A New Kind of Fuzzy Sublattice (Ideal, Filter) of A Lattice

Transcription

1 nernaional Journal of Fuzzy Sysems Vol 3 No March 2 55 A New Kind of Fuzzy Sublaice (deal Filer) of A Laice B Davvaz O Kazanci Absrac Our aim in his paper is o inroduce sudy a new sor of fuzzy sublaice (ideal filer) of a laice called ( -fuzzy sublaice (ideal filer) These fuzzy sublaices (ideals filers) are characerized by heir level sublaices (ideals filers) Finally we give a generalizaion of ( -fuzzy sublaices (ideals filers) Keywords: fuzzy se fuzzy poin laice fuzzy sublaice ( -fuzzy sublaice level se nroducion The heory of fuzzy ses which was inroduced by Zadeh [22] is applied o many mahemaical branches Rosenfeld [6] inspired he fuzzificaion of algebraic srucures inroduced he noion of fuzzy subgroups Das [8] characerized fuzzy subgroups by heir level subgroups n [5] Liu applied he concep of fuzzy ses o he heory of rings inroduced eamined he noion of a fuzzy ideal of a ring also see [] A new ype of fuzzy subgroup (viz ( -fuzzy subgroup) was inroduced in an earlier paper of Bhaka Das [3] by using he combined noions of ``belongingness" ``quasicoincidence" of fuzzy poins fuzzy ses n fac ( -fuzzy subgroup is an imporan useful generalizaion of Rosenfeld's fuzzy subgroup This concep has been sudied furher in [ ] Also a generalizaion of Rosenfeld's fuzzy subgroup Bhaka Das's fuzzy subgroup is given in [2] Algebraic srucures play a prominen role in mahemaics wih wide ranging applicaions in many disciplines such as heoreical physics compuer sciences conrol engineering informaion sciences coding heory opological spaces he like This provides sufficien moivaion o researchers o review various conceps resuls from he realm of absrac algebra in he broader framework of fuzzy seing One Corresponding Auhor: B Davvaz is wih he Deparmen of Mahemaics Yazd Universiy Yazd ran davvaz@yazduniacir; bdavvaz@yahoocom O Kazanci is wih he Deparmen of Mahemaics Karadeniz Technical Universiy 68 Trabzon Turkey kazancio@yahoocom of he srucures ha are mos eensively used discussed in he mahemaics is applicaions is laice heory As i is well known i is considered as a relaional ordered srucure on one h as an algebra on he oher h Our aim in his paper is o inroduce sudy a new sor of fuzzy sublaice (ideal filer) of a laice called ( -fuzzy sublaice (ideal filer) These fuzzy sublaices (ideals filers) are characerized by heir level sublaices (ideals filers) Finally we give a generalizaion of ( -fuzzy sublaices (ideals filers) 2 Definiion of a laice n order o make his paper self-sufficien we recall some basic definiions resuls The definiions may be found in references [6] Definiion 2: A nonempy se L ogeher wih wo binary operaions on L is called a laice if i saisfies he following ideniies: L: ( y y ( y y L2: ( ( y z) ( z ( ( y z) ( z L3: ( ( L4: ( ( ( ( The operaion is called mee he operaion is called join Le L be he se of proposiions le denoe he connecive ``or" denoe he connecive ``" Then L o L4 are well-known properies from proposiional logic A binary relaion defined on a se A is a parial order on he se A if he following condiions hold idenically in A : (i) a a (ii) a b b a imply a = b (iii) a b b c imply a c f in addiion for every a b A (iv) a b or b a hen we say is a oal order on A A nonempy se wih a parial order on i is called a parially ordered se or more briefly a pose Le B be a subse of a pose A An elemen c in A is an upper bound for B if a c for every a in B An 2 TFSA

2 56 nernaional Journal of Fuzzy Sysems Vol 3 No March 2 elemen c in A is he leas upper bound of B or supremum of B (sup B ) if c is an upper bound of B a b for every a in B implies ha c b Similarly we can define wha i means for c o be a lower bound of B for c o be he greaes lower bound of B also called he infimum of B Now we consider he second approach o laices Definiion 22: A pose L is a laice if only if for every a b in L boh sup { a b} inf { a b} eis in L One can show ha he boh definiions of a laice are equivalen f L is a laice by he firs condiion hen define on L by a b if only if a = a b so b = a b f L is a laice by he second condiion hen define he operaions by a b = sup{ a b} a b = inf{ a b} Definiion 23: A disribuive laice is a laice which saisfies eiher of he disribuive laws D: ( y z) ( ( z) D2: ( y z) ( ( z) One can see a laice L saisfies D if only if saisfies D2 Eample 24: Le L = N {} where N is he se of all naural numbers an order on L is defined by a b b a ( b divides Then ( L ) is a disribuive laice where are given by a b = ( a he g c d of { a b} a b = [ a b] he l c m of { a b} A pose P is complee if for every subse A of P boh sup A inf A eis in P All complee poses are laices a laice L which is complee as a pose is a complee laice A sublaice of a laice L is a subse of L which is a laice in is own righ using he same operaions Definiion 25: f L is a laice L = / is a subse of L such ha for every pair of elemens a b in L boh a b a b are in L where are he laice operaions of L hen we say ha L wih he same operaions is a sublaice of L Definiion 26: Le L be a laice An ideal of L is a nonempy subse of L such ha () is a sublaice of L (2) for any a b L a b Since he fac ha a b for a b in he firs condiion is already implied by he second condiion we can replace he firs condiion by a weaker one: for any a b a b Anoher equivalen characerizaion of an ideal in a laice L is (i) a b a b (ii) a b L b a b Definiion 27: Le be an ideal of a laice L Some of he common ypes of ideals are defined below is a proper ideal if =/ L if L conains =/ is a prime ideal if i is proper a b implies ha eiher a or b is a maimal ideal of L if is proper he only ideal properly conains is L Eample 28: n he laice L / a \ / e \ c d Besides L {} below are all proper ideals of L M = { a c d e f } N = { b c d e f } R = { c d e f } T = { e} U = { f } Ou of hese M N S T U are prime M N are maimal An eample of a sublaice ha is no an ideal is he subse { b c d e} Definiion 29: Le L be a laice A filer of L is he dual concep of an ideal Specifically a filer F of L is a subse of L such ha () F is a sublaice of L (2) for any a F b L a b F The firs condiion can be replaced by a weaker one: for any a b F a b F An equivalen characerizaion of a filer F in a laice L is (i) a b F a b F (ii) a F b L a b b F Consider he posiive inegers wih mee join defined by he greaes common divisor he leas common muliple operaions Then he posiive even numbers / / \ b \ f S = { d e f }

3 B Davvaz O Kazanc: A new kind of fuzzy sublaice (ideal filer) of a laice 57 form a filer Definiion 2: Le F be a filer of a laice Some of he common ypes of filers are defined below F is a proper filer if F =/ L if L conains F =/ F is a prime filer if i is proper a b F implies ha eiher a F or b F F is an ulrafiler (or maimal filer) of L if F is proper he only filer properly conains F is L Throughou his paper L = L denoes a laice [] is a complee laice where [] is he se of reals beween y = ma { y} y = min { y} 3 Fuzzy sublaice (ideal filer) of a laice Le X be a non-empy se A fuzzy subse μ of X is a funcion μ : X [] Le μ λ be wo fuzzy subses of X we say ha μ is conained in λ if λ( for all X f {μ i} i is a collecion of fuzzy subses of X hen we define he fuzzy subses μi μi by: i i i i ( μ )( = { μ ( } i i ( μi )( = { μi ( } i for all X A fuzzy subse μ is called monoonic [ani-monoonic] if [ ( ] whenever y Le μ be any fuzzy subse of X The se μ = { X } [] is called a level subse of μ Rosenfeld [6] applied he concep of fuzzy ses o he heory of groups defined he concep of fuzzy subgroups of a group Since hen many papers concerning various fuzzy algebraic srucures have appeared in he lieraure Liu [5] inroduced sudied he noions of fuzzy sub-rings fuzzy ideals n [7] Yuan Bo Wu Wangming inroduced he concep of fuzzy ideal of a disribuive laice Also see [2-47-2] Definiion 3 [7]: Le L be a laice μ be a fuzzy subse of L We say ha μ is a fuzzysubla ice of L if for all y L μ ( Le L be a laice funcion of a subse of L Then i χ be he characerisic χ is a fuzzy sublaice if only if is a sublaice Definiion 32: A monoonic fuzzy sublaice is called a fuzzy filer of L An ani-monoonic fuzzy sublaice is called a fuzzy ideal of L is easy o see ha A fuzzy subse μ is a fuzzy filer if only if = A fuzzy subse μ is a fuzzy ideal if only if = Eample 33: Suppose ha L is he following Boolean algebra: / \ a b \ / We define fuzzy subses μ λ η of L as follows: ) = = = ) = 3 λ() = λ( = λ( = λ() = η() = η( = η( = η() = Then () μ is a fuzzy sublaice of L (2) λ is a fuzzy ideal of L (3) η is a fuzzy filer of L Definiion 34: A fuzzy filer is prime if holds for all y L A fuzzy ideal is prime if holds for all y L We can easily obain he following saemen: A fuzzy subse μ is a fuzzy prime ideal if μ is a fuzzy prime filer of L where μ is defined by seing = for all L Eample 35: Consider he Boolean algebra L in Eample 33 Suppose ha fuzzy subses μ λ η of L are defined as follows: ) = = = ) = λ() = λ( = λ( = λ() = η() = η( = η( = η() = 2

4 58 nernaional Journal of Fuzzy Sysems Vol 3 No March 2 Then () μ is a fuzzy prime ideal of L (2) λ is a fuzzy prime filer of L (3) η is a fuzzy ideal of L bu is no a fuzzy prime ideal of L 4 ( -fuzzy sublaice (ideal filer) of a laice For any fuzzy subse μ of L he se { L < } is called he suppor of μ is denoed by supp μ A fuzzy se μ on L which akes he value (] a some L akes he value for all y L epec is called a fuzzypoin is denoed by where he poin is called is suppor poin is called is value A fuzzy poin is said o belong o (resp be quasi-coinciden wih) a fuzzy se μ wrien as μ resp qμ ) if μ ( (resp μ ( ) + > ) f μ or qμ hen we wrie ( μ The symbol q means q does no hold Definiion 4: A fuzzy subse μ of a laice L is said o be an ( - fuzzy sublaice of L if for all r (] y L (i) y r μ implies ( r ( μ (ii) y r μ implies ( r ( μ μ is called an ( - fuzzy ideal of L if μ is an ( -fuzzy sublaice of L (iii) μ y implies y qμ μ is called an ( - fuzzy filer of L if μ is an ( -fuzzy sublaice of L (iv) y μ y implies ha ( μ Theorem 42: Condiions (i)-(iv) in Definiion 4 are equivalen o he following condiions respecively () μ ( 5 (2) μ ( 5 (3) y implies 5 (4) y implies 5 for all y L Proof: (i ): Suppose ha y L We consider he following cases: ( < 5 ( 5 Case a: Assume ha μ ( < 5 which implies ha < Choose such ha < < Then y μ bu ( qμ which conradics (i) Case b: Assume ha μ ( < 5 Then 5 y 5 μ bu ( 5 qμ a conradicion Hence () holds (ii 2 ): The proof is similar o (i ) (iii 3 ): Le y L y We consider he following cases: ( μ ( < 5 ( 5 Case a: Assume ha μ ( = < 5 μ ( = r < Choose s such ha r < s < r + s < Then s μ bu y q s μ which conradics (iii) So 5 < Case b: Le 5 < μ ( f μ ( < 5 hen 5 μ bu y 5 qμ which conradics iii So 5 < (iv 4 ): The proof is similar o (iii 3) ( i): Le y r μ hen r Now we have r 5 μ ( 5 f 5 < r hen 5 μ ( which implies ha < μ ( + r f r 5 hen r μ ( Therefore ( r ( μ ( 2 ii): The proof is similar o (i ) ( 3 iii): Le μ Then Now we have 5 5 which implies ha or 5 according as 5 or 5 < Therefore y ( μ ( 4 iv): The proof is similar o ( 3 iii) The following corollary is eacly obained from Definiion 4 Theorem 42 Corollary 43: Le μ be a fuzzy subse of a laice L Then μ is an ( -fuzzy sublaice of L if only if he condiions () (2) in Theorem 42 hold μ is an ( -fuzzy ideal of L if only if μ is an ( -fuzzy sublaice he condiion (3) in Theorem 42 hold

5 B Davvaz O Kazanc: A new kind of fuzzy sublaice (ideal filer) of a laice 59 μ is an ( -fuzzy filer of L if only if μ is an ( -fuzzy sublaice he condiion (4) in Theorem 42 hold A fuzzy sublaice (according o Definiion 3) is an ( -fuzzy sublaice A fuzzy ideal (filer) (according o Definiion 32) is an ( -fuzzy ideal (filer) Bu he converse is no necessarily rue Eample 44: Consider Eample 28 Suppose ha μ is defined by ) = = = c) = 4 d) = 3 5 ) = e) = f ) = 6 Then μ is an ( -fuzzy ideal bu i is no a fuzzy ideal according o Definiion 32 Theorem 45: A non-empy subse of L is a sublaice (ideal filer) of L if only if χ is an ( -fuzzy sublaice (ideal filer) of L Proof: Assume ha is a sublaice (ideal filer) of L Then χ is a fuzzy sublaice (ideal filer) in he sense of Definiion 3 so i is an ( -fuzzy sublaice (ideal filer) For he converse only we prove he case ha χ is an ( -fuzzy ideal of L n his case for every y we have 5 = χ ( χ ( 5 χ ( so y Now le y L y Then 5 = χ ( 5 χ ( so y Definiion 46: Le L be a laice μ be a fuzzy subse of L μ is said o be an ( - fuzzy prime ideal of L if μ is an ( -fuzzy ideal of L for all (] y L ( μ implies ( μ or y ( μ μ is said o be an ( - fuzzy prime filer of L if μ is an ( -fuzzy filer of L for all (] y L ( μ implies ( μ or y ( μ Theorem 47: Le be a prime ideal (filer) of a laice L Then χ is an ( -fuzzy prime ideal (filer) of L Proof: Le be a prime ideal of L Then χ is an ( -fuzzy ideal of L Now le ( χ hen χ ( = so y Since is a prime ideal hen or y So ( μ or y ( μ Hence χ is an ( -fuzzy prime ideal The proof for he case ha is a prime filer is similar Theorem 48: Le be a non-empy subse of a laice L f χ is an ( -fuzzy prime ideal (filer) of L hen is a prime ideal (filer) of L Proof: According o Theorem 45 is an ideal of L We show ha is prime Le y L y Then χ ( = Since χ is an ( -fuzzy prime ideal ( χ for all (] hen qχ or y qχ f qχ hen < χ ( so f + y qχ hen < χ ( + so y Now we characerize ( -fuzzy sublaices (ideals filers) by heir level sublaices (ideals filers) Theorem 49: Le μ be an ( -fuzzy sublaice (ideal filer) of a laice L Then for all < 5 μ is a non-empy se or a sublaice (ideal filer) of L Conversely if μ is a fuzzy subse of L such ha μ ( /= ) is a sublaice (ideal filer) of L for all < 5 hen μ is an ( -fuzzy sublaice (ideal filer) of L Proof: Le μ be an ( -fuzzy sublaice of L < 5 f y μ hen Now we have = 5 = 5 = 5 = 5 so y μ y μ Hence μ is a sublaice of L Now le μ be an ( -fuzzy ideal of L Suppose ha μ y L y Then = 5 5 which implies ha y μ Conversely le μ be a fuzzy subse of L such ha μ (/= ) be a sublaice of L for all < 5 For every y L we can wrie = 5 = 5

6 6 nernaional Journal of Fuzzy Sysems Vol 3 No March 2 Then μ y μ so y μ y μ Therefore 5 5 Now le μ be an ideal of L We suppose ha y L y We have = 5 so μ Since μ is an ideal of L y we obain ha y μ Therefore ( ) μ y or 5 The proof of he filer case is similar Naurally a corresponding resul should be considered when μ is a sublaice (ideal filer) of L for all (5] Theorem 4: Le μ be a fuzzy subse of a laice L Then μ (/= ) is a sublaice (ideal filer) of L for all (5] if only if () μ ( 5 (2) μ ( 5 Moreover μ ( /= ) is an ideal of L for all (5] if only if μ saisfies he condiions () (2) saisfies he following condiion: (3) y implies μ ( 5 Finally μ ( /= ) is a filer of L for all (5] if only if μ saisfies he condiions () (2) saisfies he following condiion: (4) y implies μ ( 5 Proof Suppose ha he condiions () (2) hold We show ha μ ( /= ) is a sublaice of L for all (5] Le (5] y μ Then 5 < < 5 5 < < 5 so 5 < μ ( 5 < μ ( which imply ha y μ y μ Therefore μ is a sublaice of L Moreover suppose ha condiion (3) holds We show ha μ is an ideal of L Le (5] μ y We have 5 < μ ( 5 which implies ha 5 < μ ( or y μ Conversely le μ be a fuzzy subse of L such ha μ (/= ) be a sublaice of L for all 5 < (): Suppose ha for some y L 5 < = hen (5] μ ( < μ y μ Since y μ μ is a sublaice so y μ or μ ( which is a conradicion wih μ ( < Hence () holds (2): The proof is similar o () Now le μ be an ideal of L (3): Suppose ha for some y L wih y we have y ) 5 < = Then (5] μ ( y ) < μ Since μ μ is an ideal we ge y μ or which is a conradicion Hence (3) holds The proof of he filer case is similar n [2] Yuan Zhang Ren gave he definiion of a fuzzy subgroup wih hresholds which is a generalizaion of Rosenfeld's fuzzy subgroup Bhaka Das's fuzzy subgroup Based on [2] we can eend he concep of a fuzzy subgroup wih hresholds o he concep of fuzzy sublaice (ideal filer)) wih hresholds in he following way: Definiion 4: Le α β [] α < β Le μ be a fuzzy subse of a laice L Then μ is called a fuzzy sublaice wih hresholds of L if for all y L () ) β α (2) ) β α Moreover μ is a fuzzy ideal wih hresholds of L if only if μ saisfies he condiions () (2) saisfies he following condiion: (3) y implies ) β α Finally μ is a fuzzy filer wih hresholds of L if only if μ saisfies he condiions () (2) saisfies he following condiion: (4) y implies y ) β α f μ is a fuzzy sublaice (ideal filer) wih hresholds of L hen we can conclude ha μ is an ordinary fuzzy sublaice (ideal filer) when α = β = ; μ is an ( -fuzzy sublaice (ideal filer) when α = β = 5 Eample 42: Consider Eample 28 le α =/5 β =3/5 Suppose ha μ is defined by μ () = μ ( = 6

7 B Davvaz O Kazanc: A new kind of fuzzy sublaice (ideal filer) of a laice 6 2 μ ( = 5 3 )= c μ ( d)= 5 5 ) = e) = μ ( f) = 6 Then μ is a fuzzy ideal wih hresholds of L bu i is no a fuzzy ideal according o Definiion 32 also i is no an ( -fuzzy ideal according o Definiion 4 Now we characerize fuzzy sublaice (ideal filer) wih hresholds by heir level sublaices (ideals filers) Theorem 43: A fuzzy subse μ of a laice L is a fuzzy sublaice (ideal filer) wih hresholds of L if only if μ (/= ) is a sublaice (ideal filer) of L for all ( α β ] Proof: The proof is similar o he proofs of Theorems mplicaion-based fuzzy subsublaice (ideal filer) Logic sared as he sudy of language in argumens persuasion i can be used o judge he correcness of a chain of reasoning in a mahemaical proof for insance The goal of he heory is o reduce principles of reasoning o a code The ``ruh" or ``falsiy" assigned o a proposiion is is ruh-value n fuzzy logic a proposiion may be rue or false or an inermediae ruh-value such as maybe rue The senence ``Bijan is a all man" is an eample of a fuzzy proposiion n pracice one would subdivide he uni inerval ino finer divisions or work wih a coninuous ruh-domain n daily conversaion mahemaics senences are conneced wih he words or if-hen (or implies) if only if These are called connecives A senence which is modified by he word ``no" is called he negaion of he original senence The word ``" is used o join wo senences o form he conjuncion of he wo senences The word ``or" is used o join wo senences o form he disjuncion of he wo senences From wo senences we may consruc one of he form ``f hen "; his is called an implicaion Some operaors like in fuzzy logic are also defined by using ruh ables he eension principle can be applied o derive definiions of he operaors n he fuzzy logic ruh value of fuzzy proposiion P is denoed by [P] n he following we display he fuzzy logical corresponding se-heoreical noions used in his paper: [ F] = F( ; [ F] = F( ; [ P Q] = min{[ P][ Q]}; [ P Q] = ma{[ P][ Q]}; [ P Q] = min{ [ P] + [ Q]} ; [ P( ] = inf[ P( ]; P if only if [P] = for all valuaions The ruh valuaion rules given in he above are hose in he ukasiewicz sysem of coninuous-valued logic A funcion :[] [] [] is called fuzzy implicaion if i is monoonic wih respec o boh variables (separael fulfils he binary implicaion ruh able: () = () = () = () = By monooniciy ( = ( ) = for all [] is decreasing wih respec o he firs variable ( () < () ) is increasing wih respec o he second variable ( () < () ) Of course various implicaion operaors have been defined We only show a selecion of hem in he ne able α denoes he degree of ruh (or degree of membership) of he premise β he respecive values for he consequence he resuling degree of ruh for he implicaion m ( α β ) = ma{ α min{ α β}} a ( α β ) = min{ α + β} if α β g ( α β ) = β if α > β if α β cg ( α β ) = α if α > β if α β gr ( α β ) = if α > β b ( α β ) = ma{ α β} if α β gg ( α β ) = β if α > β α The ``qualiy" of hese implicaion operaors could be evaluaed eiher empirically or aiomaically n he following definiion we considered he definiion of implicaion operaor in he ukasiewicz sysem of coninuous-valued logic Definiion 5: A fuzzy subse μ of a laice L saisfies: () for any y L

8 62 nernaional Journal of Fuzzy Sysems Vol 3 No March 2 y μ] [ y μ ] ] (2) for any y L y μ] [ y μ ] ] hen μ is called a fuzzifying sublaice of L μ is called a fuzzifying ideal of L if μ is a fuzzifying sublaice of L (3) for any y L y ] [ y μ ] ] μ is called a fuzzifying filer of L if μ is a fuzzifying sublaice of L (4) for any y L [ [ y μ] y ] [ μ ] ] Clearly Definiion 5 is equivalen o Definiions 3 32 Therefore a fuzzifying sublaice is an ordinary fuzzy sublaice n [2] he concep of -auology is inroduced ie P if only if [ P] for all valuaions Based on [2] we can eend he concep of implicaion-based fuzzy subgroup o he concep of implicaion-based fuzzy sublaice (ideal filer) in he following way: Definiion 52: Le μ be a fuzzy subse of a laice L (] is a fied number f () for any y L y μ] [ y μ] ] (2) for any y L y μ] [ y μ] ] hen μ is called a -implicaion-based fuzzy sublaice of L Moreover if μ saisfies he condiion: (3) for any y L y ] [ y μ] ] hen μ is called a -implicaion-based fuzzy ideal of L μ is called a -implicaion-based fuzzy filer of L if μ is a -implicaion-based fuzzy sublaice of L (4) for any y L [ [ y μ] y ] [ μ] ] Now le be an implicaion operaor Then Corollary 53: μ is a -implicaion-based fuzzy sublaice of a laice L if only if (i) ( μ ( ) for all y L (ii) ( μ ( ) for all y L Le μ be a fuzzy subse of a laice L hen we have he following resuls: Theorem 54: () Le = gr Then μ is a 5-implicaion-based fuzzy sublaice (ideal filer) of L if only if μ is a fuzzy sublaice (ideal filer) wih hresholds ( α = β = ) of L (2) Le = g Then μ is a 5-implicaion-based fuzzy sublaice (ideal filer) of L if only if μ is a fuzzy sublaice (ideal filer) wih hresholds ( α = β = 5 ) of L (3) Le = cg Then μ is a 5-implicaion-based fuzzy sublaice (ideal filer) of L if only if μ is a fuzzy sublaice (ideal filer) wih hresholds ( α = 5 β = ) of L Proof: We prove (2) for he sublaice case he proofs of oher pars are similar Suppose ha μ is a 5-implicaion-based fuzzy sublaice of L Then by Corollary 53 we have (i) g ( μ ( ) 5 for all y L (ii) g ( μ ( ) 5 for all y L According o definiion of g from (i) we obain μ ( or 5 < Then μ ( 5 which implies ha μ ( 5 From (ii) we obain μ ( or 5 < Then μ ( 5 which implies ha μ ( 5 Therefore μ is a sublaice wih hresholds The converse is clear 6 Conclusions n he sudy of he properies common o all algebraic srucures (such as groups rings ec) even some of he properies ha disinguish one class of algebras from anoher laices ener in an essenial naural way To invesigae he srucure of a laice i is clear ha sublaice (ideal filer) play an imporan role The concep of fuzzy sublaice (ideal filer) was already sudied by Bo e al [7] Koguep e al [4] Swamy Viswanadha Raju [7] Tepavacevic Trajkovski [8] ohers n his paper we have given he concep of ( -fuzzy sublaice (ideal filer) on a laice

9 B Davvaz O Kazanc: A new kind of fuzzy sublaice (ideal filer) of a laice 63 using fuzzy poins Aferwards we have also esablished correlaions beween hese conceps fuzzy sublaice (ideal filer) The main resuls are Theorems We have given some eamples ha show his new sor of fuzzy sublaice (ideal filer) is a generalizaion of ordinary fuzzy sublaice (ideal filer) Finally we have invesigaed he concep of implicaion-based fuzzy sublaice (ideal filer) is our hope ha his work would serve as a foundaion for furher sudy of he laice heory n our fuure work we will consider he ( a -fuzzy sublaice (ideal filer) in laices where a b are any one of he q q or q Acknowledgmen The auhors would like o epress heir sincere hanks o he referees for heir valuable commens suggesions which help a lo for improving he presenaion of his paper References [] S-Z Bai Pre-semicompac L-subses in fuzzy laices nernaional Journal of Fuzzy Sysems vol 2 no pp [2] S K Bhaka ( -fuzzy normal quasinormal maimal subgroups Fuzzy Ses Sys vol 2 pp [3] S K Bhaka P Das On he definiion of a fuzzy subgroup Fuzzy Ses Sys vol 5 pp [4] S K Bhaka P Das ( -fuzzy subgroups Fuzzy Ses Sys vol 8 pp [5] S K Bhaka P Das Fuzzy subrings ideals redefined Fuzzy Ses Sys vol 8 pp [6] G Birkhoff Laice heory American Mahemaical Sociey Providence R 984 [7] Y Bo Wu Wang Ming Fuzzy ideals on a disribuive laice Fuzzy Ses Sys vol 35 pp [8] P Das Fuzzy groups level subgroups J Mah Anal Appl vol 85 pp [9] B Davvaz P Corsini Generalized fuzzy hyperideals of hypernear-rings many valued implicaions Journal of nelligen Fuzzy Sysems vol 7 no 3 pp [] B Davvaz ( -fuzzy subnear-rings ideals Sof Compuing vol pp [] B Davvaz P Corsini Redefined fuzzy Hv-submodules many valued implicaions nformaion Sciences vol 77 pp [2] B Davvaz V Leoreanu-Foea On a produc of fuzzy Hv-submodules nernaional Journal of Fuzzy Sysems vol no 2 pp [3] Y B Jun Y Xu J Ma Redefined fuzzy implicaive filers nformaion Sciences vol 77 pp [4] B B N Koguep C Nkuimi C Lele On fuzy prime ideals of laices SAMSA Journal of Pure Applied Mahemaics vol 3 pp - 28 [5] W J Liu Fuzzy invarian subgroups fuzzy ideals Fuzzy Ses Sys vol 8 pp [6] A Rosenfeld Fuzzy groups J Mah Anal Appl vol 35 pp [7] U M Swamy D Viswanadha Raju Fuzzy ideals congruences of laices Fuzzy Ses Sys vol 95 pp [8] A Tepavacevic G Trajkovski L-fuzzy laices: an inroducion Fuzzy Ses Sys vol 23 pp [9] S Yamak O Kazancı B Davvaz Divisible pure inuiionisic fuzzy subgroups heir properies nernaional Journal of Fuzzy Sysems vol no 4 pp [2] Y-Q Yin X-K Huang D-H Xu F Li The Characerizaion of h-semisimple hemirings nernaional Journal of Fuzzy Sysems vol no 2 pp [2] X H Yuan C Zhang Y H Ren Generalized fuzzy groups many valued applicaions Fuzzy Ses Sys vol 38 pp [22] L A Zadeh Fuzzy ses nform Conrol vol 8 pp [23] J Zhan B Davvaz Sudy of fuzzy algebraic hypersysems from a general viewpoin nernaional Journal of Fuzzy Sysems vol 2 no pp