)-interval valued fuzzy ideals in BF-algebras. Some properties of (, ) -interval valued fuzzy ideals in BF-algebra, where

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1 Inernaonal Journal of Engneerng Advaned Researh Tehnology (IJEART) ISSN: , Volume-, Issue-4, Oober 05 Some properes of (, )-nerval valued fuzzy deals n BF-algebras M. Idrees, A. Rehman, M. Zulfqar, S. Muhammad Absra. In hs paper, we nrodue he onep of (, ) -nerval valued fuzzy deals n BF-algebra, where, are any one of, q, q, q nvesgae some of her relaed properes. We prove ha every ( q, q) -nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal We show ha when an (, q) -nerval valued fuzzy deal of a BF-algebra X s an (, ) -nerval valued fuzzy deal We also prove ha he nerseon unon of any famly of (, q) -nerval valued fuzzy deals of a BF-algebra X s an (, q) -nerval valued fuzzy deal Key words: BF-algebra; (, ) -nerval valued fuzzy deals; (, q) -nerval valued fuzzy deal. I. INTRODUCTION The onep of BF-algebra was frs naed by Walendzak [5] n 007. The heory BF-algebra was furher enrhed by many auhors [5, 9, 4]. The fuzzy ses, proposed by Zadeh [] n 965, has provded a useful mahemaal ool for desrbng he behavor of sysems ha are oo omple or ll defned o adm prese mahemaal analyss by lassal mehods ools. Eensve applaons of fuzzy se heory have been found n varous felds, for eample, arfal nellgene, ompuer sene, onrol engneerng, eper sysem, managemen sene, operaon researh many ohers. The onep was appled o he heory of groupods groups by Rosenfeld [], where he nrodued he fuzzy subgroup of a group. A new ype of fuzzy subgroup, whh s, he (, q) subgroup, was nrodued by Bhaka Das [3] by usng he ombned noons of belongngness quas-ondene of fuzzy pons fuzzy ses, whh was nrodued by Pu Lu []. Mural [0] proposed he defnon of fuzzy pon belongng o a fuzzy subse under a naural equvalene on fuzzy subses. I was found ha he mos vable generalzaon of Rosenfeld s fuzzy subgroup s (, q) subgroup. Bhaka [-] naed he oneps of ( q) -level subses, (, q) normal, quas-normal mamal subgroups. Many researhers ulzed hese oneps o generalze some oneps of algebra (see [4, 7, 8, 6, 3-33]). In [6], Davvaz suded (, q) subnearrngs deals. In [-3], Jun defned he noon of (, ) subalgebras/deals n BCK/BCI-algebras. The onep of (, ) posve mplave deal n BCK-algebras was naed by Zulfqar n [3]. In [4], Jun defned (, q) subalgebras n BCK/BCI-algebras. In [3], Zulfqar nrodued he noon of sub-mplave (, ) deals n BCH-algebras. Currenly, Zulfqar Shabr defned he onep of posve mplave (, q) deals ( (, q) deals, fuzzy deals wh hresholds) n BCK-algebras n [33]. The heory of nerval valued fuzzy ses was proposed fory year ago as a naural eenson of fuzzy ses. Inerval valued fuzzy se was nrodued by Zadeh [8], where he value of he membershp funon s nerval of numbers nsead of he number. The heory was furher enrhed by many auhors [4, 7-8, 0, 5-9, 3, 9-30]. In [4], Bswas defned nerval valued fuzzy subgroups of Rosenfeld s naure, nvesgaed some elemenary properes. Jun, nrodued he onep of nerval valued fuzzy subalgebras/deals n BCK-algebras [0]. In [5], Laha e al. naed he noon of nerval valued (, ) subgroups. In [6], Ma e al. defned he onep of nerval valued (, q) deals of pseudo MV-algebras. In [7-8], Ma e al. suded (, q) -nerval valued fuzzy deals n BCI-algebras. Mosafa e al. naed he noon of nerval valued fuzzy KU-deals n KU-algebras [9]. In [3], Saed defned he onep of nerval valued fuzzy BG-algebras. Zhan e al. [30] naed he noon of nerval valued (, q) flers of pseudo BL-algebras. In he presen paper, we defne he onep of (, ) -nerval valued fuzzy deals n BF-algebra, where, are any one of, q, q, q nvesgae some of her relaed properes. We prove ha every ( q, q) -nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal We show ha when an (, q) -nerval valued fuzzy deal of a BF-algebra X s an (, ) -nerval valued fuzzy deal We also prove ha he nerseon unon of any famly of (, q) -nerval valued fuzzy deals of a BF-algebra X s an (, q) -nerval valued fuzzy deal. PRELIMINARIES Throughou hs paper X always denoe a BF-algebra whou any spefaon. We also nlude some bas aspes ha are neessary for hs paper.

2 Some properes of (, ) -nerval valued fuzzy deals n BF-algebras A BF-algebra X [5] s a general algebra (X,, 0) of ype (, 0) sasfyng he followng ondons: (BF-) = 0 (BF-) 0 = (BF-3) 0 ( y) = (y ) for all, y X. We an defne a paral order on X by y f only f y = 0. Defnon.. [5] A nonempy subse S of a BF-algebra X s alled a subalgebra of X f sasfes y S, for all, y S. Defnon.. [5] A non-empy subse I of a BF-algebra X s alled an deal of X f sasfes he ondons (I) (I), where (I) 0 I, (I) y I y I mply I, for all, y X. We now revew some nerval-valued fuzzy log oneps. Frs, we denoe by = [ -, + ] a losed nerval of [0, ], where denoe by H[0, ] he se of all suh losed nervals of [0, ]. Defne on H[0, ] an order relaon by () () = = = (3) (4) p = [p -, p + ], whenever 0 p (5) rma{, d } = [ma{, d }, ma{, (6) rmn{, d } = [mn{, d }, mn{, (7) rnf = [ I, I ] (8) rsup = [ I, I ] Where = [, ], d = [ d, d }] d }] d ] H[0, ], I., H[0, ] wh s a omplee lae, wh = rmn, = rma, 0 = [0, 0] = [, ] beng s leas elemen he greaes elemen, respevely. An nerval valued fuzzy se of a unverse X s a funon from X no he un losed nerval [0, ], ha s : X H[0, ], () H[0, ], where for eah X ( ) [ ( ), ( )] H[0, ]. For an nerval valued fuzzy se n a BF-algebra X [0, 0] < [, ], he rsp se = { X () } s alled he level subse of. We also noe ha, sne every [0, ] s n orrespondene wh he nerval [, ] H[0, ], follows ha a fuzzy se s a parular ase of nerval-valued fuzzy se. Frs we noe ha an nerval-valued fuzzy se of a BF-algebra X s a par of fuzzy ses (, ) of X suh ha () (), for all X. If C, D are wo nerval-valued fuzzy ses of a BF-algebra X, hen we defne C D f only f for all X, C () D (), C = D f only f for all X, C () = D (). C () C () = Also, he unon nerseon are defned as follows: D () D () If C D are wo nerval-valued fuzzy ses of a BF-algebra X, where C () = [ C (), C ()], D () = [ D (), all X, hen ( C D )() = C () D () = [ma{ D ()}, ma{ C (), D ()}] ( C D )() = C () D () = [ma{ D ()}, ma{ C (), C () = [ - D ()}] C (), - C ()] D ()], for C C (), (), where he operaon s he omplemen of nerval-valued fuzzy se By he jon of wo nerval-valued fuzzy ses, we know C C () = [ma{ ma{ C (), - C ()}]. C (), - C ()}, Defnon.3. An nerval valued fuzzy se of a BF-algebra X s alled an nerval valued fuzzy deal of X f sasfes he ondons (F) (F), where (F) (0) (), (F) () ( y) (y), for all, y X. Theorem.4. An nerval valued fuzzy se of a BF-algebra X s an nerval valued fuzzy deal of X f only f, for every [0, 0] < [, ], s an deal Proof. Assume ha s an nerval valued fuzzy deal of X le H[0, ] be suh ha. (0) ( ), we have 0. Le, y X be suh ha y y. ( y) (y). I follows from F ha () ( y) (y) =

3 . Hene s an deal Inernaonal Journal of Engneerng Advaned Researh Tehnology (IJEART) ISSN: , Volume-, Issue-4, Oober 05 Conversely, suppose ha s an deal of X for all [0, 0] < [, ]. Assume ha here es a X suh ha (0) < (a). Le (0) = [ (0), (0) If we ake (0) < ] (a) = [ ( a), ( a) (a) ] (0) < (a). = [, ] = ( (0) ( a)), [, ] = [ ( (0) ( a)), ( (0) (0)) ]. Hene (0) < < (a) (0) < < (a). Ths mples ha (0) = [ (0), (0) ] < [, ] < [ ( a), ( a) ]. Ths shows ha 0, whh s onradon. Therefore (0) () for all X. Now, le us suppose here are a, b X suh ha (a) < (a b) (b). Le Pu = [ (a) = [ (a b) = [, (b) = [ (a), (a)], (a b), (a b)] (b), (b)] ] = ( (a) + ( (a b) (b))) (a) < (a) < we wll have (a) = [ [ (a b) Therefore Bu (a b) = [ < < (a b) (a b) (a), (a)] < [, (b), a (a b). (b) (b) ] < (b)]. (a b), (a b)] > whh mples ha (b) = [ a b (b), (b)] > b Ths leads o a onradon. Hene () ( y) (y). s an nerval valued fuzzy deal 3. (, ) -INTERVAL VALUED FUZZY IDEALS IN BF-ALGEBRA In hs seon, we defne he onep of (, ) -nerval valued fuzzy deals n a BF-algebra nvesgae some of her properes. Throughou hs paper X wll denoe a BF-algebra, are any one of, q, q, q unless oherwse spefed. An nerval valued fuzzy se of a BF-algebra X havng he form ( [0, 0]) (y) = [0, 0] f y f y s sad o be an nerval valued fuzzy pon wh suppor value s denoed by pon. An nerval valued fuzzy s sad o belong o (resp., quas-onden wh) an nerval valued fuzzy se, wren as (resp. q ) f () (resp. () + > [, ]). By q ( q ) we mean ha or q ( q ). In wha follows le denoe any one of, q, q, q q unless oherwse spefed. To say ha no hold. means ha does Defnon 3.. An nerval valued fuzzy se of a BF-algebra X s alled an (, ) subalgebra of X, where q, f sasfes he ondon, y ( y) for all [0, 0] <, [, ], y X. Le be an nerval valued fuzzy se of a BF-algebra X suh ha () [0.5, 0.5] for all X. Le X [0, 0] < [, ] be suh ha q. () () + > [, ]. I follows ha () = () + () () + > [, ]. 3

4 Some properes of (, ) -nerval valued fuzzy deals n BF-algebras Ths mples ha () > [0.5, 0.5]. Ths means ha { Therefore, he ase = omed. q } =. q n he above defnon s Defnon 3.. An nerval valued fuzzy se of a BF-algebra X s alled an (, ) -nerval valued fuzzy deal of X, where q, f sasfes he ondons (A) (B), where (A) (B) 0, ( y), y, for all [0, 0] <,, [, ], y X. Theorem 3.3. For any nerval valued fuzzy se of a BF-algebra X, he ondon (F) (F3) are equvalen o he ondons: (C) 0, (D) ( y), y, for all [0, 0] <,, [, ], y, z X. Proof. (F) (C) ha, ha s by (F) so 0. (C) (F) 0 ( ) Sne Le X [0, 0] < [, ] be suh ( ), so we have (F) (D) ha By (F) (). (0) (), for X. by hypohess (0) (). Le, y X [0, 0] <, [, ] be suh ( y) y. ( y) (y). () ( y) (y).. (D) (F) Noe ha for every, y X, we have ( y) ( y ) Hene by hypohess Ths mples ha y. ( y ). ( y) ( y) () ( y) (y). Theorem 3.4. Every ( q, q) -nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal Proof. Le be an ( q, q) -nerval valued fuzzy deal Le X [0, 0] < [, ] be suh ha. q so 0 q. Le, y X [0, 0] <, [, ] be suh ha ( y) ( y) Ths mples ha q Therefore s an (, q) y. y q. q. -nerval valued fuzzy deal Theorem 3.5. An nerval valued fuzzy se of a BF-algebra X s an (, q) -nerval valued fuzzy deal of X f only f sasfes he ondons (E) (F), where (E) (0) () [0.5, 0.5], (F) () ( y) (y) [0.5, 0.5], for all, y X. Proof. Suppose s an (, q) deal Le X be suh ha (0) () [0.5, 0.5]. If () [0.5, 0.5], hen (0) (). Sele [0, 0] < [0.5, 0.5] suh ha whh s a onradon. (0) (). bu 0 q, -nerval valued fuzzy If () [0.5, 0.5], hen (0) [0.5, 0.5]. Ths mples ha [0.5, 0.5] bu 0 [0.5, 0.5] q. Agan a onradon. Hene (0) () [0.5, 0.5], for all X. 4

5 Inernaonal Journal of Engneerng Advaned Researh Tehnology (IJEART) ISSN: , Volume-, Issue-4, Oober 05 Now we show ha ondon (F) holds. On he onrary assume ha here es, y X suh ha () ( y) (y) [0.5, 0.5]. If ( y) (y) [0.5, 0.5], hen () ( y) (y). Sele [0, 0] < [0.5, 0.5] suh ha () ( y) (y). ( y) y bu whh s a onradon. If ( y) (y) [0.5, 0.5], hen Ths mples () [0.5, 0.5]. q, ( y) [0.5, 0.5] y [0.5, 0.5] bu [0.5, 0.5] q. Agan a onradon. Hene () ( y) (y) [0.5, 0.5]. Conversely, assume ha sasfes he ondons (E) (F). Le X [0, 0] < [, ] be suh ha. (). By ondon (E), we have (0) () [0.5, 0.5] [0.5, 0.5]. If [0.5, 0.5], hen (0). Ths mples 0. If > [0.5, 0.5], hen (0) [0.5, 0.5]. Ths mples (0) + > [0.5, 0.5] + [0.5, 0.5] = [, ], ha s, 0 q. Hene 0 q. Le, y X [0, 0] <, [, ] be suh ha ( y) y. ( y) (y). By ondon (F), we have () ( y) (y) [0.5, 0.5] [0.5, 0.5]. If [0.5, 0.5], hen (). Ths mples. If > [0.5, 0.5], hen Ths mples.e., Hene () [0.5, 0.5]. () + > [0.5, 0.5] + [0.5, 0.5] = [, ], q. q. Ths shows ha s an (, q) deal -nerval valued fuzzy Theorem 3.6. An nerval valued fuzzy se of a BF-algebra X s an nerval valued fuzzy deal of X f only f s an (, ) -nerval valued fuzzy deal Proof. Suppose s an nerval valued fuzzy deal of X for X [0, 0] < [, ]. (). By Defnon.3, (0) (), we have ha s (0), 0. Le, y X [0, 0] <, r [, ] be suh ha ( y r. y) ( y) (y) r. By Defnon.3, we have () ( y) (y) r. Ths mples ha r. Ths shows ha s an (, ) -nerval valued fuzzy deal Conversely, assume ha s an (, ) -nerval valued fuzzy deal Suppose here ess X suh ha (0) (). Sele [0, 0] < [, ] suh ha Hene (0) (). bu 0, whh s a onradon. (0) (), for all X. Now suppose here es, y X suh ha () ( y) (y). Sele [0, 0] < [, ] suh ha () ( y) (y). ( y) y bu, whh s a onradon. Hene () ( y) (y). Ths shows ha s an nerval valued fuzzy deal Theorem 3.7. Every (, ) -nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal Proof. Sraghforward. Corollary 3.8. Every nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal 5

6 Some properes of (, ) -nerval valued fuzzy deals n BF-algebras Proof. By Theorem 3.6, every nerval valued fuzzy deal of a BF-algebra X s an (, ) -nerval valued fuzzy deal Hene by above Theorem 3.7, every nerval valued fuzzy deal of X s an (, q) -nerval valued fuzzy deal of X. Ne we show ha when an (, q) -nerval valued fuzzy deal of a BF-algebra X s an (, ) -nerval valued fuzzy deal Theorem 3.9. Le be an (, q) -nerval valued fuzzy deal of a BF-algebra X suh ha () [0.5, 0.5] for all X. s an (, ) Proof. Le X [0, 0] < [, ] be suh ha Sne we have. (). (0) () [0.5, 0.5] = () -nerval valued fuzzy deal 0. Now le, y X [0, 0] <, [, ] be suh ha ( y) y. ( y) (y). I follows from Theorem 3.5(F) ha () ( y) (y) [0.5, 0.5] = ( y) (y).. Therefore s an (, ) -nerval valued fuzzy deal Corollary 3.0. Le be an (, q) -nerval valued fuzzy deal of a BF-algebra X suh ha () [0.5, 0.5] for all X. s an nerval valued fuzzy deal Theorem 3.. Le I be an deal of a BF-algebra X. he nerval valued fuzzy se of X defned by () = [0.5, 0.5] f I [0, 0] oherwse, s an (, q) -nerval valued fuzzy deal Proof. Le I be an deal Le X [0, 0] < [, ] be suh ha. () > [0, 0]. () [0.5, 0.5]. Ths mples ha I. Sne I s an deal So 0 I. Hene (0) [0.5, 0.5]. If [0.5, 0.5], hen Ths mples ha 0. If > [0.5, 0.5], hen (0) [0.5, 0.5]. (0) + [0.5, 0.5] + [0.5, 0.5] > [, ] so 0 q. Ths mples ha 0 q. Le, y X [0, 0] <, r [, ] be suh ha ( y) y r. ( y) > [0, 0] (y) r > [0, 0]. ( y) [0.5, 0.5] (y) [0.5, 0.5]. Ths mples ha ( y) I y I. Sne I s an deal So I. Hene () [0.5, 0.5]. If r [0.5, 0.5], hen () [0.5, 0.5] r. Ths mples ha r If r > [0.5, 0.5], hen. () + r [0.5, 0.5] + [0.5, 0.5] > [, ] so r q. q r Hene s an (, q) X.. -nerval valued fuzzy deal of Theorem 3.. Le I be a non-empy subse of a BF-algebra X. I s an deal of X f he nerval valued fuzzy se of X defned by () = [0.5, 0.5] f I [0, 0] oherwse, s an ( q, q) -nerval valued fuzzy deal 6

7 Inernaonal Journal of Engneerng Advaned Researh Tehnology (IJEART) ISSN: , Volume-, Issue-4, Oober 05 Proof. Le I be an deal Le X [0, 0] < [, ] be suh ha q. () + > [, ] So I. Sne I s an deal So 0 I. Hene (0) [0.5, 0.5]. If [0.5, 0.5], hen Ths mples ha If > [0.5, 0.5], hen (0) [0.5, 0.5]. 0 λ. (0) + [0.5, 0.5] + [0.5, 0.5] > [, ] so 0 q. Ths mples ha 0 q. Le, y X [0, 0] <, r [, ] be suh ha ( q y r q. y) ( y) + > [, ] (y) + r > [, ]. So ( y) I y I. Sne I s an deal So I. () [0.5, 0.5]. If r [0.5, 0.5], hen So If r > 0.5, hen () [0.5, 0.5] r. r. () + r [0.5, 0.5] + [0.5, 0.5] > [, ] so r q. r q Hene s an ( q, q) X.. -nerval valued fuzzy deal of Theorem 3.3. Le I be a non-empy subse of a BF-algebra X. I s an deal of X f he nerval valued fuzzy se of X defned by () = [0.5, 0.5] f I [0, 0] oherwse, s an ( q, q) -nerval valued fuzzy deal Proof. Le I be an deal Le X [0, 0] < [, ] be suh ha Ths mples ha If q q. Ths mples ha. or q. () + > [, ]. Ths mples ha I. Sne I s an deal So 0 I. Hene (0) [0.5, 0.5]. If [0.5, 0.5], hen Ths mples ha 0. If > [0.5, 0.5], hen (0) [0.5, 0.5]. (0) + [0.5, 0.5] + [0.5, 0.5] > [, ] so 0 q. Ths mples ha 0 q. Le, y X [0, 0] <, r [, ] be suh ha ( y) q y r q. Ths mples ha ( y) or ( y) q If y) y r or y r q. ( q y r q. Ths mples ha ( y) + > [, ] (y) + r > [, ]. So ( y) I y I. Sne I s an deal So I. () [0.5, 0.5]. If r [0.5, 0.5], hen So () [0.5, 0.5] r. r If r > [0.5, 0.5], hen. () + r [0.5, 0.5] + [0.5, 0.5] > [, ] so r q. q r. Hene s an ( q, q) deal -nerval valued fuzzy Theorem 3.4. The nerseon of any famly of (, q) -nerval valued fuzzy deals of a BF-algebra X s an (, q) -nerval valued fuzzy deal 7

8 Some properes of (, ) -nerval valued fuzzy deals n BF-algebras Proof. Le { } I be a famly of (, q) -nerval valued fuzzy deals of a BF-algebra X X. So for all I. ( I (0) () [0.5, 0.5] )(0) = I ( I I ( ( = ( I (0)) )(0) ( I Le, y X. Sne eah valued fuzzy deal So () [0.5, 0.5]) )() [0.5, 0.5]. )() [0.5, 0.5]. s an (, q) () ( y) (y) [0.5, 0.5] for all I. ( )() = ( I I Therefore ( I Hene, deal ( I = ( I )() ( I I ()) ( y) (y) [0.5, 0.5]) )( y) ( I )( y) ( I -nerval )(y) [0.5, 0.5]. )(y) [0.5, 0.5]. s an (, q) -nerval valued fuzzy Theorem 3.5. The unon of any famly of (, q) -nerval valued fuzzy deals of a BF-algebra X s an (, q) -nerval valued fuzzy deal Proof. Le { } I be a famly of (, q) -nerval valued fuzzy deals of a BF-algebra X X. So for all I. ( )(0) = ( I I ( I (0) () [0.5, 0.5] ( I = ( I (0)) () [0.5, 0.5]) )() [0.5, 0.5]. )(0) ( I Le, y X. Sne eah valued fuzzy p-deal So () for all I. ( )() = ( I I )() [0.5, 0.5]. s an (, q) ( y) (y) [0.5, 0.5] ()) -nerval Therefore ( I Hene, deal ( I = ( I )() ( I I ( y) (y) [0.5, 0.5]) )( y) ( I )( y) ( I )(y) [0.5, 0.5] )(y) [0.5, 0.5]. s an (, q) -nerval valued fuzzy 6. CONCLUSION To nvesgae he sruure of an algebra sysem, we see ha he nerval valued fuzzy deals wh speal properes always play a fundamenal role. In hs paper, we nrodue he onep of (, ) -nerval valued fuzzy deals n BF-algebra, where, are any one of, q, q, q nvesgae some of her relaed properes. We prove ha every ( q, q) -nerval valued fuzzy deal of a BF-algebra X s an (, q) -nerval valued fuzzy deal We show ha when an (, q) -nerval valued fuzzy deal of a BF-algebra X s an (, ) -nerval valued fuzzy deal We also prove ha he nerseon unon of any famly of (, q) -nerval valued fuzzy deals of a BF-algebra X s an (, q) -nerval valued fuzzy deal We beleve ha he researh along hs dreon an be onnued, n fa, some resuls n hs paper have already onsued a foundaon for furher nvesgaon onernng he furher developmen of nerval valued fuzzy BF-algebras her applaons n oher branhes of algebra. In he fuure sudy of nerval valued fuzzy BF-algebras, perhaps he followng ops are worh o be onsdered: () To haraerze oher lasses of BF-algebras by usng hs noon; () To apply hs noon o some oher algebra sruures; (3) To onsder hese resuls o some possble applaons n ompuer senes nformaon sysems n he fuure. REFERENCES [] S. K. Bhaka, ( q) -level subses, Fuzzy Ses Sysems, 03 (999), [] S. K. Bhaka, (, q) normal, quasnormal mamal subgroups, Fuzzy Ses Sysem, (000), [3] S. K. Bhaka P. Das, (, q) subgroups, Fuzzy Ses Sysem, 80 (996), [4] R. Bswas, Rosenfeld s fuzzy subgroups wh nerval valued membershp funons, Fuzzy Ses Sysem, 63 (994), [5] A. K. Dua D. Hazarka, (, q) deals of BF-algebra, 8

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