Characterization of Gamma Hemirings by Generalized Fuzzy Gamma Ideals

Transcription

1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 10, Issue 1 (June 2015), pp Applicaions and Applied Mahemaics: An Inernaional Journal (AAM) Characerizaion of Gamma Hemirings by Generalized Fuzzy Gamma Ideals Muhammad Gulisan, Muhammad Shahzad, Sarfraz Ahmed, and Mehwish Ilyas Deparmen of Mahemaics Hazara Universiy Mansehra, Pakisan Received: March 5, 2014; Acceped: Ocober 30, 2014 Absrac This paper has explored heoreical mehods of evaluaion in he idenificaion of he boundedness of he generalized fuzzy gamma ideals. A funcional approach was used o underake a characerizaion of his srucure leading o a deerminaion of some ineresing gamma hemirings heoreic properies of he generaed srucures. Gamma hemirings are he generalizaion of he classical agebraic srucure of hemirings. Our aim is o exend his idea and, o inroduce he concep of generalized fuzzy gamma ideals, generalized fuzzy prime (semiprime) gamma ideals, generalized fuzzy h -gamma ideals and generalized fuzzy k - gamma ideals of gamma hemirings and relaed properies are invesigaed. We have shown ha inersecion of any family of generalized fuzzy (lef, righ) h - gamma ideals k -gamma ideals of a hemiring is a generalized fuzzy (lef, righ) h -gamma ideal k -gamma ideal of H. Similarly we proved ha he inersecion of any family of generalized fuzzy prime (resp. semiprime) gamma ideals of H is a generalized fuzzy prime (resp. semiprime) gamma ideal of H. We have proved ha a fuzzy subse of H is fuzzy h -gamma ideal k -gamma ideal if and only if is a generalized fuzzy h -gamma ideal k -gamma ideal of H. Furher level cus provide a useful linkage bewean he classical se heorey and he fuzzy se heorey. Here we use his linkage o invesigae some useful aspecs of gamma hemirings and characerize he gamma hemmirings hrough level cus in erms of generalized fuzzy (lef, righ, prime, semiprime) gamma ideals of gamma hemirings. We have also used he concep of suppor of a fuzzy se in order o obain some ineresing resuls of gamma hemirings using he generalized fuzzy (lef, righ, prime, semiprime) gamma ideals of hemirings. Keywords: -Hemirings,,, -fuzzy -ideals;, -fuzzy prime -ideals;, -fuzzy h - -ideals;, -fuzzy k - -ideals;, fuzzy h - -ideal;, fuzzy k- -ideal MSC 2000 No.: 20M10 and 20N semi prime;

2 496 Muhammad Gulisa e al. 1. Inroducion Hemirings provide a common generalizaion of rings and disribuive laices ha arise in such diverse areas of mahemaics such as combinaorics, funcional analysis, graph heory, auomaa heory, formal language heory, mahemaical modeling of quanum physics and parallel compuaion sysem Aho e al.(1979), Glazek (2002), Golan (1999) and Hebisch and Weiner (1998). The concep of fuzzy ses was firs inroduced by Zadeh (1965). Researchers used hese noions in various algebraic srucures; see Yaqoob (2013), Yaqoob e al. (2012), Yaqoob e al. (2012), Yaqoob e al. (2013) and Yaqoob e al. (2013). As a generalizaion of -rings and of semirings he concep of -semirings was inroduced by Rao (1995). Ideals of semirings play an imporan role in srucure heory and are useful for many purposes. However, hey do no in general coincide wih he usual ring ideals if a semiring is a ring and, for his reason, heir use is somewha limied in rying o obain analogues of ring heorems for semirings. Many resuls in rings apparenly have no analogues in semirings using only ideals. In his regard Henriksen (1958) defined a new class of ideals namely k -ideals in semirings. However a sill more resriced class of ideals in hemirings has been given by Iizuka (1959). The definiion of ideal in any commuaive semiring coincides wih Iizuka's definiion provided ha S is a hemiring, and i is called h -ideal. Abdullah (2013) and Abdullah e al. (2011) inroduced he idea of, - Inuiionisic fuzzy ideals of hemirings and N-dimensional, -fuzzy H-ideals in hemirings, respecively. Recenly he concep of prime (semiprime) fuzzy h -ideals in -hemirings were inroduced by Sardar and Mandal (2009) and Sardar and Mandal (2011). For more abou, - fuzzy ideals, Gulisan e al. (in press) and Zadeh (1965). In his paper, we characerize he differen ypes of, -fuzzy -ideals of -hemirings. 2. Preliminaries We will recall some useful definiions and resuls which will be helpful in furher pursui of his sudy. A semiring is an algebraic sysem ( R,, ) consising of a non-empy se R ogeher wih wo binary operaions called addiion and muliplicaion (denoed in he usual manner) such ha ( R, ) and ( R,. ) are semigroups conneced by he following disribuive laws: a( b c) ab ac and ( b c) a ba ca, valid for all a, b, c R. An elemen 0 R is called a zero of R if a 0 0 a a and 0aa0 0, for all a R and an elemen 1 R is called he ideniy of R if 1a a1 a, for all a R. A semiring wih zero and a commuaive addiion is called a hemiring. A non-empy subse A of a semiring R is called a subsemiring of R if i is closed wih respec o he addiion and muliplicaionin R. A non-empy subse I of a semiring R is is lef (righ) ideal if i is closed wih respec o he addiion and RI I ( IR I). If A is lef and righ ideal of R hen i is an ideal of R. An ideal I of R is called a lef, righ k -ideal of R if for any a, b I and all x R from x a b x I. lef, righ h -ideal An ideal I of R is called a

3 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 497 of R if for any a, b I and x, y R from x a y b y x I. lef, righ h -ideal of R is lef, righ k -ideal of R bu converse is no rue. An ideal I of R is called prime 2 x I Every (semiprime) if xy I, implies x I or y I x I. h of R is 2 called prime (semiprime) if i is prime (semiprime) as an ideal of R, i.e if xy I x I implies x I or y I x I. Definiion 1. (Sardar and Mandal, 2009) A k -ideal -ideal Le H and be wo addiive commuaive semigroups wih zero. Then H is called a - hemiring if here exiss a mapping H H H x,, y x y, saisfying he following condiions: i a b c a c b c, ii a b c a b a c, iii a b a b ab, iv a bc a b c, v 0 a0 a0, H H H vi a0 b 0 b0 a, for a, b, c H and,. given by Throughou his paper H will denoe a -hemiring. A nonempy subse A of H is called a - subhemiring of H if AA A and AA A. Definiion 2. (Rao, 1995) By a fuzzy subse of H we mean any map f : H [0,1]. Definiion 3. (Sardar and Mandal, 2009) A fuzzy subse of a -hemiring H is called a fuzzy -subhemiring of H if for all x, y H, we have () i ( x y) min{ ( x), ( y)}, ( ii ) ( x y) min{ ( x), ( y)}.

4 498 Muhammad Gulisa e al. Definiion 4. (Rao, 1995) A fuzzy subse of a -hemiring H is called a fuzzy lef (righ) -ideal of H if for all x, y H i saisfies: () i ( x y) min{ ( x), ( y)}, ( ii ) ( x y) ( y) ( ( x y) ( x) ). Noe ha a fuzzy -ideal of H can be defined as a fuzzy subse of H for which all x, y H, saisfies ( x y) min{ ( x), ( y)} and ( x y) max{ ( x), ( y)}. Every fuzzy (lef, righ) -ideal is a fuzzy -subhemiring. Definiion 5. (Rao, 1995) A fuzzy lef, righ -ideal of H is called a fuzzy lef, righ k - -ideal if and fuzzy lef, righ h - -ideal of H if for all x, y, z, a, b H. Definiion 6. (Rao, 1995) x y z x min y, z, x a z b z x min a, b, A fuzzy -ideal of H is called prime (semiprime) if ( x y) ( x) 3., -Fuzzy -Ideals or y x x x ( ) ( ), for all x, y H and. Le H be a -hemiring and {, q, q}, {, q, q, q} unless oherwise specified. Definiion 7. A fuzzy subse of a -hemiring H is called, -fuzzy righ (resp. lef) -ideal of H if i x, y implies a b x y min,, ab

5 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 499 ii x, y implies a b x y min ab, ab, (0,1] and,. ( resp. ab y x for all x, y H and min, ) A fuzzy subse of a -hemiring H is called, -fuzzy -ideal of H if i is boh a fuzzy righ and a fuzzy lef -ideal of H. Theorem 1. Le be a non-zero, -fuzzy (lef, righ) -ideal of H. Then, he se is a (lef, righ) -ideal of H. Le x, y, x x As so, H H { x H x 0}, hen x 0 and y 0. Assume ha x y 0. If, q, y. y x y 0 min x, y and x y x y a conradicion. Hence, Le x, y H hen So, min, 0 1 1, x y min x, y, for every, q, q, q x y 0,, ha is, x y H., hen, H and. Then, x 0 and assume ha xy 0. If, q x, bu xy 0 min x, x and xy min x, x x x y f min x, x and y x min x, x f, for every, q, q, q, a conradicion. Hence, xy 0, ha is, x y H and y x H. Thus, he se

6 500 Muhammad Gulisa e al. is an (lef, righ) -ideal of H. H { x H x 0} Theorem 2. Le I be a lef resp. righ -ideal of H and le be a fuzzy subse in H such ha, Then, i is an ii is a q, iii is a i, q x -fuzzy lef q-fuzzy lef q, q -fuzzy lef 0, if xh I, 0.5, if x I. resp. righ -ideal of H, resp. righ -ideal of H, resp. righ -ideal of H. Le x, y H, and 1, 2 (0,1] such ha x and y. Then and y. So x y 0.5. Thus, if x y 0.5 min 1, 2 and so x y min. If 1, 2 x y min 1, and so x y min, q. Therefore, min, Thus, x, y I and so x y I. Le x, y H, and (0,1] such ha x Consequenly, xy 0.5. Thus, if 0.5, hen x y , hen x y an, q -fuzzy lef -ideal of H. min, 0.5, hen min, 0.5, hen x y qf.. Then, x. Thus, x I and so x y I. and so xy. If q xy q. Thus, is and so xy. Therefore, Similarly yx q. Hence, is an, q ii -fuzzy -ideal of H. yq. Then, 1 y 1. As x y I. So, x y 0.5. If min 1, x y 0.5 min 1, 2 y min,. If min 1, Le x, y H, and 1, 2 (0,1] such ha xq and and, hen, hen

7 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 501 x y min 1, and so x y min, q. Therefore, min, Le x, y H, and (0,1] such ha xq. Then, 1 x y q. x. Thus, x I and so x y I and y x I. Consequenly, xy 0.5, and y x 0.5. xy 0.5 and so xy. If 0.5, hen x y xy q. Therefore, xy q. Thus, is a q, q-fuzzy lef -ideal of H. Similarly yx q. Hence is a q, q ( iii ) Follows from i and ii. Definiion 8. -fuzzy -ideal of H. Thus, if 0.5, hen and so An, -fuzzy -ideal of H is called an, -fuzzy k - -ideal of H if x a b, ar and bs implies xmin rs, and if x a y b y, ar and bs implies xmin rs, for all a, b, x, yh, and, (0,1] Theorem 3. rs hen i is called, -fuzzy h - -ideal of H. Le be a non-zero, -fuzzy h - -ideal k- -ideal of H. Then, he se is a -ideal of H. H { x H x 0} Sraighforward. Theorem 4. Le be a non-zero, -fuzzy h - -ideal k- -ideal of H. Then, he se is an h - -ideal ( k - -ideal) of H. H { x H x 0}

8 502 Muhammad Gulisa e al. By Theorem 3, he se a 0, b 0, hen H { x H x 0} is an h - -ideal of H. Le a, b, so and x, y H be such ha a x y b y. a x, b x, bu xmin a, b Also aq 1 and bq 2, bu 1 x 0, ha is x H Le x 0. If, q f and for every, q, q, q x for every, q, q, q. Thus, he se H { x H x 0} H,, a conradicion., a conradicion. Hence, is an h - -ideal of H. For he proof of, -fuzzy k - -ideal pu y 0 in he above resul. Theorem 5. Le I be an h - -ideal k- -ideal of H and le be a fuzzy subse in H such ha, Then, x 0, if xh I, 0.5, if x I. i is an, q -fuzzy h - -ideal - -ideal k of H, ii is a q, q-fuzzy h - -ideal - -ideal k of H, iii is an q, q -fuzzy h - -ideal - -ideal k of H. i By Theorem 2, is an, q -fuzzy -ideal of H. Le us assume ha I is an h - -ideal of H and le a, b, x, y H such ha x a y b y. Le a, b a 1 and b. 2 So by hypohesis a, b I. x 0.5. If min 1, 2 0.5, hen clearly x min 1, 2 min, 0.5. Then, x min, 1, is a, q-fuzzy h - -ideal of H. ii and iii Easy o prove. and so min 1, 2. for some 1, 2 (0,1]. So As I is an h - -ideal, also x I. Hence, so xmin, and if x q Hence, xmin, q. Thus,

9 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 503 If we pu 0 Theorem 6. y we ge he required resul for, q -fuzzy k - -ideal of H. A fuzzy subse of H is an, -fuzzy -ideal of H if and only if i is a fuzzy lef, righ -ideal of H. Sraighforward. Corollary 1. A fuzzy subse of H is fuzzy h - -ideal -ideal k- -ideal of H. Sraighforward. Lemma ideal k if and only if is an, For any fuzzy subse of H he following condiions are equivalen: i x 1, y 2 implies x y, min, q ii x y x y min,,0.5, for all x, y H. fuzzy h - i ii Le x, y H and le us consider he case when min x, y 0.5. For equaion ( ii ) is valid. Now if here exis such ha x y min x, y x y min x, y,

10 504 Muhammad Gulisa e al. which implies ha x, y, x y min x, y, bu. x y q Bu, his is a conradicion. Now consider he case when min x, y 0.5. As x y 0.5, x, y, Thus, ii Le Thus, and so we have x y bu x y 0.5 q, which is again a conradicion. We have 0.5. i x, y, hen This means ha x y x y 0.5 min,,0.5, for all x, y H. x y min x, y,0.5 min,,0.5. x y min, for min, 0.5, x y min, for x y q. min 1, 2 min, 0.5. Lemma 2. For any fuzzy subse of H he following condiions are equivalen. i x, y H ii y x x Imply yx q, min,0.5, For all x, y H and. Sraighforward.

11 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 505 Lemma 3. For any fuzzy subse of H he following condiions are equivalen: i x, y H ii x y x xy Imply, q min,0.5, For all x, y H and. Sraighforward. Lemma 4. For any fuzzy subse of H he following condiions are equivalen: i x, y H yx xy Imply q, and ii x y x y q for all x, y H and, min max{, },0.5, For all x, y H and. Sraighforward. Lemma 5. Le be a fuzzy subse of H and x, y, a, b H be such ha x a y b y. Then, he following condiions are equivalen: i min, a, b x q, ii x a b min,,0.5. i ii Le x a b x min a, b and here exis such ha x min a, b., and x q, which is conradicion o hypohesis. So a b Le x, y, a, b H be such ha x a y b y. a b for x 0.5, min,,0.5. Since for This implies min, 0.5. Also we have a0.5, b0.5 and x0.5 q, which is conradicion o hypohesis. So

12 506 Muhammad Gulisa e al. x a b min,,0.5. ii i Le x a y b y and a, b for some x, y, a, b H and 1, 2 (0,1]. Then, x min a, b,0.5 min 1, 2,0.5. Thus, x min 1, 2 for and x 0.5, for min, 0.5. x q Theorem 7. A fuzzy subse of H is an Therefore, min 1, 2., q -fuzzy lef saisfies x y min x, y,0.5 and y x x. Sraighforward. Theorem 8. A fuzzy subse of H is an, q -fuzzy -ideal of H if and only if is a -ideal of H for all Le be an Then., q ; : U xh x -fuzzy -ideal of H. Assume ha x, y U ; min, 0.5, resp. righ -ideal of H if and only if i x y min x, y,0.5 min,0.5, which implies x y U ;. Moreover, for all x U ; min,0.5, for all x, y H and for some and y H we have x y min x,0.5 min,0.5, and similarly yx, which implies ha x y y x U ; :, ;. Hence, U xh x

13 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 507 is a -ideal of H for all Conversely, assume ha for every 0 0.5, ; : U xh x is a -ideal of H. Then, x y x y and here exis (0,1] such ha min,,0.5, for all x, y H. Le x y x y min,,0.5 x y x y min,,0.5. This implies ha x, y U ;, bu x y U ;, which is conradicion o hypohesis. So x y x y min,,0.5 and in a similar way x y min x,0.5 and y x y Hence, is an, q -fuzzy -ideal of H. Corollary 2. A fuzzy subse of H is an only if, q min,0.5 for all x, y H. -fuzzy (lef, righ) h - -ideal - -ideal ; : U xh x is a (lef, righ) h - -ideal k- -ideal of H for all Le be an, q -fuzzy (lef, righ) h - -ideal - -ideal for some a, b U ; x y ideal of H. If x a y b y Thus, k of, and, H. Then, x min a, b,0.5 x U ;. k of H if and H hen U; is a -

14 508 Muhammad Gulisa e al. ; : U xh x is a (lef, righ) h - -ideal k- -ideal of H for all Conversely, assume ha ; : U xh x is a (lef, righ) h - -ideal k- -ideal of H for all Then, by above heorem is an, q fuzzy -ideal of H. Le us assume ha x a y b y for some ab,, x, y H. If x a b min,,0.5, hen here exis some (0,1] such ha which implies a, b U ; bu x U Thus, is an Theorem 9., q x a b min,,0.5, ;, which is conradicion o hypohesis. Hence, x a b min,,0.5. -fuzzy (lef, righ) h - -ideal - -ideal The inersecion of any family of fuzzy (lef, righ) -ideals of H., q k of H. - -fuzzy (lef, righ) -ideals of H is an, q Le i : i be a fixed family of, q -fuzzy (lef, righ) -ideals of H and be he inersecion of his family. We have o show ha x y min, q and x y y x q for all x, y and y H. Le us assume ha min, x y q for some x, y H and 1, 2 (0,1]. Then,

15 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 509 x y min, and x y Thus, x y 0.5. As each : i is an, q can be divided ino wo disjoin pars: i min, 1. -fuzzy -ideal of H, so he family and i : i x y min, 1 x y x y : min, and min, 1. 2 i i i which is a conradicion. If i x y min 1, 2 for all i, hen also x y min 1, 2, So for some i, we have i x y min 1, 2 and i x y min, 0.5, whence min, 0.5, for all. 2 x and y i i i x x y for all. min, 1. Thus, i x y for In he similar way i 0.5 i Now le us assume ha i 0.5 some i. Le 0,0.5 such ha, hen i x 0.5 and 0.5, bu i x y and i x y 1. So x y q, no possible because for all x, y In he similar way, we can easily prove ha i y ha is which is H we have x y 0.5. Therefore, x y q i. min 1, 2 x y y x q for all x, y and y H. Hence, inersecion of any family of fuzzy (lef, righ) -ideals of H. Corollary 3., q - -fuzzy (lef, righ) -ideals of H is an, q The inersecion of any family of, q -fuzzy (lef, righ) h - -ideals k- -ideals of H is an, q k- -ideals of H. -fuzzy (lef, righ) h - -ideals Sraighforward. Definiion 9. For any fuzzy subse of H we define Q x H x q, : and

16 510 Muhammad Gulisa e al. Theorem 10. x H : x q for any (0,1]. A fuzzy subse of H is an, q -fuzzy -ideal of H if and only if xh : x q is a -ideal of H for all Le be an, q -fuzzy -ideal of H. Assume ha xy for some (0,1]. Then,, x 1 or x 1 and y 1 or y 1. As of H is an, q -fuzzy -ideal, so we have So ha x y. for any.. Hence, Also x y min x, y,0.5 min,0.5. x y min x, y,0.5 min,0.5, xy Thus, and in he similar way, is a -ideal of H for all Conversely, le xh : x q x H : x q be a -ideal of H for all If here exis some (0,1] such ha x y x y min,,0.5. yx for all x, y H and Then. bu x y, xy,, which is a conradicion. Hence,

17 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 511 and in he similar way x y x y min,,0.5 x y min x, y,0.5, y x x y and. Hence, is an, q -fuzzy -ideal of H. Theorem 11. A fuzzy subse of H is an only if, q -fuzzy min,,0.5 for all x, y H lef, righ h - -ideal k- -ideal of H if and is a lef, righ h - -ideal K- -ideal of H for all Le Then, be an, q -fuzzy lef, righ h - -ideal k- -ideal of H. xh : x q is a -ideal of H for all Assume ha x a y b y for some x, y H and ab. Then, a 1 or a 1 and b 1 or b 1., we have -fuzzy, q Since is an lef, righ h - -ideal k- -ideal of H, x x y min,,0.5. This implies x min,0.5. Indeed for x, we obain a is a conradicion. So x. Now if 0.5, hen q or b q, which x,0.5, ha is x. Hence

18 512 Muhammad Gulisa e al.. x If 0.5, Therefore,. hen x, Thus if x x This shows ha xh : x q is a lef, righ h - -ideal k- -ideal of H for all Conversely, if xh : x q and xq. is a lef, righ h - -ideal k- -ideal of H for all Then, i is an, q -fuzzy -ideal of H. Le a, b, x, y H be such ha x a y b y. There exis some such ha x min x, y,0.5. This implies 0.5, x 1, a, b. Thus, x Q,, x U, and a, bu,, which conradics o he given hypohesis. Hence, x min x, y,0.5, which shows ha is an, q -fuzzy lef, righ h - -ideal k- -ideal of H. Lemma 6. Le be an arbirary se defined on H and x H. x U, s for all s. Sraighforward. Theorem 12. Le A, where F 0,0.5 be he collecion of lef F Then x if and only if x U,, righ h - -ideals of H such ha H F Ai and for s, F, s if and only if A As. Then, a fuzzy se defined by x sup F x A is an, q -fuzzy lef, righ h - -ideal of H. To prove he heorem i will be sufficien o show ha each non-empy U, is a lef, righ h - -ideal of H. Le us consider he wo cases:

19 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 513 i s F s sup, ii s F s sup. If we consider he firs case So, xu, x A, for all s x A. s s s U, A, which is a lef, righ h - -ideal of H. Now considering he second case we have o show ha U, A. For i le x A x A, for some s, which implies ha s s s s s s x s x A U s s s,. Now for he inverse inclusion le us consider x s As x As here exis 0 x A s, hen. for all sup s F s, such ha, F. Hence, x As s, which shows ha if s Thus x and so x U,. Hence, U, s As. Also A is a lef righ h - -ideal of H. Hence, U, is a lef righ i is easy o show ha s s h - -ideal of H. 3. Prime, -fuzzy -ideals Definiion 10. An, -fuzzy -ideal of H is called semiprime if for all x H, and (0,1], x x x. An, -fuzzy -ideal of H is called prime if for all x, y H, and (0,1], x y x or y. An, -fuzzy h - -ideal k- -ideal of H is called prime (semiprime) if i is prime (semiprime) as an, -fuzzy -ideal.

20 514 Muhammad Gulisa e al. Proposiion 1. An, q -fuzzy -ideal of H is prime if and only if x y x y max, min,0.5 for all x, y H and. Le be an, q -fuzzy prime -ideal of H. Le for any x, y H and here exis (0,1] such ha x y x y max, min,0.5. This means ha x y H bu x q and y q, which is a conradicion o given hypohesis. Hence, Conversely, assume ha Then, x y x y max, min,0.5, for all x, y H and. x y x y max, min,0.5, for all x, y H and. x y max x, y,0.5 min,0.5. For 0.5 we have ha eiher x or y. I means ha eiher x or y. 0.5, we have max x, y 0.5, i.e, eiher x or y 1. Thus, eiher x Corollary 4. q or y q. Therefore, is a fuzzy prime. An, q -fuzzy h - -ideal k- -ideal of H is prime if and only if Sraighforward. x y x y max, min,0.5, for all x, y H and. For

21 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 515 Theorem 13. An, q -fuzzy -ideal of H is prime if and only if for all each non-empy U, is a prime -ideal of H. Le us suppose ha is an, q -fuzzy -ideal of H and le (0,0.5]. Then, each nonempy U, So, of H. is a -ideal of. H By Proposiion 1, for each xy U, we have max x, y min x y,0.5 min,0.5. x or y. Thus, x U, or y U,. Hence, U, Conversely le us assume ha U, is a prime -ideal is a prime -ideal of H for each (0,0.5], hen i is obviously is an, q -fuzzy -ideal of H. Le xy and 0.5. Then, xy U,. So eiher x U, or y U,. This implies ha x 0.5 and xy, we have xy 0.5. Thus, xy U x U,0.5 or y U,0.5 xq or yq. Hence, is an, q ideal of H. Corollary 5. An, q. Therefore, or y or,0.5, when -fuzzy - k of H is prime (resp. semiprime) if and only if for -fuzzy h - -ideal - -ideal each non-empy U, is a prime (resp. semiprime) h - -ideal - -ideal all H. Sraighforward. Theorem 14. k of A non-empy subse I of H is a prime -ideal if and only if a fuzzy subse of H such ha x 0, if xh I, 0.5, if x I,

22 516 Muhammad Gulisa e al. is an, q -fuzzy prime -ideal of H. Le I of H is a prime -ideal. Then, is an, q -fuzzy -ideal of. hen x y 0.5, y. x and consequenly. Hence, q or y. H. H If xy, x y I So x I or y I. Thus, x or q Therefore, is an, q -fuzzy prime -ideal of Conversely, assume ha is an, q -fuzzy prime -ideal of H and x y I. Then, xy 0.5. Thus, xy, when x q y x or x 1, and whence we ge x 0.5. from y Theorem 15. q we ge y I. So x yi x I or y I. or q. In he case x q we ge This means ha x I. Similarly, The inersecion of any family of, q -fuzzy prime (resp. semiprime) -ideals of H is an, q -fuzzy prime (resp. semiprime) -ideal of H. Sraighforward. Theorem 16. A fuzzy subse of H is an, q -fuzzy prime -ideal of H if and only if each is a prime -ideal of H for all xh : x q Le be an, q -fuzzy prime -ideal of H. Then, for (0,0.5] is each non-empy a -ideal of H. Now le xy. Bu Q, U, x y Q, or xy U,. Firs consider he case when x y Q, \ U,. Then, xy 1 and xy. Hence, for x y 0.5, we ge

23 AAM: Inern. J., Vol. 10, Issue 1 (June 2015) 517 max x, y min x y,0.5 x y 1. Hence, or y Q x Q,,. Now if xy 0.5 we have xy 0.5. Hence, max x, y min x y, Thus, So or y Q x Q,,. or y. x y Q, \ U, x Now le xy U,. In his case x y. Thus, If 0.5, hen and consequenly Therefore, So in any case Hence, for 0.5 we have x y x y max, min, or y U x U, max x, y 0.5 max x, y 1.,. or y Q x Q xy,,. eiher x or y.

24 518 Muhammad Gulisa e al. Hence, xh : x q is a prime -ideal of H for all Conversely, assume ha is a prime -ideal of H for all ideal of H. As xh : x q Then, i is obvious ha is an, q xh : x q -fuzzy - is a prime -ideal of H so from x y U,, we ge eiher x or y. q or y q. Hence, is an, q -fuzzy prime -ideal of H. implies ha x Corollary 6. This A fuzzy subse of H is an, q -fuzzy prime h - -ideal (resp. prime k - -ideal) of H if and only if each is a prime h - -ideal resp. prime k- -ideal : x H x q of H for all Sraighforward. 4. Conclusion -hemirings is ageneralizaion of he classical agebraic srucure of hemirings. Our aim has been o exend his idea and, o inroduce he concep of fuzzy -ideals,, -fuzzy prime (semiprime) -ideals,, -fuzzy h - -ideals and, -fuzzy k - -ideals of -hemirings, and relaed properies have been invesigaed. In fuure we will focus on he following. (i) A possible exension of his idea in order o inroduce new ypes of -ideals such as cubic -ideals, Inerval valued -ideals and, q -cubic- -ideals. (ii) Also o develop a new heory relaed o all ypes ( -cubic lef ideals, -cubic righ ideals, -cubic bi-ideals, -cubic inerior ideals) -hemirings.

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