-intuitionistic (fuzzy ideals, fuzzy soft ideals) of subtraction algebras

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1 Songlanaarin J. Sci. Technol. 7 (4) Jul. ug. 05 hp:// Original ricle On q inuiionisic (fuzz ideals fuzz sof ideals) of subracion algebras Madad Khan Bijan Davvaz Naveed Yaqoob Muhammad Gulisan 4 * Mohammed M. Khalaf Deparmen of Mahemaics COMSTS Insiue of Informaion Technolog bboabad Paisan. Deparmen of Mahemaics Yazd Universi Yazd Iran. Deparmen of Mahemaics College of Science in lzulfi Majmaah Universi lzulfi Saudi rabia. 4 Deparmen of Mahemaics Hazara Universi Mansehra Paisan. Received: 7 Februar 05; cceped: 9 pril 05 bsrac The inen of his aricle is o sud he concep of an q fuzz sof ideal of subracion algebras o inroduce some relaed properies. 00 MS Classificaion: 06F5 0G Kewords: Subracion algebras q inuiionisic fuzz (sof) subalgebras q inuiionisic fuzz (sof) ideals. inuiionisic fuzz ideal q inuiionisic. Inroducion Schein (99) cogiaed he ssem of he form ( X ; \) where X is he se of funcions closed under he composiion " " of funcions ( hence X is a funcion semigroup) he se heoreical subracion "\ " ( hence X \ is a subracion algebra in he sense of bbo (969). He proved ha ever subracion semigroup is isomorphic o a difference semigroup of inverible funcions. He suggesed a problem concerning he srucure of muliplicaion in a subracion semigroup. I was eplained b Zelina (995) he had solved he problem for subracion algebras of a special pe nown as he aomic subracion algebras. The noion of ideals in subracion algebras was inroduced b Jun e al. (005). For deailed sud of subracion algebras see (Ceven Ozur 009 Jun e al. 007). * Corresponding auhor. address: gulisanmah@hu.edu.p The mos appropriae heor for dealing wih uncerainies is he heor of fuzz ses developed b Zadeh (965). Since hen he noion of fuzz ses is acivel applicable in differen algebraic srucures. The fuzzificaion of ideals in subracion algebras were discussed in Lee Par (007). anassov (986) inroduced he idea of inuiionisic fuzz se which is more general one as compared o a fuzz se. Bhaa Das (996) inroduced a new pe of fuzz subgroups ha is he q fuzz subgroups. Jun e al. (0) inroduced he noion of q fuzz subgroup. In fac he q fuzz subgroup is an imporan generalizaion of Rosenfeld s fuzz subgroup. Shabir e al. (00) characerized semigroups b ( q )fuzz ideals also see Shabir Mahmood (0 0). Molodsov (999) inroduced he concep of sof se as a new mahemaical ool for dealing wih uncerainies ha are free from he difficulies ha have roubled he usual heoreical approaches. Molodsov poined ou several direcions for he applicaions of sof ses. presen wor on he sof se heor is progressing rapidl. Maji e al. (00)

2 466 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) described he applicaion of sof se heor o a decisionmaing problem. Maji e al. (00) also sudied several operaions on he heor of sof ses. Maji e al. (00b 004) sudied inuiionisic fuzz sof ses. The noion of fuzz sof ses as a generalizaion of he sard sof ses was inroduced b Maji (00a) an applicaion of fuzz sof ses in a decisionmaing problem was presened. hmad e al. (009) have inroduced arbirar fuzz sof union fuzz sof inersecion. gunoglu e al. (009) inroduced he noion of fuzz sof group sudied is properies. Yaqoob e al. (0) sudied he properies of inuiionisic fuzz sof groups in erms of inuiionisic double norm. Jun e al. (00) inroduced he noion of fuzz sof BCK/BCIalgebras (closed) fuzz sof ideals hen derived heir basic properies. Williams Saeid (0) sudied fuzz sof ideals in subracion algebras. Recenl Yang (0) have sudied fuzz sof semigroups fuzz sof (lef righ) ideals have discussed fuzz sof image fuzz sof inverse image of fuzz sof semigroups (ideals) in deail. Liu Xin (0) sudied he idea of generalized fuzz sof groups fuzz normal sof groups. In his aricle we sud he concep of q inuiionisic fuzz (sof) ideals of subracion algebras. Here we consider some basic properies of q inuiionisic fuzz (sof) ideals of subracion algebras.. Preliminaries In his secion we recall some of he basic conceps of subracion algebra which will be ver helpful in furher sud of he paper. Throughou he paper X denoes he subracion algebra unless oherwise specified. Definiion. (Jun e al. 005) nonemp se X ogeher wih a binar operaion is said o be a subracion algebra if i saisfies he following: S S S z z for all z X. The las ideni permis us o omi parenheses in epression of he form z. The subracion deermines an order relaion on X : a b a b 0 where 0 a a is an elemen ha does no depends upon he choice of a X. The ordered se X ; is a semiboolean algebra in he sense of bbo (969) ha is i is a mee semi laice wih zero in which ever inerval 0a is a Boolean algebra wih respec o he induced order. Here a b a ( a b); he complemen of an elemen b 0 a is a b is denoed b / ; b c 0 a ; hen b if / / / b c b c a b a c a a b a b a c In a subracion algebra he following are rue see (Jun e al. 005) : a. a a 0 for all z X a 4 ( ) a 5 ( ) ( ) a 6 ( ( )) a 7 ( ) ( z ) z a8 if onl if w for some w X a9 implies z z z z a 0 z implies z a z z a z z z. Definiion. (Jun e al. 005) nonemp subse of a subracion algebra X is called an ideal of X denoed b X : if i saisfies: b a for all a X b for all a b whenever a b eiss in X hen a b. Proposiion. (Jun e al. 005) nonemp subse of a subracion algebra X is called an ideal of X if onl if i saisfies: b 0 b 4 for all X for all Proposiion.4 (Jun e al. 005) Le X be a subracion algebra X. If w X is an upper bound for hen he elemen w (( w ) ) is he leas upper bound for. Definiion.5 (Jun e al. 005) Le Y be a nonemp subse of X hen Y is called a subalgebra of X if Y whenever Y. Definiion.6 (Lee Par 007) Le f be a fuzz subse Then f is called a fuzz subalgebra of X if i saisfies: (FS) f ( ) min{ f ( ) f ( )} whenever X. Definiion.7 (Lee Par 007) fuzz subse f is said o be a fuzz ideal of X if saisfies: (FI) f ( ) f ( ) (FI) If here eiss hen f ( ) min{ f ( ) f ( )} for all X.

3 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) q inuiionisic fuzz ideals In his secion we will discuss some properies relaed o q inuiionisic fuzz ideals of subracion algebras. Definiion. n inuiionisic fuzz se in X is an objec of he form {( ( ) ( )) : X} where he funcion : X [0] : X [0] denoe he degree of membership degree of nonmembership of each elemen X 0 ( ) ( ) for all X. For simplici we will use he smbol ( ) for he inuiionisic fuzz se {( ( ) ( )) : X}. We define 0( ) 0 ( ) for all X. Definiion. Le X be a subracion algebra. n inuiionisic fuzz se {( ( ) ( )) : X} of he form if 0 if is said o be an inuiionisic fuzz poin wih suppor value is denoed b. fuzz poin is said o inuiionisic belongs o (resp. inuiionisic quasicoinciden) wih inuiionisic fuzz se = {( ( ) ( )) : X} wrien resp. q if ( ) ( ) resp. ( ) ( ) B he smbol q we mean ( ) ( ) where 0. We use he smbol implies ( ) [ ] implies ( ) in he whole paper. Definiion. n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz subalgebra of X if q 4 4 for all X 4 (0). OR n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz subalgebra of X if i saisf he following condiions (i) q for all X (0] (ii) X 4 4 [0). 4 q for all Eample.4 Le X {0 a b} be a subracion algebra wih he following Cale able 0 a b a a 0 a b b b 0 Le us define he inuiionisic fuzz se ( ) of X as X 0 a b ( ) ( ) is an hen ( ) subalgebra q 0.5 inuiionisic fuzz Definiion.5 n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz ideal of X if i saisf he following condiions (i) X q (ii) If here eis hen 4 q 4 for all X 4 (0). OR n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz ideal of X if i saisf he following condiions (i) X q (ii) If here eis in X hen q for all X (0] (iii) X q (iv) If here eis 4 4 q in X hen for all X [0). 4 Eample.6 Le X {0 a b} be a subracion algebra wih he Cale able define in Eample.4 le us define an inuiionisic fuzz se ( ) of X as X 0 a b ( ) ( )

4 468 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) is an hen ( ) q 0. inuiionisic fuzz ideal Theorem.7 n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz subalgebra of X if onl if ( ) min{ ( ) } ( ) ma{ ( ) }. Proof. Le ( ) be an q inuiionisic fuzz subalgebra of X assume on conrar ha here eis some (0] r [0) such ha ( ) min{ ( ) } ( ) r ma{ ( ) }. This implies ha ( ) ( ) ( ) q which is a conradicion. Hence ( ) min{ ( ) }. lso from ( ) r ma{ ( ) } we ge r r ( ) r ( ) r r q ( ) which is a conradicion. Hence ( ) ma{ ( ) }. ( ) min{ ( ) } Conversel le ( ) ma{ ( ) }. Le for all X (0] 4 for all X 4 [0). This implies ha. 4 Consider ( ) min ( ) ( ) min. hen If ( ). So ( ) ( ) which implies ha ( ). q If hen ( ). So ( ). Thus ( ). q lso consider ( ) ma{ ( ) } ma{ }. 4 hen 4 q q If ( ). So ( ) ( ) which implies ha [ ]. If hen ( ) 4. So 4 [ ]. Thus q Hence ( ) is an inuiionisic fuzz subalgebra Theorem.8 n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz ideal of X if onl if ( ) min{ } (i) ( ) ma{ } (ii) ( ) min{ ( ) } (iii) (iv) ( ) ma{ ( ) }. Proof. The proof is similar o he proof of he Theorem.7. Proposiion Ever q inuiionisic fuzz ideal of X is an q inuiionisic fuzz subalgebra of X bu converse is no rue. Eample.9 Le X {0 a b} be a subracion algebra wih he Cale able define in Eample.4 le X 0 a b ( ) ( ) hen b Theorem.7 ( ) q inuiionisic fuzz subalgebra Bu b Theorem.8 we observe ha ( ) is no a q 0.4 inuiionisic fuzz ideal s a b is an ( a b) (0 ((0 b) a))) (0) 0.9 min{ ( ) } min{ } 0..

5 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) Definiion.0 Le ( ) is an inuiionisic fuzz se Define he inuiionisic level se as { X ( ) ( ) where (0 ] [ )}. Theorem. n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz subalgebra of X if onl if is a subalgebra be an q Proof. Le ( ) inuiionisic fuzz subalgebra Suppose ha hen ( ) ( ) ( ) ( ). Since ( ) is an q inuiionisic fuzz subalgebra of X so ( ) min{ ( ) } min{ } min{ } ( ) ma{ ( ) } ma{ } ma{ }. Thus ( ). Hence is a subalgebra Conversel assume ha is a subalgebra ssume on conrar ha here eis some X such ha ( ) min{ ( ) } ( ) ma{ ( ) }. Choose (0 ] [ ) such ha ( ) min{ ( ) } ( ) ma{ ( ) }. This implies ha ( ) which is a conradicion o he hpohesis. Hence ( ) min{ ( ) } ( ) ma{ ( ) }. Thus ( ) q inuiionisic fuzz subalgebra is an Theorem. n inuiionisic fuzz se ( ) of X is said o be an q inuiionisic fuzz ideal of X if onl if is an ideal Proof. The proof is similar o he proof of he Theorem.. Definiion. Le X be a subracion algebra X an q inuiionisic characerisic funcion where follows: { ( ) ( ) S} are fuzz ses respecivel defined as if : X [0] ( ) : 0 if if : X [0] ( ) :. if Lemma.4 For a nonemp subse of a subracion algebra X we have (i) is a subalgebra of X if onl if he characerisic inuiionisic se of in X is an q inuiionisic fuzz subalgebra (ii) is an ideal of X if onl if he characerisic inuiionisic se of in X is an q inuiionisic fuzz ideal Proof. The proof is sraighforward. 4. q inuiionisic fuzz sof ideals Molodsov defined he noion of a sof se as follows. Definiion 4. (Molodsov 999) pair ( F ) is called a sof se over U where F is a mapping given b F : P( U ). In oher words a sof se over U is a paramerized famil of subses of U. The class of all inuiionisic fuzz ses on X will be denoed b IF( X ). Definiion 4. (Maji e al. 00b) Le U be an iniial universe E be he se of parameers. Le E. pair ( F ) is called an inuiionisic fuzz sof se over U where F is a mapping given b F : IF( U ). In general for ever F [ ] F [ ] F is [ ] an inuiionisic fuzz se in U i is called inuiionisic fuzz value se of parameer Definiion 4. Le U be an iniial universe E be a se of parameers. Suppose ha B E ( F ) ( G B) are wo inuiionisic fuzz sof ses we sa ha ( F ) is an inuiionisic fuzz sof subse of ( G B) if onl if () B () for all F [ ] is an inuiionisic fuzz subse of G [ ] ha is for all U ( ) ( ) F [ ] G [ ] ( ) ( ). F [ ] G [ ] This relaionship is denoed b ( F ) ( G B). Definiion 4.4 Le ( F ) ( G B) be wo inuiionisic fuzz sof ses over a common universe U. Then

6 470 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) ( F ) ( G B) is defined b ( F ) ( G B ) ( B ) where [ ] F [ ] G [ ] for all ( ) B ha is [ ] ( ) ( ) ( ) ( ) for all ( ) B U. F [ ] G [ ] F [ ] G [ ] Definiion 4.5 Le ( F ) ( G B) be wo inuiionisic fuzz sof ses over a common universe U. Then ( F ) ( G B) is defined b ( F ) ( G B ) ( B ) where [ ] F [ ] G [ ] for all ( ) B ha is [ ] ( ) ( ) ( ) ( ) for all ( ) B U. F [ ] G [ ] F [ ] G [ ] Definiion 4.6 Le ( F ) ( G B) be wo inuiionisic fuzz sof ses over a common universe U. Then he inersecion ( C) where C B for all C U ( ) if F B [ ] ( ) ( ) if B [ ] G [ ] ( ) ( ) if B F [ ] G [ ] ( ) if F B [ ] ( ) ( ) if B [ ] G [ ] ( ) ( ) if B. F [ ] G [ ] We denoe i b ( F ) ( G B) ( C). Definiion 4.7 Le ( F ) ( G B) be wo inuiionisic fuzz sof ses over a common universe U. Then he union ( C) where C B for all C U ( ) if F B [ ] ( ) ( ) if B [ ] G [ ] ( ) ( ) if B F [ ] G [ ] ( ) if F B [ ] ( ) ( ) if B [ ] G [ ] ( ) ( ) if B. F [ ] G [ ] We denoe i b ( F ) ( G B) ( C). In conras wih he above definiions of IFsof se union inersecion we ma someimes adop differen definiions of union inersecion given as follows. Definiion 4.8 Le ( F ) ( G B) be wo IFsof ses over a common universe U B. Then he biinersecion of ( F ) ( G B) is defined o be he fuzz sof se ( C) where C B [ ] F [ ] G for all C. This is denoed b ( C) ( F ) ( G B). Definiion 4.9 Le ( F ) ( G B) be wo IFsof ses over a common universe U B. Then he biunion of ( F ) ( G B) is defined o be he fuzz sof se ( C) where C B [ ] F [ ] G for all C. This is denoed b ( C) ( F ) ( G B). Definiion 4.0 Le ( F ) be an IFsof se over X. Then ( F ) is called an q subracion algebra if F F F is an q inuiionisic fuzz sof inuiionisic fuzz subalgebra of X for all. Eample 4. Le X {0 a b} be a subracion algebra wih he following Cale able 0 a b a a 0 a b b b 0 le U is he se of cellular brs of mobile companies in he mare. Le {aracive epensive cheap} is a parameer space = {aracive cheap}. Define F aracive { a0.80. b0.70. } F cheap { a b0.70. }. Le 0.4 hen F sof subalgebra is an q 0.4 inuiionisic fuzz Definiion 4. n inuiionisic fuzz sof se F of X is called an q inuiionisic fuzz sof subalgebra of X if for all F F F is an q inuiionisic fuzz subalgebra of X if ( ) min{ ( ) } (i) F F F (ii) F F F for all X. ( ) ma{ ( ) } Definiion 4. n inuiionisic fuzz sof se F of X is called an q inuiionisic fuzz sof ideal of X if for all F F F is an q inuiionisic fuzz sof ideal of X if ( ) min{ } (i) F F ( ) ma{ } (ii) F F

7 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) ( ) min{ ( ) } (iii) F F F (iv) F F F for all X. ( ) ma{ ( ) } Proposiion 4.4 n q inuiionisic fuzz sof ideal of X is an q inuiionisic fuzz sof subalgebra of X bu converse is no rue. Eample 4.5 From Eample 4. F inuiionisic fuzz sof ideal s is no an q ( ) (0 ((0 ) ))) aracive aracive aracive (0) 0.9 F a b b a F F min{ aracive ( ) aracive ( ) } F a F b 0. min{ } for 0.4. be wo Theorem 4.6 Le F G B q inuiionisic fuzz sof subalgebras (resp. ideals) Then F G B is also an q inuiionisic fuzz sof subalgebra (resp. ideal) be wo Proof. Le F G B q inuiionisic fuzz sof ideals We now ha F G B B where [ ] F [ ] G [ ] for all ( ) B ha is [ ]( ) for all X. Le X we have F [ ] G [ ] F [ ] G [ ] F G F G min{ } min{ } F G min{ } F G min{ } F G F G F G lso we have ma{ } ma{ } F G ma{ } F G ma{ }. F G F G F G min{ } min{ } F F G [ ] G [ ] min{ } F [ ] G [ ] F [ ] G [ ] min{ } F G F G F G F G Hence F G B ma{ } ma{ } F F G G ma{ } F G F G ma{ }. F G F G is also an q inuiionisic fuzz sof ideal The oher case can be seen in a similar wa. be wo Theorem 4.7 Le F G B q inuiionisic fuzz sof subalgebras (resp. ideals) Then F G B is also an q inuiionisic fuzz sof subalgebra (resp. ideal) Proof. The proof is similar o he proof of he Theorem 4.6. Theorem 4.8 Le F be an inuiionisic fuzz sof se of X. Then F is an q inuiionisic fuzz sof subalgebra (resp. ideal) of X if onl if F F { X where (0 ] X. Proof. Le F F[ ] F [ ] [ )} is a sof subalgebra (resp. ideal) of be an q F hen F F F F [ ). Since F is an q subalgebra Le inuiionisic fuzz sof (0 ] where inuiionisic fuzz sof subalgebra of X so min{ } F F F min{ } ma{ } F F F ma{ }. Thus F. Hence F is a sof subalgebra

8 47 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) Conversel assume ha F is a sof subalgebra Le here eis some r (0] s [0) such ha min{ } F r F F min{ }. F s F F This implies ha F which is conradicion. Thus min{ } F F F ma{ }. F F F Hence F is an q inuiionisic fuzz sof subalgebra Proposiion 4.9 Ever q inuiionisic fuzz sof ideal F of X saisfies he following (i) (0) min{ ( ) } F [ ] F [ ] (0) ma{ ( ) } for all X. (ii) F [ ] F [ ] Proof: B leing in he Definiion 4. we ge he required proof. Lemma 4.0 If an q inuiionisic fuzz sof se F of X saisfies he followings (i) (0) min{ ( ) } F [ ] F [ ] (ii) (0) ma{ ( ) } F [ ] F [ ] ( z) min{ ( z) } (iii) F[ ] F[ ] F[ ] ( z) ma{ ( z) } (iv) F[ ] F[ ] F[ ] Then we have a ( ) min{ ( a) } F [ ] F [ ] ( ) ma{ ( ) a } for all a z X. F[ ] F [ ] Proof. Le a X a. Consider ( ) ( 0) b ( a) F [ ] F [ ] min{ ( 0) } b (iii) F a a F min{ (0) a } b a iff a 0 F F min{ ( a) } b (i). F [ ] lso consider ( ) ( 0) b ( a) F [ ] F [ ] ma{ ( 0) } b (iv) F a a F ma{ (0) a } b a iff a 0 F F ma{ ( a) } b (ii). F [ ] This complee he proof. Theorem 4. n q inuiionisic fuzz sof se F of X saisfies he condiions (i)(iv) of he Lemma 4.0 if onl if F is an q inuiionisic fuzz sof ideal i.e Proof. Suppose ha F F saisfies F F he condiions (i)(iv) of he Lemma 4.0. Le X hen b using (a) we have. Now b he use of Lemma 4.0 ( ) min{ ( ) } F F ( ) F ma{ ( ) } for all X. F lso ( ) min{ ( ) } ( ) F F F ma{ ( ) } whenever eiss in X b using F he Lemma 4.0 we have ( ) min{ ( ) ( ) } F F F ( ) ma{ ( ) ( ) } for all X. F F F Hence F is an q inuiionisic fuzz sof ideal Conversel assume ha F is an q inuiionisic fuzz sof ideal Condiions (i) (ii) direcl follows from he Proposiion 4.9 le z X. Pu z in (a) (a4) We ge from (a) z z from (a4)we have ( (( z)) z. Which indicaes ha is an upper bound for z ( (( z)). B using he Proposiion.4 we have z ( (( z)) z ((( z) (( z) )) ( ( ( z)))) z. Thus z F z z F ( ) ( ( (( )) ) min{ ( ( ) } F z z F min{ z } b S ( a). F F Which is (iii). nd

9 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) F z F z z ( ) ( ( (( )) ) ma{ ( ( ) } F z z F ma{ z } b S ( a). F F Which is (iv). Hence i complees he proof. Theorem 4. n q inuiionisic fuzz sof se F of X is an q inuiionisic fuzz sof ideal of X if onl if i saisfies: ( (( a) b)) min{ ( a) ( b) } (i) F F F (ii) F F F for all a b X. ( (( a) b)) ma{ ( a) ( b) } Proof. Le q inuiionisic fuzz se F F of X saisfies (i) (ii). Consider F F F ( ) ( ) ((( ) ) )) b (i) min{ ( ) ( ) } F F min{ ( ) } F F ( ) ( ) ((( ) ) )) b (ii) ma c{ ( ) ( ) } F F ma{ ( ) }. F lso suppose ha eiss in X b using he Proposiion.4 we have w (( w ) ). Now using he given condiions F F ( ) w (( w ) ) F min{ ( ) ( ) } b (i) F F ( ) w (( w ) ) F ma{ ( ) ( ) } b (ii). F F Hence F F is an q inuiionisic fuzz sof ideal Conversel le F F be an q inuiionisic fuzz sof ideal B Theorem 4. ( ) min{ (( ) ) ( ) } F z z F F ( ) ma{ (( ) ) ( ) }. F z z F F Le z ( a) b b hen ( (( ) )) min{ (( ) (( ) )) ( ) } F a b a b b F F min{ (( ) (( ) )) ( ) } b ( ) F b a b S F min{ ( a) ( b) } b puing b a in ( a4) F F ( (( ) )) ma{ (( ) (( ) )) ( ) } F a b a b b F F ma{ (( ) (( ) )) ( ) } b ( ) F b a b S F ma{ ( a) ( b) } b puing b a in ( a4). F F Hence F F saisf condiion (i) (ii). Lemma 4. n inuiionisic fuzz se F F of X is an q inuiionisic fuzz sof ideal of X if onl if fuzz ses F F are q fuzz sof ideals Proof. The proof is sraighforward. Theorem 4.4 n inuiionisic fuzz se F F of X is an q inuiionisic fuzz sof ideal of X if F F onl if F F are q inuiionisic fuzz sof ideal Proof. The proof is sraighforward. inuiion Theorem 4.5 If F F isic fuzz sof ideal of X hen is an q X { X : ( ) min{ ( ) } (0)} F F F F X { X : ( ) ma{ ( ) } (0)} F F F F are sof ideals Proof. Le a X X. Then b definiion F min{ ( ) } (0). F F Since F F is an q inuiionisic fuzz sof ideal of X so we have (i) a a ( ) min{ } 0 which F F F implies ha a X. F ( a ) ma{ a } 0 which (ii) F F F implies ha a X. F Le eiss in X we ge

10 474 M. Khan e al. / Songlanaarin J. Sci. Technol. 7 (4) (iii) ( ) min{ ( ) } 0 F F F F which implies ha X F ma{ ( ) } 0 F F F X. Hence ( ) F which implies ha F X { X : ( ) min{ ( ) } (0)} F F F F X { X : ( ) ma{ ( ) } (0)} F F F F are sof ideals be wo Theorem 4.6 Le F G B q inuiionisic fuzz sof subalgebras (resp. ideals) Then so is F G B. be wo Proof. Le F G B q inuiionisic fuzz sof subalgebras We now ha F G B C where C B. Now for an C X we consider he following cases Case : For an B we have ( ) ( ) min{ ( ) } F F F [ ] min{ ( ) } [ ] ( ) ( ) ma{ ( ) } F F F [ ] ma{ ( ) }. [ ] Case : For an B we have ( ) ( ) min{ ( ) } G G G [ ] min{ ( ) } [ ] ( ) ( ) ma{ ( ) } G G G [ ] ma{ ( ) }. [ ] Case : For an B we have F [ ] G [ ]. F [ ] G [ ] nalogous o he proof of Theorem 4.6. Hence F G B is an q inuiionisic fuzz sof subalgebra be wo Theorem 4.7 Le F G B q inuiionisic fuzz sof subalgebra (resp. ideals) Then so is F G B. Proof. The proof is similar o he proof of he Theorem 4.6. cnowledgemens We highl appreciae he deailed valuable commens of he referees which greal improve he quali of his paper. References bbo J.C Ses laices Boolean algebra lln Bacon Boson. hmad B. har K Mappings on fuzz sof classes. dvances in Fuzz Ssems. 6. anassov K.T Inuiionisic fuzz ses. Fuzz ses Ssem gunoglu. gun H Inroducion o fuzz sof groups. Compuers Mahemaics wih pplicaions Bhaa S.K. Das P fuzz subgroups. Fuzz Ses Ssems Ceven Y. Ozur M Some resuls on subracion algebras. Haceepe Journal of Mahemaica Saisics Jun Y.B. Kim H.S. Roh E.H Ideal heor of subracion algebras. Scieniae Mahemaicae Japonicae Jun Y.B. Kim H.S On ideals in subracion algebras. Scieniae Mahemaicae Japonicae Jun Y.B. Kang M.S. Par C.H. 0. Fuzz subgroups based on fuzz poins. Communicaions of he Korean Mahemaical Socie Jun Y.B. Lee K.J. Par C.H. 00. Fuzz sof se heor applied o BCK/BCIalgebras. Compuers Mahemaics wih pplicaions Lee K.J. Par C.H Some quesions on fuzzificaions of ideals in subracion algebras. Communicaions of he Korean Mahemaical Socie Liu Y. Xin X. 0. General fuzz sof groups fuzz normal sof groups nnals of Fuzz Mahemaics Informaics Maji P.K. Ro.R. Biswas R. 00. n applicaion of sof ses in a decision maing problem Compuers Mahemaics wih pplicaions Maji P.K. Biswas R. Ro.R. 00. Sof se heor. Compuers Mahemaics wih pplicaions Maji P.K. Biswas R. Ro.R. 00a. Fuzz sof ses. Journal of Fuzz Mahemaics

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