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Ordered semigroups characterized by their (2;2 _q)-fuzzy bi-ideals

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Using the idea of a quasi-coincedence of a fuzzy point with a fuzzy set, the concept of an (�;�)-fuzzy bi-ideal in ordered semigroups is introduced, which is a generalization of the concept of a fuzzy bi-ideal in ordered semigroups and some
  Ordered semigroups characterized by their( 2 ; 2 _ q)-fuzzy bi-ideals Young Bae Jun and Asghar Khan  Department of Mathematics EducationGyeongsang National UniversityChinju 660-701, Koreae-mail: skywine@gmail.com, akmath2005@yahoo.comMuhammad ShabirDepartment of MathematicsQuaid-i-Azam UniversityIslamabad, Pakistane-mail: mshabirbhatti@yahoo.co.ukDecember 7, 2008 Abstract Using the idea of a quasi-coincedence of a fuzzy point with a fuzzy set,the concept of an ( ;  )-fuzzy bi-ideal in ordered semigroups is introduced,which is a generalization of the concept of a fuzzy bi-ideal in orderedsemigroups and some interesting characterizations theorems are obtained.A special attention is given to ( 2 ; 2 _ q)-fuzzy bi-ideals. AMS Mathematics Subject Classi…cation: 06F05, 20M12, 08A72Keywords: Fuzzy algebra; Belong to; Quasi-coincident with;  ( ;  ) -fuzzybi-ideal 1 Introduction The fundamental concept of a fuzzy set, introduced by L. A. Zadeh, provides anatural frame-work for generalizing several basic notions of algebra. The studyof fuzzy sets in semigroups was introduced by Kuroki  [19  21] . A systematic  I would like to thank the Higher Education Commission of Pakistan for …nancial supportunder grant no. I-8/HEC/HRD/2007/182. 1  exposition of fuzzy semigroups was given by Mordeson et al., where one can …ndtheoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy …nitestate machines and fuzzy languages. The monograph by Mordeson and Malikdeals with the application of fuzzy approach to the concepts of automata andformal languages. Murali  [25]  proposed the de…nition of a fuzzy point belongingto a fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, played a vital role to generate somedi¤erent types of fuzzy subgroups. Bhakat and Das  [2 ;  3]  gave the conceptsof   ( ;  ) -fuzzy subgroups by using the "belongs to" relation  ( 2 )  and "quasi-coincident with" relation  ( q  )  between a fuzzy point and a fuzzy subgroup, andintroduced the concept of an  ( 2 ; 2 _ q  ) -fuzzy subgroup. In  [4] ;  ( 2 ; 2 _ q  ) -fuzzysubrings and ideals are de…ned. In  [5]  Davvaz de…ne  ( 2 ; 2 _ q  ) -fuzzy subnearringand ideals of a near ring. In  [1]  Bhakat de…ne  ( 2 _ q  ) -level subset of a fuzzyset. In  [6]  Jun and Song initiated the study of   ( ;  ) -fuzzy interior ideals of a semigroup. In  [7]  Kazanci and Yamak study  ( 2 ; 2 _ q  ) -fuzzy bi-ideals of asemigroup. The concept of a fuzzy generalized …lters was introduced by Ma etal. in [22] and characterized  R 0 -algebras in terms of this notion.Algebraic structures play a prominent role in mathematics with wide rangingapplications in many disciplines such as theoritical physics, computer sciences,control engineering, information sciences, coding theory, topological spaces andthe like. This provides su¢cient motivation to researchers to review variousconcepts and results from the realm of abstract algebra in broader frameworkof fuzzy setting.Our aim in this paper is to introduce and study the new sort of fuzzy bi-idealscalled ( ;  )-fuzzy bi-ideals and to study some interesting characterizations of ordered semigroups in terms of   ( ;  ) -fuzzy bi-ideals. Special concentrationis given to  ( 2 ; 2 _ q ) -fuzzy bi-ideals and some characterizations of regular andintra-regular ordered semigroups are obtained by using  ( 2 ; 2 _ q ) -fuzzy bi-ideals. 2 Preliminaries An ordered semigroup is an ordered set  S   at the same time a semigroup suchthat  a;b  2  S;a    b  = )  xa    xb  and  ax    bx  for all  x  2  S  .Let  ( S;:;  )  be an ordered semigroup. For  A    S  , we denote ( A ] :=  f t  2  S  j t    h  for some  h  2  A g : For  A;B    S;  we denote,  AB  :=  f ab j a  2  A;b  2  B g :  Let  A;B    S  . Then A    ( A ] ,  ( A ]( B ]    ( AB ] , and  (( A ]] = ( A ] .Let  S   be an ordered semigroup and  ; 6 =  A    S  . Then  A  is called a  subsemi-group  of   S   if   A 2   A . A subsemigroup  A  of an ordered semigroup  S   is called a bi-ideal   of   S   if (1)  ASA    A  and (2)  ( 8 x  2  S  )( 8 y  2  A ) ( x    y  = )  x  2  A )[9] . De…nition 1  (cf. [13] ) An element   z  of an ordered semigroup  S   is called left (resp. right) zero if for all   x  2  S  ,  zx  =  x  ( resp.  xz  =  z ) . An ordered semigroup S   is called left(resp. right) zero if every element of   S   is left(resp. right) zero. 2  An ordered semigroup  S   is called  regular   if for every  a  2  S  , there exists x  2  S   such that  a    axa .Equivalent de…nitions: [15 ; 16](1) ( 8 a  2  S  )( a  2  ( aSa ]) . (2) ( 8 A    S  )( A    ( ASA ]) .An ordered semigroup  S   is called intra-regular if for every  a  2  S   there exist x;y  2  S   such that  a    xa 2 y .Equivalent de…nitions: [16](1) ( 8 a  2  S  )( a  2  ( Sa 2 S  ]) : (2) ( 8 A    S  )( A    ( SA 2 S  ]) .Let  S   be an ordered semigroup by a  fuzzy subset   A  of   S  , we mean a function A  :  S   !  [0 ; 1] .Let  A  be a fuzzy subset of   S  , then  A  is called a  fuzzy subsemigroup  [9]  of   S  if  ( 8 x;y  2  S  )( A ( xy )    min f A ( x ) ;A ( y ) g ) : De…nition 2  (cf.  [9] ) Let   S   be an ordered semigroup and   A  a fuzzy subset of  S  . Then   A  is called a fuzzy bi-ideal of   S   if: ( B 1 )  ( 8 x;y  2  S  ) ( x    y  = )  A ( x )    A ( y )) . ( B 2 )  ( 8 x;y  2  S  ) ( A ( xy )   min f A ( x ) ;A ( y ) g ) : ( B 3 )  ( 8 x;y;z  2  S  )( A ( xyz )   f A ( x ) ;A ( z ) g ) .Let  S   be an ordered semigroup and  A  a fuzzy subset of   S  , then for all t  2  (0 ; 1] , the set U  ( A ; t ) :=  f x  2  S  j A ( x )    t g is called a  level subset   of   S  . Theorem 3  Let   ( S;:;  )  be an ordered semigroup and   A  a fuzzy subset of   S  .Then   A  is a fuzzy bi-ideal of   S   if and only if the level subset   U  ( A ; t )( 6 =  ; )  is a bi-ideal of   S   for all   t  2  (0 ; 1] . Let  S   be an ordered semigroup and  ; 6 =  B    S  . Then the  characteristic  function    B  of   B  is de…ned as follows:  B  :  S   !  [0 ; 1] j x  !   B ( x ) :=   1  if   x  2  B , 0  if   x = 2  B: Lemma 4  (cf.  [9 , Theorem   1] ). Let   S   be an ordered semigroup and   B  a non-empty subset of   S  . Then   B  is a bi-ideal of   S   if and only if    B  is a fuzzy bi-ideal of   S  . 3  For  a  2  S;  de…ne A a  :=  f ( y;z )  2  S    S  j a    yz g [10] : For fuzzy subsets  A 1  and  A 2  of   S  , de…ne A 1   A 2  :  S   !  [0 ; 1] j a  ! 8<:_ ( y;z ) 2 A a min f A 1 ( y ) ;A 2 ( z ) g  if   A a  6 =  ; 0  if   A a  =  ; We denote by  F  ( S  )  the set of all fuzzy subsets of   S  . One can easily seethat the multiplication “  ” on  F  ( S  )  is well de…ned and associative. The orderrelation “  ” on  F  ( S  )  is de…ned as follows: A 1    A 2  if and only if   A 1 ( x )    A 2 ( x )  for all  x  2  S: Clearly  ( F  ( S  ) ;  ;  )  is an ordered semigroup.For a nonempty family of fuzzy subsets  f A i g i 2 I , of an ordered semigroup  S  ,the fuzzy subsets [ i 2 I  A i  and \ i 2 I  A i  of   S   are de…ned as follows: [ i 2 I  A i  :  S   !  [0 ; 1] j a  ! [ i 2 I  A i ! ( a ) :=  sup i 2 I  f A i ( a ) g  and \ i 2 I  A i  :  S   !  [0 ; 1] j a  ! \ i 2 I  A i ! ( a ) :=  inf  i 2 I  f A i ( a ) g : If   I   is a …nite set, say  I   =  f 1 ; 2 ;:::;n g , then clearly [ i 2 I  A i ( a ) =  max f A 1 ( a ) ;A 2 ( a ) ;:::;A n ( a ) g  and \ i 2 I  A i ( a ) =  min f A 1 ( a ) ;A 2 ( a ) ;:::;A n ( a ) g : Proposition 5  ( cf.  [9 , Proposition   5]) . Let   ( S;:;  )  be an ordered semigroupand   A;B    S  . Then  (i)  A    B  if and only if    A     B : (ii)   A  \  B  =   A \ B : (iii)   A    B  =   ( AB ] : 3 ( ;  )-fuzzy bi-ideals In what follows let  S   denote an ordered semigroup and  ;   denote any one of  2 ; 2 _ q ; 2 ^ q unless otherwise speci…ed.Every fuzzy bi-ideal of   S   is an ( ;  )-fuzzy bi-ideal of   S   as shown in thefollowing Theorem4  Theorem 6  For any fuzzy subset   A  of   S  , the conditions   ( B 1 ) ,  ( B 2 )  and   ( B 3 ) are equivalent to the conditions   ( B 4 ) ;  ( B 5 )  and   ( B 6 ) , where   ( B 4 ) ,  ( B 5 )  and   ( B 6 ) are given as follows: ( B 4 ) ( 8 x;y  2  S  ) ( 8 t  2  (0 ; 1])( x    y; y t  2  A  = )  x t  2  A ) . ( B 5 ) ( 8 x;y  2  S  )( t;r  2  (0 ; 1])( x t ;y r  2  A  = )  ( xy ) min f t;r g  2  A ) .( B 6 ) ( 8 x;y;z  2  S  )( t;r  2  (0 ; 1])( x t ;y  2  S;z r  2  A  = )  ( xyz ) min f t;r g  2  A ). Proof.  ( B 1 ) = )  ( B 4 ) . Let  x;y  2  S  , and  t  2  (0 ; 1]  be such that  x    y; y t  2  A .Then  A ( y )    t . Since  x    y , we have  A ( x )    t  by ( B 1 ). Hence  x t  2  A . ( B 4 ) = )  ( B 1 ) . Assume  ( B 1 )  is not valid. Then there exist  x;y  2  S   suchthat  x    y  and  A ( x )  < A ( y ) . Hence  A ( x )  < t    A ( y )  for some  t  2  (0 ; 1]  and so y t  2  A  but  x t  = 2  A , a contradiction. Hence  ( B 1 )  is valid. ( B 2 ) = )  ( B 5 ) . Let  x;y  2  S   and  t;r  2  (0 ; 1]  be such that  x t ;y r  2  A . Then A ( x )    t  and  A ( y )    r . By  ( B 2 ) , we have  A ( xy )   min f A ( x ) ;A ( y ) g  min f t;r g ; it follows that  ( xy ) min f t;r g  2  A: ( B 5 ) = )  ( B 2 ) . Let  x;y  2  S:  Since  A A ( x )  2  A  and  A A ( y )  2  A . By  ( B 5 )  wehave  ( xy ) min f A ( x ) ;A ( y ) g  2  A;  it follows that  A ( xy )   min f A ( x ) ;A ( y ) g . ( B 3 ) = )  ( B 6 ) . Let  x;y;z  2  S   and  t;r  2  (0 ; 1]  be such that  x t ,  z r  2  A . Then A ( x )    t and A ( z )    r . By  ( B 3 ) , we have A ( xyz )   min f A ( x ) ;A ( z ) g  min f t;r g ; it follows that  ( xyz ) min f t;r g  2  A: ( B 6 ) = )  ( B 3 ) . Let  x;y;z  2  S:  Since  A A ( x )  2  A  and  A A ( z )  2  A . By  ( B 5 )  wehave  ( xyz ) min f A ( x ) ;A ( z ) g  2  A;  it follows that  A ( xyz )   min f A ( x ) ;A ( z ) g .Let  A  be a fuzzy subset of   S   such that  A ( x )    0 : 5  for all  x  2  S  . Let  x  2  S  and  t  2  (0 ; 1]  be such that  x t  2 ^ q A:  Then  A ( x )    t  and  A ( x )+ t >  1 :  It followsthat 1  < A ( x ) + t    A ( x ) + A ( x ) = 2 A ( x ) ; so that  A ( x )  >  0 : 5 . This means that  f x t j x t  2 ^ qA g  =  ; : De…nition 7  A fuzzy subset   A  of   S   is called an   ( ;  ) -fuzzy bi-ideal of   S  , where    6 = 2 ^ q;  if it satis…es: ( B 7 ) ( 8 x;y  2  S  )( 8 t  2  (0 ; 1])( x    y ,  y t A  = )  x t A ) . ( B 8 ) ( 8 x;y  2  S  )( 8 t;r  2  (0 ; 1])( x t ;y r A  = )  ( xy ) min f t;r g A ) . ( B 9 ) ( 8 x;y;z  2  S  )( 8 t;r  2  (0 ; 1])( x t ;z r A  = )  ( xyz ) min f t;r g A ) . Example 8  Consider a set   S   =  f a;b;c;d;e g  with the following multiplication "  : " and order relation "   "  : : a b c d ea a d a d db a b a d dc a d c d ed a d a d de a d c d e 5
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