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78:2 (2016) 179–191 | www.jurnalteknologi.utm.my | eISSN 2180–3722 |

Jurnal

Teknologi Full Paper

B I - IDEALS OF ORDERED SEMIGROUPS BASED ON THE INTERVAL - VALUED FUZZY POINT

Hidayat Ullah Khan

a,b

, Nor Haniza Sarmin

a*

, Asghar Khan

c

, Faiz Muhammad Khan

d

a

Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor, Malaysia

b

Department of Mathematics, University of Malakand, at Chakdara, District Dir (L), Khyber Pakhtunkhwa, Pakistan

c

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan

d

Department of Mathematics, University of Swat, Khyber Pakhtunkhwa, Pakistan

Article history Received 29 December 2014 Received in revised form

9 March 2015 Accepted 15 January 2016

*Corresponding author nhs@utm.my

Graphical abstract Abstract

Interval-valued fuzzy set theory (advanced generalization of Zadeh's fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with (

k~

q ) relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued ( , q~)

k

 -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued ( , q~)

k

 -fuzzy bi-ideals are described. It is shown that an interval-valued )

q ,

( ~

k

 -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi- ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued

) q ,

( ~

k

 -fuzzy bi-ideal are considered.

Keywords: Interval-valued fuzzy bi-ideals, Interval-valued (,q)-fuzzy bi-ideal, Interval- valued ( , q~)

k

 -fuzzy bi-ideal, Interval-valued fuzzifying bi-ideal, t~-implication-based interval-valued fuzzy bi-ideal

Abstrak

Teori set kabur bernilai-selang (pengitlakan lanjutan bagi set kabur Zadeh) adalah teori yang lebih menyeluruh yang dapat mengendalikan masalah-masalah dunia sebenar dengan lebih tepat berbanding teori set kabur biasa. Dalam kertas kerja ini, kami memperkenalkan idea kuasi-kebetulan teritlak (

k~

q ) bagi suatu titik kabur bernilai-selang dengan suatu set kabur bernilai-selang. Malahan, konsep baru ini juga adalah bentuk yang lebih menyeluruh bagi kuasi-kebetulan suatu titik kabur bernilai-selang dengan suatu set kabur bernilai-selang.

Dengan menggunakan idea yang baru didefinisikan ini, konsep bagi suatu dwi-unggulan kabur-( , q~)

k

 bernilai-selang telah diperkenalkan. Sebagai tambahan, beberapa pencirian bagi dwi-unggulan kabur-( , q~)

k

 bernilai-selang telah diterangkan. Telah ditunjukkan bahawa suatu dwi-unggulan kabur-( , q~)

k

 bernilai-selang adalah suatu dwi- unggulan kabur dengan diberikan satu syarat. Akhir sekali, konsep bagi dwi-unggulan kabur bernilai-selang yang berasaskan implikasi, pengitlakan bagi suatu dwi-unggulan kabur

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bernilai selang dan suatu dwi-unggulan kabur- ( , q~)

k

 bernilai-selang telah dipertimbangkan.

Kata kunci: Dwi-unggulan kabur bernilai-selang, Dwi-unggulan kabur-(,q)bernilai-selang, Dwi-unggulan kabur-( , q~)

k

 bernilai-selang, Dwi-unggulan pengkaburan bernilai-selang, Dwi-unggulan kabur bernilai-selang berasaskan implikasi-~t

© 2016 Penerbit UTM Press. All rights reserved

1.0 INTRODUCTION

In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages, computer arithmetics, and error-correcting codes. The fundamental concept of a fuzzy set, introduced by L. A. Zadeh [1], provides a natural frame-work for generalizing several basic notions of algebra. Moreover, the study of fuzzy sets in semigroups was introduced by Kuroki [2-4]. Likewise, a systematic exposition of fuzzy semigroups was given by Mordeson et al. [5], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines and fuzzy languages. In addition, the monograph by Mordeson and Malik [6] dealing with the application of fuzzy approach to the concepts of automata and formal languages. Moreover, Murali [7] proposed the definition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on fuzzy subset. Besides, the idea of quasi-coincidence of a fuzzy point with a fuzzy set [8], played a vital role to generate some different types of fuzzy subgroups. Furthermore, Bhakat and Das [9-10] gave the concepts of (,)-fuzzy subgroups by using the “belongs to” relation () and

“quasi-coincident with” relation (q) between a fuzzy point and a fuzzy subgroup, and introduced the concept of an (,q)-fuzzy subgroup. Jun and Song [11] initiated the study of

,

-fuzzy interior ideals of a semigroup. In addition, Kazanci and Yamak [12]

studied (,q)-fuzzy bi-ideals of a semigroup and Shabir et al. [13] studied characterization of regular semigroups by (,)-fuzzy ideals. Moreover, Jun et al.

[14] discussed generalization of an (,)-fuzzy ideals of a BCK/BCI-algebra. In addition, Shabir and Khan [15]

characterized different classes of ordered semigroups by the properties of fuzzy quasi-ideals. Further, by applying fuzzy soft set theory the notions of fuzzy left (right, bi- and quasi-) ideals of type (,q) [16] are introduced in ordered semigroups. For further study on generalized fuzzy sets in ordered semigroups the reader is referred to [17-21]. The concept of a fuzzy bi- ideal in ordered semigroups was first introduced by Kehayopulu and Tsingelis in [22], where some basic properties of fuzzy bi-ideals were discussed.

Furthermore, using the idea of a quasi-coincidence of a fuzzy point with a fuzzy set, Khan et al. 23 introduced the concept of interval-valued (,)-fuzzy bi-ideals in an ordered semigroup.

In this paper, we present a more general form of the idea presented in [23]. This new generalization is called an interval-valued (,qk~)-fuzzy bi-ideal of an ordered semigroup. By constructing suitable examples, it is shown that there are interval-valued ( , q~)

k

 -fuzzy

bi-ideals which are not interval-valued (,q)-fuzzy bi- ideals. In addition, ordered semigroups are characterized by the properties of this new concept.

Further, a condition for an interval-valued ( , q~)

k

 -

fuzzy bi-ideal to be an interval-valued fuzzy bi-ideal is provided. It is important to note that several results of [23] are corollaries of our results obtained in this paper, which is the important achievement of this study.

2.0 PRELIMINARIES

By an ordered semigroup (or po-semigroup) we mean a structure

S,,

in which the following are satisfied for x,a,bS:

(OS1) (S,) is a semigroup, (OS2) (S,) is a poset,

(OS3) abaxbx,xaxb.

Throughout this paper, xy is simply denoted by xy for all x,yS.

A nonempty subset A of an ordered semigroup S is called a subsemigroup of S if A2A.

A non-empty subset A of an ordered semigroup S is called a bi-ideal of S if it satisfies the following three conditions:

(b1) for all aS and bA,abaA, (b2) A2A,

(b3) ASAA.

By an interval number a~ we mean an interval ]

,

[a a where 0aa1. The set of all interval numbers is denoted by D[0,1]. The interval [a,a] can be simply identified by the number a[0,1]. We define

(3)

the following for the interval numbers ~ai[ai,ai], ]

,

~ [

i i

i b b

b for all iI:

(i) ~} [max( , ),max( , )]

~, { max

r ai biai bi ai bi , (ii) ~} [min( , ),min( , )]

~, { min

r ai biai bi ai bi , (iii) r inf~ [ , ],r sup~ [ , ]

   

a a a a a

a

I i i I i i I

i i I

i i ,

(iv) ~a1~a2a1a2 and a1a2, (v) a~1~a2a1a2 and a1a2, (vi) ~a1~a2a1a2 and a1a2, (vii) ~ [ , ]

1 ka i ka i

a

k , whenever 0k1.

Then, it is clear that (D[0,1],,,) forms a complete lattice with ~0 [0,0]

 as its least element and 1~ [1,1]

 as its greatest element.

The interval-valued fuzzy subsets provide a more adequate description of uncertainty than the traditional fuzzy subsets; it is therefore important to use interval-valued fuzzy subsets in applications. One of the main applications of fuzzy subsets is fuzzy control, and one of the most computationally intensive parts of fuzzy control is the defuzzification. Since a transition to interval-valued fuzzy subsets usually increase the amount of computations, it is vitally important to design faster algorithms for the corresponding defuzzification.

An interval-valued fuzzy subset ~: [0,1] D X

f  of X is

the set

]}

1 , 0 [ )]

( ), ( [ ,

|

~ {

D x f x f x X x

f  

where f: X[0,1] and f:X[0,1] are fuzzy subsets such that 0f(x)f(x)1 for all xX and

)]

( ), (

[f x f x is the interval degree of membership function of an element x to the set f~

. Let f~

be an interval-valued fuzzy subset of X . Then

for every ~ 1~

~0t  , the crisp set

~} )

~(

| {

~)

~;

(f t x X f x t

U    is called the level set of f~ . The reader is referred to23 for more details on operations on two interval-valued fuzzy sets of X . Note that since every a[0,1] is in correspondence with the interval [a,a]D[0,1], hence a fuzzy set is a special case of the interval-valued fuzzy set.

For any ~ [ , ]

f f

f and ~t [t,t], we define ]

) ( , ) (

~ [ )

~(

t f x t f x t x

f , for all xX . In

particular, if f(x)t1 and f(x)t1, then we write ~ ~1

)

~(

t x

f .

An interval-valued fuzzy subset f~ of an ordered semigroup S is called an interval-valued fuzzy bi- ideal23 of S if it satisfies the following for all x,y,zS:

(b4) ~( )

)

~(

y f x f y

x   ,

(b5) ~( )}

),

~( min{

r )

~(

y f x f xy

f  ,

(b6) ~( )}

),

~( min{

r )

~(

z f x f xyz

f  .

An interval-valued fuzzy subset ~f

of an ordered semigroup (S,,) of the form

 



 

, if ,

~0 (0,1], if ,

~

~ :

x y

x y D

y t f

is called an interval-valued fuzzy point with support x and interval value ~tD[0,1] and is denoted by xt~. For an interval-valued fuzzy subset ~f

of S, we say that an interval-valued fuzzy point xt~ is:

(b7) contained in f~

, denoted by xt ~f

~ , if f x t~ )

~(

 . (b8) quasi-coincident with f~

, denoted by xt ~f q

~ , if

~1 ) ~

~(

t x

f .

For an interval-valued fuzzy point xt~ and an interval- valued fuzzy subset f~

of a set S, we say that:

(b9) xt ~f q

~ if xt f~

~ or xt ~f q

~ .

(b10) xt f~

~ if xt ~f

~ does not hold for {,q,q}.

3.0 GENERALIZATION OF INTERVAL-VALUED

) q ,

(

-FUZZY BI-IDEALS

Throughout this paper, S is an ordered semigroup and let ~ [ , ]

k k

k denote an arbitrary element of D[0,1) unless otherwise specified. For an interval-valued fuzzy point

x

t~ and an interval-valued fuzzy subset f~

of S, we say that

(c1) xt kf~ q~

~ if 1~

~>

) ~

~(

k t x

f   , where ftk1,

1

t k

f .

(c2) x f

t k

q~~

~ if xt ~f

~ or x f

t k

q~~

~ .

(c3) xt ~f

~ if xt ~f

~ does not hold for any  in }

q , q ,

{ ~ ~

k k 

 .

We emphasize here that the interval-valued fuzzy subset ~( ) [ ( ), ( )]

x f x f x

f must satisfy the following condition:







 

  



 

  

X x x

f x k f

k

k x k

f x f

all for )]

( ), ( 2 [

,1 2 1

or

2 ,1 2 )] 1 ( ), ( [

(E)

In what follows, we emphasize that all the interval- valued fuzzy subsets of X must satisfy the condition (E) and any two elements of D[0,1] are comparable unless otherwise specified.

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3.1 Theorem Let ~f

be an interval-valued fuzzy subset of S. Then the following are equivalent:

(1)

~

2 ,1

 

1k

D t

 ~)

~; (

~)

~;

(f t U f t

U  is a bi-ideal of S.

(2) f~ satisfies the following assertions for all x,y,zS: (2.1) ~( ) r max

~( ),[ , 2 ]

1 2 1

y f y f x k k

x ,

(2.2) r min

~( ),~( )

r max

~( ),[ , 2 ]

1 2 1

f xy k k y

f x

f ,

(2.3) r min

~( ),~( )

r max

~( ),[ , 2 ]

1 2 1

f xyz k k z

f x

f .

Proof. Assume that ~)

~; (f t

U is a bi-ideal of S for all





   ,1 2

~ 1 k

D

t with ~)

~; (f t

U . If there exist a,bS with b

a such that the condition (2.1) is not valid, then

~( ),[ , ]

.

max r )

~(

2 1 2 1

f a k k

b

f In which it follows that





   ,1 2 ) 1

~( k

D b

f and ~( ))

~, (f f b U

b . However,

)

~( )

~( b f a

f  which implies that ~( ))

~, (f f b

aU , a

contradiction. Hence (2.1) is valid.

Suppose that (2.2) is false, that is

~( ),~( )

r max

~( ),[ , ]

,

min

~ r

2 1 2 1

f a f c f ac k k

s

for some a,cS. Then ~

,1

2 1k

D

s and ~,~) ( ,c U f s

a  but

~)

~, (f s

acU , a contradiction, and so (2.2) holds for all S

y x,  .

Assume that there exist a,b,cS such that

~( ),~( )

r max

~( ),[ , ]

.

min

~ r

2 1 2 1

f a f c f abc k k

r

This implies ~

,1

2 1k

D

r and ~,~)

( ,c U f r

a  but

~)

~, (f r

abcU , this is impossible and therefore

~( ),~( )

r max

~( ),[ , ]

min

r f x f zf xyz 12k 12k , for all x,y,zS.

Conversely, assume that f~

satisfies all the three conditions (2.1), (2.2) and (2.3). Suppose that

~)

~; (f t

U for all ~

,1

2 1k

D

t . Let x,yS be such that y

x and ~)

~; (f t U

y . Then f y ~t )

~(

 , and so by (2.1);

~( ),[ , ]

~( ) ~ [ , ]

max

r 2

1 2 1 2

1 2

1k kf ytk k

x

f .

Hence f x t~ )

~(

 , that is ~)

~; (f t U

x .

If ~)

~; ( ,y U f t

x  , it follows from (2.2) that

~( ),[ , ]

r min

~( ),~( )

~ [ , ],

max

r f xy 12k 12kf x f yt12k 12k in which it follows that f xy t~

)

~(  i.e., ~)

~; (f t

xyU .

Take ~)

~; ( ,y U f t

x  , it follows from (2.3) that

~( ),[ , ]

r min

~( ),~( )

~ [ , ].

max

r f xyz 12k 12kf x f zt12k 12k

This implies f xyz t~ )

~(

 i.e., ~)

~; (f t

xyzU . Therefore

~)

~; (f t

U is a bi-ideal of S for all ~

2 ,1

1k

D

t with

 )

;~ (~f t

U .

Taking ~ [0,0]

k in Theorem 3.1 the following corollary arises.

3.2 Corollary

The following are equivalent for every interval-valued fuzzy subset ~f of S:

(1) The level subset ~)

~; (f t

U is a bi-ideal of S for all ]

1 , 5 . 0

~tD( , whenever ~)

~; (f t

U .

(2) The interval-valued fuzzy subset f~

satisfies the following assertions for all x,y,zS:

(2.1). ~f(y)r max

~f(x),[0.5,0.5]

for xy,

(2.2). r min

~f(x),~f(y)

r max

~f(xy),[0.5,0.5]

,

(2.3). r min

~f(x),~f(z)

r max

~f(xyz),[0.5,0.5]

.

3.3 Definition

An interval-valued fuzzy subset f~ of S is called an interval-valued (,qk~)-fuzzy bi-ideal of S if it satisfies the following conditions for all x,y,zS and for all

] 1 , 0

~ (

~,

~,

2

1 t D

t

t  :

(i) ~ ~)

( )

(xy y~tfx~tqk~f ,

(ii) x f y f xy q f

t k t t

t

) ~

~ (

~ ,

} ~ ,~ min{~

~ r

~

2 1 2

1    ,

(iii) xt f zt f xyz t t qk~f )

~ (

~ ,

} ~ ,~ min{~

~ r

~

2 1 2

1    .

The following example is constructed to support the newly defined notion of interval-valued (,qk~)-fuzzy bi-ideals in ordered semigroups.

3.4 Example

Consider the ordered semigroup S

a,b,c,d,e

with the following order relation “” and multiplication Table 3.1:

. ) , ( ), , ( ), , ( ), , ( ), , ( ), , (

), , ( ), , ( ), , ( ), , ( ), , ( ), , (

: 









e e e d d d e c c c e b

d b b b e a d a c a a a

Table 3.1 Multiplication table of S

e d c d a e

d d a d a d

e d c d a c

d d a b a b

d d a d a a

e d c b

a

(5)

Define an interval-valued fuzzy subset ~: [0,1] D S

f  by





. if

], 30 . 0 , 15 . 0 [

, if

], 35 . 0 , 25 . 0 [

}, , { if ], 45 . 0 , 35 . 0 [

, if

], 60 . 0 , 40 . 0 [ )

~(

e x

d x

c b x

a x x

f

Then using Definition 3.3, f~

is an interval-valued )

,

( ~

qk

 -fuzzy bi-ideal of S for all ~ [0.3,0.5]

k .

The necessary and sufficient conditions for an interval-valued fuzzy subset to be an interval-valued

) ,

( ~

qk

 -fuzzy bi-ideal are given in the following result.

3.5 Theorem

An interval-valued fuzzy subset f~

of S is an interval- valued ( , q~)

k

 -fuzzy bi-ideal of S if and only if:

(1) ~f(x)r min

f~(y),[k, 2k]

for xy

1 2

1 .

(2) ~( ) r min

~( ),~( ),[ , 2 ]

1 2 1

f x f y k k xy

f .

(3) ~( ) r min

~( ),~( ),[ , 2 ]

1 2 1

f x f z k k xyz

f .

Proof. Suppose that f~

is an interval-valued ( , q~)

k

 -

fuzzy bi-ideal of S. Let x,yS such that xy. If )

~( )

~( y f x

f  , then ~ ~( )

)

~(

y f t x

f   for some ~

,1

2 1k

D

t . It follows that yt ~f

~ , but xt f~

~ and xt k~f q~

~ . Therefore, f

xt k

q ~

______

~

~ , a contradiction and hence ~( ) )

~( y f x

f  .

Now if ~( ) [ , ]

2 1 2 1

k k y

f , then y k k ~f

] ,

[12 12  and so f

x k k k

q~~

] ,

[12 12  that is ~( ) [ , ]

2 1 2 1

k k x

f or

k x

f k k ~

1~ ] , [ )

~(

2 1 2

1  

. Hence, ~( ) [ , ]

2 1 2 1

k k x

f ,

otherwise,

k x

f k k k k k k ~

1~ ] , [ ] , [ ] , [ )

~(

2 1 2 1 2 1 2 1 2 1 2

1    

, again a contradiction. Consequently,

~( ),[ , ]

min r )

~(

2 1 2 1

f y k k

x

f for all x,yS with xy. Let x,yS be such that r min

~f(x),~f(y)

[12k,12k].

We claim that ~f(xy)r min

~f(x),~f(y)

. If not, then

~( ),~( )

min

~ r )

~(

y f x f t

xy

f   for some ~

,1

2 1k

D

t . It follows that xt f~

~ and yt ~f

~ , but xy t ~f )

( ~ and f

xy t k

q ~ )

( ~ ~ , a contradiction. Thus

~( ),~( )

min r )

~(

y f x f xy

f  for all x,yS with

~( ),~( )

[ , ]

min

r 2

1 2 1

k k y

f x

f .

If r min

~( ),~( )

[ , 2 ] 1 2 1

k k y

f x

f , then x k k f~

] ,

[12 12  and f

y k k ~

] ,

[ 2

1 2

1  . By using Definition 3.3 (ii), it implies that

 

f

xy

xy k k k k k k k~

q )

( )

( ~

] , [ ], , [ min r ] ,

[12 1212 12 12 12  .

Hence ~( ) [ , ]

2 1 2 1

k k xy

f or f xy k k k~

~1 ] , [ )

~(

2 1 2

1  

. Now if ~( ) [ , ]

2 1 2 1

k k xy

f , then

k xy

f k k k k k k ~

1~ ] , [ ] , [ ] , [ )

~(

2 1 2 1 2 1 2 1 2 1 2

1    

, a contradiction and hence ~( ) [ , ]

2 1 2 1

k k xy

f .

Consequently, ~( ),[ , ]}

),

~( min{

r )

~(

2 1 2 1

f x f y k k xy

f for all

S y x,  .

Assume that x,y,zS be such that

~( ),~( )

[ , ]

min

r 2

1 2 1

k k z

f x

f and claim that

~( ),~( )

min

~ r )

~(

z f x f s

xyz

f   for some ~

,1

2 1k

D

s . It follows that xs ~f

~ and zs ~f

~ , but xyz s ~f )

( ~ and k

s s xyz

f ~

~1 2~ ) ~

~(

 , i.e., xyz sqk~f )

( ~ ~ , a contradiction

and hence ~( )}

),

~( min{

r )

~(

z f x f xyz

f  for all x,y,zS with r min

~( ),~( )

[ , 2 ]

1 2 1

k k z

f x

f .

If r min

~f(x),~f(z)

[12k,12k], then x k k f

~

] ,

[12 12  and f

z k k ~

] ,

[12 12  and by Definition 3.3 (iii),

 

f

xyz

xyz k k k k k k k~

q )

( )

( ~

] , [ ], , [ min r ] ,

[ 2

1 2 1 2 1 2 1 2

1 2

1  .

In which it follows that ~( ) [ , ]

2 1 2 1

k k xyz

f or

k xyz

f k k ~

~1 ] , [ )

~(

2 1 2

1  

. If ~( ) [ , ]

2 1 2 1

k k xyz

f , then

k xyz

f k k k k k k ~

1~ ] , [ ] , [ ] , [ )

~(

2 1 2 1 2 1 2 1 2 1 2

1    

, a contradiction and hence ~( ) [ , ]

2 1 2 1

k k xyz

f .

Consequently,

~( ),~( ),[ , ]

min r )

~(

2 1 2 1

f x f z k k xyz

f ,

for all x,y,zS.

Conversely, let ~f be an interval-valued fuzzy subset of S that satisfies the Conditions (1), (2) and (3).

Let x,yS and ~ (0,1] D

t be such that xy and f

yt ~

~ . Then f y t~ )

~(

 , and so

 

 





 

].

,

~ [ if ], , [

], ,

~ [ if

~,

] , [

~, min r

] , [ ),

~( min r )

~(

2 1 2 1 2

1 2 1

2 1 2 1 2 1 2 1

2 1 2 1

k k k

k

k k k k

k k

t t t

t y f x

f

It follows that xt f~

~ or x f

t k

q~~

~ i.e., x f

t k

q~~

~ . Let x,yS and ~,~ (0,1]

2

1 t D

t  be such that xt ~f

1

~ and f

yt ~

2

~ . Then ~1 )

~( t x

f  and ~2

)

~( t y

f  . It follows from (2) that

(6)

 

 





 

], , [

~}

~, min{

r if ], , [

], , [

~}

~, min{

r if

~},

~, min{

r

] , [

~,

~, min r

] , [ ),

~( ),

~( min r )

~(

2 1 2 1 2 2 1

1 2 1

2 1 2 1 2 1 2

1

2 1 2 1 2 1

2 1 2 1

k k k

k

k k k

k

k k

t t

t t t

t t t

y f x f xy

f

This implies xy f

t k t

q ~ )

( ~

~}

~, { min

r 1 2  .

If xt ~f

1

~ and zt ~f

2

~ for some x,y,zS and ]

1 , 0

~ (

~,

2

1 t D

t  , then ~1

)

~( t x

f  and ~2

)

~( t z

f  . It follows from (3) that

 





 

], , [

~}

~, min{

r if ], , [

], , [

~}

~, min{

r if

~},

~, min{

r

} ] , [

~,

~, min{

r

] , [ ),

~( ),

~( min r )

~(

2 1 2 1 2 2 1

1 2 1

2 1 2 1 2 1 2

1

2 1 2 1 2 1

2 1 2 1

k k k

k

k k k

k

k k

t t

t t t

t t t

z f x f xyz

f

This implies, xyz t t f~ )

( ~}

~, { min

r 1 2  or xyz f

t k t

q ~ )

( ~

~}

~, { min

r 1 2 and

consequently, xyz f

t k t

q ~ )

( r min{~1,~2} ~ .

The following corollary arises by taking ~k[0,0] in Theorem 3.5.

3.6 Corollary

An interval-valued fuzzy subset f~

of S is an interval- valued (,q)-fuzzy bi-ideal of S if and only if for all

S z y

x, ,  the following assertions hold:

(1) ~( ),[0.5,0.5]}) min{

r )

~( ( )

(xy f xf y .

(2) ~( ),[0.5,0.5]} ),

~( min{

r )

~(

y f x f xy

f  .

(3) ~( ),[0.5,0.5]} ),

~( min{

r )

~(

z f x f xyz

f  .

Clearly every interval-valued ( , ~) qk

 -fuzzy bi-ideal is an interval-valued fuzzy generalized bi-ideal of type

) q ,

( ~

k

 . The following example illustrates that its converse is not true in general.

3.7 Example

Consider the ordered semigroup S{a,b,c,d} with the order relation “” and multiplication given in Table 3.2:

)}

, ( ), , ( ), , ( ), , ( ), , {(

: aa bb cc d d ab

Table 3.2 Multiplication table of S

b b a a d

a b a a c

a a a a b

a a a a a

d c b a .

Define an interval-valued fuzzy subset ~: [0,1] D S

f  by





. if ], 0 , 0 [

, if ], 8 . 0 , 7 . 0 [

, if ], 0 , 0 [

, if ], 9 . 0 , 8 . 0 [ )

~(

d x

c x

b x

a x x

f

Then ~f

is interval-valued (,qk~)-fuzzy generalized bi-ideal. However, ccb but

~( ) [0.7,0.8],[ , ]

min r ] 0 , 0 [ )

~( )

~(

2 1 2 1

f b f c k k

cc

f for all

0,1

~ D

k and hence Theorem 3.5 (2) is not satisfied.

Therefore, f~ is not an interval-valued ( , q~)

k

 -fuzzy

bi-ideal.

Since, every interval-valued fuzzy bi-ideal of S is an interval-valued ( , q~)

k

 -fuzzy bi-ideal of S for some

0,1

~ D

k . Therefore, the following example is constructed to show that there exists ~

0,1

D

k such that, the interval-valued fuzzy subset f~

of S is an interval-valued ( , q~)

k

 -fuzzy bi-ideal of S but not an interval-valued fuzzy bi-ideal of S.

3.8 Example

The interval-valued ( , q~)

k

 -fuzzy bi-ideal f~

of S in Example 3.4 is not an interval-valued fuzzy bi-ideal of

S. Since ~( )

] 45 . 0 , 35 . 0 [ )

~(

c f b

f   and

)

~( )

~( ] 35 . 0 , 25 . 0

[  f df cb but on the other hand;

~( )

r min

[0.35,0.45],[0.35,0.45]

[0.35,0.45].

),

~( min

r f c f b  

It follows that that ~f(cb)r min

~f(c),~f(b)

.

In the next result the condition under which an interval-valued ( , ~)

qk

 -fuzzy bi-ideal is an interval- valued fuzzy bi-ideal is provided.

3.9 Theorem

If for all xS the value of the interval-valued fuzzy subset f~

of S is less than the interval [12k,12k]. Then every interval-valued ( , ~)

qk

 -fuzzy bi-ideal f~ of S is an interval-valued fuzzy bi-ideal of S.

Proof. Consider ~f

be an interval-valued ( , ~) qk

 -

fuzzy bi-ideal of S and ~( ) [ , ]

2 1 2 1

k k x

f for all xS. Let S

y

x,  be such that xy, then by Theorem 3.5 (1)

 

since~( ) [ , ]

,

)

~(

] , [ ),

~( min r )

~(

2 1 2 1

2 1 2 1

Rujukan

DOKUMEN BERKAITAN

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