Lower and Upper Approximation of Fuzzy Ideals in a Semiring

Transcription

1 nernaional Journal of Scienific & Engineering eearch, Volume 3, ue, January-0 SSN Lower and Upper Approximaion of Fuzzy deal in a Semiring G Senhil Kumar, V Selvan Abrac n hi paper, we inroduce he rough fuzzy ideal of a emiring We alo inroduce and udy rough fuzzy prime ideal of a emiring ndex Term Semiring, lower approximaion, upper approximaion, fuzzy ideal, fuzzy prime ideal, rough ideal NTODUCTON T he fuzzy e inroduced by LAZadeh [6] in 965 and he rough e inroduced by Pawlak [] in 98 are generalizaion of he claical e heory Boh hee e heorie are new mahemaical ool o deal he uncerain, vague, imprecie and inexac daa n Zadeh fuzzy e heory, he degree of memberhip of elemen of a e play he key role, wherea in Pawlak rough e heory, he equivalence clae of a e are ued o define he lower and upper approximaion of a e oenfeld [3] applied he noion of fuzzy e o group and inroduced he noion of fuzzy ubgroup Afer hi paper, many reearcher applied he heory of fuzzy e o everal algebraic concep uch a ring, field, vecor pace, ec The noion of rough ubgroup wa inroduced by Biwa and Nanda [] The concep of rough ideal in a emigroup wa inroduced by Kuroki in [] BDavvaz [3], [], [4] udied he roughne in many algebraic yem uch a ring, module, n-ary yem, H -group, ec Oman Kazanci and BDavvaz V [0] inroduced he rough prime and rough primary ideal in commuaive ring and alo dicued he roughne of fuzzy ideal in ring The roughne of ideal in BCK algebra wa conidered by YB Jun in [8] n [4] he preen auhor have udied rough ideal in emiring n hi paper, we inroduce he concep of rough fuzzy ideal of a emiring Alo we udy he noion of rough fuzzy prime ideal in a emiring CONGUENCE N SEMNGS Definiion A emiring i a nonempy e on which operaion G Senhil Kumar Deparmen of Mahemaic, Faculy of Engineering and Technology, SM Univeriy, Kaankulahur, Chennai , ndia genhilkumar77@gmailcom V Selvan Deparmen of Mahemaic, K M Vivekananda College, Chennai , ndia venelvan@yahoocoin, venelvan@gmailcom of addiion and muliplicaion have been defined uch ha he following condiion are aified (i), i a commuaive monoid wih ideniy elemen 0; (ii), i a monoid wih ideniy elemen ; (iii) Muliplicaion diribuive over addiion from eiher ide; (iv) 0r0 r0, for all r Throughou hi paper denoe a emiring Definiion [6] Le be an equivalence relaion on, hen i called a congruence relaion if ab, implie a x, b x, x a, x b, ax, bx and xa, xb for all x Theorem 3 [6] Le be a congruence relaion on, hen ab, and cd, imply a c, b d and ac, bd for all a, b, c, d Lemma 4 [6] Le be a congruence relaion on a emiring f a, b hen a b a a b b a b ab / a a, b b ab (i) /, (ii) Definiion 5 A congruence relaion on i called complee if (i) a b a b / a a, b b and (ii) ab ab / a a, b b for all a, b Definiion 6 A ideal of a emiring i a nonempy ube of aifying he following condiion: (i) f a, b hen a b (ii) f a and r hen ar, ra A ideal of a emiring define an equivalence relaion on, called he Bourne relaion, given by r r if and only if here exi elemen a and a of aifying r a r a The relaion i an congruence relaion on [6], [7] JSE 0 hp://wwwijerorg

2 nernaional Journal of Scienific & Engineering eearch, Volume 3, ue, January-0 SSN We denoe he e of all equivalence clae of elemen of under hi relaion by / and we will denoe he equiva- r lence cla of an elemen r of by Throughou hi paper denoe he Bourne congruence relaion induced by an ideal of a emiring Definiion 7 An ideal of a emiring i called a -ideal if ra implie r for each r and each a 3 LOWE AND UPPE APPOXMATON OF A FUZZY DEAL N A SEMNG A mapping : 0, i called a fuzzy ube of A fuzzy ube of a emiring i called a fuzzy ideal of if i ha he following properie: x y x y (i) (ii) xy x y A fuzzy ideal of i aid o be normal if 0 Definiion 3 A fuzzy ideal of a emiring i aid o be prime if i no a conan funcion and for any wo fuzzy ideal and of, implie eiher or Definiion 3 Le be he Bourne congruence relaion on induced by and be a fuzzy ube of Then we define he fuzzy e and a x a x x = a x = a a follow: The fuzzy e and he -lower and The pair, are repecively called -upper approximaion of he fuzzy e fuzzy e wih repec o i called a rough if Theorem 33 For every approximaion pace, and every fuzzy ube, of, we have: () () =, where i he characeriic funcion of empy e (3) =, where i he characeriic funcion of (4) f (5) f hen hen (6) = (7) = (8) = (9) = (0) = () () (3) = Proof i raighforward Theorem 34 Le be he Bourne congruence relaion on induced by he ideal of f i a fuzzy ideal of, hen i a fuzzy ideal of Proof We have, a xy b x c y b x c y x y = a b c b c b c b x c y x y Hence, x y x y Alo we have, a xy b x c y xy = a b x c y bc b c b c b x c y x y Hence, xy x y JSE 0 hp://wwwijerorg

3 nernaional Journal of Scienific & Engineering eearch, Volume 3, ue, January-0 3 SSN Therefore, i a fuzzy ideal of Theorem 35 Le be he Bourne congruence relaion on induced by he ideal of f i a fuzzy ideal of, hen i a fuzzy ideal of Proof We have, a xy b x c y b x c y x y = a bc b c b x c y b c x y Hence, x y x y Alo we have, xy = a a xy b x c y b x c y bc b c b x c y b c x y Hence, xy x y Therefore, i a fuzzy ideal of Le be a fuzzy ube of and, rough fuzzy e f and, hen, a are fuzzy ideal of a rough fuzzy ideal of x x x for all x f and are fuzzy ideal of a emiring of, hen i alo a fuzzy ideal of Corollary 36 f i a fuzzy ideal of hen, i a rough fuzzy ideal of Corollary 37 f and are fuzzy ideal of hen, i a rough fuzzy ideal of Le be a fuzzy ube of Then he e and x x where 0, x x are called repecively, -level ube and -rong level ube of Theorem 38 [7] Le be a fuzzy ube of Then i a fuzzy ideal of iff and are, if hey are non-empy, ideal of for every [0,] Theorem 39 Le be a congruence relaion on f i a fuzzy ube of and 0,, hen (i) (ii) Proof (i) We have (ii) We have x x a x x a a, a x a, a x x a x x a x a, for ome a x Le and be wo fuzzy ube of The incluion i defined by x x by for all x, and i defined JSE 0 hp://wwwijerorg

4 nernaional Journal of Scienific & Engineering eearch, Volume 3, ue, January-0 4 SSN x a, for ome a x x Lemma 30 f i a normal fuzzy ideal of a emiring and if hen Proof Since, we have x x x Tha i, a x, a x x, a x a x n paricular, 0, 0 a a A i a normal fuzzy ideal of, Hence, Thu, a 0, a 0 0, The convere of he above leema i rue if i a -ideal of Lemma 3 f i a ideal of uch ha Proof Since, we have x, x hen By Theorem 33, So, o complee he proof we need o how ha ie, we need o how ha, x x x ie, we need o how ha, x Cae (i) Le x and a Then a i x i, for ome i, i x a a x a i x i min x, i x n paricular,, a x a x Thi implie ha x x Cae (ii) Le x, hen x We have x 0 Since i a ideal, 0 We have, x a a x a x a Hence in hi cae alo, Combining Cae (i) and (ii), we ge 4 OUGH FUZZY PME DEALS An ideal P of a emiring i prime if and only if whenever HK P, for ideal H and K of, we mu have eiher H P or K P Theorem 4 Le be a fuzzy prime ideal of a emiring (i) f i a complee congruence relaion on and, hen i a lower rough fuzzy prime ideal of (ii) f i a complee congruence relaion on hen i a Proof upper rough fuzzy prime ideal of (i) Since i a fuzzy prime ideal, 0, i, if i i nonempy, a prime ideal of By Theorem 35 [4], we obain ha if i i nonempy, i a prime ideal of Hence by Theorem 39, i a prime ideal of and hence by Theorem 38, i a fuzzy prime ideal of (ii) can be een in a imilar way Theorem 4 Le be a complee congruence relaion on a emiring Then i a lower (an upper) rough fuzzy prime ideal iff for 0,, are, if hey are nonempy, lower [upper] rough prime ideal of Proof i raighforward Theorem 43 [4] Le f be an ono homomorphim of a emiring o a emiring and le be a congruence relaion on and A be a ube of Then (i) a, b / f a, f b i a congruence relaion on (ii) f i complee and f i one o one, hen i complee (iii) f A f A (iv) f A f A f f i one-o-one, hen f A f A JSE 0 hp://wwwijerorg

5 nernaional Journal of Scienific & Engineering eearch, Volume 3, ue, January-0 5 SSN Theorem 44 Le f be a urjecive homomorphim of a emiring o a emiring Le be a complee congruence relaion on and be a fuzzy ube of f x, y / f x, f y, hen (i) only if f of i a fuzzy ideal (fuzzy prime ideal) of if and i a fuzzy ideal (fuzzy prime ideal) (ii) f f i one o one, hen (fuzzy prime ideal) of if and only if f Proof fuzzy ideal (fuzzy prime ideal) of i a fuzzy ideal i a (i) By Theorem 35 [4] we obain ha i a fuzzy ideal (fuzzy prime ideal) of iff, if i i nonempy, an ideal (prime ideal) of, for every 0, We have by Theorem 39, Thu we obain ha iff f i an ideal (prime ideal) of Since f f, f f f Therefore, f every 0, Thu, f i an ideal (prime ideal) of i an ideal (prime ideal) of for i a fuzzy ideal (fuzzy prime ideal) of (ii) f f i one o one, hen Thi proof i imilar o ha of (i) f A f A [] BDavvaz, oughne in ring, nform Sci, vol 64, pp 47-63, 004 [3] BDavvaz, A new view of approximaion in Hv-group, Sof Compu, 0(), pp , 006 [4] BDavvaz, oughne baed on fuzzy ideal, nform Sci, vol 76, pp , 006 [5] BDavvaz, MMahdavipour, oughne in module, nform Sci, vol 76, pp , 006 [6] UHebich, HJWeiner, Semi ring - Algebraic Theory and Applicaion in Compuer Science, World Scienific Serie in Algebra, [7] Jonahan S Golan, Semiring and heir applicaion, Kluwer Academic Publiher, 999 [8] YBJun, oughne of ideal in BCK-algebra, Sci Mah Japonica, vol 57(), pp 65-69, 003 [9] OKazanci, BDavvaz, On he rucure of rough prime (primary) ideal and rough fuzzy prime (primary) ideal in commuaive ring, nformaion Science, vol 78, pp , 008 [0] NKuroki, ough ideal in emigroup, nform Sci, vol 00, pp 39-63, 997 [] NKuroki, PPWang, The lower and upper approximaion in a fuzzy group, nform Sci, vol 90, pp 0-30, 996 [] ZPawlak, ough e, n J nf Comp Sci, vol, pp , 98 [3] Aoenfeld, Fuzzy group, J Mah Anal Appl, vol 35, pp 5-57, 97 [4] VSelvan, GSenhil Kumar, ough ideal in emiring - ubmied for publicaion [5] QMXiao, ZLZhang, Neighbourhood operaor yem and aproxi maion, nform Sci, vol 44, pp 0-7, 00 [6] LAZadeh, Fuzzy e, nform Conrol, vol 8, pp , CONCLUSON The heory of emiring ha wide applicaion in everal area uch a opimizaion heory, dicree even dynamical yem, auomaa heory, formal language heory and parallel compuing The heory of fuzzy e and rough e alo ha many applicaion in he above area n hi paper, we developed he concep of a rough fuzzy ideal of a emiring We cerainly hope ha our work will be very ueful boh in he heoreical and applicaion apec We alo propoe o work furher on hi area o bring ou many more inereing properie of rough fuzzy ideal in emiring EFEENCES [] Biwa, SNanda, ough group and rough ubgroup, Bull Polih Acad Sci Mah, vol 4, pp 5-54, 994 JSE 0 hp://wwwijerorg