Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] Fuzz v-submodules in Γ-v-modules Avind Kuma Sinha 1, Manoj Kuma Dewangan 2 Depamen of Mahemaics NIT Raipu, Chhaisgah, India Absac In his pape, we inoduced he concep of Γ-v-modules, which is a genealizaion of Γ -modules vmodules. The noion of, q -fuzz v-submodules of a Γ-v-module is povided some elaed popeies ae invesigaed. Kewods peopeaion, Algebaic hpesucue, Γ-v-module,, q -fuzz v-submodule. I. INTRODUCTION The algebaic hpesucue is a naual genealizaion of he usual algebaic sucues which was fis iniiaed b Ma [1]. Afe he pioneeing wok of F. Ma, algebaic hpesucues have been developed b man eseaches. A sho eview of which appeas in [2]. A ecen book on hpesucues [3] poins ou hei applicaions in geome, hpegaphs, bina elaions, laices, fuzz ses ough ses, auomaa, cpogaph, codes, median algebas, elaion algebas, aificial inelligence pobabiliies. Vougiouklis [4] inoduced a new class of hpesucues so-called v-sucue, Davvaz [5] suveed he heo of v-sucues. The v-sucues ae hpesucues whee equali is eplaced b nonemp inesecion. The concep of fuzz ses was fis inoduced b Zadeh [6] hen fuzz ses have been used in he econsideaion of classical mahemaics. In paicula, he noion of fuzz subgoup was defined b Rosenfeld [7] is sucue was heeb invesigaed. Liu [8] inoduced he noions of fuzz subings ideals. Using he noion of belongingness quasi-coincidence q of fuzz poins wih fuzz ses, he concep of, -fuzz subgoup whee, of, q, q, q wih q subgoup is he noion of ae an wo was inoduced in [9]. The mos viable genealizaion of Rosenfeld s fuzz, q -fuzz subgoups, he deailed sud of which ma be found in [10]. The concep of an, q -fuzz subing ideal of a ing have been inoduced in [11] he concep of, q -fuzz subnea-ing ideal of a nea-ing have been inoduced in [12]. Fuzz ses hpesucues inoduced b Zadeh Ma, especivel, ae now sudied boh fom he heoeical poin of view fo hei man applicaions. The elaions beween fuzz ses hpesucues have been alead consideed b man auhos. In [13-15], Davvaz applied he concep of fuzz ses o he heo of algebaic hpesucues defined fuzz v-subgoups, fuzz v-ideals fuzz vsubmodules, which ae genealizaions of he conceps of Rosenfeld s fuzz subgoups, fuzz ideals fuzz submodules. The concep of a fuzz v-ideal v-subing has been sudied fuhe in [16, 17].The concep of (, q)-fuzz subhpequasigoups of hpequasigoups was inoduced b Davvaz Cosini [18]; also see [19-24]. As is well known, he concep of Γ ings was fis inoduced b Nobusawa in 1964, which is a genealizaion of he concep of ings. Davvaz e.al. [25] inoduced he noion of, q -fuzz v-ideals of a Γ-v-ing. This pape coninues his line of eseach fo, q -fuzz v-submodules of a Γ-v-module. The pape is oganized as follows: In Secion 2, we ecall some basic definiions of v-modules. In Secions 3 4, we inoduce he concep of Γ-v-modules pesen some opeaions of fuzz ses in Γ-v-modules. In Secion 5, b using a new idea, we inoduce invesigae he (, q)-fuzz v-submodules of a Γ-v-module. he inoducion of he pape should explain he naue of he poblem, pevious wok, pupose, he conibuion of he pape. The conens of each secion ma be povided o undes easil abou he pape. II. BASIC DEFINITIONS We fis give some basic definiions fo poving he fuhe esuls. Definiion 2.1[27] Le X be a non-emp se. A mapping : X [0, 1] is called a fuzz se in X. The complemen of c, denoed b, c is he fuzz se in X given b x 1 x x X. Definiion 2.2[28] Le G be a non emp se : G G ( G) be a hpeopeaion, whee ( G) is he se of all he non-emp subses of G. Whee A B a b, A, B G. aa, bb The is called weak commuaive if x x, x, G. Page 34
Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] The is called weak associaive if ( x ) z x ( z), x,, z G. A hpesucue G, is called an v-goup if (i) is weak associaive. (ii) a G = G a = G, a G (Repoducion axiom). Definiion 2.3[28] An v-ing is a ssem ( R,, ) wih wo hpeopeaions saisfing he ing-like axioms: i ( R, ) is an v-goup, ha is, (( x ) z) ( x ( z)) x, R, a R R a R a R; ii ( R, ) is an v-semigoup; iii () is weak disibuive wih espec o ( ), ha is, fo all ( x( z)) ( x z), (( x ) z) ( x z z). x,, z R Definiion 2.4[26] Le R be an v-ing. A nonemp subse I of R is called a lef (esp., igh) v-ideal if he following axioms hold: (i) ( I, ) is an v-subgoup of ( R, ), (ii) RI I (esp., I R I ). Definiion 2.5[26] Le ( R,, ) be an v ing a fuzz subse of R. Then is said o be a lef (esp., igh) fuzz v -ideal of R (1) min{ ( ), ( )} inf{ ( ) : },, if he following axioms hold: x z z x R x, a R (2) Fo all hee exiss R such ha xa min{ ( ), ( )} ( ), a x x, a R (3) Fo all hee exiss z R such ha xz a min{ ( a), ( x)} ( z), (4) ( ) inf{ ( z) : z x } ( x) inf{ ( z) : z x } x, R. especivel {x X (x) } Le μ be a fuzz subse of a non-emp se X le (0, 1]. The se is called a level cu of μ. Theoem 2.2[15] Le R be an v-ing μ a fuzz subse of R. Then μ is a fuzz lef (esp., igh) v-ideal of R if onl ( ) if fo eve (0, 1], is a lef (esp., igh) v-ideal of R. When μ is a fuzz v-ideal of R, is called a level v-ideal of R. The concep of level v-ideal has been used exensivel o chaaceize vaious popeies of fuzz v-ideals. III. Γ-V-MODULES The concep of Γ -ings was inoduced b Nobusawa in 1964. Definiion 3.1[29] Le (R, +) (Γ,+) be wo addiive abelian goups. Then R is called a Γ -ing if he following condiions ae saisfied fo all x,, z R fo all α, β Γ, (1) xα R, (2) (x + )αz = xαz + αz, x(α + β) = xα + xβ, xα( + z) = xα + xαz, (3) xα(βz) = (xα)βz. Definiion 3.2[25] Le (R, ) (Γ, ) be wo v-goups. Then R is called a Γ v-ing if he following condiions ae saisfied fo all x,, z R fo all α, β Γ, 1 x R, 2 (x ) z (xz z), x( ) (x ), x ( z) (x xz), 3 x ( z) (x) z. follows, unless ohewise saed, (R,, Γ) alwas denoes a Γ v-ing. Definiion 3.3[25] A subse I in R is said o be a lef (esp., igh) v-ideal of R if i saisfies (1) (I, ) is an v-subgoup of (R, ), (2) xα I (esp., αx I) fo all x R, I α Γ. In wha Page 35
Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] I is said o be an v-ideal of R if i is boh a lef a igh v-ideal of R. Now, we inoduce he concep of Γ v-modules as follows. Definiion 3.4 Le R,, be a - v -ing, ove R f : R M M if hee exiss a, b R, m1, m2 M,,, we have a b m a b m 1 a m m a m a m ; 1 2 1 2 2 a b m a m b m ; 1 1 1 3 a m a m am ; 1 1 1 4. 1 1 (he image of,, m M be a canonical v -goup. M is called a - being denoed b m Thoughou his pape, R M ae a - v -ing - v Definiion 3.5 A subse A in M is said o be a - v -submodule of M if i saisfies he following condiions: 1 A, is a v -subgoup of M, ; 2 x A, fo all R, x A. 4. Fuzz ses in Γ-v-modules v -module ) such ha fo all -module, especivel, unless ohewise specified. Le X be a non-emp se. The se of all fuzz subses of X F X. is denoed b Fo an A X fuzz subse A of X is defined b if x A, A x 0 ohewise, fo all x X. In paicula, when = 1, A ; is said o be he chaaceisic funcion of A, denoed b A x. is said o be a fuzz poin wih suppo x value is denoed b quasicoinciden wih) a fuzz se x 1 o x q., hen we wie, x wien as xq (esp., ) if A fuzz poin (0, 1], he when A = {x}, A x is said o belong o (esp., be x (esp., x 1 ). If x F X, Fo is said o have he sup-pope if fo an non-emp subse A of X, hee exiss x A x. A Nex we define a new odeing elaion elaion, as follows: F X Fo an,, q if onl if define he elaion on F(X) as follows: q F on X,, F X, q Fo an if onl if M 1 2 In wha follows, unless ohewise saed, n 1, 2,... n 0,1, all q means x x implies,,..., such ha which is called he fuzz inclusion o quasi-coincidence q. q does no hold q q fo all x X 0,1. And we whee n is a posiive inege, will denoe 1 2... n fo q implies is no ue. Lemma 4.1[30] Le X be a non-emp se μ, ν F(X). Then μ qν if onl if ν(x) M (μ(x), 0.5) fo all x X. Lemma 4.2[30] Le X be a non-emp se μ, ν, ω F(X) be such ha μ qν qω. Then μ qω. Cleal, Lemma 4.1 implies ha μ ν if onl if M(μ(x), 0.5) = M(ν(x), 0.5) fo all x X μ, ν F(X), i follows fom Lemmas 4.1 4.2 ha is an equivalence elaion on F(X). Now, le us define some opeaions of fuzz subses in a Γ v-module M. Definiion 4.3. Le μ, ν F(M) α Γ. We define fuzz subses b Page 36
Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] z M x, Γ. z I is woh noing ha fo an 1 2 1 2 Lemma 4.4. Le,,, F R q 1 1 2 2 fo all α Γ. z M x, zx F M, x M, s s s x s M, s. such ha 1 q2 q. 1 2 especivel, fo all z M α 1 q Then 1 1 2 2 2 1 1 q2 2. FR, Lemma 4.4 indicaes ha he equivalence elaion is a conguence elaion on Γ. Lemma 4.5. Le μ, ν, ω F(R). Then 1,. 3, 2,. F R, fo all α fo all α Γ. 4, 5 fo all α Γ. fo all α Γ. Poof. The poof of (1) (4) is saighfowad. We show (5). Le x R α Γ. If x z fo all, z R. Then x 0 x. Ohewise, we have x M, z M, M p, q xz xz zpq,,,, M p q M a b x z, zpq xa b, a p, b q M, z x. xz ence. IV. This complees he poof. (, Q)-FUZZY V-SUBMODULES OF A Γ V-MODULE In his secion, using he new odeing elaion on F (M), we define invesigae (, q)-fuzz lef (igh) vsubmodules of Γ v-module. Definiion 5.1. A fuzz subse μ of M is called an (, q)-fuzz lef (esp., igh) v-submodule if i saisfies he following condiions: q, (F1a) (F2a) as implies ha hee exiss M such ha x a, q, M s z q, M, s (F3a) as implies ha hee exiss z M such ha xz a (F4a) R q (esp., R q ) fo all α Γ. A fuzz subse μ of M is called an (, q)-fuzz v-submodule of M if i is boh an (, q)-fuzz lef an (, q)-fuzz igh v-submodule of M. Befoe poceeding, le us fis povide some auxilia lemmas. Lemma 5.2. Le μ F (M). Then (F1a) holds if onl if one of he following condiions holds: (F1b) (F1c) z s M s implies, z M x,,0.5 q fo all x, M, s (0, 1]. fo all x, M. Page 37
Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] Poof. (F1a) (F1b) Le x, M, s (0, 1] be such ha s. Then fo an z x, we have z M a, b M M, s. ence M, s za b M s so z, q. I follows fom (F1a) ha M, s q. (F1b) (F1c) Le x, M. If possible, le z M be such ha z x μ (z) < = M (μ(x), μ(), 0.5). Then x, μ (z) + < + 1, ha is, zq, a conadicion. ence (F1c) is valid. (F1c) (F1a) Fo if possible, le xq, Then μ(x) < μ(x) < 0.5. If z fo some, z M, b (F1c), we have 0.5 > μ(x) M (μ(), μ(z), 0.5), which implies μ(x) M(μ(), μ(z)). x M a, b x x, ence we have xa b xa b a conadicion. ence (F1a) is saisfied. Lemma 5.3. Le μ F (M). Then (F2a) holds if onl if he following condiion holds: (F2b) fo all x, a M hee exiss M such ha x a M (μ (a), μ(x), 0.5) μ(). Poof. (F2a) (F2b): Suppose ha x, a M. We conside he following cases: (a) M (μ(x), μ (a)) < 0.5, (b) M (μ(x), μ (a)) 0.5. Case a: Assume ha fo all wih x a, we have μ () < μ (x) μ (a). Choose such ha μ() < < M (μ(x), μ (a)). a q, bu Then which conadics (F2a). Case b: Assume ha fo all wih x a x, we have μ() < M (μ(x), μ (a), 0.5). Then 0.5, a0.5, bu 0.5 q, which conadics (F2a).. (F2b) (F2a): Le a Then μ(x) μ (a). Now, fo some wih x a, we have μ () M(μ(a), μ(x), 0.5) M(,, 0.5). If M (, ) > 0.5, hen μ() 0.5 which implies μ() + M (, ) > 1. If M (, ) 0.5, hen μ() M (, ). q., M Theefoe, ence (F2a) holds. Lemma 5.4. Le μ F(M). Then (F3a) holds if onl if he following condiion holds: (F3b) fo all x, a M hee exiss z M such ha xz a M (μ(a), μ(x), 0.5) μ(z). Poof. I is simila o he poof of Lemma 5.3. Lemma 5.5. Le μ F(M) α Γ. Then (F4a) holds if onl if one of he following condiions holds: x implies x1 q q (F4b) (esp., x 1 ) fo all x,, z M (0, 1]. (z) M( (),0.5) (z) M( (x),0.5)) z z (F4c) (esp., fo all x, M α Γ. Poof. The poof is simila o he poof of Lemma 5.2. Le μ be a fuzz subse (0, 1]. Then he se [ ] {x R x q } is called he q-level subse of μ. The nex heoem povides he elaionship beween (, q)-fuzz lef (esp., igh) v-submodules of Γ v-module cisp lef (esp., igh) v-submodules of Γ v-module. F(M). Theoem 5.6. Le Then (1) μ is an (, q) ( ) -fuzz lef (esp., igh) v-submodule of M if onl if is a lef (esp., igh) vsubmodule of M fo all (0,0.5]. (2) μ is an (, q) -fuzz lef (esp., igh) v-ideal of R if onl if [ ] ([ ] ) is a lef (esp., igh) v-ideal of R fo all (0,1]. x Page 38
Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] Poof. We onl show (2). Le μ be an (, q) -fuzz lef v-submodule of M x, [ ] fo some (0,1]. Then μ(x) o μ(x) > 1 μ() o μ() > 1. Since μ is an (, q) -fuzz lef v-submodule of M, we have μ(z) M(μ(x), μ(), 0.5) fo all z. We conside he following cases: (a) (0,0.5]. (b) (0.5,1]. Case a: Since (0,0.5]. We have 1 0.5 so μ(z) M(,, 0.5) = o μ(z) M(, 1, 0.5) = o μ(z) M(1, 1, 0.5) = 0.5 fo all z. ence z fo all z. Case b: Since (0.5,1]. We have 1 < 0.5 < so μ(z) M(,, 0.5) = 0.5 o μ(z) > M(, 1, 0.5) = 1 o μ(z) > M(1, 1, 0.5) = 1 fo all z. ence z q fo all z. Thus in an case, z [ ] fo all z.,z [ ] Similal we can show ha hee exis such ha x a xz a fo all x,a [ ] ha x [ ] fo all x M, [ ]. [ ] Theefoe, is a lef v-submodule of M. F(M) Convesel, le [ ] is a lef v-submodule of M fo all (0,1]. x, M. Le If hee exiss z M such ha μ(z) < = M(μ(x), μ(), 0.5), hen x, [ ] bu z [ ], a conadicion. Theefoe, μ(z) M(μ(x), μ(), 0.5) fo all z. Similal we can show ha hee exis,z M x a, x z a, such ha μ() M(μ(x), μ(a), 0.5) μ(z) M(μ(x), μ(a), 0.5) ha μ(xα) M(μ(), 0.5) fo all x, M. Theefoe, μ is an (, q) -fuzz lef v-submodule of M b Lemmas 5.2 5.5. The case fo (, q) -fuzz igh v-submodules of M can be similal poved. REFERENCES [1] F. Ma, Su une genealizaion de la noion de goupe, in: 8h Congess Mah. Scianaves, Sockholm, 1934, pp. 45 49. [2] P. Cosini, Polegomena of pegoup Theo, second ed., Aviani Edio, 1993. [3] P. Cosini, V. Leoeanu, Applicaions of pesucue Theo, in: Advances in Mahemaics (Dodech), Kluwe Academic Publishes, Dodech,2003. [4] T. Vougiouklis, The fundamenal elaion in hpeings. The geneal hpefield, in: Poc. 4 h Inenaional Congess on Algebaic pesucues Applicaions, Xanhi, 1990, Wold Sci. Publishing, Teaneck, NJ, (1991), 203-211. [5] B. Davvaz, A bief suve of he heo of v-sucues, in: Poc. 8h In. Congess AA, Geece Spanids Pess, 2003, pp. 39 70. [6] L. A. Zadeh, Fuzz ses, Infom. Conol, 8 (1965), 338-353. [7] A. Rosenfeld, Fuzz goups, J. Mah. Anal. Appl. 35 (1971) 512 517. [8] W.J. Liu, Fuzz invaian subgoups fuzz ideals, Fuzz Ses Ssems 8 (1982) 133 139. [9] S.K. Bhaka, (, q) -fuzz nomal, quasinomal maximal subgoups, Fuzz Ses Ssems 112 (2000) 299 312. [10] S.K. Bhaka, P. Das, (, q) -fuzz subgoups, Fuzz Ses Ssems 80 (1996) 359 368. [11] S.K. Bhaka, P. Das, Fuzz subings ideals edefined, Fuzz Ses Ssems 81 (1996) 383 393., q [12] B. Davvaz, -fuzz subnea-ings ideals, Sof Compuing, 10 (2006), 206-211. [13] B. Davvaz, Fuzz v -goups, Fuzz Ses Ssems, 101 (1999), 191-195. [14] B. Davvaz, Fuzz v -submodules, Fuzz Ses Ssems, 117 (2001), 477-484. [15] B. Davvaz, On v -ings fuzz v -ideals, J. Fuzz Mah., 6 (1998), 33-42. [16] B. Davvaz, T-fuzz v -subings of an v-ing, J. Fuzz Mah., 11 (2003), 215-224. [17] B. Davvaz, Poduc of fuzz v -ideals in v -ings, Koean J. Compu. Appl. Mah., 8 (2001), 685-693. [18] B. Davvaz, P. Cosini, Genealized fuzz sub-hpequasigoups of hpequasigoups, Sof Compu. 10 (11) (2006) 1109 1114. [19] B. Davvaz, P. Cosini, Genealized fuzz hpeideals of hpenea-ings man valued implicaions, J. Inell. Fuzz Ss. 17 (3) (2006) 241 251. [20] B. Davvaz, P. Cosini, On (α, β)-fuzz v-ideals of v -ings, Ian. J. Fuzz Ss. 5 (2) (2008) 35 47. Page 39
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