More General Forms of Interval Valued Fuzzy Filters

110

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups Bijan Davvaz, Asghar Khan, Nor Haniza Sarmin, and Hidayatullah Khan Abstract1

fuzzy set, introduced by Zadeh [30], provides a natural frame-work for generalizing several basic notions of In mathematics, an ordered semigroup is a algebra. The study of fuzzy sets in semigroups was semigroup together with a partial order that is introduced by Kuroki [18, 19, 20]. Also see [1, 4, 8, 10, compatible with the semigroup operation. Ordered 15, 24, 26]. A systematic exposition of fuzzy semigroups semigroups have many applications in the theory of was given by Mordeson et al. [21], where one can find sequential machines, formal languages, computer theoretical results on fuzzy semigroups and their use in arithmetics, and error-correcting codes. In this article, fuzzy coding, fuzzy finite state machines and fuzzy we try to obtain a more general form than interval languages. The monograph by Mordeson and Malik [22] valued (,  q ) -fuzzy left (right) filters in ordered deals with the application of fuzzy approach to the semigroups. The notion of an interval valued concepts of automata and formal languages. Murali [23] ( ,  q k )-fuzzy left (right) filter is introduced, and proposed the definition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on fuzzy subset. several properties are investigated. Characterizations The idea of quasi-coincidence of a fuzzy point with a of an interval valued ( ,  q k )-fuzzy left (right) fuzzy set, played a vital role to generate some different filter are established. A condition for an interval types of fuzzy subgroups. Bhakat and Das [2, 3] gave the valued ( ,  q k )-fuzzy left (right) filter to be an concepts of  ,   -fuzzy subgroups by using the interval valued fuzzy left (right) filter is provided. belongs to relation (  ) and quasi-coincident with Using implication operators and the notion of relation (q) between a fuzzy point and a fuzzy subgroup, implication-based interval valued fuzzy left (resp. and introduced the concept of an , q  -fuzzy right) filters, characterizations of an interval valued subgroup. Many researchers used the idea of generalized fuzzy left (resp. right) filter and an interval valued fuzzy sets and gave several results in different branches ( ,  q k )-fuzzy left (resp. right) filter are considered. of algebras. In [12], Jun and Song initiated the study of Keywords: semigroup; filter; ordered semigroup; fuzzy  ,   -fuzzy interior ideals of a semigroup. In [14], Kazanci and Yamak studied , q  -fuzzy bi-ideals of set; interval valued fuzzy set. a semigroup. In [7], Davvaz and Mozafar studied the 1. Introduction notion of (,  q) -fuzzy Lie subalgebras and ideals. Also see [6, 27, 31]. Shabir et al. [25], studied Interval-valued fuzzy sets were proposed as a natural characterization of regular semigroups by  ,   -fuzzy extension of fuzzy sets. Interval valued fuzzy sets were ideals. Jun et. al [13], discussed a generalization of an introduced independently by Zadeh [29], GrattanGuiness [9], John [11], in the same year, where the value , q  -fuzzy ideals of a BCK/BCI -algebra. In of the membership functions are intervals of numbers mathematics, an ordered semigroup is a semigroup instead of the numbers. The fundamental concept of a together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal Corresponding Author: Bijan Davvaz is with the Department of Malanguages, computer arithmetics, design of fast adders thematics, Yazd University, Yazd, Iran. and error-correcting codes. The concept of a fuzzy filter E-mail: [email protected] Asghar Khan is with the Department of Mathematics, Abdul Wali in ordered semigroups was first introduced by Khan University Mardan, Khyber Pakhtoon Khwa, Pakistan. Kehayopulu and Tsingelis in [13], where some basic E-mail: [email protected] properties of fuzzy filters and prime fuzzy ideals were Nor Haniza Sarmin and Hidayatullah Khan are with the Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi discussed. A theory of interval valued fuzzy generalized sets on ordered semigroups can be developed. Using the Malaysia 81310 UTM Johor Bahru, Johor, Malaysia. Email: [email protected], [email protected] idea of a quasi-coincidence of an interval valued fuzzy Manuscript received 3 Feb. 2013; revised 13 June 2013; accepted 21 point with an interval valued fuzzy set, Khan et al. [16] June. 2013.

© 2013 TFSA

111

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

introduced the concept of an interval valued  ,   -fuzzy bi-ideals in an ordered semigroup. They introduced a new sort of interval valued fuzzy bi-ideals, called interval valued  ,   -fuzzy bi-ideals, and studied interval valued , q  -fuzzy bi-ideals. They provided different characterizations of bi-ideals of ordered semigroups in terms of , q  -fuzzy filters, and extended their study in , q  -fuzzy left (right) and , q  -fuzzy bi-filters and investigated different characterizations of left (right) and bi-filters of ordered semigroups in terms of , q  -fuzzy left (right) filters and , q  -fuzzy bi-filters. They also introduced the concepts of fuzzy left (right and bi)-filters with thresholds. In this paper, we try to have more general form of an interval valued , q  -fuzzy left (right) filter of an ordered semigroup. We introduced the notion of an interval valued ,  qk  -fuzzy left (right) filter of an ordered semigroup, and gave examples which are interval valued ,  qk  -fuzzy left (right) filter but not

interval valued , q  -fuzzy left (right) filter. We discuss characterizations of interval valued ,  qk  -fuzzy left (right) filters in ordered

semigroups. We provided a condition for an interval valued ,  qk  -fuzzy left (right) filter to be an interval valued fuzzy left (right) filter. We finally considered characterizations of an interval valued fuzzy left (resp. right) filter and an interval valued ,  qk  -fuzzy left (resp. right) filter by using implication operators and the notion of implication-based interval valued fuzzy left (resp. right) filters. The important achievement of the study with an interval valued ,  qk  -fuzzy left (resp. right) filter is

that the notion of an interval valued , q  -fuzzy left (resp. right) filter is a special case of an interval valued ,  qk  -fuzzy left (resp. right) filter, and thus several results in the paper [17] are corollaries of our results obtained in this paper.

2. Preliminaries By an ordered semigroup (or po-semigroup) we mean a structure S ,,   in which the following are satisfied: (OS1) ( S ,) is a semigroup, (OS 2) ( S , ) is a poset, (OS 3) x, a, b  S  a  b  a  x  b  x, x  a  x  b . Remark 1 [5]: We know in the concepts of S -language and S -automation over a finite non-empty set X , we

need an ordered semigroup. Indeed, let X be a non-empty finite set and X * be the free semigroup generated by X with identity  . Let ( S ,, ) be an ordered semigroup. A function f from X * into S is called an S -language over X . An S -automation over X is a 4 -tuple A = ( R, p, h, g ) , where R is a finite non-empty set, p is a function from R  X  R into S , and h and g are functions from R into S . Therefore, the results obtained in this paper are fundamental for the advanced study on fuzzy ordered semigroups and fuzzy automata. In what follows, 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 F of an ordered semigroup S is called a left (resp. right) filter of S if it satisfies b1 a  S  b  F  a  b  b  F  , b2 a, b  S  a, b  F  ab  F  , b3 a, b  S  ab  F  a  F resp.b  F  If F is both a left filter and a right filter of S , we say that F is a filter of S . A fuzzy subset  of an ordered semigroup S is called a fuzzy left (resp. right) filter [4] of S if it satisfies:  b4   x, y  S  x  y   x     y  ,

 b5  x, y  S   xy   min{ x ,  ( y )} ,  b6   x, y  S  (   x   resp.  y      xy )

By an interval number a we mean an interval [a  , a  ] where 0  a   a   1. The set of all interval nubers is denoted by D[0,1]. The interval [a, a ] can be simply identified by the number a  [0,1]. For the interval numbers a = [a  , a  ] , b = [b  , b  ]  D [0,1], i

i

i

i  I , we define for every

i

iI



i

i





( rmax ai , bi 



[max(ai , bi ), max(ai , bi )] ), ( rmin ai , bi  [min(ai , bi ),  i

 i

min( a , b )] ),

rinf a i =  a i , a i  ,    i I  i I  rsup a i =  a i , a i     i I  i I  and put • a1  a2  a1  a2 and a1  a2 , • a1 = a2  a1 = a2 and a1 = a2 , • a1 < a2  a1  a2 and a1  a 2 , • kai = [kai , kai ], whenever 0  k  1.

112

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

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 part 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 F = {x  X | ( x,[ F  ( x), F  ( x)])  D [0,1]}, 



where F and F are two fuzzy subset such that F  ( x)  F  ( x) for all x  X . Let F be an interval valued fuzzy subset of X . Then, for every [0, 0] < t  [1,1], the crisp set U ( F ; t ) = {x  X | F  x   t} is called the level set of F Note that since every a  [0,1] is in correspondence with the interval [a, a]  D [0,1] , hence a fuzzy set is a particular case of the interval valued fuzzy sets. For any F = [ F  , F  ] and t = [t  , t  ], we define F ( x)  t = [ F  ( x)  t  , F  ( x)  t  ] , for all x  X . In particular, if F  ( x)  t  > 1 and F  ( x)  t  > 1, we write F ( x)  t > [1,1] . An interval valued fuzzy subset F of a set S of the form t  D(0,1] if y = x, F ( y ) :=  if y  x, [0, 0] is called an interval valued fuzzy point with support x and value t and is denoted by xt . For an interval valued fuzzy subset F of a set S , we say that an interval valued fuzzy point xt is b7 contained in F denoted by x  F , if F ( x)  t.

b8

t

quasi-coincident with F denoted by x q F if  t

F ( x)  t > [1,1]. For an interval valued fuzzy point xt and an interval valued fuzzy subset F of a set S , we say that b9 x   q F if x  F or x q F

b10

t

t

t

xt  F if xt F does not hold for   , q,  q .

3. Generalizations of interval valued , q  -fuzzy filters In what follows, let S be 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 and an interval valued fuzzy subset F of S ,  t

we say that c1 xt q 



k

F

F  x   t  k > [1,1],

if 





where



F  t  k > 1 and F  t  k > 1. c2 xt   q k F if xt  F or xt q k F if xt F does not c3 xt  F

hold

for

  { q k ,  q k }. 3.1 Theorem: Let F be an interval valued fuzzy subset of S . Then, the following are equivalent: 









 1  k    U F ; t    U F ; t 

,1   . (1)  t  D   2    is a left (right) filter of S    (2) F satisfies the following assertions:  x  y  F  x     (2.1)  x, y  S   1  k  1  k     ,   ,    r max  F  y  ,  2     2    r min  F  x  , F  y          F xy ,   (2,2)  x, y  S   ,     r max  1  k  1  k     ,    2     2    F  x  ,      2.3 x, y  S  ( F  xy    r max  1  k  1  k     ,    2     2     1  k  1  k     resp. r max F y , ,      . 2     2   Proof: Assume that U  F ; t  is a filter of S for all

 1  k  with U F ; t   . If there exist t  D  ,1  2  a, b  S such that the condition (2.1) is not valid, that is, there exist a, b  S with a  b such that





 ~ 1  k  1  k    F  a  > r max  F  b  ,  , . 2     2 ~ 1 k Then, F a   D( ,1] and a  U ( F ; F  a ) . But 2 implies that a F  y  < F  a  b U F ; F  a  ,





113

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

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



~

~



~

 1  k  with t  D  ,1  2  U F ; t  . Similarly, we can obtain the desired result

is

a

filter



s := r min F  a  F  c 

 ~ 1  k  1  k    > r max  F  ac  ,  ,  2     2 for some a,c. Then, s  D  1  k ,1 and a, c    2 U ( F ; s). But ac U ( F ; s ) since F  ac  < s. That is a

of

for

S

all



for the right case. If we take k = [0, 0] in Theorem 3.1, then we have the following corollary. 3.2 Corollary [14, Theorem 3.3]: Let F be an interval valued fuzzy subset of S . Then, the following are equivalent:





 U F ; t  

 .   U F ; t is a left (right) filter of S   

contradiction, and so (2.2) holds. Assume that there exist a, b  S such that

(1)  t  D (0.5,1] 

1  k  1  k     F  ab  > r max  F  a  ,  ,  2     2 Then, F  ab   D  1  k ,1 and ab  U F ; F  ab  . But   2 a U F ; F  ab  , which is impossible. Therefore,

(2) F satisfies the following assertions:   (2.1)  x, y  S  x  y  F ( x)  r max F ( y ),[0.5, 0.5]









 1  k  1  k    F  ab   r max  F  a  ,  ,   for all a, b  S. 2    2  Conversely, assume that F satisfies the three conditions (2.1), (2.2) and (2.3). Suppose that 1 k  U F ; t   for all t  D  ,1 . Let x, y  S be  2    such that x  y and x U F ; t . Then, F  x   t









and so ~ 1  k  1  k    max  F  y ,  ,  2   2  1  k 1  k  ,  F x   t >  . 2   2 is y U ( F ; t ). F  y   t, that ~

Hence,







~

If

x, y U F ; t , it follows from (2.2) that  ~ 1  k  1  k    r max  F  xy  ,  ,   2    2 





~ ~ ~ 1  k  1  k   r min F  x  , F  y   t >  , 2   2 so that F  xy   t i.e., xy U ( F ; t). Let x, y  S be

such that xy  U  F ; t  . Then, F  xy   t and thus ~ 1  k  1  k    , r max  F  x ,   2   2 

~ ~ 1  k  1  k   ,  F xy   t >  . 2   2 Hence F  x   t i.e., x U F ; t . Therefore, U ( F ; t)













(2.2)  x, y  S 

 r min F ( x), F ( y)  r max F ( xy),[0.5, 0.5] , 2.3  x, y  S  F  xy    r max F  x  ,  0.5,0.5  resp. r max F  y  , 0.5, 0.5 .

3.3 Definition: An interval valued fuzzy subset F of S is called an (,  q k ) -fuzzy left (resp. right) filter of S if it satisfies the following conditions: c4 x  y, xt  F  yt  q k F ,



c5 c6







 xt , F , yt  F  2  1 ,   ( xy ) r min (t ,t }   q k F    1 2   ( xy )t  F  xt   q k F , (resp. yt   q k F ,





for all x, y  S and t, t1 , t2  D(0,1]. An (,  q k ) -fuzzy left (resp. right) filter of S with k = [0, 0] is called an (,  q) -fuzzy left (resp. right) filter of S . 3.4 Example: Consider the ordered semigroup S = {a, b, c, d , e, f } with the multiplication given in Table 1and order relation “  ”: Table 1. Multiplication table for S.

. a b c  d  e  f  a a b b  d  e  f  b b b b  b  b  b  c b b b  b  b  b  d d b b  d  e  f  e e f f  e  e  f  f f f f  f  f  f   :=  a , a ,  ,  b , b  ,  c , c  ,  d , d  ,  e , e  ,  f , f  ,  a , d  ,  a , e  ,  d , e  ,  b , f  ,  c , f  ,  c , e  ,  f , e  . Let F be an interval valued fuzzy subset defined by

114

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

0.9, 0.99] 0.2, 0.3]  F  x  =  0.8, 0.9] 0.3, 0.4] Then, F is an interval valued

if x = a if x  {b, c, f } if x = d if x = e.

, q

[0.4,0.5]

1  k  1  k   r min F ( x), F  y  <  , . We claim that 2   2 F ( xy)  r min F ( x), F  y  . If not, then F ( xy ) < t 





 -fuzzy

filter of S. 3.5 Theorem: An interval valued fuzzy subset F of S is an interval valued ,  q k  -fuzzy left (resp. right) filter of S if and only if:   F ( x),  1  x, y  S   F ( y)  r min 1  k  1  k    with x   2 , 2      

2  x, y  S  3 x, y  S 

k

1 k 1 k   , 2   2 

1 k 1 k   , 2   2 

1  k  1  k   , F  y    2   2

or

1  k 1  k   F  y    ,  > [1,1]  k . 2   2 

1  k  1  k   1  k  1  k    1  k  1  k   F  y    , < ,  , 2   2 2   2 2   2 = [1,1]  k, a contradiction.

 F ( x),  Consequently, F ( y)  r min  1  k  1  k    for all  2 , 2    

with

x  y.

Let

k





for

x, y  S

all

x, y  S

with





1 k  1 k     , 2   2 

,

then

x

1 k  1 k     , 2   2 

be such that



 F

and

so

 F . Using (c5), we have

 xy 1k  ,1k   = xy  2

2  

 1 k  1 k   1 k  1 k    ,  r min   , , 2  2 2   2    

  q k F ,

and so 1  k  1  k   F ( xy )   , 2   2

or 1  k  1  k   F ( xy )   , > [1,1]  k. 2   2     If F ( xy ) < 1  k , 1  k  , then 2   2 1  k  1  k   1  k  1  k   1  k  1  k   F ( xy )   , , ,  [1,1]  k, i.e., y  q  F . t

k

Hence, yt   q k F . Let x, y  S and t1 , t2  D(0,1] be such that xt  F and yt  F . Then, F ( x)  t1 and 2

1

F ( x)  t2 . It follows from (2) that  F ( x), F  y  ,   1  k  1  k      F ( xy )  r min  1  k  1  k     r min t1 , t2 ,  ,  2   ,  2     2 2  

so that ( xy )    F or r mint1 ,t2 

 

~ ~ ~ 1  k  1  k   ~ ~  F  xy   r min t 1 , t 2   ,  r min t 1 , t 2   2     2 ~  1  k  1  k   1  k  1  k    > , , = [1,1]  k ,   2   2 2   2  q F . Therefore, that is ( xy ) r mint1 ,t2  k ( xy ) r min t ,t   q k F . Let x, y  S and t  D(0,1] be  1 2 such that ( xy )t  F .Then, F ( xy)  t which implies from (3) that

1  k  1  k      1  k  1  k    F  x   r min  F ( xy ),  , ,   r min t,  .  2   2      2  2

Thus, xt  F or 1  k  1  k    , F  x   t     t > [1,1]  k , 2 2   i.e., xt q k F . Hence, xt   q k F . Therefore, F is a

left

, q  -fuzzy k

filter of S. Similarly, we can

obtain the desired result for the right case. If we take k = [0, 0] in Theorem 3.5, then we have the following corollary. 3.6 Corollary [14, Theorem 4.3]: An interval valued fuzzy subset F of S is an interval valued , q  -fuzzy left (resp. right) filter of S if and only

if: 1 x, y  S 

 x  y  F ( y)  r min F ( x),[0.5, 0.5] . 2  x, y  S   F ( xy )  r min F ( x), F  y  ,[0.5, 0.5] . 



(3) x, y  S  F ( x )  resp. F ( y )   r min  F ( xy ),[0.5, 0.5] .

Obviously, every fuzzy left (resp. right) filter of S is an ,  q k  -fuzzy left (resp. right) filter of S for some k  D (0,1] . The following example shows that there exists k  D(0,1] such that (i) F is an ,  q k  -fuzzy left (resp. right) filter of S. (ii) F is not a fuzzy left (resp. right) filter of S . 3.7 Example: The interval valued , q[0.4,0.5]  -fuzzy

left (right) filter F of S in Example 3.4 is not an interval valued fuzzy left (right) filter of S since a  e and F  a  = [0.9, 0.99] > [0.3, 0.4] = F (e) . We give a condition for an interval valued , q k  -fuzzy left (resp. right) filter to be an interval valued fuzzy left (resp. right) filter. 3.8 Theorem: Let F be an interval valued , q k  -fuzzy left (resp. right) filter of S. If ~ 1  k  1  k    , F x  <   for all x  S , then F is an 2   2 interval valued fuzzy left (resp. right) filter of S. Proof: It is straightforward by Theorem 3.5. 3.9 Corollary [14, Theorem 4.9]: Let F be an interval valued ,  q k  -fuzzy left (resp. right) filter of S . If

F  x  <  0.5, 0.5 for all x  S . Then, F is an interval

valued fuzzy left (resp. right) filter of S. Proof: It follows from Theorem 3.8 by taking k = [0, 0]. 3.10 Theorem: Let S be an ordered semigroup. If [0, 0]  k < r < [1,1] . Then, every interval valued

, q  -fuzzy k

left (resp. right) filter is an interval

valued ,  q r  -fuzzy left (resp. right) filter. Proof. It is straightforward. Let S = {a, b, c, d , e, f } be an ordered semigroup which is considered in Example 3.4. Let F be an interval valued fuzzy subset of S defined by 0.7,0.8] if x = f , ~  F x  := 0.4,0.5] if x  {a, d , e}, 0.2,0.3] if x  {b, c}.   Then, F is an interval valued ,  q  -fuzzy left k

116

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

filter of S with k  [0.2, 0.2] . Note that f  e and 1  k F  e  = [0 .4 , 0 .5 ] < [0 .4 3 , 0 .5 3 ] =   2  1  k  1  k     = r min  F  f  ,  ,  2    2 



1 k  2  

,

where k = [0.14, 0.6]. Thus, F does not satisfy the first condition of Theorem 3.5, and so F is not an interval valued ,  q k  -fuzzy left filter of S for k = [0.14, 0.6]. This shows that the converse of Theorem 3.10 is not true. 3.11 Theorem: For an interval valued fuzzy subset F of S , the following are equivalent: 1 F is an interval valued ,  qk  -fuzzy left (resp.

right) filter of S .

2 





 U F ; t         1 k     ; t . ,1 t D U F           2    is a left (right) filter of S   

Assume



F

that



interval valued , qk  -fuzzy left filter of S and let t  D  0, 1  k  2   be such that U  F ; t   . Using Theorem 3.5 (1), we Proof:

is

an

have  1  k  1  k    F  y   r min  F  x  ,  ,  2     2 for any x, y  S with x  y and x  U F ; t . It





follows that    1  k 1  k    F  y   r min t ,  , =t 2     2 so that y  U F ; t . Let x, y  U F ; t . Then, F  x   t and F  y   t. Theorem 3.5(2) induces that









~ ~  ~ 1  k  1  k    F  xy   r min  F  x  , F  y  ,  ,  2    2 

Thus,

 ~ 1  k  1  k    ~ ,  r min t ,   = t . 2     2 xy  U F ; t . Now, let xy  U F ; t . Then,









F  xy   t. Theorem 3.5 (3) ~







and therefore



x  U F ; t



for

 1  k  is a filter of S. t  D  0, , 2   Conversely, let F be an interval valued fuzzy subset of S such x  U F ; t is non-empty and is a left filter





of S for all t  D  0, 1  k  . If there exist a, b  S 2   with a  b and ~  ~ 1  k  1  k    , F  b  < r min  F  a  ,  , 2    2    ~ ~ ~   then F b  < t b  r min  F  a  , 1  k , 1  k   2    2  for some tb  D  0, 1  k  and so b  U  F ; tb  . This is 2   a contradiction. Therefore, F  y   r min

  1  k  1  k    for all x, y  S with x  y . F x ,     2 , 2     

Assume that ~ ~  ~ 1  k  1  k    F  ab  < r min  F  a  , F  b  ,  , . 2    2  ~ ~ ~  ~ 1  k  1  k    F  ab  < t  r min  F  a  , F  b  ,  ,  2    2 

Then,

for some t  D  0, 1  k  . It follows that a  U  F ; t  2   and b  U  F ; t  , but ab  U  F ; t  . This is impossible, and thus 1  k  1  k     F  xy   r min  F  x  , F  y  ,  ,  2    2  for all x, y  S. Suppose that ~  ~ 1  k  1  k    F  a  < r min  F  ab  ,  ,  2    2 

for some a, b  S . Then, there exists ta  D  0, 1  k  2   such that  ~ 1  k  1  k    F  a  < ta  r min  F  ab  ,  , . 2    2  Hence,

 1  k 1  k   F  x   r min  F  xy  ,  ,  2     2    ~ 1  k 1  k   ~ ,  r min t ,   = t . 2     2 ~



x U F ; t

Thus,

 ab t  F and at F . F  a   ta < 2ta  [1,1]  k, a

Also,

a

ata   q k F , a contradiction.

Therefore,

i.e. ,

ata q k F . Thus,

117

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

1  k  1  k     ~ F  x   r min  F  xy  ,  ,  2    2  for all x, y  S. Using Theorem 3.5, we conclude that F is an interval valued ,  q  -fuzzy left filter of k

S. For the right case, it is checked by the similar way. Taking k = [0, 0] in Theorem 3.11 induces the following corollary. 3.12 Corollary [14, Theorem 4.10]: For an interval valued fuzzy subset F of S , the following are equivalent: 1 F is an interval valued , q  -fuzzy left (resp. right) filter of S .

2    t  D (0 , 0 .5 ]   U  F ; t     U  F ; t   .    is a left (righ t) filter o f S 

3.13 Theorem: For any left (resp. right) filter F of S , let F be an interval valued fuzzy subset of S defined by ~ ~ t 1 if x  F (3.1) F  x  = ~ t 2 otherwise, where t1   k, 1,1 and t2   0,0 , k . Then, F is   an interval valued , qk  -fuzzy left (resp. right)





 F if r  t2 , k   (3.2) U F ; r =     S if r  0, 0 , t ,   2   which is a left (resp. right) filter of S . It follows from Theorem 3.11 that F is an interval valued , q k  -fuzzy left (resp. right) filter of S .



3.14 Corollary [14]: For any left (resp. right) filter F of S , let F be an interval valued fuzzy subset of S defined by ~ ~ t 1 if x  F F  x  = ~ t 2 otherwise,

where t1   0.5, 0.5 , 1,1 and t2   0, 0 ,  0.5, 0.5 . Then, F is an , q  -fuzzy left (resp. right) filter of S. For any interval valued fuzzy subset F of S and t  D (0,1], we consider four subsets: Q F ; t := x  S | x qF and



 F  t

   :=  x  S | x   qF  . t

t

   :=  x  S | x   q F  .



k  F   t

It

is

clear

t



that





k

 F   := U F ; t t

 Q  F ; t   k

and



 k  F  :=  F   Q k F ; t . t t

3.15 Theorem: If F is an interval valued , q k  -fuzzy left ( resp. right) filter of S , then        1  k    Q k F ; t    Q k F ; t .  t  D ,1      2    is a left (resp. right) filter of S    

Assume

that





F

is



an



valued , q k  -fuzzy left filter of S. Let t  D  1 2 k ,1  

Proof:

interval

be such that Q k  F ; t   . Let x  Q k  F ; t  and y  S be such that x  y. Then, F  x   t > [1,1]  k. 



By means of Theorem 3.5 (1), we have  1  k  1  k     , F  y   r min  F  x  ,   2    2     1  k  1  k     x   1  k , 1  k  , , if F   2 2  2   2  =    F x   x  < 1  k , 1  k  , if F     2   2 



filter of S . Proof: Note that





 Q k F ; t := x  S | xt q k F and

> [1,1]  t  k,  and so y  Q k  F ; t  . F  x   t > [1,1]  k

Let

and then





 x, y  Q k F ; t .

Then,

F  y   t > [1,1]  k. It

follows from Theorem 3.5 (2) that 1  k  1  k   F  xy   r m in  F  x  , F  y  ,  , 2   2   r min F  x  , F  y   =    1  k , 1  k    2   2





and so xy  Q Q

k

 F ; t  .

k



    

1  k  1  k   if r min F  x  , F  y  <  , , 2   2 1  k  1  k   if r min F  x  , F  y    , , 2   2









> [1,1]  t  k, F ; t . Let x, y  S such that xy 



Then, F  xy   t > [1,1]  k and so

 1  k  1  k  F  x   r m in  F  xy  ,  , 2  2   1  k  1  k   ,  2   2 =   F x    

    

1  k  1  k   if F  x y    , , 2   2 1  k  1  k   if F  x y  <  , , 2   2

118

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

> [1,1]  t  k ,

Hence, x  Q  F ; t  . Therefore, Q  F ; t  is a left k

k

filter of S . Similarly, we can obtain the desired result for the right case. 3.16 Corollary: If F is an interval valued , q  -fuzzy left ( resp. right) filter of S , then





 Q k F ; t    .  t  (0.5,1]     Q k F ; t is a left (resp.right) filter of S   







 k       1 r    Q F;t   .  t  D  2 ,1        Q k F ; t is a left (resp.right) filter of S    



F  y   t

and

1  k  1  k   , t   2   2

If







k y  U F ; t   F   . t

hence

and If

1  k  1  k   t >  , , then 2   2  1  k  1  k   1  k  1  k    , , F  y   t >    = [1,1]  k , 2   2 2   2

3.17 Corollary: Let F be an interval valued , q k  -fuzzy left ( resp. right) filter of S , if [0, 0]  k < r < [1,1], then



1  k  1  k   , . F  y    2   2

which implies that







k  y  Q k F ; t   F   . Therefore, t





k k  F  satisfies the condition (b1). Let x, y   F  . t t Then, x  U F ; t or xt q k F and y  U  F ; t  or





yt q k F that is, F  x   t or F  x   t > [1,1]  k and F  y   t or F  y   t > [1,1]  k. We consider the



Proof. It is straightforward by Theorem 3.10 and 3.15. 3.18 Theorem: For any interval valued fuzzy subset F following four cases. i  If F  x   t and F  y   t. of S , the following are equivalent: 1 F is an interval valued ,  q k  -fuzzy left ( resp. ii  If F  x   t and F  y   t > [1,1]  k. right) filter of S. iii  If F  x   t > [1,1]  k and F  y   t. k   F     iv  If F  x   t > [1,1]  k and F  y   t > [1,1]  k.    t    k For the case i  , Theorem 3.5 (2) implies that .  2   t  D(0,1]    F  t       x y   r m i n  F  x  , F  y  ,  1  k , 1  k   F is a left (resp.right)     2 2      filter of S      1  k  1  k    ,  r m i n  t ,   Proof: Assume that F is an interval valued 2 2     , qk  -fuzzy left filter of S and let t  D(0,1] such         

  =  t  



k k that  F   . Let x   F  and y  S be such that t t  or x  Q k F ; t , i.e., x  y . Then, x  U F ; t









F  x   t or F  x   t > [1,1]  k. Using Theorem 3.5

(1), we get  1  k  1  k    F  y   r min  F  x  ,  , . 2    2 

F

x  >

1  k  2 



,

1 k 2



 . 

The first case implies

from (3.4) that F  y   F  x  . Thus, if F  x   t , then F  y   t





so that xy  U  F ; t  or

so



t

  second case assume that t >  1  k , 1  k  . Then,   2 2



t

Combining

the

second

case

and

(3.4)

induces



t  k [ 1 , 1 ]  t

[ 1 ,1 ]  
 , , 2   2 2   2 = [1 ,1 ],

If y  U F ; t   F   . t F  x   t > [1,1]  k, then F  y   t  F  x   t > [1,1]  k and

1 k if t >   2 1  k  if t    2

F  x y   t  k

(3.3)

  We consider two cases: F  x    1  k , 1  k  and 2   2

1 k 1 k ,  2 2 

and so

 1  k  2 



,

1 

k 2



  

119

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

which implies that

 1  k  1  k     F  xy   r min  F  x  , F  y  ,  ,  2    2     1  k  1  k     ,  > [1,1]  t  k  r min  F  y  ,  2    2    =   F x  t     

 1  k  1  k     if r min  F  y  ,  ,  2    2   F  x  ,  1  k  1  k     if r min  F  y  ,  ,  2    2  > F  x  .

Thus, xy  U  F ; t  Q k  F ; t  =  F  . Suppose that  k



t

1  k 1  k  , t . Then, 2   2  1  k  1  k    , F  xy   r min  F  x  , F  y  ,   2    2  

~



  1  k  1  k      min , r F x      2 , 2    t       =  F y > [1,1]  t  k     

 1  k  1  k    if r min F  x  ,  ,  2     2  F  y  ,

k

1  k  1  k   , then t >  , 2   2 1  k  1  k   [1,1]  t  k <  ,  . Hence, 2 2  

1  k  1  k   [1 ,1 ] t k    2 , 2 ,    







t

Then,

~ 1  k  1  k   , F  xy     2   2     and F  xy  > 1  k , 1  k  . 2   2 The first case implies from (3.5) that F  x   F  xy  . Thus, if F  xy   t, then F  x   t and so x 



then that

k xt q k F , i.e., x  Q k  F ; t    F   . Combining the t

    second case and (3.5) induces F  xy   1  k , 1  k  . 2   2     If t  1  k , 1  k  , then F  x   t and hence 2   2

1  k  1  k   then t >  , , 2   2 1  k  1  k   1  k  1  k    F  x   t >  ,    2 , 2  = [1,1]  k , 2 2    



  1  k  1  k    r m in  F  x  , F  y  ,  ,  2     2 1  k  1  k    =  ,  > [ 1 , 1 ]  t  k 2  2 

 k    F   . t  



k xy   F   .

that

or F  xy   t > [1,1]  k. It follows from Theorem 3.5 (3) that 1  k  1  k     (3.4) , F  x   r min  F  xy  ,   . 2    2 





k x U F ; t   F   . t

 xy  

   F ; t  

such

.



similar result for the case iii . For the final case, if

k

be

k t

F  xy   t



t

xy  Q

x, y  S

Let

k

  1  k  1  k      k If F  xy   t > [1,1]  k, if r min F  x  ,  ,   U F ; t   F  t . 2 2     F  x   t  F  xy   t > [1,1]  k which implies  > F  y.



whenever r min  F  x  , F  y 

 Q  F ; t  =  F 

We consider two cases:

and thus xy  U  F ; t  Q k  F ; t  =  F  . We have 

F



xy  U F ; t



 

If



k  which implies that x  Q k  F ; t    F  . Therefore,  t

k



k  F  satisfies the condition (b3). Therefore,  F  is a t t left filter of S. Conversely, suppose that (2) is valid. If there exist such that and a, b  S ab

1  k  1  k     F  b  < r min  F  a  ,  , , 2    2 

then

1  k  1  k     F  b  < tb  r min  F  a  ,  ,  2    2 

for some







 1  k  . It follows that a  U F ; t   F  k tb  D  0 , b   tb 2  

but





 b  Q k F ; t .

F  b   tb < 2tb  [1,1]  k ,

Also, and

we so

btb q k F ,

have i.e.,

120

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013







k  b  Q k F ; t . Therefore, b   F   , a contradiction. t      Hence, F  y   r min  F  x  , 1  k , 1  k   for all 2    2  x, y  S with x  y . Suppose that there exist a, b  S such that

  1  k  1  k    F  a b  < t  r m in  F  a  , F  b  ,  , . 2    2 

for

 1 k  t  D  0, 2  

some







.

It





follows

that



k k a  U F ; t   F   and b  U F ; t   F   , so from t t

(b2)



k ab   F   .

that

t

F  ab   t > [1,1]  k,

a

F  ab   t

Thus,

contradiction.

or

Therefore,

~  ~ 1  k  1  k    F  xy   r min  F  x  , F  y  ,  ,   for all 2 2     x, y  S. Assume that there exist a, b  S such that

1  k  1  k     F  a  < r min  F  ab  ,  ,  . Then, 2     2 F

for

a 
[1,1]  k , a

Therefore,

F  x   r min

 ~ 1  k  1  k    ,  F  xy  ,    for all x, y  S. Using 2    2  Theorem 3.5, we conclude that F is an interval valued , q k  -fuzzy left filter of S. Similarly, we obtain

the right case. 3.19 Corollary: For any interval valued fuzzy subset F of S , the following are equivalent: 1 F is an interval valued , q  -fuzzy left ( resp. right) filter of S ,

2 

  F  k       t   t  D(0,1]  .     F  k is a left (resp.right) filter of S     t 

An interval valued fuzzy subset F of S is said to be proper if Im( F ) has at least two elements. Two interval valued fuzzy subsets are said to be equivalent if they have same family of level subsets. Otherwise, they

are said to be non-equivalent. 3.20 Theorem: Let F be an interval valued (,  q k ) -fuzzy left (resp. right) filter of S such that  1  k  1  k    #  F  x  | F  x  <  ,   2. 2    2  Then, there exist two proper non-equivalent interval valued (,  q k ) -fuzzy left (resp. right) filters of S such that F can be expressed as the union of them. Proof:      Let  F  x  | F  x  < 1  k , 1  k   = {t1 , t2 ,..., tn }, 2    2  where t1 > t2 > ... > tn and n  2. Then, the chain of (,  q ) -level left (resp. right) filters of F is k

 [ F ]k

1 k  1 k     , 2   2 

    [ F ]tk  [ F ]tk2  ...[ F ]tkn = S .

Let  and  be interval valued fuzzy subsets of S defined by t1 if x  [ ]tk , 1  t2 if x  [ ]tk \ [ ]tk , 2 1   x  =  ...    k  k tn if x  []tn \ []tn1 , and    x  if x  [  ] k  , 1 k 1 k      , 2   2       k if x  [  ] tk2 \ [  ] k  , 1 k 1 k       , 2   2  x =      k k   if x  [  ] t2 \ [  ] t1 ,  t3  ...     tn if x  [  ] tkn \ [  ] tkn 1 ,  respectively, where t < k < t . Then,  and  are 3

2

interval valued (,  q k ) -fuzzy left (resp. right) filters of S , and  ,   F . The chains of (  q k ) -level left (resp. right) filters of  and  are, respectively, given by    [  ] tk1  [  ] tk2  . . .  [  ] tkn and [  ] 1k  k  [  ] tk1  . . .  [  ] tkn . 



2



Therefore,  and  are non-equivalent and clearly F =   . This completes the proof.

121

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

4. Implication-based interval valued fuzzy left (resp. right) filters

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables or terms of the linguistic variable truth. Some operators, for example ,, ,  in fuzzy logic are also defined by using truth tables and the extension principle can be applied to derive definitions of the operators. In fuzzy logic, the truth value of fuzzy proposition  is denoted by [ ]. For a universe U of discourse, we display the fuzzy logical and corresponding set-theoretical notations used in this paper (4.1) x  [ F ] = F  x  , [   ] = min{[],[ ]}, [   ] = min{1,1  []  [ ]}, [  x ] = inf[  x ], xU

(4.2) (4.3) (4.4)

 if and only if [] = 1 for all valuation. (4.5) The truth valuation rules given in (4.3) are those in the Ł ukasiewicz system of continuous-valued logic. Of course, various implication operators have been defined. We show only a selection of them in the following. (a) Gaines-Rescher implication operator I GR  : 1 if a  b, I GR a, b  =  0 otherwise. (b) Gödel implication operator I G  : 1 if a  b, I G a, b  =  b otherwise. (c) The contraposition of Gödel implication operator I cG  :

an interval valued fuzzifying left (resp. right) filter is an ordinary interval valued fuzzy left (resp. right) filter. In [17], the concept of t -tautology is introduced, i.e., for all valuations ~

~

~

|  if and only if [ ]  t ,

(4.6)  4.2 Definition: An interval valued fuzzy subset F of S and t  D(0,1] is called a t -implication-based interval valued fuzzy left (resp. right) filter of S if it satisfies: ~ ~ (d4) x, y  S  ( x  y   |  x  F    y  F  ,     t     (d5) x, y  S  ~ ~ ~      | rmin x  F ,  y  F     xy  F  ,         t (d6) x, y  S  ~ ~ ~     |  xy  F    x  F  resp. y  F   .         t  Let I be an implication operator. Clearly, F is a t -implication-based interval valued fuzzy left (resp. right) filter of S if and only if it satisfies: ~ ~ ~ (d7) x, y  S   x  y  I  F  x , F  y   t ,    





~ ~ ~ ~  (d8) x, y  S   I  r min F  x  , F  y  , F  xy    t  ,    

(d9) x, y  S   I  F  xy , F  x   t   resp. F  xy , F  y   t . ~

 

~

~

~







~

~





4.3 Theorem: For any interval valued fuzzy subset F of S , we have, if a  b, 1 If I = IGR , then F is a 0.5 -implication-based 1 I cG a, b  =  interval fuzzy left (resp. right) filter of S if and only if 1  a otherwise. Ying [28] introduced the concept of fuzzifying topology. F is an interval valued fuzzy left (resp. right) filter of We can expand his/her idea to ordered semigroups, and S . we define an interval valued fuzzifying left (resp. right) 1 k (2) If I = I G , then F is a -implication-based filters as follows. 2 4.1 Definition: An interval valued fuzzy subset F of interval valued fuzzy left (resp. right) filter of S if and S is called an interval valued fuzzifying left (resp. right ) only if F is an interval valued ,  q -fuzzy left  k  filter of S if it satisfies the following conditions: (resp. right) filter of S . ~ ~   (d1) x, y  S   x  y    x  F    y  F   , 1 k       (3) If I = I cG , then F is a -implication-based    2 ~ ~ ~ interval valued fuzzy left (resp. right) filter of S if and (d2) x, y  S   r min   x  F  ,  y  F     xy  F   ,                only if F satisfies the following conditions: ~ ~ ~    ~ 1  k  1  k    (d3) x, y  S    xy  F    x  F  resp. y  F   .   3.1 x  y  r max F y , ,                2        2 Obviously, the conditions (d1), (d2) and (d3) are ~  r min  F ( x), 1,1, equivalent to (b4), (b5) and (b6), respectively. Therefore,  

122

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

conclude that F is an interval valued ,  q k  -fuzzy

 ~ 1  k  1  k    r max  F  xy  ,  ,   2    2 

3.2

~ ~  r min  F ( x), F  y , 1,1,     ~ ~ 3.3 r max  F  x  , 1  k , 1  k    r min F ( xy), 1,1 2     2  1  k  1  k      ~ ,  resp.max  F  y  ,    2      2  ,   ~  r min F ( xy ), 1,1    for all x, y  S. Proof: 1 Straightforward. 2  Assume that F is a 1  k - implication2 based interval valued fuzzy left filter of S . Then, (i) x, y  S 







left filter of S. Conversely, suppose that F is an interval valued , q k  -fuzzy left filter of S. Let x, y  S such that x  y. Using Theorem 3.5 (1), we have ~ ~  I G  F x , F  y   

 1  k  1  k   , [1,1]   2   2     ~  ~ if r min  F  x  , 1  k , 1  k   = F  x  ,  2     2  ~ 1  k  1  k    = , F  y   2   2   1  k  1  k     ~  if r min  F  x  ,  ,   2    2   1  k  1  k    =  2 , 2 .    



~ ~  1  k  1  k     x  y  I G  F  x , F  y    ,  ,  2      2  (ii) x, y  S 



From Theorem 3.5 (2), if ~ ~ ~ 1  k  1  k     ~ r min  F  x  , F  x  ,  ,  = r min{F  x  , F  y },  2    2 



~ ~ ~    1  k  1  k    ,  I G  r min F  x  , F  y  , F  xy      , 2    2     ~ ~    (iii) x, y  S   I G  F xy , F x   1  k , 1  k  .   2     2  Let x, y  S be such that x  y. Using (i) we have

then

And so

~

~

(ii),

~

~  ~ 1  k  1  k    F ( y )  r min  F ( x),  , . 2    2 

Hence, we





~

~

~

From



F  xy   r min F  x  , F  y 

get



or

~ ~ ~ 1  k  1  k   Thus, r min F  x  , F  y  > F  xy    , . 2   2 ~ ~  ~ 1  k  1  k    F  xy   r min  F  x  , F  y  ,  ,  . Case (iii) 2    2 

implies

that

~

~

F  x   F  xy 

or

~

~

F  xy  > F  x  

~ 1  k  1  k   and so F  x   r min  2 , 2 ,   1  k  1  k     ~ ,  F  xy  ,    . Using Theorem 3.5, we 2    2 

~

~







~ ~ ~   I G  r min F  x  , F  y  , F  xy   = [1,1]     1  k 1  k  , .  2   2

1  k  1  k   F  y   F x  or F  x  > F  y    , . 2   2 ~



~

F  xy   r min F  x  , F  y  .

If ~ ~ 1  k  1  k    r min  F  x , F  y ,  ,  2   2  1  k  1  k   = , , 2   2

  ~   then F  xy   1  k , 1  k  and thus 2   2 ~ ~ ~   1  k  1  k   I G  r min F  x  , F  y  , F  xy     , . 2     2 From Theorem 3.5 (3), if  ~ 1  k  1  k    ~ r min  F  xy  ,  ,   = F  xy  , 2    2  then





123

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

~ 1  k  1  k   ~  , I G  F  xy , F x  = [1,1]   . 2     2 Otherwise,   ~ ~  1  k 1  k  , I G  F  xy , F  x    . 2     2

Therefore,

1 k -implication-based fuzzy 2 left filter of S . Similarly, we obtain the right case.

If r min F  x  , F  y  ,[1,1] = r min F  x  , F  y  , then

~

~

Suppose that F satisfies (3.1), (3.2) and (3.3). Let be such that x  y. In (3.1), if x, y  S ~

r min{F  x  ,[1,1]} = [1,1]

,

then

 ~ 1  k  1  k    r max  F  y  ,  ,   = [1,1] 2    2  ~

and

hence



~

~

1  k  1  k    ~  ~ (4.7) r max  F  y  ,  ,   F  x. 2     2   ~ ~ ~   If F  y  > 1  k , 1  k  in (4.7), then F  y   F  x  2   2 and thus ~ 1  k  1  k   ~  I cG  F  x , F  y  = [1,1]   , . 2     2

in

(4.7),

then



In (3.2), if r min F  x  , F  y  ,[1,1] = [1,1] , then  ~ 1  k  1  k    r max  F  xy  ,  ,   = [1,1] 2    2 



~

then





~ ~ 1  k  1  k   r min F  x  , F  y    , . 2   2 Therefore, ~ ~ ~   I cG  r min F  x  , F  y  , F  xy   = [1,1]     1  k 1  k   , 2   2





~

~

whenever F  xy   r min{F  x  , F ~  y }, and





~ ~ ~   I cG  r min F  x  , F  y  , F  xy      ~ ~ 1  k 1  k   = [1,1]  r min F  x  , F  y    , 2   2



~

 1  k  1  k   , if   y     x  1,1]    2 2    =   ~ 1,1]  F  x   1  k , 1  k  otherwise.    2   2

~







~

~

whenever F  xy  < r min{F  x  , F  y }. Now, if

~ ~  I cG  F x , F  y   



~

~



and so F  xy  = [1,1]  r min F  x  , F  y  .



1  k  1  k    1  k  1  k   in (4.8),  ~ r max  F  xy  ,  , , = 2    2 2   2 

and

If F x  < [1,1], then

~

~

~ 1  k  1  k   F  xy    ,  2   2

~ 1  k  1  k   ~  , I cG  F  x , F  y  = [1,1]   . 2     2

~



~ ~  ~ 1  k  1  k    r max  F  xy  ,  ,   r min F  x  , F  y  .  2    2  Thus, if

~

1  k  1  k   If , F y    2   2 ~ 1  k  1  k   F x    ,  . Hence, 2   2



~

F  y  = [1,1]  F  x . Therefore,

~



1  k  1  k    , . 2   2

Consequently, F is a

3



~ ~ ~   I cG  r min F  x  , F  y  , F  xy   = [1,1]  

 ~ 1  k  1  k    ~ r max  F  xy  ,  ,   = F  xy  2    2  ~



~

~



in (4.8), then F  xy   r min F  x  , F  y 





and so

~ ~ ~   I cG  r min F  x  , F  y  , F  xy   = [1,1]  

1  k  1  k    , . 2   2 ~

In (3.3), if F  xy  = [1,1] , then 1  k  1  k     ~ r max  F  x  ,  ,   = [1,1] and hence 2    2 

124

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

Therefore,

~ 1  k  1  k   ~  I cG  F  xy , F  x  = [1,1]   , . 2     2

~ 1  k  1  k    ~  ~  r max  F  y  ,  , F x = r min{ F    x  ,[1,1]}.  2     2

~

If F xy  < [1,1], then  ~ 1  k  1  k    ~ r max  F  x  ,  ,    F  xy  . 2    2 

(4.9)

If

~



~ 1  k  1  k   ~  , I cG  F  xy , F  x  = [1,1]   . 2     2

 ~ 1  k  1  k    1  k  1  k   r max  F  x  ,  , ,  =   2    2 2   2 

in

  ~   (4.9), then F  xy   1  k , 1  k  which implies that 2   2 ~ ~  I cG  F  xy , F  x    ~ ~  1  k  1  k   , if F ( x)  F xy , 1,1]    2    2 . =   ~  1,1]  F  xy   1  k , 1  k  otherwise.    2   2 1 k Consequently, F is a -implication-based 2 interval valued fuzzy left filter of S . is a Conversely assume that F 1 k -implication-based interval valued fuzzy left filter 2 of S . Then, (i) x, y  S  ~ ~  1  k  1  k     x  y  I cG  F  x , F  y    ,  ,  2      2  (ii) x, y  S  ~ ~ ~  1  k  1  k    I ( r min{ F x , F y }, F xy ) ,         cG   , 2   2    ~ ~    (iii) x, y  S   I G  F  xy , F  x   1  k , 1  k  .   2     2  Let x, y  S be such that x  y. (iv) implies that   ~ ~   ~  I cG  F  x , F  y  = [1,1] or [1,1]  F  x   1  k , 1  k  2     2

that

F x   F  y 

or

~ 1  k 1  k  F x    , . 2   2 

~



~

~

~



  ~ ~   or [ 1,1]  r min F  x  , F  y   1  k , 1  k  . 2   2 Hence, 1  k  1  k     ~ r max  F  y  ,  ,   2     2

in (4.9), then F  x   F xy . Hence,

~



i.e., r min{F  x  , F  y }  F  xy  ,

 ~ 1  k 1  k   ~ r max  F  x  ,  ,  = F  x 2    2 

so



~



If

From (v), we have ~ ~   I cG  r min F  x  , F  y  ,   xy   = [1,1],  



~ ~ ~ ~  min  F  x , F  y  = min{F  x , F  y , [1,1]}   for all x, y  S. Finally, (vi) implies that ~ ~  I cG  F  x , F  y  = [1,1] or   ~ ~ ~ 1  k  1  k   [1,1]  F  xy    ,  so that F  xy   F x  2   2

  ~   or F  xy   1  k , 1  k . Therefore, 2   2 1  k  1  k    ~  ~ r max  F  x  ,  ,   F  xy  2    2  ~

 min{F  xy , [1,1]}. The right case is similar to the left case. This completes the proof. 4.4 Corollary: If I = I G , then any interval valued fuzzy subset F of S is a 0.5 -implication-based interval

valued fuzzy left (resp. right) filter of S if and only if F is an interval valued , q  -fuzzy left (resp. right) filter of S . 4.5 Corollary: If I = I cG , then any interval valued fuzzy subset F of S is a 0.5 -implication based interval valued fuzzy left (resp. right) filter of S if and only if F satisfies the following condition: 3.1





~





~

x  y  r max F  y  ,[0.5, 0.5]  r min F ( x),[1,1] ,

3.2 





~



~

~



r max F  xy  ,[0.5, 0.5]  r min F ( x), F  y  ,[1,1] ,

3.3



~



~

r max F  x  ,[0.5, 0.5]  r min{F ( xy ),[1,1]}

Bijan Davvaz et al.: More General Forms of Interval Valued Fuzzy Filters of Ordered Semigroups

~   ~   resp.r max  F  y , [0.5,0.5]  r min{F ( xy ), [1,1]}     for all x, y  S.

5. Conclusions

[9]

[10]

In 1975, Zadeh [29] introduced the notion of interval fuzzy sets as an extension of fuzzy sets in which the value of the membership degrees are interval of numbers instead of numbers. On the other hand, semigroups are important in many areas of mathematics, for example, coding and language theory, automata theory, combinatorics and mathematical analysis. So, a theory of interval-valued fuzzy sets on ordered semigroups can be developed. Since the concept of fuzzy filters of ordered semigroups play an important role in the study of ordered semigroup structure, we studied a more general form of interval valued (,  q ) -fuzzy left (right) filters in ordered semigroups and we established characterizations of an interval valued ( ,  q )-fuzzy  k

left (right) filter. Moreover the obtained results can be applied to semirings, near-rings, hemirings, rings and etc.

[11] [12] [13] [14] [15] [16]

References [1] [2] [3]

[4] [5]

[6] [7]

[8]

M. Akram, A. B. Saeid, K. P. Shum, and B. L. Meng, “Bipolar fuzzy K-algebras,” Int. J. Fuzzy Syst., vol. 12, no. 3, pp. 252-258, 2010. S. K. Bhakat and P. Das, “On the definition of a fuzzy subgroup,” Fuzzy Sets and Systems, vol. 51, pp. 235-241, 1992. S. K. Bhakat and P. Das, “ , q  - fuzzy subgroup,” Fuzzy Sets and Systems, vol. 80, pp. 359-368, 1996. B. Davvaz and A. Khan, “Generalized fuzzy filters in ordered semigroups,” Iranian Journal of Science & Technology, vol. A1, 77-86, 2012. B. Davvaz and A. Khan, “Characterizations of regular ordered semigroups in terms of (  ,  )-fuzzy generalized bi-ideals,” Information Sciences, vol. 181, pp. 1759-1770, 2011. B. Davvaz and O. Kazanci, “A new kind of fuzzy sublattice (ideal, filter) of a lattice,” Int. J. Fuzzy Syst., vol. 13, no. 1, pp. 55-63, 2011. B. Davvaz and M. Mozafar, “ (, q) -fuzzy Lie subalgebra and ideals,” Int. J. Fuzzy Syst., vol. 11, no. 2, pp. 123-129, 2009. B. Davvaz and V. Leoreanu-Fotea, “On a product of fuzzy H v -submodules,” Int. J. Fuzzy Syst., vol.

[17] [18] [19] [20] [21] [22] [23] [24]

[25]

125

10, no. 2, pp. 119-122, 2008. I. Grattan-Guiness, “Fuzzy membership mapped onto interval and many-valued quantities,” Z. Math. Logik. Grundladen Math., vol. 22, pp. 149-160, 1975. H. Hedayati, “Intutionistic ( S , T ) - fuzzy (1,2) ideals of semigroups with interval valued membership functions,” Int. J. Fuzzy Syst., vol. 14, no. 1, pp. 154-159, 2012. K. U. John, “Intervall-wertige Mengen,” Math. Nach., vol. 68, pp. 115-132, 1975. Y. B. Jun and S. Z. Song, “Generalized fuzzy interior ideals in semigroups,” Inform. Sci., vol. 176, pp. 3079-3093, 2006. Y. B. Jun, K. J. Lee, and C. H. Park, “New types of fuzzy ideals in BCK/BCI -algebras,” Comput. Math. Appl., vol. 60, pp. 771-785, 2010. O. Kazanci and S. Yamak, “Generalized fuzzy bi-ideals of semigroup,” Soft. Computing, vol. 12, pp. 1119-1124, 2008. N. Kehayopulu and M. Tsingelis, “Fuzzy sets in ordered groupoids,” Semigroup Forum, vol. 65, pp. 128-132, 2002. A. Khan, Y. B. Jun, and M. Shabir, “Ordered semigroups characterized by interval valued , q  - fuzzy bi-ideals,” Journal of Intelligent & Fuzzy Systems, DOI: 10.3233/IFS-2012-0614, IOS Press. A. Khan, B. Davvaz, and I. A. Shah, “Generalized interval valued fuzzy filters in ordered semigroups,” Submitted. N. Kuroki, “Fuzzy bi-ideals in semigroups,” Comment. Math. Univ. St. Paul., vol. 28, no. 1, pp. 17-21, 1980. N. Kuroki, “On fuzzy ideals and fuzzy bi-ideals in semigroups,” Fuzzy Sets and Systems, vol. 5, no. 2, pp. 203-215, 1981. N. Kuroki, “On fuzzy semigroups,” Inform. Sci., vol. 53, no. 3, pp. 203-236, 1991. J. N. Mordeson, D. S. Malik, and N. Kuroki, “Fuzzy Semigroups,” Springer, Berlin, 2003. J. N. Mordeson and D. S. Malik, “Fuzzy Automata and Languages,” Chapman & Hall/CRC, Boca Raton, FL, 2002. V. Murali, “Fuzzy points of equivalent fuzzy subsets,” Inform. Sci., vol. 158, pp. 277-288, 2004. S. K. Sardar, B. Davvaz, S. K. Majumder, and S. Kayal, “ ( ,  ) -Fuzzy Subsemigroup (Bi-ideal) in  - Semigroups,” International J. Fuzzy Systems, vol. 14, no. 4, pp. 557-567, 2012. M. Shabir, Y. B. Jun, and Y. Nawaz, “Characterizations of regular semigroups by ( ,  ) -fuzzy ideals,” Comput. Math. Appl. , vol.

126

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013

59, pp. 161-175, 2010. [26] S. Yamak, O. Kazanci, and B. Davvaz, “Divisible and pure intuitionistic fuzzy subgroups and their properties,” Int. J. Fuzzy Syst., vol. 10, no. 4, pp. 298-307, 2008. [27] Y. Q. Yin and X.-K. Huang, “Fuzzy roughness in hyperrings based on a complete residuated lattice,” Int. J. Fuzzy Syst., vol. 13, no. 3, pp. 185-194, 2011. [28] M. S. Ying, “A new approach for fuzzy topology (I),” Fuzzy Sets and Systems, vol. 39, pp. 303-321, 1991. [29] L. A. Zadeh, “The concept of linguistic variable and its application to approximate reasoning I,” Inform. Sci., vol. 8, pp. 199-249, 1975. [30] L. A. Zadeh, “Fuzzy sets,” Infotmation and Control, vol. 8, pp. 338-353, 1965. [31] J. Zhan and B. Davvaz, “Study of fuzzy algebraic hypersystems from a general viewpoint,” Int. J. Fuzzy Syst., vol. 12, no. 1, pp. 73-79, 2010. Bijan Davvaz took his B.Sc degree in Applied Mathematics at Shiraz University in 1988 and his M.Sc. degree in Pure Mathematics at Tehran University in 1990. In 1998 he received his Ph.D. in Mathematics at Tarbiat Modarres University. He is a member of Editorial Boards of more than 20 Mathematical Journals. He is author of around 240 research papers, especially on algebraic hyperstructures and their applications. Moreover, he published four books in algebra. He is currently Professor of Mathematics at Yazd University in Iran. Asghar Khan has got his M. Phil and Ph. D degrees in 2003 and 2009, respectively, under the supervision of Prof. Dr. Muhammad Shabir and Prof. Dr. Young Bae Jun. He has started his professional carear in 2009, in the department of Mathematics, CIIT, Abbottabad, Pakistan. Currently he is an Assistant Professor in the Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan. He is working in the field of fuzzy algebra. His main interest areas are fuzzy sets, rough sets, soft sets and applications. Nor Haniza Sarmin is an Associate Professor in the Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM) Johor Bahru, Johor. Presently, she is the Deputy Dean (Admission & Customer Relation) at School of Graduate Studies (SPS), UTM. She started her professional life at UTM as a lecturer in May 1991.

She received her BSc (Hons), MA and PhD (Mathematics) from Binghamton University, Binghamton, New York, USA. Her specialization of research is in Group Theory and Its Applications. Beside this, she is also working on DNA Splicing System. The author is very active in publications. Along with others, she has translated a book title “Elementary Linear Algebra” by Howard Anton and she wrote a book titled “Aljabar Moden”, published by Penerbit UTM Press. She has also written more than 250 research papers in indexed/ non-indexed journals and proceedings mainly in the area of Group Theory and Splicing System. She is currently involved as a reviewer in several international journals of high repute. She has also supervised more than 70 undergraduate students for their final year project, and 30 students at the postgraduate level. Hidayat Ullah Khan is Lecturer in the Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa Pakistan. He started his professional life as a lecturer in the department of Mathematics, Univerity of Malakand, in 2005. Presently, he is a PhD student in the Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM) Johor Bahru, Johor under the supervision of Associate Professor Dr. Nor Haniza Sarmin and Assistant Professor Dr. Asghar Khan.