Bi-ideals of ordered semigroups based on the interval-valued fuzzy point

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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

[email protected]

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

 

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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

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 ( , q~)

k

 

 -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,bS:

(OS1) (S,) is a semigroup,

(OS2) (S,) is a poset,

(OS3) abaxbx,xaxb.

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

all x,yS.

A nonempty subset A of an ordered semigroup S is called a subsemigroup of S if A2A.

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,abaA,

(b2) A2A, (b3) ASAA.

By an interval number a~ we mean an interval ]

,

[a a where 0aa1. The set of all interval

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the following for the interval numbers ~a_i[a_i,a_i], ]

, [ ~ _  

i i

i b b

b for all iI:

(i) r max{~ai,b~i} [max(a i,b i),max(a i,b i)]

  



 ,

(ii) r min{a~i,b~i} [min(a i,b i),min(a i,b i)]

  



 ,

(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~a2  a1a2 and a1a2,

(v) _a~₁_~_a₂ _

2 1 

 __a

a and a1a2,

(vi) ~a1~a2  a1a2 and a1a2,

(vii) ka~₁[kai,kai], 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 ~f:XD[0,1] 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 xX 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 ~0~t 1~, 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 x f t x f t x

f , for all xX . In

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

write ~f(x)~t ~1.

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_,_z__S_:

(b4) xy~f(x)f~(y),

(b5) f~(xy)r min{~f(x),~f(y)},

(b6) f~(xyz)r min{~f(x),~f(z)}.

An interval-valued fuzzy subset ~f of an ordered semigroup (S,,) of the form

 

  

  



, if , 0

~ (0,1], if , ~

: ~

x y

x y D

t y f

is called an interval-valuedfuzzy point with support

x

and interval value ~tD[0,1] and is denoted by xt~. For

an interval-valued fuzzy subset ~f of S, we say that an interval-valued fuzzy point x_t~ is:

(b7) contained in f~, denoted by x~_t~f , if f~(x)t~. (b8) quasi-coincident with f~, denoted by x~_tq~f , if

1 ~ ~ ) ( ~



t

x

f .

For an interval-valued fuzzy point x_t~ and an

interval-valued fuzzy subset f~ of a set S, we say that: (b9) x~_tq~f if x_t~f~ or x f

t

~ q

~ .

(b10) x_t~ ~f if x~_t~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) x_t~q_k~f~ if ~f(x)t~k~>~1, where ftk1, 1

 

  

 _t _k

f .

(c2) x f

k t

~ q~

~ if x f

t

~

~ or x f

k t

~ q~ ~ .

(c3) x_t~ ~f if x_t~~fdoes not hold for any  in

} q , q ,

{ ~ ~

k k 

 .

We emphasize here that the interval-valued fuzzy subset ~f(x)[f(x),f(x)] must satisfy the following

condition:

       

 

   



  

   



  



   

  



X x x

f x f k k

k k x

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

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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 

2 1   

 _f _xyz k k

z f x

f .

Proof. Assume that U(~f;~t) is a bi-ideal of S for all

     

 ,1

2 1

~ k

D

t with U(~f;t~). If there exist a,bS with b

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



~( ),[ , ]



.

max r ) ( ~

2 1 2 1  

 k k

a f b

f In which it follows that

     

 ,1

2 1 ) (

~ k

D b

f and bU(~f,~f(b)). However, )

( ~ ) ( ~

b f a

f  which implies that aU(~f,~f(b)), a

contradiction. Hence (2.1) is valid. Suppose that (2.2) is false, that is



2 1 2 1   



 _f _a _f _c _f _abc k k

r

This implies ~_r_D



1₂k,1



and _a_,_c_U₍~_f_,~_r₎ but )

~ , ~ (f r U

abc , this is impossible and therefore



~( ),~( )



r max



~( ),[ , ]



min

r 1₂

2 1   

 _f _xyz k k

z f x

f ,

for all x,y,zS.

Conversely, assume that f~ satisfies all the three conditions (2.1), (2.2) and (2.3). Suppose that





) ~ ; ~ (f t

U for all ~_t__D



1₂k,1



_{. Let} _x_,_y__S_{be such that}

y

x and yU(~f;~t). Then ~f(y)~t , and so by (2.1);



~( ),[ , ]



~( ) ~ [ , ]

max r

2 1 2 1 2

1 2

1k k __f _y __t _ k k

x

f .

Hence f~(x)t~, that is xU(~f;~t).

If x,yU(~f;~t), it follows from (2.2) that



~( ),[ , ]



r min



~( ),~( )



~ [ , ],

max

r f xy 1₂k 1₂k  f x f y t  1₂k 1₂k in which it follows that ~f(xy)t~ i.e., xyU(~f;~t).

Take x,yU(~f;~t), it follows from (2.3) that



~( ),[ , ]



r min



~( ),~( )



~ [ , ].

max

r f xyz 1₂k 1₂k  f x f z t  1₂k 1₂k

This implies f~(xyz)t~ i.e., xyzU(~f;~t). Therefore )

~ ; ~ (f t

U is a bi-ideal of S for all ~_t_D



1₂k,1



with





) ~ ; ~ (f t

U .

Taking k~[0,0] 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 U(~f;~t) is a bi-ideal of S for all ]

1 , 5 . 0 (

~_t__D _{, whenever} __

) ~ ; ~ (f t

U .

(2) The interval-valued fuzzy subset f~ satisfies the following assertions for all x,y,zS:

,

(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 ( , ~)

k

q

 

 -fuzzy bi-ideal of S if it satisfies the following conditions for all x,y,zS and for all

] 1 , 0 ( ~ , ~ , ~

2 1 t D

t

t  :

(i) (xy) (y_t~f~x_t~q_k~f~),

(ii) x f y f xy q f

k t t t

t

~ )

( ~ , ~

~ } ~ , ~ min{ r ~

~

2 1 2

1    ,

(iii) x_t ~f ,z_t ~f (xyz) _t _t q_k~~f

} ~ , ~ min{ r ~

~

2 1 2

1    .

The following example is constructed to support the newly defined notion of interval-valued ( , ~)

k

q

 

 -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 ~f:SD[0,1] 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

) ,

( ~

k

q

 

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

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

) ,

( ~

k

q

 

 -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_, k_]



_forxy

2 1 2

1 _.

(2) ~( ) r min



~( ),~( ),[ ,1₂ ]



2 1   

 _f _x _f _y k k

xy

f .

(3) ~( ) r min



~( ),~( ),[ ,1₂ ]



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,yS such that xy. If )

( ~ ) ( ~

y f x

f  , then f~(x)t~f~(y) for some _t~__D



1₂k,1



_{. It}

follows that y_t~~f , but x f

t

~

~ and x_t~q_k~~f . Therefore, f

x

k t

~ q

______ ~

~ , a contradiction and hence ~f(x)~f(y).

Now if ~( ) [ ,1₂ ]

2 1   

 k k

y

f , then y k k f

~

] , [ 2

1 2

1   and so

f x

k

k k

~ q~ ] , [ 1₂

2

1   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,yS with xy. Let x,yS be such that r min



~( ),~( )



[ , ]

2 1 2 1   

 k k

y f x

f .

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 ~



2 ,1



1k D

t  . It

follows that x_t~f~ and y f

t

~

~ , but xy f

t

~ )

( ~ and

f xy

k t

~ q )

( ~ ~ , a contradiction. Thus



~( ),~( )



min r ) ( ~

y f x f xy

f  for all x,yS with



~( ),~( )



[ , ]

min r

2 1 2 1   

 k k

y f x

f .

If r min



~( ),~( )



[ ,1₂ ]

2 1   

 k k

y f x

f , then x k k f

~

] , [ 1₂

2

1   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

2 1 2 1 2 1 2

1 2

1        .

Hence ~( ) [ , ]

2 1 2 1   

 k k

xy

f or f~₍xy₎ _[ k _, k _] ~₁ k~

2 1 2

1 _ _

 _  _ 

.

Now if ~( ) [ ,1₂ ]

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 ~( ) [ ,1₂ ]

2 1   

 k k

xy

f .

Consequently, ~f(xy)r min{f~(x),f~(y),[1₂k,1₂k]} for all S

y x,  .

Assume that x,y,zS 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 _s~_D



1₂k,1



. It

follows that x_s~~f and z~_s~f , but (xyz)~_s~f and

k s s xyz

f( ) ~ 2~ ~1 ~ ~

  

 , i.e., xyz sq_k f

~ )

( ~ ~ , a contradiction

and hence f~(xyz)r min{~f(x),f~(z)} for all x,y,zS

with r min



~( ),~( )



[ ,1₂ ]

2 1   

 k k

z f x

f .

If r min



~( ),~( )



[ , ]

2 1 2 1   

 k k

z f x

f , then x _k _k f~

] , [ 1₂

2

1   and

f

z _k _k ~

] , [ 1₂

2

1   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 ~( ) [ ,1₂ ]

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,zS.

Conversely, let ~f be an interval-valued fuzzy subset of S that satisfies the Conditions (1), (2) and (3). Let x,yS and ~tD(0,1] be such that xy and

f

y_t~~. 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 x_t~f~ or x f k t

~ q~

~ i.e., x f k t

~ q~ ~ .

Let x,yS and t~1,~t2D(0,1] be such that xt f

~

1

~ and

f y_t ~

2

~ . Then ~f(x)~t₁ and 2

~ ) ( ~

t y

(6)









   

  

 

  



  



 

  



  



 

], , [ } ~ , ~ min{ r if ], , [

], , [ } ~ , ~ min{ r if }, ~ , ~ min{ r

] , [ , ~ , ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2 1 2

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

y f x f xy

f

This implies xy f

k t t

~ q )

( ~

} ~ , ~ { min

r 1 2  . If x_t ~f

1

~ and z_t ~f

2

~ for some x,y,zS and ]

1 , 0 ( ~ , ~

2 1 t D

t  , then 1

~ ) ( ~

t x

f  and ~f(z)~t₂. It follows from (3) that





   

  

 

  



  



 

  



  



 

], , [ } ~ , ~ min{ r if ], , [

], , [ } ~ , ~ min{ r if }, ~ , ~ min{ r

} ] , [ , ~ , ~ min{ r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2 1 2

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

z f x f xyz

f

This implies, (xyz) _t ~_t_} f~ , ~ { min

r 1 2  or xyz t t kf

~ q )

( ~

} ~ , ~ { min

r 1 2 and

consequently, xyz f

k t t

~ q )

( _r_min_{~_,~_} ~

2

1  .

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) (xy)(~f(x)r min{~f(y),[0.5,0.5]}).

(2) ~f(xy)r min{~f(x),f~(y),[0.5,0.5]}.

(3) ~f(xyz)r min{~f(x),f~(z),[0.5,0.5]}.

Clearly every interval-valued ( , ~)

k

q

 

 -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 ~f:SD[0,1] by

      

   



. if ], 0 , 0 [

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

, if ], 0 , 0 [

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

~

D

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



0,1



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 f~(b)[0.35,0.45]~f(c) and

) ( ~ ) ( ~ ] 35 . 0 , 25 . 0

[  f d f 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 ( , ~)

k

q

 

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

3.9 Theorem

If for all xS the value of the interval-valued fuzzy subset f~ of S is less than the interval [ ,1₂ ]

2

1k k . Then

every interval-valued ( , ~)

k

q

 

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

Proof. Consider ~f be an interval-valued ( , ~)

k

q

 



-fuzzy bi-ideal of S and ~( ) [ ,1₂ ]

2 1  

 k k

x

f for all xS. Let

S y

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







since~( ) [ , ]



, )

( ~

] , [ ), ( ~ min r ) ( ~

2 1 2 1

   

   

 



k k k k

y f y f

y f x

f

(7)

If x,yS, then ~( ) [ ,1₂ ]

2 1  

 k k

x

f and ~( ) [ ,1₂ ]

2 1 

 k k

y f

this implies r min



~( ),~( )



[ , ]

2 1 2 1   

 k k

y f x

f and Theorem

3.5 (2) implies















since r min ~( ),~( ) [ , ]



, )

( ~ ), ( ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2 1 2 1

   

   

 



k k k k

y f x f

y f x f xy

f

that is ~f(xy)r min



~f(x),~f(y)



. Finally, for x,y,zS using Theorem 3.5 (3)







~( ),~( )





since~( ) [ , ]



, min

r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2 1 2 1

S x x

f z

f x f

z f x f xyz

f

k k k k

  

 

   

   

in which it follows that ~f(xyz)r min



~f(x),~f(z)



for all

S z y

x, ,  . The above discussion shows that f~ is an interval-valued fuzzy bi-ideal of S.

By taking ~k[0,0] in the Theorem 3.9, reduced to the

following corollary.

3.10 Corollary

An interval-valued (,q)-fuzzy bi-ideal ~f of S is ordinary interval-valued fuzzy bi-ideal of S, if

] 5 . 0 , 5 . 0 [ ) ( ~



x

f for all xS.

The following result shows that intersection of any finite collection of interval-valued ( , q~)

k  

 -fuzzy bi-ideal of an ordered semigroup

S

is an interval-valued

) q ,

( ~

k

 

 -fuzzy bi-ideal.

3.11 Theorem

If

 

fi _i__I

~

is a collection of interval-valued ( , q~)

k

 



-fuzzy bi-ideal of S, then _i

I i

f

~



 is an interval-valued

) q ,

( ~

k  

 -fuzzy bi-ideal of S.

Proof. Let ~f_i be an interval-valued ( , q~)

k

 

 -fuzzy bi-ideal of S for all iI and a,bS with ab. Consider









. ] , [ ), ( ~ min r

] , [ ), ( ~ min r

) ( ~ ) ( ~

2 1 2 1

   

  

 

 

 

  

 

k k i I i

k k i I

i i I i i I i

b f

b f a f a f

 

Next we take a,bS and consider









. ] , [ ), ( ~ ), ( ~ min r

] , [ ), ( ~ ), ( ~ min r

) ( ~ ) ( ~

2 1 2 1

   

  

 

 

 

  



  

 

k k i I i i I i

k k i i I

i i I i i I i

b f a f

b f a f ab f ab f

  

Finally, if a,b,cS, then









. ] , [ ), ( ~ ), ( ~ min r

] , [ ), ( ~ ), ( ~ min r

) ( ~ ) ( ~

2 1 2 1

   

  

 

 

 

  



  

 

k k i I i i I i

k k i i I

i i I i i I i

c f a f

c f a f abc f abc f

  

Consequently by Theorem 3.5 _i

I i

f

~



 is an

interval-valued ( , q~)

k

 

Based on the level subsets, the following three theorems establish links between interval-valued

) ,

( ~

k

q

 

 -fuzzy bi-ideals and ordinary interval-valued fuzzy bi-ideals.

3.12 Theorem

For an interval-valued fuzzy subset ~f of S, the following assertions are equivalent:

(1) An interval-valued fuzzy subset ~f of S is an interval-valued ( , ~)

k

q

 

(2) The level subset U(~f;~t) is a bi-ideal of S for all

   

 



2 1 , 0

~ k

D

t whenever, U(~f;~t).

Proof. Assume that U(~f;~t) for all ~_t_D



1₂k,1



and _f~

be an interval-valued ( , ~)

k

q

 

 -fuzzy bi-ideal of S. If

S y

x,  with xy such that yU(~f;~t), then by

Theorem 3.5 (1),









, ~

] , [ , ~ min r

] , [ ), ( ~ min r ) ( ~

2 1 2 1

t t

y f x

f

k k

  

 

 

in which it follows that xU(~f;~t).

If x,yU(~f;~t), then ~f(x)~t and ~f(y)~t and Theorem 3.5 (2) induces that









. ~

] , [ , ~ , ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1

t t t

y f x f xy

f

k k

  

 

 

Thus xyU(~f;~t).

(8)









. ~

] , [ , ~ , ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1

t t t

z f x f xyz

f

k k

  

 

 

In which it follows that xyzU(~f;~t) . Consequently, )

~ ; ~ (f t

U is a bi-ideal of S for all _t~_D



1₂k,1



.

Conversely, let f~ be an interval-valued fuzzy subset of

S such U(f~;~s) is a bi-ideal for all ~_t__D



1₂k,1



_{. If}

there exist a,bS with ab and such that



~( ),[ , ]



min r ) ( ~

2 1 2 1   

 _f _b k k

a

f ,

then ~f(a)~t r min





_s _f _a _f _b k k

ab

f for some



,1



~

2 1k D

s_  _{. It follows that}_a_,_b__U₍~_f_;~_s₎_but_ab__U₍~_f_;~_s₎_. This is impossible because U(~f;~s) is a bi-ideal and thus



~( ),~( ),[ , ]



min r ) ( ~

2 1 2 1   

 _f _x _f _y k k

xy

f for all x,yS.

Let there exist a,b,cS such that



~( ),~( ),[ , ]



min r ) ( ~

2 1 2 1   

 k k

c f a f abc

f . Then



~( ),~( ),[ , ]



min r ~ ) ( ~

2 1 2 1   



_r _f _a _f _c k k

abc

f for some

] 2 1 , 0 (

~ k

r  . It follows that a,cU(~f;r~) but )

~ ; ~ (f r U

abc , a contradiction and hence



~( ),~( ),[ , ]



min r ) ( ~

2 1 2 1   

 k k

z f x f xyz

f for all x,y,zS.

Using Theorem 3.1 it is concluded that f~ is an interval-valued ( , ~)

k

q

 

Theorem 3.12 induces the following corollary by taking

] 0 , 0 [ ~_

k .

3.13 Corollary

If ~f is an interval-valued fuzzy subset of S, then the following assertions are equivalent:

(1) f~ is an interval-valued (,q)-fuzzy bi-ideal of S. (2) For all ~t(0,0.5] the non-empty level subset U(~f;~t)

is a bi-ideal of S.

3.14 Theorem

If f~ is an interval-valued ( , ~)

k

q

 

 -fuzzy bi-ideal of S,

then the non-empty level subset (~;~)

~

t f

Qk is a bi-ideal

of S for all ~



2 ,1



1k D t  .

Proof. Assume that ~f is an interval-valued ( , ~)

k

q

 



-fuzzy bi-ideal of S. Let ~



2 ,1



1k D

t  be such that





) ~ ; ~ (

~

t f

Qk . If (~;~)

~

t f Q

y k and xS be such that

y

x , then f~(x)t~~1k~ and by Theorem 3.5 (1),





, ~ ~ 1 ~

], , [ ) ( ~ if ), ( ~

], , [ ) ( ~ if ], , [

] , [ ), ( ~ min r ) ( ~

2 1 2 1

2 1 2 1 2

1 2 1

2 1 2 1

k t

y f y

f

y f y

f x

f

k k

k k k

k

k k

  

   

  



 

  



 

 

  



 

in which it follows that (~;~)

~

t f Q x k .

Let , (~;~)

~

t f Q y

x  k , then f~(x)t~1~k~ and k

t y

f( ) ~ 1~ ~ ~

 

 . It follows from Theorem 3.5 (2) that

















, ~ ~ 1 ~

], , [ ) ( ~ ), ( ~ min r if ], , [

], , [ ) ( ~ ), ( ~ min r if , ) ( ~ ), ( ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2

1 2 1

2 1 2 1 2

1 2 1

k t

y f x f

y f x f y

f x f

y f x f xy

f

k k k

k

k k k

k

  

   

  



  



  



  



  



this shows that xy_Q~k(~f;~t)_.

If , (~;~)

~

t f Q z

x  k , then ~f(x)t~~1~k and k

t z

f( ) ~ 1~ ~ ~

 

 . It follows from Theorem 3.5 (3) that

















, ~ ~ 1 ~

], , [ ) ( ~ ), ( ~ min r if ], , [

], , [ ) ( ~ ), ( ~ min r if , ) ( ~ ), ( ~ min r

] , [ ), ( ~ ), ( ~ min r ) ( ~

2 1 2 1 2

1 2 1

2 1 2 1 2

1 2 1

k t

z f x f

z f x f z

f x f

z f x f xyz

f

k k k

k

k k k

k

  

   

  



  



  



  



  



hence (~;~)

~

t f Q

xyz k . From the above discussion it is

concluded that the level subset (~;~)

~

t f

Qk is a bi-ideal of S.

The following corollary is derived from Theorem 3.14 by taking k~[0,0].

3.15 Corollary

If f~ is an interval-valued (,q)-fuzzy bi-ideal of S and Q(~f;~t) for all t~



0.5,1



, then Q(~f;~t) is a

bi-ideal of S.

3.16 Theorem

An interval-valued fuzzy subset ~f of S is an interval-valued fuzzy ( , q~)

k  

 -fuzzy bi-ideal if and only if k t f

~ ~

] ~ [

(9)

Proof. Assume that f~ is an interval-valued ( , q~)

k  



-fuzzy bi-ideal of S and f k_t  ~ ~

] ~

[ for all ~tD(0,1]. Let

k t f y ~ ~ ] ~ [

 and xS be such that xy. Then yU(~f;t~)

or (~;~)

~

t f Q

y k i.e., f~(x)~t or f~(y)~t ~1k~ and by Theorem 3.5 (1), we have ~( ) r min



~( ),[ , ]



2 1 2 1   

 _f _y k k

x

f .

The following two cases are considered here: (i). ~( ) [ ,1₂ ]

2 1 

 k k

y

f ,

(ii). ~( ) [ ,1₂ ]

2 1 

 k k

y

f .

If ~f(y)t~, then the first case implies that t

y f x

f( ) ~( ) ~ ~



 and therefore k

t f t f U x ~ ~ ] ~ [ ) ~ ; ~ ( 

 and if

k t y

f( ) ~ 1~ ~ ~

 

 , then ~f(x)~t ~f(y)~t ~1k~, it follows

that k

t k f t f Q x ~ ~ ~ ] ~ [ ) ~ ; ~ ( 

 . Combining the second case with Theorem 3.5 (1) induces ~( ) [ ,1₂ ]

2 1 

 k k

x

f . If

] , [ ~ 2 1 2 1  

 k k

t , then ~f(x)t~ and hence k t f t f U x ~ ~ ] ~ [ ) ~ ; ~ ( 

 . Now, if ~ [ ,1₂ ]

2 1 

 k k

t , then

k t

x

f₍ ₎ ~ _[ k _, k _] _[ k _, k _] ₁~ ~

~ 2 1 2 1 2 1 2

1   



 _ _  _ _

, it follows that

k t k f t f Q x ~ ~ ~ ] ~ [ ) ~ ; ~ (   .

If k

t f y x ~ ~ ] ~ [

,  , then f~(x)t~ or f~(x)t~1~k~, f~(y)~t or ~f(y)t~~1k~. The following four cases are considered here:

(i) If ~f(x)t~ and ~f(y)~t . (ii) If f~(x)~t and f~(y)~t 1~k~. (iii) If f~(x)~t 1~k~ and f~(y)~t . (iv) If f~(x)t~1~k~ and ~f(y)t~~1k~. Using Case (i) in Theorem 3.5 (2) leads to









                             ], , [ ~ 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 t y f x f xy f

it follows that k

t f t f U xy ~ ~ ] ~ [ ) ~ ; ~ ( 

 or k

t

k _f _t _f

Q xy ~ ~ ~ ] ~ [ ) ~ ; ~ (  

and hence k

t f xy ~ ~ ] ~ [  .

For the second if ~ [ ,1₂ ]

2 1  

 k k

t , then

] , [ ~ ~

1 1₂

2 1  

 

_t _k k k _{and Theorem 3.5 (2) it implies that}

















                                   ), ( ~ ] , [ ), ( ~ min r if , ~ ) ( ~ ), ( ~ ] , [ ), ( ~ min r if , ~ ~ 1 ~ ] , [ ), ( ~ min r ] , [ ), ( ~ ), ( ~ min r ) ( ~ 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 x f y f t x f x f y f k t y f y f x f xy f k k k k k k k k

in which it follows that k

t k f t f Q t f U xy ~ ~ ~ ] ~ [ ) ~ ; ~ ( ) ~ ; ~ ( 

  . On

the other hand if

~

[

,

2

]

1 2 1   



k k

t

, then

















                                   ), ( ~ ] , [ ), ( ~ min r if , ~ ~ 1 ~ ) ( ~ ), ( ~ ] , [ ), ( ~ min r if , ~ ] , [ ), ( ~ min r ] , [ ), ( ~ ), ( ~ min r ) ( ~ 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 y f x f k t y f y f x f t x f y f x f xy f k k k k k k k k

it follows that k

t k f t f Q t f U xy ~ ~ ~ ] ~ [ ) ~ ; ~ ( ) ~ ; ~ ( 

  . The similar

result can be obtained for Case (iii).

For the final case, if ~ [ ,1₂ ]

2 1 

 k k

t , then

] , [ ~ ~

1 1₂

2 1 

 

_t _k k k _. _Hence



f x f y



t k

xy

f( ) r min ~( ),~( ),[ k , k ] 1 ~ ~ ~

2 1 2

1 _ _ _

 _  _ 

, thus

k t k f t f Q xy ~ ~ ~ ] ~ [ ) ~ ; ~ ( 

 . If ~ [ ,1₂ ]

2 1 

 k k

t , then

















                                   ], , [ ) ( ~ ), ( ~ min r if , ~ ~ 1 ~ ) ( ~ ), ( ~ min r ], , [ ) ( ~ ), ( ~ min r if , ~ ] , [ ] , [ ), ( ~ ), ( ~ min r ) ( ~ 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 k k k k k k k k y f x f k t y f x f y f x f t y f x f xy f

this implies k

t k f t f Q t f U xy ~ ~ ~ ] ~ [ ) ~ ; ~ ( ) ~ ; ~ (    .

Let k

t f z x ~ ~ ] ~ [

,  , then x,zU(f~;~t) or , (~;~)

~

t f Q z

x  k that is

t x f( ) ~ ~

 or f~(x)~t 1~k~ and ~f(z)~t or

k t z

f( ) ~ 1~ ~ ~

 

 . In this regard the following four cases are considered:

(i) If ~f(x)t~ and ~f(z)~t . (ii) If ~f(x)~t and ~f(z)~t 1~k~. (iii) If f~(x)t~1~k~ and ~f(z)~t . (iv) If ~f(x)t~~1k~ and ~f(z)t~~1k~. For the case (i), Theorem 3.5(3) implies that









                             ], , [ ~ 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 t z f x f xyz f

in which it follows that k

t k f t f Q t f U xyz ~ ~ ~ ] ~ [ ) ~ ; ~ ( ) ~ ; ~ (    .

For the second case assume that ~ [ ,1₂ ]

2 1  

 k k

t , then

] , [ ~ ~

1 1₂

2 1 

 

_t _k k k _{. From Theorem 3.5 (3) it can be}