1, , , ,n n n nM y y kt M y y t+ -≥ , for every 0t > , 0n > . Therefore, by Lemma (2.12),

{ }ny is a Cauchy sequence in X . As X is complete, { }ny finds a limit z in X .

Consequently, all the subsequences { }2nAx , { }2nSx , { }2 1nBx - , { }2 1nTx - converge to

z . As [ ],A S is reciprocal continuous and compatible (by Lemma (2.15) and (3.5.2)),

we get 2nASx AzÆ , 2nSAx SzÆ , ( )2 2, , 1n nM ASx SAx t Æ , for every 0t > .

Therefore, by (2.17.1), we get ( ), , 1M Az Sz t = , for every 0t > and so Az Sz= . By a similar argument, we get Bz Tz= .


Now, ( ) ( ) ( ) ( ){ }2 2 2 2, , min , , , , , , , ,n n n nM Ax Bz kt M Ax Sx t M Bz Tz t M Sx Tz t≥ , by

(3.5.4). Letting nÆ• , we get ( ) ( ) ( ){ (, , min , , , , , , ,M z Bz kt M z z t M Bz Tz t M z≥

)} ( ) ( ), , , , ,Tz t M z Tz t M zBz t= = , for every 0t > . Therefore, we get z Bz= and thus

z Bz Tz= = . Again, ( ) ( ) ( ){ ( ), , , , min , , , , ,M Az z kt M Az Bz kt M Az Sz t M Bz Tz t= ≥

( )} ( ) ( ), , , , , , ,M Sz Tz t M Sz Tz t M Az z t= = . Therefore, Az z= and so Az Bz= = Sz Tz z= = , a common fixed point of A , B , S and T . To show uniqueness, let w XŒ such that Aw Bw Sw Tw w= = = = . Now, ( ), ,M z w kt ( ), ,M Az Bw kt=

( ) ( ) ( ){ }min , , , , , , , ,M Az Sz t M Bw Tw t M Sz Tw t≥ ( ){ ( )min , , , , , ,M z z t M w w t=

( )}, ,M z w t ( ), ,M z w t= , for every 0t > . Therefore z w= and so the common fixed point is unique.

Corollary 3.6. Let A , B be two continuous self maps of a complete fuzzy metric space ( ), ,X M * such that ( ) ( ) ( ) ( ){ }, , min , , , , , , , ,M Ax By kt M Ax x t M By y t M x y t≥ , for every ,x y XŒ , 0t > and some 0 1k< < . Then A and B have a unique common fixed point.

Proof. Taking S T I= = , the identity map of X , the result immediately follows from the Theorem 3.5.

With A B= , we get the following Corollary.

Corollary 3.7. Let A be a continuous self map of a complete fuzzy metric space ( ), ,X M * satisfying ( ) ( ) ( ) ( ){ }, , min , , , , , , , ,M Ax Ay kt M Ax x t M Ay y t M x y t≥ , for all ,x y XŒ , 0t > and some 0 1k< < . Then A has a unique fixed point.

We now deduce the fuzzy Banach contraction Theorem due to Grabiec [8] as a Corollary.


that ( ) ( ), , , ,M Ax Ay kt M x y t≥ , for all ,x y XŒ , 0t > and some 0 1k< < . Then A has a unique fixed point.

Proof. It follows immediately from Corollary 3.7 as such a mapping is obviously continuous.

Example 3.9. Let [ ]0,1X = and M be the standard fuzzy metric on X induced by

( ),d x y x y= - . Now ( ), ,X M * is a complete fuzzy metric space. Let A , B , S , T be self maps of X given by 81Ax x= , 27Bx x= , 9Sx x= , 3Tx x= . Then A , B , S , T satisfy all the conditions of the Theorem 3.5 and have a unique common fixed point 0 XŒ . It may be verified that ( ) ( ), , , ,M Ax By kt M Sx Ty t≥ , for all ,x y XŒ ,

0t > with 1 3k = . So the condition (3.5.4) is satisfied.


Remarks. The Theorem 3.1 and 3.5 remain true if the condition “ ( ), ,X M * is a complete fuzzy metric space” is replaced by “ the range of one of the maps is a complete subspace of X ”. Also the Theorem 3.1 cannot be deduced as a particular case of the Theorem 3.5. Moreover, the t -norm * we have used is an arbitrary continuous t -norm.

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Los Angeles

Bipolar Fuzzy Structure of BG -subalgebras

Tapan Senapati

Department of Mathematics, Padima Janakalyan Banipith Kukrakhupi, Paschim Medinipur-721517, India.


Abstract: Based on the concept of bipolar fuzzy set, a theoretical approach of the BG -

subalgebras is established. Some characterizations of bipolar fuzzy BG -subalgebras of BG -algebras are given.

Key words and phrases: BG -algebra, BG -subalgebra, bipolar fuzzy BG -subalgebra.

1. Introduction

Since the fuzzy set was introduced by Zadeh [22], many new approaches and theories treating imprecision and uncertainty have been proposed, such as the generalized theory of uncertainty ( )GTU introduced by Zadeh [23] and the intuitionistic fuzzy sets introduced by Atanassov [2] and so on. Among these theories, a well-known extension of the classic fuzzy set is bipolar fuzzy set theory, which was pioneered by Zhang [25]. Since then, many researchers have investigated this topic and obtained some meaningful conclusions [3,4,24,26].

In traditional fuzzy sets the membership degree range is [ ]0,1 . The membership degree is the degree of belonging ness of an element to a set. The membership degree 1 indicates that an element completely belongs to its corresponding set, the membership degree 0 indicates that an element does not belong to the corresponding set and the membership degree on the interval ( )0,1 indicate the partial membership to the corresponding set. Sometimes, membership degree also means the satisfaction degree of elements to some property corresponding to a set and its counter property. Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership degree range is increased form the interval [ ]0,1 to the interval [ ]1,1- . In a bipolar fuzzy set the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degrees on ( ]0,1 indicate that elements somewhat satisfy the property

210 Tapan Senapati

and the membership degrees on [ ]1,0- indicate that elements somewhat satisfy the implicit counter-property. Although bipolar fuzzy sets and intuitionistic fuzzy sets look similar to each other, they are essentially different sets [10]. In many domains, it is important to be table to deal with bipolar information. It is noted that positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. BCK -algebras and BCI -algebras [ ]5,14 are two important classes of logical

algebras introduced by Imai and Iseki. It is known that the class of BCK -algebra is a proper subclass of the class of BCI -algebras. Neggers and Kim [12] introduced a new notion, called a B -algebras which is related to several classes of algebras of interest such as BCI BCK -algebras. Kim and Kim [7] introduced the notion of BC -algebras, which is a generalization of B -algebras. Ahn and Lee [1] fuzzified BG -algebras. Saeid [13] introduced fuzzy topological BG -algebras. Senapati et al. [17, 20, 21] done lot of works on B -algebras. They also [15, 16, 18, 19] presented the concept and basic properties of intuitionistic fuzzy subalgebras, intuitionistic L -fuzzy ideals, interval-valued intuitionistic fuzzy subalgebras, interval-valued intuitionistic fuzzy closed ideals of BG -algebras. Lee [8] introduced the notion of bipolar fuzzy subalgebras and ideals in BCK BCI -algebras. The concept of bipolar valued fuzzy translation and bipolar valued fuzzy S -extension of a bipolar valued fuzzy subalgebra in BCK BCI -algebra was introduced by Jun et al. [6]. Motivated by this, in this paper, the notions of bipolar fuzzy BG -subalgebras is introduced and their properties are investigated.

2. Preliminaries

In this section, some elementary aspects that are necessary for this paper are included. A BG -algebra is an important class of logical algebras introduced by Kim and Kim [7] and was extensively investigated by several researchers. This algebra is defined as follows.

A non-empty set X with a constant 0 and a binary operation * is called to be BG -algebra [7] if it satisfies the following axioms

F1. 0x x* = F2. 0x x* =

F3. ( ) ( )0x y y x* * * = , for all ,x y XŒ . In any BG -algebra X , the following hold (see [7]):

(i) the right cancellation law holds in X , i.e., x y z y* = * implies x z= ,

(ii) ( )0 0 x x* * = for all x XŒ ,

(iii) if 0x y* = , then x y= for any ,x y XŒ , (iv) if 0 0x y* = * , then x y= for any ,x y XŒ ,

(v) ( )( )0x x x x* * * = for all x XŒ .

A non-empty subset S of a BG -algebra X is called a subalgebra [1] of X if x y S* Œ for any ,x y SŒ .

Bipolar Fuzzy Structure of BG -subalgebras 211

Definition 2.1 [22] (Fuzzy set). Let X be the collection of objects denoted generally by x , then a fuzzy set A in X is defined as ( ){ }, :AA x x x Xm= Œ , where ( )A xm is

called the membership value of x in A and ( )0 1A xm£ £ . Combined the definitions of BG -subalgebra over crisp set and the idea of fuzzy set

Ahn and Lee [1] defined fuzzy BG -subalgebra, which is defined below.

Definition 2.2 [1] (Fuzzy BG -subalgebra). Let m be a fuzzy set in a BG -algebra.

Then m is called a fuzzy BG -subalgebra of X if ( ) ( ) ( ){ }min ,x y x ym m m* ≥ for all

,x y XŒ , where ( )xm is the membership value of x in X .

Definition 2.3 [25] (Bipolar fuzzy set). Let X be a nonempty set. A bipolar fuzzy set j in X is an object having the form ( ) ( ){ }, , :x x x x Xj j j+ -= Œ where

[ ]: 0,1Xj + Æ and [ ]: 0,1Xj - Æ are mappings.

We use the positive membership degree ( )xj + to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set j and the negative

membership degree ( )xj - to denote the satisfaction degree of an element x to some

implicit counter-property corresponding to a bipolar fuzzy set j . If ( ) 0xj + π and

( ) 0xj - = , it is the situation that x is regarded as having only positive satisfaction for

j . If ( ) 0xj + = and ( ) 0xj - π , it is the situation that x does not satisfy the property of j but somewhat satisfies the counter property of j . It is possible for an element x

to be such that ( ) 0xj + π and ( ) 0xj - π when the membership function of the property overlaps that of its counter property over some portion of X .

Definition 2.4 [9]. Let ( )1 1 1,j j j+ -= and ( )2 2 2,j j j+ -= be two bipolar fuzzy sets on

X . Then the intersection and union of 1j and 2j is denoted by 1 2j j« and 1 2j j» respectively and is given by

( ) ( ) ( ){ }1 2 1 2 1 2 1 2 1 2, min ,maxj j j j j j j j j j+ + - - + + - -« = « « = « «

( ) ( ) ( ){ }1 2 1 2 1 2 1 2 1 2, max , , min ,j j j j j j j j j j+ + + +» = » » = .

3. Main results

For the sake of simplicity, we shall use the symbol ( ),j j j+ -= for the bipolar fuzzy

subset ( ) ( ){ }, , :x x x x Xj j j+ -= Œ . Throughout this paper, X always means a

BG -algebra without any specification.

212 Tapan Senapati

Definition 3.1. Let ( ),j j j+ -= be a bipolar fuzzy set in X , then the set j is

bipolar fuzzy BG -subalgebra over the binary operator * if it satisfies the following conditions:

(IBS1) ( ) ( ) ( ){ }min ,x y x yj j j+ + +* ≥

(IBS2) ( ) ( ) ( ){ }max ,x y x yj j j- - -* £ for all ,x y XŒ . We consider an example of bipolar fuzzy BG -subalgebra below.

Example 3.2. Let { }0,1, 2,3,4,5X = be a BG -algebra with the following Cayley table:

0 1 2 3 4 50 0 5 4 3 2 11 1 0 5 4 3 22 2 1 0 5 4 33 3 2 1 0 5 44 4 3 2 1 0 55 5 4 3 2 1 0

*

Let ( ),j j j+ -= be a bipolar fuzzy set in X defined by ( ) { }0.55, if 0, 2,40.46, otherwise

xxj + Ï Œ

=ÌÓ

and ( ) { }0.42, if 0, 2, 40.36, otherwise.

xxj - Ï- Œ

= Ì-Ó

All the conditions of Definition 3.1 have been satisfied by the set j . Thus

( ),j j j+ -= is a bipolar fuzzy BG -subalgebra of X .

Proposition 3.3. If ( ),j j j+ -= is a bipoar fuzzy BG -subalgebra in X , then for all

x XŒ , ( ) ( )0 xj j+ +≥ and ( ) ( )0 xj j- -£ . Thus, ( )0j + and ( )0j - are the upper

bounds and lower bounds of ( )xj + and ( )xj - respectively.

Proof. Let x XŒ . Then ( ) ( ) ( ) ( ){ } ( )0 min ,x x x x xj j j j j+ + + + += * ≥ = and

( ) ( ) ( ) ( ){ } ( )0 max ,x x x x xj j j j j- - - - -= * £ = .

Theorem 3.4. Let ( ),j j j+ -= be a bipolar fuzzy BG -subalgebra of X . If there

exists a sequence nx in X such that ( )lim 1n nxj +Æ• = and ( )lim 1n nxj -

Æ• = - , then

( )0 1j + = and ( )0 1j - = - .


Proof. By Proposition 3.3, ( ) ( )0 xj j+ +≥ for all x XŒ , therefore ( ) ( )0 nxj j+ +≥

for every positive integer n . Consider, ( ) ( )1 0 lim 1n nxj j+ +Æ•≥ ≥ = . Hence,

( )0 1j + = .

Again, by Proposition 3.3, ( ) ( )0 xj j- -£ for all x XŒ , thus ( ) ( )0 nxj j- -£ for

every positive integer n . Now, ( ) ( )1 0 lim 1n nxj j- -Æ•- £ £ = - . Hence, ( )0 1j - = - .

Proposition 3.5. If a bipolar fuzzy set ( ),j j j+ -= in X is a bipolar fuzzy BG -

subalgebra, then ( ) ( )0 x xj j+ +* ≥ and ( ) ( )0 x xj j- -* £ for all x XŒ .

Proof. For all x XŒ , ( ) ( ) ( ){ } ( ) ( ){ }0 min 0 , min ,x x x x xj j j j j+ + + + +* ≥ = * ≥

( ) ( ){ } ( ){ } ( )min min , ,x x x xj j j j+ + + += and ( ) ( ) ( ){ }0 max 0 ,x xj j j- - -* £ =

( ) ( ){ } ( ) ( ){ } ( ){ } ( )max , max max , ,x x x x x x xj j j j j j- - - - - -* £ = .

For any elements x and y of X , let us write n x y’ * for ( )( )( )x x x y* * * where x occurs n times.

Theorem 3.6. Let ( ),j j j+= be a bipolar fuzzy BG -subalgebra of X and let

nŒ (the set of natural numbers). Then (i) ( ) ( )n x x xj j+ +’ * ≥ , for any odd number n ,

(ii) ( ) ( )n x x xj j- -’ * £ , for any odd number n ,

(iii) ( ) ( )n x x xj j+ +’ * = , for any even number n ,

(iv) ( ) ( )n x x xj j- -’ * = , for any even number n .

Proof. Letx XŒ and assume that n is odd. Then 2 1n p= - for some positive integer p . We prove the theorem by induction. Now ( ) ( ) ( )0x x xj j j+ + +* = ≥ and ( )x xj - *

( ) ( )0 xj j- -= £ . Suppose that ( ) ( )2 1p x x xj j+ - +’ * ≥ and ( ) ( )2 1p x x xj j- - -’ * £ . Then by assumption,

( )( ) ( ) ( )( )( ) ( ) ( )2 1 1 2 1 2 1 2 1p p p px x x x x x x x x x xj j j j j+ -+ + + + - + - +’ * = ’ * = ’ * * * = ’ * ≥

and ( )( ) ( ) ( )( )( ) ( ) ( )2 1 1 2 1 2 1 2 1p p p px x x x x x x x x x xj j j j j+ -- - + - - - - -’ * = ’ * = ’ * * * = ’ * £ ,

which proves (i) and (ii). Proofs are similar for the cases (iii) and (iv). The intersection of two bipolar fuzzy BG -subalgebras is also a bipolar fuzzy BG -

subalgebra, which is proved in the following theorem.

Theorem 3.7. Let ( )1 1 1,j j j+ -= and ( )2 2 2,j j j+ -= be two bipolar fuzzy BG -sub-

algebras of X . Then 1 2j j« is a bipolar fuzzy BG -subalgebra of X .

214 Tapan Senapati

Proof. Let 1 2,x y j jŒ « . Then 1,x y jŒ and 2j . Now, ( ) { (1 2 1minx y xj j j+ + +« * =

) ( )} ( ) ( ){ } ( ) ( ){ }{ }2 1 1 2 2, min min , ,min ,y x y x y x yj j j j j+ + + + +* * ≥ = {{ ( )1min min ,xj +

( )} ( ) ( ){ }} ( ) ( ){ }2 1 2 1 2 1 2, min , min ,x y y x yj j j j j j j+ + + + + + += « « and ( )1 2 x yj j- -« * =

( ) ( ){ } ( ) ( ){ } ( ) ( ){ }{ }1 1 1 1 2 2max , max max , ,max ,x y x y x y x yj j j j j j- - - - - -* * ≥ = max

( ) ( ){ } ( ) ( ){ }{ } ( ) ( ){ }1 2 1 2 1 2 1 2max , ,max , max ,x x y y y yj j j j j j j j- - - - - - - -= « « . Hence,

1 2A A« is a bipolar fuzzy BG -subalgebra of X .

The above theorem can be generalized as follows.

Theorem 3.8. Let { }1, 2,3,i ij = be a family of bipolar fuzzy BG -subalgebra of

X . Then ij∩ is also a bipolar fuzzy BG -subalgebra of X , where ( )(min ,i i xj j +=∩

( ))max i xj - .

The union of any set of bipolar fuzzy BG -subalgebras need not be a bipolar fuzzy BG -subalgebra which is shown in the following example.

Example 3.9. Let { }0,1, 2,3,4,5X = be a BG -algebra with the following Cayley table:

0 1 2 3 4 50 0 2 1 3 4 51 1 0 2 5 3 42 2 1 0 4 5 33 3 4 5 0 1 24 4 5 3 2 0 15 5 3 4 1 2 0

*

Let ( ),j j j+ -= and ( ),y y y+ -= be two bipolar fuzzy sets in X defined by

( ) { }{ }

0.77, if 0,40.26, if \ 0,4

xx

x Xj + Ï ŒÔ= Ì ŒÔÓ

, ( ) { }{ }

0.52, if 0,30.13, if \ 0,3

xx

x Xj - Ï- ŒÔ= Ì- ŒÔÓ

,

( ) { }{ }

0.81, if 0,50.36, if \ 0,5

xx

x Xy + Ï ŒÔ= Ì ŒÔÓ

and ( ) { }{ }

0.44, if 0,40.18, if \ 0, 4

xx

x Xy - Ï- ŒÔ= Ì- ŒÔÓ

.

Then ( ),j j j+ -= and ( ),y y y+ -= are bipolar fuzzy BG -subalgebras of X . But

the union of j y» is not a bipolar fuzzy BG -subalgebras of X since ( )4 5j y+ +» *

( ) ( ){ }0.36 0.81 max 4 , 5j y j y+ + + += ≥/ = » » and ( )3 4 0.18 0.52j y- -» * = - £/ -

( ) ( ){ }min 3 , 4j y j y- - - -= » » .


The sets ( ) ( ){ }: 0x X xj j+ +Œ = and ( ) ( ){ }: 0x X xj j- -Œ = are denoted by Ij +

and Ij - -respectively. These two sets are also BG -subalgebra of X .

Theorem 3.10. Let ( ),j j j+ -= be a bipolar fuzzy BG -subalgebra of X , then the

sets Ij + and Ij - are BG -subalgebras of X .

Proof. Let ,x y Ij +Œ . Then ( ) ( ) ( )0x yj j j+ + += = and so, ( )x yj + * ≥ ( ){min ,xj +

( )} ( )0yj j+ += . By using Proposition 3.3, we know that ( ) ( )0x yj j+ +* = or equiva-

lently x y Ij +* Œ .

Again let ,x y Ij -Œ . Then ( ) ( ) ( )0x yj j j- - -= = and so, ( ) ( ){max ,x y xj j- -* £

( )} ( )0yj j- -= . Then by Proposition 3.3, we know that ( ) ( )0x yj j- -* = or equiva-

lently x y Ij -* Œ . Hence, the sets Ij + and Ij - are BG -subalgebras of X .

Theorem 3.11. Let B be a nonempty subset of X and ( ),j j j+ -= be a bipolar

fuzzy set in X defined by

( ) , if , otherwise

xx

lj

t+ ŒÏ

= ÌÓ

B and ( ) , if

, otherwisex

xg

jd

- ŒÏ= ÌÓ

B

for all l , t , g and [ ]0,1d Œ with l t≥ and g d£ . Then j is a bipolar fuzzy BG -subalgebra of X if and only if B is a BG -subalgebra of X . Moreover, I Ij j+ -= =B .

Proof. Let j be a bipolar fuzzy BG -subalgebra of X . Let ,x y XŒ be such that

,x y ŒB . Then ( ) ( ) ( ){ } { }min , min ,x y x yj j j l l l+ + +* ≥ = = and ( ) maxx yj - * £

( ) ( ){ } { }, max ,x yj j g g g- - = = . So x y* ŒB . Hence, B is a BG -subalgebra of X .

Conversely, suppose that B is a BG -subalgebra of X . Let ,x y XŒ . Consider two cases:

Case (i) If ,x y ŒB then x y* ŒB , thus ( ) ( ) ( ){ }min ,x y x yj l j j+ + +* = = and

( ) ( ) ( ){ }max ,x y x yj g j j- - -* = = .

Case (ii) If xœB or, y œB ,then ( ) ( ) ( ){ }min ,x y x yj t j j+ + +* ≥ = and ( )x yj - *

( ) ( ){ }max ,x yd j j- -£ = .

Hence,j is a bipolar fuzzy BG -subalgebra of X . Also, Ij + = ( ) ( ){ }, 0x X xj j+ +Œ =

( ){ },x X xj l+= Œ = = B and ( ) ( ){ }, 0I x X xj j j-- -= Œ = = ( ){ },x X xj g-Œ =

= B .

216 Tapan Senapati

Definition 3.12. Let ( ),j j j+ -= is a bipolar fuzzy BG -subalgebra of X . For

( ) [ ] [ ], 1,0 0,1s t Œ - ¥ , the set ( ) ( ){ }: :U t x X x tj j+ += Œ ≥ is called positive t -cut of

j and ( ) ( ){ }: :L s x X x sj j- -= Œ £ is called negative s -cut of j .

Theorem 3.13. If ( ),j j j+ -= is a bipolar fuzzy BG -subalgebra of X , then the

positive t -cut and negative s -cut of j are BG -subalgebras of X .

Proof. Let ( ), :x y U tj +Œ . Then ( )x tj + ≥ and ( )y tj + ≥ . It follows that ( )x yj + *

( ) ( ){ }min ,x y tj j+ +≥ ≥ so that ( ):x y U tj +* Œ . Hence, ( ):U tj + is a BG -sub-

algebra of X . Let ( ), :x y L sj -Œ . Then ( )x sj - £ and ( )y sj - £ . It follows that

( ) ( ) ( ){ }max ,x y x y sj j j- - -* £ £ so that ( ):x y L sj -* Œ . Hence, ( ):L sj - is a

BG -subalgebra of X .

Theorem 3.14. Let ( ),j j j+ -= be a bipolar fuzzy set in X , such that the sets

( ):U tj + and ( ):L sj - are BG -subalgebras of X for every ( ) [ ] [ ], 1,0 0,1s t Œ - ¥ .

Then ( ),j j j+ -= is a bipolar fuzzy BG -subalgebra of X .

Proof. Let for every ( ) [ ] [ ], 1,0 0,1s t Œ - ¥ , ( ):U tj + and ( ):L sj - are BG -sub-

algebras of X . In contrary, let 0 0,x y XŒ be such that ( ) { ( ) ( )}0 0 0 0min ,x y x yj j j+ + +* < .

Let ( )0 1xj q+ = , ( )0 2yj q+ = and ( )0 0x y tj + * = . Then { }1 2min ,t q q< . Let us con-

sider, ( ) ( ) ( ){ }1 0 0 0 01 2 min ,t x y x yj j j+ + +È ˘= * +Î ˚ . We get that ( { })1 1 21 2 min ,t t q q= + .

Therefore, { }( )1 1 1 21 2 min ,t t tq q q> = + > and { }( )2 1 1 21 2 min ,t t tq q q> = + > .

Hence, { } ( )1 2 1 0 0min , t t x yq q j +> > = * , so that ( )0 0 :x y U tj +* œ which is a con-

tradiction, since ( ) { }0 1 1 2 1min ,x tj q q q+ = ≥ > and ( ) { }0 2 1 2 1min ,y tj q q q+ = ≥ > .

This implies ( )0 0, :x y U tj +Œ . Thus ( ) ( ) ( ){ }min ,x y x yj j j+ + +* ≥ for all ,x y XŒ .

Again let 0 0,x y XŒ be such that ( ) ( ) ( ){ }0 0 0 0max ,x y x yj j j- - -* > . Let

( )0 1xj h- = , ( )0 2yj h- = and ( )0 0x y sj - * = . Then { }1 2max ,s h h> . Let us consider,

( ) ( ) ( ){ }1 0 0 0 01 2 max ,s x y x yj j j- - -È ˘= * +Î ˚ . We get that { }( )1 1 21 2 max ,s s h h= + .

Therefore, { }( )1 1 1 21 2 max ,s s sh h h< = + < and { }( )2 1 1 21 2 max ,s s sh h h< = + < . Hence,

{ } ( )1 2 1 0 0max , s s x yh h j -< < = * , so that ( )0 0 :x y L sj -* œ which is a contradiction,

since ( ) { }0 1 1 2 1max ,x sj h h h- = £ < and ( ) { }0 2 1 2 1max ,y sj h h h- = £ < . This implies

( )0 0, :x y L sj -Œ . Thus ( ) ( ) ( ){ }max ,x y x yj j j- - -* £ for all ,x y XŒ . Hence, j =

( ),j j+ - is a bipolar fuzzy BG -subalgebra of X .


Theorem 3.15. Any BG -subalgebra of X can be realized as both the positive t -cut and negative s -cut of some bipolar fuzzy BG -subalgebra of X .

Proof. LetP be a bipolar fuzzy BG -subalgebra of X and ( ),j j j+ -= be a bipolar fuzzy set on X defined by

( ) , if 0, otherwise

x Px

lj + ŒÏ

= ÌÓ

, ( ) , if 0, otherwise

x Px

tj - ŒÏ

= ÌÓ

for all [ ]0,1l Œ and [ ]1,0t Œ - . We consider the following four cases:

Case (i). If ,x y PŒ , then ( )xj l+ = , ( )xj t- = and ( )yj l+ = , ( )yj t- = .

Thus, ( ) { } ( ) ( ){ }min , min ,x y x yj l l l j j+ + +* = = = and

( ) { }max ,x yj t t t- * = = = ( ) ( ){ }max ,x yj j- - .

Case (ii). If x PŒ and y Pœ then ( )xj l+ = , ( )xj t- = and ( ) 0yj + = , ( ) 0yj - = .

Thus, ( ) { } ( ) ( ){ }0 min ,0 min ,x y x yj l j j+ + +* ≥ = = and ( ) { }0 max ,0x yj t- * £ =

( ) ( ){ }max ,x yj j- -= .

Case (iii). If x Pœ and y PŒ then ( ) 0xj + = , ( ) 0xj - = and ( )yj l+ = , ( )yj t- = .

Thus, ( ) { } ( ) ( ){ }0 min 0, min ,x y x yj l j j+ + +* ≥ = = and ( )x yj - * £ { }0 max 0,t=

( ) ( ){ }max ,x yj j- -= .

Case (iv). If x Pœ and y Pœ then ( ) 0xj + = , ( ) 0xj - = and ( ) 0yj + = , ( ) 0yj - = .

Now ( ) { } ( ) ( ){ }0 min 0,0 min ,x y x yj j j+ + +* ≥ = = and ( )x yj - * £ { }0 max 0,0= =

( ) ( ){ }max ,x yj j- - .

Therefore ( ),j j j+ -= is a bipolar fuzzy BG -subalgebra of X .

Theorem 3.16. Let P be a subset of X and ( ),j j j+ -= be a bipolar fuzzy set on

X which is given in the proof of Theorem 3.15. If the positive t -cut and negative s -cut of j are BG -subalgebras of X , then P is a bipolar fuzzy BG -subalgebra of X .

Proof. Let ( ),j j j+ -= be a bipolar fuzzy BG -subalgebra of X , and ,x y PŒ .

Then ( ) ( )x yj l j+ += = and ( ) ( )x yj t j- -= = . Thus ( ) ( ) ( ){ }min ,x y x yj j j+ + +* ≥

{ }min ,t l l= = and ( ) ( ) ( ){ } { }max , max ,x y x yj j j t t t- - -* £ = = , which imply that x y P* Œ . Hence the theorem.

Theorem 3.17. If every bipolar fuzzy BG -subalgebras ( ),j j j+ -= of X has the

finite image, then every descending chain of BG -subalgebras of X terminates at finite step.

218 Tapan Senapati

Proof. Suppose there exists a strictly descending chain 0 1 2S S S of BG -subalgebras of X which does not terminate at finite step. Define a bipolar fuzzy set

( ),j j j+ -= in X by

( ) ( ) 1

0

1 if \1 if

n n

n n

n n x S Sx

x Sj ++

•=

Ï + Œ= Ì ŒÓ ∩

and ( ) ( ) 1

0

1 if \1 if

n n

n n

n n x S Sx

x Sj +-

•=

Ï- + Œ= Ì- ŒÓ ∩

where 0,1,2,n = and 0S stands for X . Let ,x y XŒ . Assume that 1\n nx S S +Œ and 1\k ky S S +Œ for 0,1,2,n = ; 0,1, 2,k = . Without loss of generality, we may assume that n k£ . Then obviously x and ny SŒ ,so nx y S* Œ because nS is a BG -subalgebra of X . Hence,

( ) ( ) ( ) ( ){ }1 min ,x y n n x yj j j+ + +* ≥ + =

( ) ( ) ( ) ( ){ }1 max ,x y n n x yj j j- - -* £ - + = .

If 0, n nx y S•=Œ∩ , then 0n nx y S•

=* Œ∩ . Thus

( ) ( ) ( ){ }1 min ,x y x yj j j+ + +* = =

( ) ( ) ( ){ }1 min ,x y x yj j j- - -* = - = .

If 0n nx S•=œ∩ and 0n ny S•

=Œ∩ , then there exists a positive integer r such that xŒ

1\r rS S + . It follows that rx y S* Œ so that

( ) ( ) ( ) ( ){ }1 min ,x y r r x yj j j+ + +* ≥ + =

( ) ( ) ( ) ( ){ }1 max ,x y r r x yj j j- - -* £ - + = .

Finally suppose that 0n nx S•=Œ∩ and 0n ny S•

=œ∩ . Then 1\s sy S S +Œ for some posi-tive integer s . It follows that sx y S* Œ , and hence

( ) ( ) ( ) ( ){ }1 min ,x y s s x yj j j+ + +* ≥ + =

( ) ( ) ( ) ( ){ }1 max ,x y s s x yj j j- - -* £ - + = .

This proves that ( ),j j j+ -= is a bipolar fuzzy BG -subalgebras with an infinite number of different values, which is a contradiction. This completes the proof.

4. Conclusions and future work

In this paper, notion of bipolar fuzzy BG -subalgebras in BG -algebra are introduced and investigated some of their useful properties. It is my hope that this work would other foundations for further study of the theory of BG -algebras. In my future study of fuzzy structure of BG -algebra, may be the following topics should be considered: (i) to find bipolar fuzzy closed ideals in BG -algebra, (ii) to find the relationship between bipolar


fuzzy BG -subalgebras and closed ideals in BG -algebra, (iii) to find bipolar fuzzy translation in BG -algebra.

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__________________________ Received February, 2014 *Corresponding author


Los Angeles

Singular Fuzzy Submodules

Mrinal C. Kalita

Department of Mathematics, Pandu College, Guwahati-781012 INDIA E-mail: [email protected]

Helen K. Saikia*

Department of Mathematics, Gauhati University, Guwahati-781014 INDIA


Abstract: In this paper we introduce the notion of singular fuzzy submodules of modules in

terms of fuzzy essentiality. We investigate various characteristics of such submodules.

Keywords: Fuzzy submodule, essential fuzzy submodule, fuzzy annihilators, singular submodule.

1. Introduction

In 1965, Lotfi Zadeh introduced the notion of fuzzy sets and since then this concept has been applied to many algebraic structures like groups, rings, modules, topologies and so on. Algebraic structures play an important role in Mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory and topological spaces and so on. Rosenfeld [16] was the first one to define the concept of fuzzy subgroups of a group. Since then many generalizations of this fundamental concept have been done (especially, in the last few decades). Naegoita and Ralescu [12] applied the concept of fuzzy set to the theory of modules and Pushkav [14] defined t -norm based fuzzy submodules. Pan [13] studied fuzzy finitely generated modules and quotient modules. Mukherjee et al [10], Kumar at al [4] studied various aspects of fuzzy submodules. In [19] Sidky introduced the notion of radical of a fuzzy submodule and also defined primary fuzzy submodule. Saikia and Kalita introduced the concept of fuzzy annihilators of fuzzy subsets of modules in [17] and consequently in [18] they have defined and characterized essential submodules. Recently, Rahman et al defined and characterized fuzzy small submodules, the dual notion of essential submodules in [15]. The notion of singularity plays a very important role in the study of algebraic structures. It was remarked by

222 Mrinal C. Kalita and Helen K. Saikia

Miguel Ferrero and Edmund R. Puczylowski in [8] that studying properties of rings one can usually say more assuming that the considered rings are either singular or nonsingular. The same holds for modules. It has motivated us to study the fuzzy aspects of singularity. The concept of essentiality leads to the notion of singularity in rings and modules. In this paper using the concept of fuzzy essentiality we attempt to define singular fuzzy submodules and investigate different characteristics of such submodules.

2. Basic definitions and notations

Throughout this paper R denotes a commutativering with unity and M denotes an R -module. The zero elements of R and M are 0 and q respectively. The class of fuzzy subsets of X is denoted by [ ]0,1 X .

Let [ ]0,1 Rm Œ . Then m is called a fuzzy ideal of R if it satisfies

(i) ( ) ( ) ( ) ,x y x y x y Rm m m- ≥ Ÿ " Œ

(ii) ( ) ( ) ( ) ,xy x y x y Rm m m≥ ⁄ " Œ .

The class of all fuzzy ideals of R is denoted by ( )FI R . Let [ ]0,1 Mm Œ . Then m is called a fuzzy sub module of M if it satisfies:

(i) ( ) 1m q =

(ii) ( ) ( ) ( ) ,xy x y x y Mm m m≥ Ÿ " Œ

(iii) ( ) ( ) ,rx x r R x Mm m≥ "Œ Œ .

The class of fuzzy sub modules is denoted by ( )F M .

Let [ ]0,1 Xm Œ . Then a fuzzy point tx , x XŒ , ( ]0,1F Œ is defined as the fuzzy

subset tx of X such that ( )tx x t= and ( ) 0tx y = for all { }y X xŒ . We write tx mŒ . If and only if tx mŒ .

Let [ ]0,1 Mms Œ . Then the sum of m and s defined as

( )( ) ( ) ( ){ }, ,x y z y z M x y zm s m s+ = Ÿ Œ = +� .

Let [ ]0,1 Mm Œ and [ ]0,1 Rs Œ then the product of m and s is defined as

( )( ) ( ) ( ){ }, ,x r m r R m M x rmsm s m= Ÿ Œ Œ =� .

A fuzzy sub-module m of M is called essential fuzzy sub-module of M , denoted by

e Mm Õ if for every non zero fuzzy sub-module s of M , qm s c« π . A fuzzy ideal m of R with ( )0 1m = is called an essential fuzzy ideal of R if it is essential as a fuzzy submodule of R considered as a module over itself. In other words m is an essential fuzzy ideal of R if and only if 0m s c« π , for every non-zero fuzzy ideal s of R .

Singular Fuzzy Submodules 223

Let [ ]0,1 Mm Œ . Then annihilator of m , denoted by ( )ann m is defined as

( ) [ ]{ }ann 0,1 ,R

qm h h hm c= Œ Õ∪

It is seen that ( ) [ ]ann 0,1 Rm Œ . Let a MŒ a qπ and g be an essential fuzzy submodule of M . We define a fuzzy

set [ ]0,1 Rs Œ by [ ]{ }0,1 ,R

pas m m m g= Œ Õ∪ .

Let d be a fuzzy submodule of M . Let us define a fuzzy subset ( )Z d of M by

( ) [ ]{ }0,1 , , , for some essential fuzzy ideal ofMZ Rqd m m m d md c s= Œ Õ Õ∪ .

3. Preliminaries

In this section we discuss some preliminary results needed for the sequel.

Lemma 3.1. If [ ]0,1 Mm Œ then

(i) ( )0 annc mÕ .

(ii) ( ) [ ]{ }ann , 0,1 ,r r R ra a qm a m c= Œ Œ Õ∪ .

(iii) ( )ann qm m cÕ .

Lemma 3.2. A submodule A is essential in M if and only if Ac is fuzzy essential in M .

The above result also holds for essential ideal of rings. Thus, R being an essential ideal of R , Rc is a fuzzy essential ideal in R .

Lemma 3.3. ( ) { , , for some essential fuzzy idealZ m m ma a a qd d s c= Œ Õ∪

} of Rs .

Proof. Clearly, { }, , for some essential fuzzy ideal m m ma a a qd s c sŒ Õ { }g gÕ

[ ]0,1 MŒ , g dÕ , 0gs cÕ , for some essential fuzzy ideal }s . Therefore, {m ma a∪

}, , for some essential fuzzy ideal ma qd s c sŒ Õ { } [ ]0,1 Mg gÕ Œ∪ , g dÕ , 0gs cÕ ,

for some essential fuzzy ideal }s ( )Z d=

Let [ ]0,1 Mg Œ be such that qgs cÕ for some essential fuzzy ideal s of R . Let

m MŒ be such that ( )mg a= . Now ( )( ) { ( ) ( ) ; ,m x m s y x sy s M ya as s= Ÿ = Œ Œ�

} ( ) ( ){ }R m z x mzg s= Ÿ =� ( ) ( ){ } ( )( ); ,s y x sy s M y R xg s g s£ Ÿ = Œ Œ = Õ�


( )x x Mqc " Œ . Thus ma qs cÕ . So, ( ) { , for someZ m m ma a a qd d s cÕ Œ Õ∪

} essential fuzzy ideal s . Hence we get the result. The following can be proved in a similar manner.

Lemma 3.4. ( ) ( ){ , , for some essential fuzzy ideal Z F M qd m m ms c s= Œ Õ∪

}of R .

Lemma 3.5. Let m and n be fuzzy ideals of R such that m nÕ and m is an essential fuzzy ideal of R . Then n is also an essential fuzzy ideal of R .

4. Main results

We now present our main results

Theorem 4.1. ( )Z d is a fuzzy submodule of M .

Proof. Clearly Rc is an essential fuzzy ideal of R . Since RJ qc c c= ,so ( )Zqc dÕ .

Now ( )( ) ( )( ) ( ){(1 2 1 1 1 1 1, , for some essentialZ m Z m m qd d g g d g s cŸ = Õ Õ� fuzzy

}) ( ){ }(1 2 2 2 2 2 2ideal , for some essential fuzzy ideal m qs g g d g s c sŸ Õ Õ� (= �

( ) ( ){ 1 1 2 2 1 2 1 1 2 2, , , ,m m q qg g g g d g s c g s cŸ Õ Õ Õ for some essential fuzzy ideals

}1 2 and s s ( )( ) ( )( ){ }1 2 1 1 2 2 1 2 1 1 2 2, , ,m m q qg g g g g g d g s c g s c£ + Ÿ + Õ Õ Õ £�

( )( ) ( )( ) ( ) ( ){ 1 2 1 2 1 2 1 2 1 1 2 1 1 2m mg g g g s s g s s g s s+ - + « Õ « + «� 1 1 2 2g s g sÕ +

q q qc c cÕ + Õ ( ){ ( ) ( )(1 2 1 2 1 2 ,m m qg g g g s c= + - + Õ� for some essential fuzzy

}ideal s ( )( )1 2Z m md= - 1 2,m m M" Œ . Now ( )( ) ( ){ ,Z sr srd g g d gs= Õ�

}for some essential fuzzy ideal qc sÕ ( ){ , ,m qg g d gs c≥ Õ Õ� for some essential

} fuzzy ideal s ( )( )RZ rc= ,s M r R" Œ Œ . Therefore, ( )Z d is a fuzzy submodule of M .

Theorem 4.2. For [ ]0,1 Mma Œ , ( )m Za dŒ if and only if ( )ann ma is an essential

fuzzy ideal of R .

Proof. Let ( )m Za dŒ . Then ma qs cÕ , for some essential fuzzy ideal s of R .

So, ( )ann mas Õ . Now s being an essential fuzzy ideal of R , by Lemma 3.5,

( )ann ma is also an essential fuzzy ideal of R .

Conversely, let ( )ann ma be an essential fuzzy ideal of R . Now, by Lemma 3.1,

( )annm ma a qcÕ . By definition of ( )Z d , ( )m Za dÕ .

Singular Fuzzy Submodules 225

Theorem 4.3. If 1d and 2d are fuzzy submodules of M with 2 1d dÕ then

( ) ( )2 2 1Z Zd d d= « .

Proof. Let ( )2m Za dŒ . Then 2ma dŒ and ma qs cÕ for some essential fuzzy

ideal s of R . So 1ma dŒ and ma qs cÕ imply ( )1m Za dŒ . Therefore 2ma dŒ

( )1Z d« .

Conversely let ( )1 2m Za d dŒ « . Then 2ma dŒ and ma qs cÕ for some essential

fuzzy ideal s . So ( )2m Za dŒ . Consequently ( ) ( )2 2 1Z Zd d d= « .

Definition 4.4. A fuzzy submodule d of M is singular or non singular according as ( )Z d d= or qc .

Lemma 4.5. If ( )F Md Œ , then ( )Z d is singular.

Proof. ( )( ( ){ }, for some essential fuzzy ideal Z Z m m Z ma a a qd d s c s= Œ Õ∪

( ){ }m m Za a d= Œ∪ ( )Z d=

Theorem 4.6. (1) Let 2d be any fuzzy submodule of a non singular (singular) fuzzy submodule 1d . Then 2d is also non singular (singular).

(2) If 1d is an essential extension of a non singular fuzzy submodule 2d , 1d is non singular.

Proof. (1) By Theorem 4.3, ( ) ( )2 2 1Z Zd d d= « . 1d being non singular implies

( )1Z qd c= . So ( )2 2Z q qd d c c= « = . Therefore 2d is also non singular. Again 1d is

singular implies ( )1 1Z d d= . So ( )2 2 1 2Z d d d d= « = . Therefore 2d is singular.

(2) 2d non singular implies ( )2Z qd c= .

Now ( ) ( )2 2 1Z Zd d d= « implies ( )2 1Z qd d c« = . But 1d is an essential extension

of 2d . So ( )1Z qd c= . Therefore 1d is non singular.

Theorem 4.7. The sum (direct or not) of two singular fuzzy submodules of a fuzzy submodule is again singular.

Proof. Let 1d and 2d be two singular fuzzy submodules such that 1d dÕ and

2d dÕ . Then ( )1 1Z d d= , ( )2 2Z d d= . Now ( ) ( )1 1Z Zd d d dÕ fi Õ ( )1 Zd dfi Õ

and 2d dÕ ( ) ( )2Z Zd dfi Õ ( )2 Zd dfi Õ \ ( )1 2 Zd d d+ Õ By Lemma 4.5, ( )Z d

is singular. So 1 2d d+ is a fuzzy submodule of a singular fuzzy submodule ( )Z d . Hence by Theorem 4.6, 1 2d d+ is singular.

5. Conclusions


The study of fuzzy substructures of algebraic structures has attracted many researchers in the last few decades. In our attempt to investigate rings with finiteness conditions, we have introduced essential fuzzy ideals and annihilators of fuzzy subsets. In this paper these two notions help us to develop the concept of singularity which in turn will lead us to many aspects of modules with chain conditions on various fuzzy substructures.

References

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(series A) 64 (1998), 195-209. [9] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World scientific, (1998). [10] T. K. Mukherjee and M. K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets Syst., 21 (1987), 99-104. [11] Gabriel Navarro, Oscar Cortadellas and F. J. Lobillo, Prime fuzzy ideals over non-commutative rings,

Fuzzy Sets and Systems, 199 (2012), 108-120. [12] C. V. Negoita and D. A. Ralescu, Application of fuzzy sets in system analysis, Birkhauser, Basel,

(1975). [13] F. Z. Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems, 21 (1985), 105-115. [14] S. G. Pushkov, Fuzzy modules with respect to a t -norm and some of their properties, Journal of

Mathematical Sciences, 154 No.3 (2008), 374-378. [15] Saifur Rahman and H. K. Saikia, Fuzzy small submodules and Jacobson L radical, International

Journal of Mathematics and Mathematical Sciences, Vol 2011, 12 pages (2011) doi.org/10.1155/2011/980320.

[16] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. [17] H. K. Saikia and M. C. Kalita, On annihilators of fuzzy subsets of modules, International Journal of

Algebra, Vol.3, no.10 (2009), 483-488. [18] H. K. Saikia and M. C. Kalita, On fuzzy essential submodules, The Journal of Fuzzy Mathematics, 17

(1) (2009), 109-117. [19] F. I. Sidky, On radicals of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and Systems,

119 (2001), 419-425. [20] U. M. Swami and K. L. N. Swami, Fuzzy prime ideals of rings, J. Math. Anal. Appl., 134 (1988), 94-

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__________________________ Received March, 2014


Los Angeles

Generalized Regular Fuzzy Closed Sets and Their Applications

B. Bhattacharya

Department of Mathematics, NIT Agartala, Tripura, 799055, India

Email: [email protected]

J. Chakraborty

Department of Mathematics, NIT Agartala, Tripura, 799055, India Email: [email protected]

Abstract: This paper introduces a new class of sets called generalized regular fuzzy closed sets

which is a stronger form of generalized fuzzy closed sets. Basic properties of generalized regular fuzzy closed sets are analyzed. Generalized fuzzy closed sets lie between generalized regular fuzzy closed sets and regular generalized fuzzy closed sets. Generally fuzzy closed set is not a generalized regular fuzzy closed set. But in this paper it is shown that closure of any fuzzy open set is always a generalized regular fuzzy closed set. With the help of generalized regular fuzzy closed sets, the concept of generalized regular fuzzy continuity and fuzzy generalized regular closed irresolute mapping are introduced. Latter on the interrelationship between generalized regular fuzzy continuity, generalized fuzzy continuity and regular generalized fuzzy continuity are also discussed. Different kinds of contra continuity based on fuzzy closed set, regular generalized fuzzy closed set, generalized regular fuzzy closed set are defined and few related results are shown. The decomposition of fuzzy homeomorphism is discussed via generalized fuzzy homeomorphism, generalized regular fuzzy homeomorphism and regular generalized fuzzy homeomorphism.

Key words: Generalized regular fuzzy closed set, generalized regular fuzzy open set, generalized

regular fuzzy continuous function, generalized regular fuzzy contra continuity, generalized regular fuzzy homeomorphism.

1. Introduction and preliminaries

228 Baby Bhattacharya and Jayasree Chakraborty

The concept of generalized closed set and generalized open set was first introduced by N. Levin [6] in an ordinary topological space. Latter on N. Palaniappan [7] studied the concept of regular generalized closed set in a topological space. In fuzzy topological space(fts) G. Balasubramanian, and P. Sundaram [2] first introduced the concept of generalized fuzzy closed set and generalized fuzzy continuous mapping and their properties.

Jin Han Park and Jin Keun Park [8] intorudced weaker form of generalized fuzzy closed set and generalized fuzzy continuous mapping i.e. regular generalized fuzzy closed set and generalizations of fuzzy continuous functions. In this paper we define and study another generalization of fuzzy closed set i.e. generalized regular fuzzy closed set which is the stronger form of the previous two generalizations. Section 2 is devoted to generalize regular fuzzy closed sets and their properties. Section 3 is contributed to generalize regular fuzzy open sets and their properties. In Section 4 we introduce generalized regular fuzzy continuous functions and fuzzy generalized regular closed irresolute functions and investigate interrelation between them. In Section 5 we introduce generalized regular fuzzy contra continuity. Lastly in Section 6 we introduce generalized regular fuzzy homeomorphism, regular generalized fuzzy homeomorphism, generalized regular fuzzy homeomoprhism, generalized fuzzy closed homeomorphism, regular generalized fuzzy closed homeomorphism and generalized regular fuzzy closed homeomorphism.

Throughout this paper, simply by X and Y we shall denote fts’s ( ),X t and ( ),Y s

and :f X YÆ will mean that f is a function from ( ),X t to ( ),Y s . For a fuzzy set

A of X , ( )cl A , ( )int A and 1 A- will denote the closure of A , the interior of A and the complement of Arespectively, whereas the constant fuzzy sets taking on the values 0 and 1 on X are denoted by 0X and 1X respectively. A fuzzy set A of a fts is called

fuzzy regular open [1], if ( )( )int clA A= . For any fuzzy subset A , ( ) {Cl :R A G= �

}, is a fuzzy regular closed subset of A G G X£ . Before entering into our work, we recall the following definitions which are prerequisite for this paper.

Definition 1.1 [1]. A fuzzy set l of a fuzzy space X is called a fuzzy regular closed est of X if Cl int l l= .

Definition 1.2 [2]. Let X be a fuzzy topological space. A fuzzy set l in X is called generalized fuzzy closed set (in short gfc ) ( )cl l m¤ £ whenever l m£ and m is fuzzy open.

Definition 1.3 [8]. A fuzzy set A in a fts X is called regular generalized fuzzy closed (in short, rgf -closed) if ( )cl A m£ whenever A U£ and U is fuzzy regular open. A fuzzy set A is called regular generalized fuzzy open (in short, rgf -open) if its compliment 1 A- is rgf -closed.

Definition 1.4 [2]. A function :f X YÆ is called generalized fuzzy continuous (in short gf -continuous) if the inverse image of every fuzzy closed set in Y is gf -closed in X .

Generalized Regular Fuzzy Closed Sets and Their Applications 229

Definition 1.5 [8]. A function :f X YÆ is called regular generalized fuzzy continuous (in short rgf -continuous) if the inverse image of every fuzzy closed set in Y is rgf -closed in X .

Definition 1.6 [2]. A function ( ) ( ): , ,f X Yt sÆ is called fuzzy generalized closed

irresolute (in short fgc -irresolute) if ( )1f V- is generalized fuzzy closed in ( ),X t for

every generalized fuzzy closed set V in ( ),Y s .

Definition 1.7 [8]. A function ( ) ( ): , ,f X Yt sÆ is called fuzzy regular generalized

closed irresolute (in short frgc-irresolute) if ( )1f V- is regular generalized fuzzy closed

in ( ),X t for every regular generalized fuzzy closed set V in ( ),Y s .

Definition 1.8 [1]. A mapping ( ) ( ): , ,f X X Y Yt tÆ form a fuzzy space X to

another fuzzy space Y is called fuzzy almost continuous mapping if ( )1f Xl t- Œ for each fuzzy regular open set l of Y .

2. Generalized regular fuzzy closed sets

Definition 2.1. A fuzzy set A in a fts X is called generalized regular fuzzy closed set (in short, grf -closed) if ( )clR A U£ whenever A U£ and U is fuzzy open.

Theorem 2.2. If A and B are grf -closed sets then A B⁄ is also grf -closed set.

Proof. Let A B U⁄ £ and U be fuzzy open. Then ,A B U£ and thus ( )clR A U£ ,

( )clR B U£ . Hence ( )clR A B U⁄ £ from the fact that ( ) ( ) (cl cl clR A R B R A⁄ =

)B⁄ .

Remark 2.3. Intersection of any two grf -closed sets need not be a grf -closed set.

Example 2.4. Let { }, ,X a b c= , { }1 10 ,1 ,X X At = where ( )1 0.4A a = , ( )1 0.5A b = ,

( )1 0.6A c = . ( )2 0.4A a = , ( )2 0.7A b = , ( )2 0.6A c = , ( )3 0.9A a = , ( )3 0.5A b = , ( )3A c 0.8= . Here 2A and 3A are two grf -closed sets but 2 3A AŸ is not grf -closed set.

Proposition 2.5. Every fuzzy regular closed set is grf -closed set.

Proof. It is obvious. But the converse is not true as shown in the following example.


( )1 0.6A c = . ( )2 0.4A a = , ( )2 0.5A b = , ( )2 0.3A c = . Here 2A is not fuzzy regular closed though it is grf -closed set.


Proposition 2.7. Every grf -closed set is rgf -closed set.

Proof. It is obvious from the fact that ( ) ( )cl clA R A£ . But the converse may not be true as shown in the following example.

Example 2.8. Let { }, ,X a b c= , { }1 10 ,1 ,X X At = , where ( )1 0.4A a = , ( )1 0.7A b = ,

( )1 0.3A c = . The fuzzy set 2A defined as ( )2 0.3A a = , ( )2 0.3A b = , ( )2 0.2A c = is rgf -closed set but not grf -closed set.

Proposition 2.9. Every grf -closed set is fuzzy generalized closed set.

Proof. Let A X£ be grf -closed set. Let A U£ where U is fuzzy open. So ( )clR A

U£ . Since every fuzzy regular closed set is fuzzy closed set so ( ) ( )cl clA R A U£ £ . But the converse is not true in general as shown in the following example.


( )1 0.9A c = . Then 11 A- is fgc set but it is not grf -closed set.

Theorem 2.11. If A be any fuzzy open and grf -closed set, then A is fuzzy regular closed and hence fuzzy clopen.

Proof. If A is fuzzy open and grf -closed set then ( )clR A A£ . This implies A is regular closed. Hence A is fuzzy clopen.

Theorem 2.12. If A is fuzzy preclosed and fuzzy semi-open then it is grf -closed set.

Proof. If A is fuzzy preclosed then ( )( )cl int A A£ and also if A is fuzzy semi-open

then ( )( )cl intA A£ . Both of this holds when A is fuzzy regular closed. It implies that

A is grf -closed set.

Theorem 2.13. If A and B be any two fuzzy regular closed subsets of ( ),X t then

A B⁄ is also grf -closed subsets of ( ),X t .

Proof. If A and B be any two fuzzy regular closed subsets of ( ),X t then A B⁄ is

also fuzzy regular closed subsets of ( ),X t . And every fuzzy regular closed set is grf -

closed et. So A B⁄ is grf -closed subsets of ( ),X t .

Theorem 2.14. The closure of a fuzzy open set is grf -closed set.

Proof. In [1] it is shown that if l be a fuzzy open set of a fuzzy space X then

( )( ) ( )int cl cll l£ implies that ( )( )( ) ( )cl int cl cll l£ . Now l is fuzzy open implies

that ( )( )int cl Al £ and hence ( ) ( )( )( )cl cl int cll l£ . Thus ( )cl l is fuzzy regular


closed set. And every fuzzy regular closed set is grf -closed set. Therefore the closure of a fuzzy open set is grf -closed set.

Remark 2.15. If A be any gfc set in ( ),X t and if ( ) ( )cl clA R A U£ £ where U is open then A is grf -closed set.

Theorem 2.16. If A is grf -closed set and ( )clA B R A£ £ then B is grf -closed set.

Proof. Let B U£ where U is open. Since A B£ , therefore A U£ and A is grf -closed ( )clR A U£ . But ( ) ( )cl clR B R A£ since ( )clB R A£ and also ( )clR B U£ . Hence B is grf -closed set.

3. Generalized regular fuzzy open sets

Definition 3.1. A fuzzy set A in a fts X is called generalized regular fuzzy open set (in short, grf -open) iff its compliment is generalized regular fuzzy closed.

Theorem 3.2. A fuzzy set A is grf -open iff ( )intF R A£ whenever F is fuzzy closed and F A£ .

Proof. Let A be grf -open set and F be a fuzzy closed set such that F A£ . Then 1 1A F- £ - . Where 1 F- is open. Generalized regular closedness of 1 A- implies that ( )1 int cl 1 1R A R A F- = - £ - which implies ( )intF R A£ .

Conversely, suppose that A is a fuzzy set such that ( )intF R A£ whenever F is closed and F A£ . We claim that 1 A- is grf -closed. For if 1 A U- £ where U is fuzzy open then 1 A U- £ , 1 U A- £ . Hence by assumption ( )1 intU R A- £ . i.e.

( )1 intR A U- £ . Hence ( )cl 1R A U- £ which implies 1 A- is generalized regular fuzzy closed set. Therefore A is grf -open set.

Theorem 3.3. If ( )intR A B A£ £ and A is grf -open then B is grf -open.

Proof. ( )intR A B A£ £ implies ( )1 1 1 intA B R A- £ - £ - . That is 1 1A B- £ -

( )cl 1R A£ - . Since 1 A- is grf -closed, by Theorem 2.16, 1 B- is grf -closed and B is grf -open.

Remark 3.4. (i) The union of two grf -open sets is not generally grf -open. Example 2.5 serves the purpose.

(ii) The intersection of any two grf -open sets is grf -open. From the discussion we get the following relations


Regular fuzzy closed Fuzzy closed

Generalized regularfuzzy closed

Generalized fuzzyclosed

Regular generalizedfuzzy closed

4. Generalized regular fuzzy continuous functions and fuzzy generalized regular closed irresolute functions

Definition 4.1. A function :f X YÆ is called generalized regular fuzzy continuous (in short, grf continuous) if the inverse image of every fuzzy closed set in Y is grf -closed in X .

Theorem 4.2. Every generalized regular fuzzy continuous function is generalized fuzzy continuous.


Remark 4.3. However the converse of the above theorem may not true be as shown in the following example.

Example 4.4. { }, ,X a b c Y= = , { }1 10 ,1 ,X X At = where ( )1 0.5A a = , ( )1A b 0.7= ,

( )1 0.4A c = , { }2 20 ,1 ,Y Y At = where ( )2 0.7A a = , ( )2 0.5A b = , ( )2 0.9A c = . Here

:f X YÆ defined by ( )f a b= , ( )f b a= , ( )f c c= . Here the inverse image of fuzzy closed set is generalized fuzzy closed set. Therefore it is generalized fuzzy con-tinuous function. But the inverse image of fuzzy closed set ( ) ( ) ( ){ },0.3 , ,0.5 , ,0.1a b c is not generalized regular fuzzy closed set. Thus f is not generalized regular fuzzy continuous function.

Theorem 4.5. Every generalized regular fuzzy continuous function is regular generalized fuzzy continuous function.

Remark 4.6. However the converse of the above theorem is not true as shown in the following example.

Example4.7. Let { }, ,X a b c Y= = , { }1 10 ,1 ,X X At = where ( )1 0.5A a = , ( )1 0.7A b = ,

( )1 0.6A c = . And { }2 20 ,1 ,Y Y At = where ( )2 0.7A a = , ( )2 0.6A b = , ( )2 0.5A c = .

Here :f X YÆ defined by ( )f a b= , ( )f b a= , ( )f c c= . Then f is regular genera-lized fuzzy continu-ous function but not generalized regular fuzzy continuous function. Since ( )1

21f A- - is not generalized regular fuzzy closed in ( )1,X t for a fuzzy closed

set ( )21 A- in ( )2,Y t .


Proposition 4.8. A function :f X YÆ is generalized regular fuzzy continuous as well as fuzzy continuous if the inverse image of fuzzy closed set is regular fuzzy closed set.

Theorem 4.9. Let ( ) ( ): , ,f X Yt sÆ be a function. Then following statements are equivalent:

(i) f is generalized regular fuzzy continuous.

(ii) The inverse image of each fuzzy open set in Y is grf -open in X .

Theorem 4.10. Let ( ) ( ): , ,f X Yt sÆ and ( ) ( ): , ,g Y Zs hÆ be any two function. If f is generalized regular fuzzy continuous and g is fuzzy continuous then the composition gof is generalized regular fuzzy continuous.

Proof. Let V be a fuzzy closed set in ( ),Z h . Then ( )1g V- is closed in ( ),Y s as g

is continuous. Generalized regular fuzzy continuity of f implies that ( )( )1 1f g V- - is

generalized regular fuzzy closed set in ( ),X t . That is, ( ) ( )1gof V

- is generalized

regular fuzzy closed in ( ),X t . Hence gof is generalized regular fuzzy continuous.

Remark 4.11. The composition of two generalized regular fuzzy continuous functions need not be a generalized regular fuzzy continuous function.

Example 4.12. Let { }, ,X Y Z a b c= = = , { }1 10 ,1 ,X X At = where ( )1 0.5A a = , ( )1A b

0.7= , ( )1 0.4A c = , { }2 20 ,1 ,Y Y At = where ( )2 0.5A a = , ( )2 0.5A b = , ( )2 0.5A c =

and { }3 30 ,1 ,Z Z At = where ( )3 0.5A a = , ( )3 0.3A b = , ( )3 0.6A c = . For the identity

functions :f X YÆ and :g Y ZÆ , ( ) ( )131gof A

- - is not generalized regular fuzzy

closed set in ( )1,X t for fuzzy closed set ( )31 A- in ( )3,Z t .

Definition 4.13. A function ( ) ( ): , ,f X Yt sÆ is called fuzzy regular irresolute if

( )1f V- is fuzzy regular open (resp. closed) in ( ),X t for every fuzzy regular open

(resp. closed) set V in ( ),Y s .

Definition 4.14. A function ( ) ( ): , ,f X Yt sÆ is called fuzzy generalized regular

closed irresolute (in short fgrc -irresolute) if ( )1f V- is generalized regular fuzzy closed

in ( ),X t for every generalized regular fuzzy closed set V in ( ),Y s .

Remark 4.15. Fuzzy generalized regular closed irresolute function and generalized regular fuzzy continuous function are independent of each other concept.

Example 4.16. Generalized regular fuzzy continuous function does not imply fuzzy generalized regular closed irresolute function.


Let { }, ,X Y a b c= = , { }1 10 ,1 ,X X At = where ( )1 0.4A a = , ( )1 0.5A b = , ( )1 0.7A c =

and { }2 20 ,1 ,Y Y At = where ( )2 0.4A a = , ( )2 0.5A b = , ( )2 0.6A c = . The identity func-tion f is generalized regular fuzzy continuous but not fuzzy generalized regular closed irresolute function. For a fuzzy set 3A in Y defined by ( )3 0.5A a = , ( )3 0.4A b = ,

( )3 0.7A c = is generalized regular fuzzy closed in ( )2,Y t , ( )13f A- is not generalized

regular fuzzy closed in ( )1,X t .

Example 4.17. Fuzzy generalized regular closed irresolute function may not be a generalized regular fuzzy continuous function.

Let { }, ,X Y a b c= = , { }1 10 ,1 ,X X At = where ( )1 0.4A a = , ( )1 0.6A b = , ( )1 0.8A c =

and { }2 20 ,1 ,Y Y At = where ( )2 0.5A a = , ( )2 0.7A b = , ( )2 0.9A c = . Here :f X YÆ

defined by ( )f a b= , ( )f b a= , ( )f c c= , is generalized regular fuzzy closed irresolute

function but not generalized regular fuzzy continuous function as ( )121f A- - is not

generalized regular fuzzy closed set in ( )1,X t .

Remark 4.18. Fuzzy generalized regular closed irresolute function and fuzzy generalized closed irresolute function are independent of each other.

Example 4.19. Fuzzy generalized regular closed irresolute function may not be a fuzzy generalized closed irresolute function.

Let { }, ,X Y a b c= = , { }1 10 ,1 ,X X At = where ( )1 0.4A a = , ( )1 0.6A b = , ( )1 0.8A c = .

{ }2 20 ,1 ,Y Y At = where ( )2 0.5A a = , ( )2 0.7A b = , ( )2 0.9A c = . Here :f X YÆ

defined by ( )f a b= , ( )f b a= , ( )f c c= . Then f is fuzzy generalized regular closed irresolute function but not fuzzy generalized closed irresolute function. For a fuzzy set

21 A- in Y defined by ( )3 0.5A a = , ( )3 0.3A b = , ( )3 0.1A c = is generalized fuzzy

closed in ( )2,Y t , ( )13f A- is not generalized fuzzy closed in ( )1,X t .

Example 4.20. Fuzzy generalized closed irresolute function may not be a fuzzy generalized regular closed irresolute function.

Let { }, ,X Y a b c= = , { }1 10 ,1 ,X X At = where ( )1 0.5A a = , ( )1 0.4A b = , ( )1 0.3A c = .

{ }2 20 ,1 ,Y Y At = where ( )2 0.6A a = , ( )2 0.5A b = , ( )2 0.7A c = . Here :f X YÆ de-

fined by ( )f a b= , ( )f b a= , ( )f c c= . Then f is fuzzy generalized closed irresolute function but not fuzzy generalized regular closed irresolute function. For a fuzzy set 3A in Y defined by ( )3 0.4A a = , ( )3 0.5A b = , ( )3 0.3A c = is generalized regular fuzzy

closed in ( )2,Y t , ( )13f A- is not generalized regular fuzzy closed in ( )1,X t .

Theorem 4.21. Let ( ) ( ): , ,f X Yt sÆ be a function. Then following statements are equivalent:

(i) f is fgrc -irresolute.


(ii) The inverse image of each grf -open set in Y is grf -open in X .

Theorem 4.22. Let :f X YÆ and :g Y ZÆ be function. If f and g both are fgrc -irresolute then the composition gof is also fgrc -irresolute.

Definition 4.23. A mapping ( ) ( ): , ,f X Yt sÆ form a fuzzy space X to another

fuzzy space Y is called generalized regular fuzzy almost continuous mapping if ( )1f l- is generalized regular fuzzy closed in X for each fuzzy regular open set l of Y .

Theorem 4.24. Every generalized regular fuzzy continuous function is generalized regular fuzzy almost continuous function.


Example 4.26. Let { }, ,X a b c Y= = , { }1 10 ,1 ,X X At = where ( )1 0.5A a = , ( )1 0.7,A b =

( )1 0.6A c = . { }2 20 ,1 ,Y Y At = where ( )2 0.7A a = , ( )2 0.6A b = , ( )2 0.5A c = . Here

:f X YÆ de-fined by ( )f a b= , ( )f b a= , ( )f c c= . Then f is generalized regular fuzzy almost continuous function but not generalized regular fuzzy continuous function. Since ( )1

21f A- - is not generalized regular fuzzy closed in ( )1,X t for a fuzzy closed

set ( )21 A- in ( )2,Y t .

Fuzzycontinuity

Generalized regularfuzzy continuity

Generalized fuzzycontinuity

Regular generalizedfuzzy continuity

5. Generalized regular fuzzy contra continuous functions

Definition 5.1. A function :f X YÆ is called generalized fuzzy contra continuous if the inverse image of every fuzzy open set in Y is gfc (generalized fuzzy closed) set in X .

Definition 5.2. A function :f X YÆ is called generalized regular fuzzy contra continuous if the inverse image of every fuzzy open set in Y is grf -closed set in X .


Definition 5.3. A function :f X YÆ is called regular generalized fuzzy contra continuous if the inverse image of every fuzzy open set in Y is rgf -closed (regular generalized fuzzy closed) set in X .

Theorem 5.4. Every generalized regular fuzzy contra continuous function is generalized fuzzy contra continuous.

Remark 5.5. However the converse of the above theorem may not be true as shown in the following example.

Example 5.6. { }, ,X a b c Y= = , { }1 10 ,1 ,X X At = where ( )1 0.6A a = , ( )1 0.7A b = ,

( )1 0.5A c = , { }2 20 ,1 ,Y Y At = where ( )2 0.4A a = , ( )2 0.3A b = , ( )2 0.5A c = . Here

:f X YÆ defined by ( )f a a= , ( )f b b= , ( )f c c= . Here the inverse image of fuzzy open set in Y is generalized closed in X . Therefore it is generalized fuzzy contra continuous. But the inverse image of fuzzy open set ( ) ( ) ( ){ },0.4 , ,0.3 , ,0.5a b c in Y is not generalized regular fuzzy closed set in X . Here f is not generalized regular fuzzy contra continuous function.

Theorem 5.7. Every generalized fuzzy contra continuous is regular generalized fuzzy contra continuous function.


Example 5.9. { }, ,X a b c Y= = , { }1 10 ,1 ,X X At = where ( )1 0.4A a = , ( )1 0.6A b = ,

( )1 0.7A c = . { }2 20 ,1 ,Y Y At = where ( )2 0.2A a = , ( )2 0.5A b = , ( )2 0.4A c = . Here :f X

YÆ de--fined by ( )f a a= , ( )f b b= , ( )f c c= . Here the inverse image of fuzzy open set in Y is regular generalized closed in X . Therefore it is regular generalized fuzzy contra con-tinuous. But the inverse image of fuzzy open set ( ) ( ) ( ){ },0.2 , ,0.5 , ,0.4a b c in Y is not generalized fuzzy closed set in. X Here f is not generalized fuzzy contra continuous function.

Theorem 5.10. Let :f X YÆ and :g Y ZÆ be function.

(i) If f is fuzzy generalized closed irresolute and g is generalized fuzzy contra continuous then gof is generalized fuzzy contra continuous function.

(ii) If f is fuzzy generalized regular closed irresolute and g is generalized regular fuzzy contra continuous then gof is generalized regular fuzzy contra continuous function.

(iii) If f is fuzzy regular generalized closed irresolute and g is regular generalized fuzzy contra continuous then gof is regular generalized fuzzy contra continuous function.


6. Generalized regular fuzzy homeomorphisms

Definition 6.1. A bijection :f X YÆ is called generalized fuzzy homeomorphism if f is generalized fuzzy continuous and generalized fuzzy open.

Definition 6.2. A bijection :f X YÆ is called regular generalized fuzzy homeomor-phism if f is regular generalized fuzzy continuous and regular generalized fuzzy open.

Definition 6.3. A bijection :f X YÆ is called generalized regular fuzzy homeomor-phism if f is generalized regular fuzzy continuous and generalized regular fuzzy open.

Theorem 6.4. Every fuzzy homeomorphism is generalized fuzzy homeomorphism.


Remark 6.5. The converse of the above theorem may not be true in general as shown in the following example.

Example 6.6. Let { }, ,X a b c= , { }, ,Y p q r= , { }1 0 ,1 ,X Xt l= where ( ) 1al = , ( )bl

0= , ( ) 0cl = and { }2 0 ,1 ,Y Yt m= where ( ) 0pm = , ( ) 1qm = , ( ) 0rm = . Here

:f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Then f is generalized fuzzy homeomorphism but f is not fuzzy homeomorphism.

Theorem 6.7. Every generalized fuzzy homeomorphism is regular generalized fuzzy homeomorphism.

Proof. By the definition, it can be proved.

Remark 6.8. The converse of the above theorem may not be true as shown in the following example.

Example 6.9. Let { }, ,X a b c= , { }, ,Y p q r= , { }1 0 ,1 ,X Xt l= where ( ) 0.4al = , ( )bl

0.7= , ( )cl 0.3= and { }2 0 ,1 ,Y Yt m= where ( ) 0.7pm = , ( ) 0.7qm = , ( ) 0.8rm = .

Here :f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Then f is regular genera-lized fuzzy homeomorphism but f is not generalized fuzzy homeomorphism.

Theorem 6.10. Every generalized regular fuzzy homeomorphism is generalized fuzzy homeomorphism.

Proof. Straight forward.

Remark 6.11. The converse of the above theorem may not be true as shown in the following example.

Example 6.12. Let { }, ,X a b c= , { }, ,Y p q r= , { }1 0 ,1 ,X Xt l= where ( ) 0.4al = ,

( ) 0.6bl = , ( )cl 0.8= , { }2 0 ,1 ,Y Yt m= where ( ) 0.5pm = , ( ) 0.7qm = , ( ) 0.3rm = .


Here :f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Thus f is generalized fuzzy homeomor-phism but f is not generalized regular fuzzy homeomorphism.

Definition 6.13. A bijection :f X YÆ is called generalized fuzzy closed homeo-morphism if f is fuzzy generalized closed irresolute and its inverse 1f - is also fuzzy generalized closed irresolute.

Theorem 6.14. Every generalized fuzzy closed homeomoprhism is generalized fuzzy homeomorphism.



( ) 0.6bl = , ( )cl 0.8= . { }2 0 ,1 ,Y Yt m= where ( ) 0.5pm = , ( ) 0.7qm = , ( ) 0.3rm = .

Here :f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Then f is generalized fuzzy

homeomorphism. But ( ) ( ) ( ){ },0.4 , ,0.5 , ,0.4p q r is a gfc in ( )2,Y t but it is not gfc in

( )1,X t . So f is not generalized fuzzy closed homeomorphism.

Definition 6.17. A bijection :f X YÆ is called regular generalized fuzzy closed homeomorphism if f is regular generalized fuzzy closed irresolute and its inverse 1f - is also regular generalized fuzzy closed irresolute.

Theorem 6.18. Every regular generalized fuzzy closed homeomorphism is regular generalized fuzzy homeomorphism.



( ) 0.6bl = , ( ) 0.3cl = and { }2 0 ,1 ,Y Yt m= where ( ) 0.7pm = , ( ) 0.7qm = , ( )rm =

0.8 . Here :f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Here f is generalized

fuzzy homeomorphism. But ( ) ( ) ( ){ },0.2 , ,0.7 , ,0.1p q r is a rgfc in ( )2,Y t . Though it is

not in ( )1,rgfc X t . So f is not regular generalized fuzzy closed homeomorphism.

Definition 6.21. A bijection :f X YÆ is called generalized regular fuzzy closed homeomorphism if f is fuzzy generalized regular closed irresolute and its inverse 1f - is also fuzzy generalized regular closed irresolute.

Theorem 6.22. Every generalized regular fuzzy closed homeomorphism is generaliz-ed regular fuzzy homeomorphism.



Example 6.24. Let { }, ,X a b c= , { }, ,Y p q r= , { }1 0 ,1 ,X Xt l= where ( ) 1al = , ( )bl

0= , ( ) 0cl = and { }2 0 ,1 ,Y Yt m= where ( ) 0pm = , ( ) 1qm = , ( ) 0rm = . Here

:f X YÆ defined by ( )f a p= , ( )f b q= , ( )f c r= . Then f is generalized fuzzy

homeomorphism. Thus ( ) ( ) ( ){ },1 , ,0 , ,0p q r is a grfc in ( )2,Y t but it is not grfc in

( )1,X t . So f is not generalized regular fuzzy closed homeomorphism.

References

[1] K. K. Azad, On fuzzy semi continuity, Fuzzy almost continuity and fuzzy weakly continuity, Jour. Math. Anal. Appl., 82 (1981), 14-32.

[2] G. Balasubramanian and P. Sundaram, On some generalizations of fuzzy continuous functions, Fuzzy Sets and Systems, 86 (1997), 93-100.

[3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [4] R. Devi, K. Balachandran and H. Maki, Semi-generalized homeomorphisms and genelraized semi

homeomorphisms in topological spaces, Indian. J. Pure Appl. Math., 26 (3) (1995), 271-284 March. [5] Y. Gnanambal, On generalized pre regular closed sets in topological spaces, Indian. J. Pure Appl.

Math., 28 (3) (1997), 351-360 March. [6] N. Levine, Generalized closed sets in topology, Rend, Circ. Math. Palermo, 19 (1970), 89-96. [7] N. Palaniappan and K. C. Rao, Regular generalized closed sets, Kyungpook Math, 33 (2) (1993), 211-

219. [8] J. H. Park and J. K. Park, On regular generalized fuzzy closed sets and generalization of fuzzy

continuous functions, Indian. J. Pure Appl. Math., 34 (7) (2003), 1013-1024 July. [9] R. K. Saraf, N. G, K. M, On fuzzy semi-pre-generalized closed sets, Bull. Malays. Math. Sci. Soc., (2)

28 (1) (2005), 19-30. [10] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.


__________________________ Received June, 2014


Los Angeles

( ), qŒŒ⁄ -fuzzy Subalgebras of Lattice Implication Algebras

Peng Deng and Yang Xu

School of Mathematics, Southwest Jiaotong University, Sichuan, Chengdu 610031, China E-mail: [email protected]; [email protected]

Abstract: In this paper we propose the concept of ( ), qŒ Œ⁄ -fuzzy subalgebras of lattice

implication algebras ( )LIAs , and discuss its equivalent characterization. Then, some related properties are investigated, such as the properties about lattice implication homomorphism of ( ), qŒŒ⁄ -fuzzy subalgebras of LIAs are given. Finally, new

operations of ( ), qŒ Œ⁄ -fuzzy subalgebras of LIAs are introduced and also discuss its related properties.

Keywords: Lattice implication algebra, ( ), qŒŒ⁄ -fuzzy subalgebra, homomorphism.

1. Introduction

In the real world there exists a lot of uncertainty information, such as incomplete information, incomparable information, fuzzy information, etc. with the development of science and technology, there are large amounts of information, how the machine was used to simulate the human brain to effectively deal with the uncertainty information was attention widely by the people. Intelligent information processing is one important research direction in artificial intelligence. Information processing dealing with certain information is based on the classical logic. However, non-classical logics including logics behind fuzzy reasoning handle information with various facets of uncertainty such as fuzziness, randomness, etc. Therefore, non-classical logic have become as a formal and useful tool for computer science to deal with uncertain information. Many-valued logic, a great extension and development of classical logic, has always been an important direction in non-classical logic. In 1993, in order to investigate a many-valued logical system whose prepositional values is given in a lattice, Xu established the lattice implication algebra by combining lattice and implication algebra, and discussed many useful structures [1-3]. Its related properties and structures are further studied [4, 17, 18].

242 Peng Deng and Yang Xu

The concept of fuzzy set was introduced by Zadeh [8, 9]. Since then this idea has been applied to other algebraic structures [6, 13-16]. Rosenfeld inspired the fuzzification of algebraic structure and introduced the notion of fuzzy subgroup [11]. Pu and Liu given the idea of fuzzy point and ‘belongingness’ and ‘quasi-coincidence’ with a fuzzy set [12]. Jun et al [19] introduced the concept of ( ), qŒŒ⁄ -fuzzy implicative filter of a lattice implication algebra. Zhan et al [7] investigated this kind of fuzzy implicative filters. Liu Yi et al [20, 21] further studied this kind of fuzzy implicative filters. Recently, Ruijuan Lv [5] introduced the concept of ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras and discussed some of their related properties.

2. Preliminaries

In this section, we give some notions, definitions and basic results which will be needed in the following discussion.

Definition 2.1 [3] (Lattice implication algebra). Let ( ), , , ,L O I⁄ Ÿ be a bounded lattice with an order-reversing involution “ ' ”, I and O the greatest and the smallest element of L respectively, and :L L LÆ ¥ Æ be a mapping ( ), , , ', , ,L O I⁄ Ÿ Æ is called a lattice implication algebra if the following conditions hold for any , ,x y z LŒ :

( )1I ( ) ( )x y z y x zÆ Æ = Æ Æ ;

( )2I x x IÆ = ;

( )3I x y y xÆ = Æ¢ ¢ ;

( )4I x y y x IÆ = Æ = implies x y= ;

( )5I ( ) ( )x y y y x xÆ Æ = Æ Æ ;

( )6I ( ) ( ) ( )x y z x z y z⁄ Æ = Æ Ÿ Æ ;

( )7I ( ) ( ) ( )x y z x z y zŸ Æ = Æ ⁄ Æ .

Theorem 2.1 [3]. Let L be a lattice implication algebra, then for any , ,x y z LŒ ;

(1) If I x IÆ = , then x I= ; (2) I x xÆ = and x O xÆ = ¢ ; (3) O x IÆ = and x I IÆ = ;

(4) ( ) ( ) ( )( )x y y z x z IÆ Æ Æ Æ Æ = ;

(5) ( ) ( )x y x y x yÆ Æ = Æ Æ¢ ¢ ;

(6) ( )( )x y x y x ¢Ÿ = Æ Æ ¢ ;

(7) ( )( )x y x y y⁄ = Æ Æ .

( ), qŒ Œ ⁄ -Fuzzy Subalgebras of Lattice Implication Algebras 243

Definition 2.2 [3] (Lattice implication subalgebra). Let L be a lattice implication algebra. S LÕ is called a lattice implication subalgebra of L , if the following conditions hold:

(1) ( ), , , 'S ⁄ Ÿ is bounded sublattice of ( ), ,L ⁄ Ÿ with an order-reversing involution ' ;

(2) If ,x y SŒ , then x y SÆ Œ .

For a fuzzy subset F of L and ( ]0,1tŒ , the crisp set ( ) ( ){ };U F t x L F x t= Œ ≥ is called the level subset of F .

A fuzzy subset F of L the form

( ) ( )0 if 0 if t y x

F yy x

Ï π == Ì πÓ

is said to be a fuzzy point with support x and value t and is denoted by tx . A fuzzy point tx is said to belong to (resp. be quasi-coincident with) a fuzzy subset

F , written by tx FŒ (resp., tx qF ) if ( )F x t≥ (resp., ( ) 1F x t+ > ). If tx FŒ or

tx qF , then we write tx qFŒ⁄ . If ( )F x t< (resp., ( ) 1F x t+ £ ), then we call tx FŒ (resp., tx qF ).

Definition 2.3 [2]. Let L be a lattice implication algebra, A LÕ , then A is a subalgebra of L if it satisfies the following conditions:

(1) O AŒ ; (2) For any ,x y AŒ , implies x y AÆ Œ .

Definition 2.4 [2]. Let L be a lattice implication algebra, ( )A F LÕ , then A is a fuzzy subalgebra of L if it satisfies the following conditions:

(1) ( ) ( )A I A O= ;

(2) For any ,x y AŒ , ( ) ( ) ( ){ }min ,A x y A x A yÆ ≥ .

Theorem 2.2 [2]. Let L be a lattice implication algebra, A is a fuzzy subalgebra of L , then for any ,x y LŒ ,

(1) ( ) ( ) ( )A I A O A x= ≥ ;

(2) ( ) ( )A x A x= ¢ ;

(3) ( ) ( ) ( ){ }min ,A x y A x A y⁄ ≥ ;

(4) ( ) ( ) ( ){ }min ,A x y A x A yŸ ≥ .

Theorem 2.3 [4]. Let 1L , 2L be lattice implication subalgebras of L , then 1 2L L« is also lattice implication subalgebra of L .


Definition 2.5 [5]. Let L be a lattice implication algebra, ( )A F LÕ , then A is an

( ), qŒŒ⁄ -fuzzy subalgebra of L if it satisfies the following conditions hold for any

,x y LŒ , ( ], 0,1r tŒ :

(1) r rx A x qAŒ fi Œ⁄¢ ;

(2) ( ) { }min ,,r t r tx y A x y qAŒ fi Æ Œ⁄ .

Theorem 2.4 [5]. Let L be a lattice implication algebra, ( )A F LÕ , then A is an

( ), qŒŒ⁄ -fuzzy subalgebra of L if and only if the following conditions hold for any

,x y LŒ , ( ], 0,1r tŒ :

(1) ( ) ( ){ }min ,0.5A x A x≥¢ ;

(2) ( ) ( ) ( ){ }min , ,0.5A x y A x A yÆ ≥ .

Definition 2.6 [6]. Let A , B are fuzzy sets of X and Y , definition the mapping [ ]: 0,1A B X Y¥ ¥ Æ , ( )( ) ( ) ( ){ }, min ,A B x y A x A y¥ = , ( ),x y X Y" Œ ¥ , then

A B¥ is the fuzzy set of X Y¥ , called direct product of A B¥ .

3. The ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras

In this section, we introduce the concepts of ( ), qŒ Œ⁄ -fuzzy subalgebras of lattice implication algebras, and discuss its equivalent characterization as well as some properties.

Definition 3.1. Let L be a lattice implication algebra, ( )A F LÕ , then A is an

( ), qŒ Œ⁄ -fuzzy subalgebra of L , if it satisfies the following conditions hold for any

,x y LŒ , ( ], 0,1r tŒ :

(1) r rx A x qAŒ fi Œ⁄¢ ;

(2) ( ) { }min , rr tx y A x qAÆ Œ fi Œ⁄ or ly qAŒ⁄ .

Theorem 3.1. Let L be a lattice implication algebra, ( )A F LÕ , then A is an

( ), qŒŒ⁄ -fuzzy subalgebra of L if and only if the following conditions hold for any

,x y LŒ , ( ], 0,1r tŒ :

(3) ( ){ } ( )max ,0.5A x A x≥¢ ;

(4) ( ){ } ( ) ( ){ }max ,0.5 min ,A x y A x A yÆ ≥ .


Proof. (1) fi (3) If there exists x LŒ such that ( ){ } ( )max ,0.5A x A x r< =¢ , then

0.5 1r< £ , rx AŒ¢ but rx AŒ . By (1), we have rx qA , and whence, ( )A x r≥ ,

( ) 1A x r+ £ . Thus 0.5r < , a contradiction.

(3) fi (1) Let rx AŒ¢ , then ( )A x r<¢ .

(a) If ( ) ( )A x A x≥¢ , then ( )A x r< , and so rx AŒ . This proves that rx qAŒ⁄ .

(b) If ( ) ( )A x A x<¢ , then by (3), we have ( )0.5 A x≥ . Putting rx AŒ , then

( )0.5 A x r≥ ≥ , it follows that ( ) 1A x r+ £ and so rx qA . This proves that rx qAŒ⁄ .

(2) fi (4). If there exists ,x y LŒ such that ( ){ }max ,0.5A x y tÆ < =

( ) ( ){ }min ,A x A y , then 0.5 1t< £ , ( )t

x y AÆ Œ , but tx AŒ , ty AŒ . By (2), we

have tx qA or ty qA . Then ( ( )A x t≥ and ( ) 1A x t+ £ ) or ( ( )A y t≥ and

( ) 1A y t+ £ ). This implies that 0.5t £ , a contradiction.

(4) fi (2). Let ( ) { }min ,r tx y AÆ Œ . Then ( ) { }min ,A x y r tÆ < .

(a) If ( ) ( ) ( ){ }min ,A x y A x A yÆ ≥ , then ( ) ( ){ } { }min , min ,A x A y r t< and

consequently, ( )A x r< or ( )A y t< . It follows that rx AŒ or ly AŒ . Thus,

rx qAŒ⁄ or ly qAŒ⁄ .

(b) If ( ) ( ) ( ){ }min ,A x y A x A yÆ < , then by (4), we have ( ) ( ){ }0.5 min ,A x A y≥ .

Putting rx AŒ and ty AŒ . Then ( ) 0.5t A y£ £ or ( ) 0.5r A x£ £ , it follows that

( ) 1t A y+ £ or ( ) 1r A x+ £ , and thus rx qAŒ⁄ or ly qAŒ⁄ .


( ), qŒ Œ⁄ -fuzzy subalgebra of L if A is a fuzzy subalgebra of L .

Proof. This theorem is an immediate consequence of Definition 2.2 and Theorem 3.1. In general, the ( ), qŒŒ⁄ -fuzzy subalgebra of L is not a fuzzy subalgebra of L , for

Example 3.1 (1).

Theorem 3.3. Let L be a lattice implication algebra, ( )A F LÕ , then A is a fuzzy

subalgebra of L if A is an ( ), qŒŒ⁄ -fuzzy subalgebra of L and ( )0.5 1A O< £ .

Proof. Let A is an ( ), qŒŒ⁄ -fuzzy subalgebra of L , then for any ,x y LŒ , such that

( ){ } ( )max ,0.5A x A x≥¢ . Hence ( ){ } ( )max ,0.5A O A I≥ , since ( )0.5 1A O< £ , then

( ) ( )A O A I≥ . And ( ){ } ( )max ,0.5A I A O≥ , then ( ) ( )A O A I£ . Hence ( )A O = ( )A I .

And for any ,x y LŒ ,such that ( ){ } ( ) ( ){ }max ,0.5 min ,A x y A x A yÆ ≥ . If ( )A x yÆ

( ) ( ){ }min ,A x A y< , then ( ) ( ){ }0.5 min ,A x A y≥ . By ( ) ( )0.5 1A O A I< = £ , a con-


tradiction. Then ( ) ( ) ( ){ }min ,A x y A x A yÆ ≥ . This proves that A is a fuzzy subalgebra of L .

Example 3.1. Let { }, , ,L O a b I= , the Hasse diagram of L be defined as Fig. 1 and

operators of L be defined in Table 1, then ( ), , , ',L ⁄ Ÿ Æ is a lattice implication algebra.

O

Fig. 1. Hasse diagram of { }, , ,L O a b I=

Table 1. Operators of { }, , ,L O a b I=

�

O a b I

O O a b I a a a I I b b I b I I I I I I

� O a b I O O O O O a O a O a b O O b b I O a b I

Æ O a b I O I I I I a b I b I b a a I I I O a b I

' O a b I I b a O

(1) Define a fuzzy subset A of L by ( ) 0.3A O = , ( ) 0.2A a = , ( ) 0.25A b = ,

( ) 0.4A I = , then it can be easily verified that A is an ( ), qŒŒ⁄ -fuzzy subalgebra of L .

But A is not a fuzzy subalgebra of L , because for ( ) ( )0.4 0.3A I A O= π = .


(2) Define a fuzzy subset C of L by ( ) 0.6C O = , ( ) 0.8C a = , ( ) 0.3C b = ( )C I

0.7= , because for ( ){ } ( )max ,0.5 0.5 0.8C b C a= < = ,then it can be easily verified that

B is not an ( ), qŒ Œ⁄ -fuzzy subalgebra of L . Because for ( ) ( )0.7 0.6C I C O= π = , then C is not a fuzzy subalgebra of L by Definition 2.4.

(3) Define a fuzzy subset B of L by ( ) 0.6B O = , ( ) 0.3B a = , ( ) 0.3B b = ,

( ) 0.6B I = , then it can be easily verified that B is an ( ), qŒŒ⁄ -fuzzy subalgebra of L . B is a fuzzy subalgebra of L by Definition 2.4.

Theorem 3.4. Let L be a lattice implication algebra, ( )A F LÕ , A is an

( ), qŒŒ⁄ -fuzzy subalgebra of L , for any ,x y LŒ , then the following statements hold.

(1) ( ){ } ( )max ,0.5A O A x≥ ;

(2) ( ){ } ( ) ( ){ }max ,0.5 min ,A x y A x A y⁄ ≥ ;

(3) ( ){ } ( ) ( ){ }max ,0.5 min ,A x y A x A yŸ ≥ .

Proof. (1)A is an ( ), qŒŒ⁄ -fuzzy subalgebra of L ,for any ,x y LŒ ,then { ( )max ,A O

} ( )0.5 A I≥ , hence ( ){ } ( ){ }max ,0.5 max ,0.5A O A I≥ ( ){ }max ,0.5A x x≥ Æ ≥

( ) ( ){ }min ,A x A x ( )A x≥ .

(2) By Theorem 3.1, ( ){ }max ,0.5A x y⁄ ( )( ){ }max ,0.5A x y y= Æ Æ {max max=

( )( ){ } },0.5 ,0.5A x y yÆ Æ ( ) ( ){ }{ }max min , ,0.5A x y A y≥ Æ ({{min max A x= Æ

) } ( ){ }},0.5 ,max ,0.5y A y ( ) ( ){ } ( ){ }{ }min min , , max ,0.5A x A y A y≥ ( ){min ,A x=

( )}A y .

(3) By Theorem 3.1 and Theorem 3.4 (2), ( ){ } ( )max ,0.5A x y A x y ¢Ÿ ≥ Ÿ , then

( ){ } ( ){ }max ,0.5 max ,0.5A x y A x y ¢Ÿ ≥ Ÿ ( ){ }max ,0.5A x y ¢= ⁄¢ ( ) ( ){ }min ,A x A y≥ ¢ ¢ ,

then ( ){ } ( ){ } ( ){ }{ }max ,0.5 min max ,0.5 , max ,0.5A x y A x A y⁄ ≥¢ ¢ ¢ ¢ ( ){min ,A x≥

( )}A y .


( ), qŒ Œ⁄ -fuzzy subalgebra of L if and only if for any ( ]0.5,1tŒ , { ( )tA x L A x= Œ

}t≥ is a subalgebra of L .

Proof. Let A be an ( ), qŒŒ⁄ -fuzzy subalgebra of L , for any , tx y AŒ , ( ]0.5,1tŒ .

By Theorem 3.4, there exist 0 tx AŒ , then ( ){ } ( )0max ,0.5A O A x t≥ = . It follows that

( ) ( )0A O A x≥ , and hence tO AŒ . Putting ( ){ } ( ) ( ){ }max ,0.5 min ,A x y A x A y tÆ ≥ = ,


then ( )A x y tÆ ≥ , and hence tx y AÆ Œ . It follows that tA is a subalgebra of L by Definition 2.3.

Conversely, if tA is a subalgebra of L , for any ( ]0.5,1tŒ . Then A is an ( ), qŒ Œ⁄ -fuzzy subalgebra of L is an immediate consequence of Definition 2.1 and Theorem 3.1.

Theorem 3.6. Let L be a lattice implication algebra, ( ),A B F LÕ , A , B are

( ), qŒ Œ⁄ -fuzzy subalgebras of L , then A B« is an ( ), qŒ Œ⁄ -fuzzy subalgebra of L .

Proof. A ,B are ( ), qŒŒ⁄ -fuzzy subalgebras of L . By Theorem 3.1, for any ,x y LŒ ,

( )( ){ }max ,0.5A B x« ¢ ( ){ } ( ){ }max ,0.5 max ,0.5A x A x= Ÿ¢ ¢ ( ) ( )A x B x≥ Ÿ (A= «

)( )B x , ( )( ){ }max ,0.5A B x y« Æ ( ) ( ){ }max ,0.5A x y B x y= Æ Ÿ Æ ({max A x=

) } ( ){ },0.5 max ,0.5y B x yÆ Ÿ Æ ( ) ( ){ } ( ) ( ){ }min , min ,A x A y B x B y≥ Ÿ ({min A=

)( ) ( )( )},B x A B y« « , then A B« is an ( ), qŒŒ⁄ -fuzzy subalgebra of L .

Corollary 3.1. Let L be a lattice implication algebra, ( )iA F LÕ ( )1, 2, ,i n= , iA is ( ), qŒŒ⁄ -fuzzy subalgebras of L ( )1,2, ,i n= , then iA∩ is an ( ), qŒŒ⁄ -fuzzy

subalgebra of L ( )1, 2, ,i n= .

Theorem 3.7. Let 1L , 2L are lattice implication algebras, 1 2:f L LÆ is a lattice implication isomorphism of 1L to 2L , 1A LŒ . If A is an ( ), qŒ Œ⁄ -fuzzy subalgebras

of 1L , then ( )f A is an ( ), qŒŒ⁄ -fuzzy subalgebras of 2L .

Proof. Since 1 2:f L LÆ is a lattice implication isomorphism of 1L to 2L , then for any 1 2 2,y y LŒ , there exists 1 2 1,x x LŒ , such that ( )1 1f x y= , ( )2 2f x y= . We have

( )( ){ } ( ) ( ){ } ( ) ( ) ( )( )1 1 1 11 1 1 1max ,0.5 max ,0.5f x y f x y

f A y A x A x f A y= =¢ ¢= ≥ =¢ ¢� � .

If there exists 1 2 2,y y LŒ , such that ( )( ){ }1 2max ,0.5f A y yÆ < ( ){ ( ) ( )1min ,f A y f A

( )}2y , thus ( )( ) ( )( ) ( ) ( )1 2 1 11 1f x yf A y y f A y A x=Æ < = � , ( )( ) ( )( )1 2 2f A y y f A yÆ < =

( ) ( )12 2f x yA x=� , ( )( ) ( ) ( )1 11 10.5

f x yf A y A x=< = � , ( )( ) ( ) ( )2 22 20.5

f x yf A y A x=< = � ,

then ( ){ } ( ) ( )1 2 1 2max ,0.5f x y y

A x x A x= ÆÆ £ =� ( )( )1 2f A y yÆ . This is a contradict-

tion by Theorem 3.1, thus ( )( ){ } ( )( ) ( )( ){ }1 2 1 2max ,0.5 min ,f A y y f A y f A yÆ ≥ .

Therefore, ( )f A is the ( ), qŒŒ⁄ -fuzzy subalgebras of 2L .

Theorem 3.8. Let 1L , 2L are lattice implication algebras, 1 2:f L LÆ is a lattice implication homomorphism of 1L to 2L , 1A LŒ , 2B LŒ . If B is an ( ), qŒŒ⁄ -fuzzy

subalgebras of 2L , then ( )1f B- is an ( ), qŒŒ⁄ -fuzzy subalgebras of 1L .


Proof. For any 1 1 2 1, ,x x x LŒ¢ , such that ( )1 1f x y= , ( )1 1f x y=¢ ¢ , because of B is the

( ), qŒŒ⁄ -fuzzy subaglebras of 2L , thus by Theorem 3.1, we have { ( )( )11max ,f B x- ¢

} ( )( ){ } ( ){ } ( ) ( )( )1 1 1 10.5 max ,0.5 max ,0.5B f x B y B y B f x= = ≥ = =¢ ¢ ( )( )11f B x- .

( )( ){ } ( )( ){ }11 2 1 2max ,0.5 max ,0.5 maxf B x x B f x x- Æ = Æ = { ( ) ( )( )1 2 ,B f x f xÆ

} ( )( ) ( )( ){ } ( )( ) ( )( ){ }1 11 2 1 20.5 min , min ,B f x B f x f B x f B x- -≥ = . Therefore, ( )1f B-

is an ( ), qŒ Œ⁄ -fuzzy subalgebras of 1L .

4. The direct product of ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras

In this section, we introduce the concepts of the direct product of ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras, and discuss their related properties.

Definition 4.1. Let 1L , 2L are lattice implication algebras, A , B are ( ), qŒ Œ⁄ -

fuzzy subalgebras of 1L and 2L , definition the mapping [ ]: 0,1A B X Y¥ ¥ Æ ,

( )( ) ( ) ( ){ }, min ,A B x y A x B y¥ = , ( ) 1 2,x y L L" Œ ¥ ,

A B¥ is called the direct product of ( ), qŒ Œ⁄ -fuzzy subalgebra of 1L and 2L .

Theorem 4.1. Let 1L , 2L are lattice implication algebras, A , B are ( ), qŒŒ⁄ -

fuzzy subalgebras of 1L and 2L , then A B¥ is an ( ), qŒŒ⁄ -fuzzy sualgebra of 1 2L L¥ .

Proof. A , B are ( ), qŒ Œ⁄ -fuzzy subalgebras of 1L and 2L , for any ( ) 1 2,x y L LŒ ¥ ,

then by Theorem 3.1 and Definition 3.2, we have ( ) ( ) ({max , ,0.5 maxA B x y AÏ ¸Ê ˆ¢¥ = ¥Ì ˝Ë ¯Ó ˛

)( ) } ( ) ( ){ }{ }, ,0.5 max min , ,0.5B x y A x B y= =¢ ¢ ¢ ¢ ( ){ }{ ( ){min max ,0.5 ,max ,A x B y¢ ¢

}} ( ) ( ){ } ( )( )0.5 min , ,A x B y A B x y≥ = ¥ . ( )( ){ } ({{max ,0.5 max minA B x y A x¥ Æ = Æ

) ( )} } ( ){ }{, ,0.5 min max ,0.5 ,y B x y A x yÆ = Æ ( ){ }} ( ){max ,0.5 min ,B x y A xÆ ≥

( ) ( ) ( )} ( )( ) ( )( ){ }, , min ,A y B x B y A B x A B y= ¥ ¥ . Then A B¥ is an ( ), qŒŒ⁄ -fuzzy subalgebra of 1 2L L¥ by Theorem 3.1.

Corollary 4.1. Let 1 2, , , nL L L are lattice implication algebras, 1 2, , , nA A A are

( ), qŒ Œ⁄ -fuzzy subalgebras of 1 2, , , nL L L , then 1 2 nA A A¥ ¥ ¥ is an ( ), qŒ Œ⁄ -fuzzy subalgebra of 1 2 nL L L¥ ¥ ¥ .


5. Conclusion

In this paper, we firstly introduced the concepts of ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras, and further discussed its equivalent characterization. Then we studied homomorphic properties of ( ), qŒ Œ⁄ -fuzzy subalgebras of lattice implication algebras. Finally we researched the intersection operations and direct product operations of ( ), qŒŒ⁄ -fuzzy subalgebras of lattice implication algebras.

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