New types of bipolar fuzzy sets in -semihypergroups

Transcription

1 Songlanaarin J. Sci. Technol. 38 (2) Mar. - pr Original rticle New tpes of bipolar fu sets in -semihpergroups Naveed Yaqoob 1 * Muhammad slam 2 Inaatur Rehman 3 Mohammed M. Khalaf 4 14 Department of Mathematics College of Science in l-zulfi Majmaah Universit l-zulfi Saudi rabia. 2 Department of Mathematics College of Science King Khalid Universit bha Saudi rabia. 3 Department of Mathematics Science College of rts pplied Sciences Dhofar Universit Oman. Received 26 June 2015; ccepted 12 September 2015 bstract The notion of bipolar fu set was initiated b Lee (2000) as a generaliation of the notion fu sets intuitionistic fu sets which have drawn attention of man mathematicians computer scientists. In this paper we initiate a stud on bipolar ( )-fu sets in -semihpergroups. B using the concept of bipolar ( )-fu sets (Yaqoob nsari 2013) we introduce the notion of bipolar ( )-fu sub -semihpergroups (-hperideals bi--hperideals) discuss some basic results on bipolar ( )-fu sets in -semihpergroups. Furthermore we define the bipolar fu subset prove that if is a bipolar ( )-fu sub -semihpergroup (resp. -hperideal bi--hperideal) of H; then is also a bipolar ( )-fu sub -semihpergroup (resp. -hperideal bi--hperideal) of H. Kewords: -semihpergroups bipolar fu sets bipolar ( )-fu -hperideals 1. Introduction Uncertaint is an attribute of information uncertain data are presented in various domains. The most appropriate theor for dealing with uncertainties is the theor of fu sets developed b Zadeh (1965) where he introduced the notion of a fu subset of a non-empt set X as a function from X to [0; 1]. Several researchers have conducted research on the generaliations of the notions of fu sets with huge applications in computer science logics man branches of pure applied mathematics. Rosenfeld (1971) defined the concept of fu group. Since then man papers * Corresponding author. address: naaqoob@mail.com have been published in the field of fu algebra; for instance Kuroi ( ) applied fu set theor to the ideal theor of semigroups. Shabir (2005) studied full fu prime semigroups. Kehaopulu Tsingelis (2007) studied fu ideals in ordered semigroups. ram Dar (2005) studied fu d-algebras. There are several inds of fu set extensions in the fu set theor; for example intuitionistic fu sets interval-valued fu sets vague sets etc. Bipolar-valued fu set is another extension of fu set whose membership degree range is different from the above extensions. Lee (2000) introduced the notion of bipolar-valued fu sets. Bipolar-valued fu sets are an extension of fu sets whose membership degree range is enlarged from the interval [0 1] to [-1 1]. In a bipolar-valued fu set the membership degree 0 indicate that elements are irrelevant to

2 120 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) the corresponding propert the membership degrees on (0; 1] assign that elements somewhat satisf the propert the membership degrees on [-1 0) assign that elements somewhat satisf the implicit counter-propert (Lee 2004). Jun par (2009) applied the notion of bipolar-valued fu sets to BCH-algebras. The introduced the concept of bipolar fu subalgebras bipolar fu ideals of a BCHalgebra. Lee (2009) applied the notion of bipolar fu subalgebras bipolar fu ideals of BCK/BCI-algebras. lso some results on bipolar-valued fu BCK/BCI-algebras were introduced b Saeid (2009). ram et al. ( a 2012b) applied bipolar fu sets to Lie algebras. The theor of hperstructure was born in 1934 when Mart (1934) defined hpergroups began to analsis their properties applied them to groups rational algebraic functions. In the following decades nowadas a number of different hperstructures are widel studied from the theoretical point of view for their applications to man subjects of pure applied mathematics b man mathematicians. Nowadas hperstructures have a lot of applications to several domains of mathematics computer science the are studied in man countries of the world. In a classical algebraic structure the composition of two elements is an element while in an algebraic hperstructure the composition of two elements is a set. recent boo on hperstructures (Corsini Leoreanu 2003) points out their applications in rough set theor crptograph codes automata probabilit geometr lattices binar relations graphs hpergraphs. Recentl nvarieh et al. (2010) Heidri et al. (2010) Mirvaili et al. (2013) Hila et al. (2012) bdullah et al. ( ) introduced the notion of -semihpergroups as a generaliation of semigroups semihpergroups of - semigroups. The proved some results in this respect presented man interesting examples of -semihpergroups. Fu set theor has been well developed in the context of hperalgebraic structure theor (Davva 2010). Several authors investigated the different aspects of fu semihpergroups fu -semihpergroups; for instance slam et al. (2012) Corsini et al. (2011) Davva ( ) Erso et al. ( a 2013b) Shabir Mahmood ( ) Yaqoob et al. (2014). Yao introduced a new tpe of fu sets called ( )- fu sets studied ( )-fu normal subfields (2005). Several authors extended Yao s idea continued their researches in appling ( )-fu sets on different algebraic structures. Coumaressane (2010) characteried near-rings b their ( )-fu quasi-ideals. Shabir et al. (2011) characteried semigroups b the properties of their fu ideals with thresholds Khan et al. (2012) characteried ordered semigroups b their ( )-fu bi-ideals. Li Feng (2013) extended the idea of ( )-fu sets in intuitionistic fu sets studied intuitionistic fu ( )-fu sets in - semigroups. Yaqoob nsari (2013) also extended the idea of ( )-fu sets in bipolar fu sets studied bipolar ( )-fu ideals in ternar semigroups. In this paper we stud bipolar ( )-fu sets in - semihpergroups introduce the notion of bipolar ( )- fu sub -semihpergroups (-hperideals bi-hperideals). 2. Preliminaries Basic Definitions In this section we will recall the basic terms definitions from the hperstructure theor bipolar fu sets. Definition 2.1 map : H H ( H ) is called hperoperation or join operation on the set H where H is a nonempt set ( H ) ( H ) \{ denotes the set of all non-empt subsets of H. hpergroupoid is a set H with together a (binar) hperoperation. Definition 2.2 hpergroupoid ( H ) which is associative that is x( ) ( x) x H is called a semihpergroup. Definition 2.3 (nvarieh et al. 2010) Let H be two non-empt sets. Then H is called a -semihpergroup if ever is a hperoperation on H i:e: x H for ever x H for ever x H we have x ( ) ( x ). If ever is an operation then H is a -semigroup. If ( H ) is a hpergroup for ever then H is called a -hpergroup. Let B be two non-empt subsets of H. Then we define B B a b a b B. Let H be a -semihpergroup. non-empt subset of H is called a sub -semihpergroup of H if x for ever x. -semihpergroup H is called commutative if for all x H we have x x. non-empt subset of a -semihpergroup H is a right (left) -hperideal of H if H ( H ) is a -hperideal of H if it is both a right a left -hperideal. non-empt subset B of a -semihpergroup H is called bi--hperideal of H if (1) BB B; (2) BHB B. Lee (2000) introduced the concept of bipolar fu set defined on a non-empt set X as objects having the form: x ( x) ( x) : x X. Where : X [01] : X [ 1 0]. The positive membership degree ( x) denotes the satisfaction degree of an element x to the propert corresponding to a bipolar fu set the negative membership degree ( x) denotes the satisfaction degree of x to some implicit counter propert of. For the sae of simplicit we shall use the smbol for the bipolar fu set x ( x) ( x) : x X.

3 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) Definition 2.4 be two bipolar fu subsets of a -semihpergroup H. Then for all x H their intersection is defined b where x ( x) ( x) : x H ( x) ( x) ( x) ( x) ( x) ( x) ( x) ( x). Their union is defined b where x ( x) ( x) : x H ( x) ( x) ( x) ( x) ( x) ( x) ( x) ( x). Definition 2.5 Let be two bipolar fu subsets of a -semihpergroup H. Then their product is defined b where x ( x) ( x) : x H ( x) ( x) sup{ ( ) ( ) if x x 0 otherwise inf { ( ) ( ) if x x for x H. 0 otherwise Definition 2.6 Let be a bipolar fu set ( s t) [ 10] [01]. Define: 1) the sets { x X ( x) t s t { x X ( x) s which are called positive t-cut of the negative s-cut of respectivel 2) the sets { x X ( x) t t s { x X ( x) s which are called strong positive t-cut of the strong negative s-cut of respectivel ( t s ) 3) the set X { x X ( x) t ( x) s is called an ( s t) -level subset of H ( t s) 4) the set X { x X ( x) t ( x) s is called a strong ( s t) -level subset of. 3. Bipolar ( )-fu -hperideals in -semihpergroups Yaqoob nsari (2013) introduced the notion bipolar ( )-fu ideals in ternar semigroups. In this section we introduce stud the notion bipolar ( )- fu -hperideals (resp. interior -hperideals bi-hperideals) in -semihpergroups discuss some related properties. In what follows let [01] be such that 0 1 [ 10] be such that 1 0. Both [01] are arbitrar but fixed. Definition 3.1 bipolar fu subset of a - semihpergroup H is called (1) a bipolar ( )-fu sub -semihpergroup of H if max inf ( x ) min{ ( ) ( ) x - - x min sup (x) max{ ( ) ( ) for all x H. (2) a bipolar ( )-fu left -hperideal of H if max inf ( x ) min{ ( ) x - - x min sup (x) max{ ( ) for all x H. (3) a bipolar ( )-fu right -hperideal of H if max inf ( x ) min{ ( ) x - - x min sup (x) max{ ( ) for all x H. Example 3.2 Let M n 01 for ever n n we define hperoperation on M as follows n x x 0 n x M. n 2 Then x M for ever n m n x M x ( x ) 0 n m x. n m n m 2 So M is a -semihpergroup (Hila bdullah 2014). Now we define a bipolar fu set on M as: if 0 x x if x where

4 122 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) if 0 x x if x where. Then b routine calculations is a bipolar ( )-fu -hperideal of M. Definition 3.3 bipolar fu subset of H is called a bipolar ( )-fu interior -hperideal of H if max inf ( a ) min{ ( ) ax a min sup ( ) max{ ( ) ax for all x H. Definition 3.4 bipolar ( )-fu sub -semihpergroup of H is called a bipolar ( )-fu bi-hperideal of H if max inf ( a ) min{ ( x ) ( ) ax a x ax min sup ( ) max{ ( ) ( ) ffor all x H. Example 3.5 Let H { x w { be the sets of binar hperoperations defined below: Clearl H is a -semihpergroup. Now we define a bipolar fu set on H as: if t { x t 0.6 if t 0.8 if t w -0.6 if t {x t = -0.8 if t= -0.9 if t=w Then b routine calculation is a bipolar ( )-fu bi--hperideal of H. Definition 3.6 bipolar ( )-fu sub -semihpergroup of H is called a bipolar (12)-fu bi-hperideal of H if ax w ( ) max inf ( a) min{ ( x) ( ) ( ) ax w ( ) min sup ( a) max{ ( x) ( ) ( ) or all x H. Theorem 3.7 bipolar fu subset of a - semihpergroup H is a bipolar ()-fu sub -semihpergroup (resp. left -hperideal right -hperideal interior - hperideal bi--hperideal (1 2)--hperideal) of H if ( ) onl if H t s is a sub -semihpergroup (resp. left - hperideal right -hperideal interior -hperideal bi-hperideal (1 2)--hperideal) of H for all ( s t) [ ) ( ]. Proof. Let be a bipolar ( )-fu sub - semihpergroup of H. Let x H ( s t) [ ) ( ] ( t s ) x H. Then ( x) t ( ) t also ( x) s ( ) s. s is a bipolar ()-fu sub -semihpergroup of H. Therefore max inf ( ) min{ ( x ) ( ) x min t t t min sup ( ) max{ ( x ) ( ) x max s s s. This implies that inf ( ) t sup ( ) s. x x ( Thus t s ) ( t s ) x H. Hence H is a sub -semihpergroup of H.

5 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) ( t s ) Conversel suppose that H is a sub -semihpergroup of H. Let x H such that max inf ( ) min{ ( x ) ( ) x min sup ( ) max{ ( x ) ( ). x Then there exist ( s t) [ ) ( ] such that max inf ( ) t min{ ( x ) ( ) x min sup ( ) s max{ ( x ) ( ). x This shows that ( x) t ( ) t inf inf ( ) t also ( x) s ( ) s sup ( ) s. x Thus x ( t s) ( t s ) x H since H is a sub -semihpergroup ( ) of H. Therefore x H t s for some x but this is a contradiction to inf ( ) t sup ( ) s. Thus x x max inf ( ) min{ ( x ) ( ) x min sup ( ) max{ ( x ) ( ). x Since is a bipolar ( )-fu sub -semihpergroup therefore max inf ( ) min{ ( x ) ( ) x min min sup ( ) max{ ( x) ( ) x max. Hence inf ( ) sup ( ). This shows that x x x for x. Hence is a sub - semihpergroup of H. The other cases can be seen in a similar wa. Theorem 3.10 non-empt subset of H is a sub -semihpergroup (resp. left -hperideal right -hperideal interior -hperideal bi--hperideal (1 2)--hperideal) of H if onl if the bipolar fu subset of H defined as follows: Hence is a bipolar ( )-fu sub -semihpergroup of H. The other cases can be seen in a similar wa. Corollar 3.8 Ever bipolar fu sub -semihpergroup (resp. left -hperideal right -hperideal interior - hperideal bi--hperideal (1 2)--hperideal) is a bipolar ()-fu sub -semihpergroup (resp. left - hperideal right -hperideal interior -hperideal bi-hperideal (1 2)--hperideal) of H with Theorem 3.9 If a bipolar fu subset is a bipolar ( )-fu sub -semihpergroup (resp. left - hperideal right -hperideal interior -hperideal bi-hperideal (1 2)--hperideal) of H. Then the set is a sub -semihpergroup (resp. left hperideal right -hperideal interior -hperideal bi-- hperideal (1 2)--hperideal) of H where { x H ( x) { x H ( x). Proof. Suppose that is a bipolar ()-fu sub -semihpergroup of H. Let x H such that x. Then ( x) ( ) ( x) ( ). is a bipolar ( )-fu sub -semihpergroup (resp. left - hperideal right -hperideal interior -hperideal bi-hperideal (1 2)--hperideal) of H. Proof. Suppose that is a sub -semihpergroup of H. Let x H be such that x then x for. Hence inf ( ) sup ( ). Therefore x x x x max inf ( ) min ( ) ( ) min sup ( ) max ( x) ( ). x If x or then min ( x) ( ) max ( x) ( ). Thus x x max inf ( ) min ( ) ( ) min sup ( ) max ( x) ( ). x

6 124 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) Consequentl is a bipolar ( )-fu sub - semihpergroup of H. Conversel: Let x. Then ( x) ( ) ( x) ( ). s is a bipolar ()-fu sub -semihpergroup of H therefore x x max inf ( ) min ( ) ( ) min min sup ( ) max ( x) ( ) x max. This implies that x. Hence is a sub -semihpergroup of H. The other cases can be seen in a similar wa. Theorem 3.11 non-empt subset of H is a sub -semihpergroup (resp. left -hperideal right -hperideal interior -hperideal bi--hperideal (1 2)--hperideal) of H if onl if is a bipolar ()-fu sub -semihpergroup (resp. left -hperideal right -hperideal interior -hperideal bi--hperideal (1 2)--hperideal) of H. Proof. Let be a sub -semihpergroup of H. Then is a bipolar fu sub -semihpergroup of H b Corollar 3.8 B is a bipolar ( )-fu sub -semihpergroup of H. Conversel let x H be such that x. Then ( x) ( ) 1 ( x) ( ) 1. Since B is a bipolar ()-fu sub -semihpergroup of H. Therefore x x max inf ( ) min ( ) ( ) min 11 min sup ( ) max ( x) ( ) x max 1 1. It implies that inf ( ) sup ( ). x x Thus x for x. Therefore is a sub -semihpergroup of H. The other cases can be seen in a similar wa. Theorem 3.12 Let be a bipolar ( )-fu left -hperideal be a bipolar ()-fu right -hperideal of a -semihpergroup H. Then bipolar ( )-fu -hperideal of H. Proof. The proof is straightforward. is a Lemma 3.13 Intersection of an famil of bipolar ( )-fu sub -semihpergroups (resp. left -hperideals right - hperideals interior -hperideals bi--hperideals (1 2)- -hperideals) of a -semihpergroup H is a bipolar ( )- fu sub -semihpergroup (resp. left -hperideal right - hperideal interior -hperideal bi--hperideal (1 2)-hperideal) of H. Proof. The proof is straightforward. Now we prove that if is a bipolar ()- fu -hperideal of H then is also a bipolar ()-fu -hperideal of H. Definition 3.14 Let be a bipolar fu subset of a -semihpergroup H (01] such that. We define the bipolar fu subset of H as follows x x for all x H. x = x Definition 3.15 Let be bipolar fu subsets of a -semihpergroup H. Then we define (1) the bipolar fu subset as follows: as follows: as follows: ( x) ( ( x) ) (x)=( (x) ) (2) the bipolar fu subset ( x) ( ( x) ) (x)=( (x) ) (3) the bipolar fu subset for all x H. ( x) ( ( x) ) (x)=( (x) ) Lemma 3.16 Let be bipolar fu subsets of a -semihpergroup H. Then the following holds: (1) (2) (3).

7 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) Proof. The proof is straightforward. Theorem 3.17 bipolar fu subset of a - semihpergroup H is a bipolar ()-fu (1) sub -semihpergroup of H if onl if (2) left -hperideal of H if onl if (3) right -hperideal of H if onl if H (4) bi--hperideal of H if onl if (5) interior -hperideal of H if onl if (6) (1 2)--hperideal of H if onl if. Proof. (1) Let be a bipolar ( )-fu sub - semihpergroup of H H. Let us suppose that x for x H. Then Hence. Conversel assume that. If there exist m n H such that x m n then x max ( x ) x ( x ) x x ) x m n m n ( ) max inf x x min ) min x ( x ) x ( x ) = x x ( ) min sup x x max. Hence is a bipolar ()-fu sub -semihpergroup of H. The proofs of (2) (3) (4) (5) (6) are similar to the proof of (1). Proposition 3.18 If is a bipolar ()-fu sub -semihpergroup (resp. left -hperideal right -hperideal interior -hperideal bi--hperideal (1 2)--hperideal) of a -semihpergroup H then is also a bipolar ( )-fu sub -semihpergroup (resp. left - hperideal right -hperideal interior -hperideal bi-hperideal (1 2)--hperideal) of H. If there do not exist an x H such that x then ( x) ( x) ( x) ( x). Proof. Suppose is a bipolar ()-fu sub - semihpergroup of H. Let x H. Then x inf ( ) x inf ( ) x inf ( ) x max inf ( )

8 126 inf ( ) x inf ( ) x ( x) ( ) ( x) ( ) N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) ( ( x) ) ( ( ) ) (( ( x) ) ) (( ( ) ) ) ( ) ( ) min x ( x ) ( ) min sup ( ) x sup ( ) x sup ( ) x sup ( ) x sup ( ) x sup ( ) x ( x) ( ) ( x) ( ) ((( ( x) ) ( ( ) )) ) x ( ( ) ( ) ) ( ) ( ) max x ( x ) ( ). Hence is a bipolar ()-fu sub -semihpergroup of H. The other cases can be seen in a similar wa. Theorem 3.19 Let be a bipolar ()-fu right -hperideal be a bipolar ( )-fu left -hperideal of H. Then. Proof. Let be a bipolar ( )-fu right - hperideal be a bipolar ()-fu left - hperideal of H. Then for all H we have ( ) ( ) ( ( ) ) ( ( ) ) (( ( ) ( )) ) ( )( ) ( ) ( ) ( ) ( ( ) ) ( ( ) ) (( ( ) ( )) ) ( )( ) ( ). If there do not exist an x H such that x then ( ) x ( x ) 0 ( x) ( x) ( ) x ( x ) 0 ( x) ( x). Hence we get. References bdullah S. slam M. nwar T note on M- hpersstems N-hpersstems in -semihpergroups. Quasigroups Related Sstems bdullah S. Hila K. slam M On bi--hperideals of -semihpergroups. U.P.B. Scientific Bulletin Series. 74(4) ram M Bipolar fu L-Lie algebras. World pplied Sciences Journal. 14(12)

9 N. Yaqoob et al. / Songlanaarin J. Sci. Technol. 38 (2) ram M. Chen W. Lin Y. 2012a. Bipolar fu Lie superalgebras. Quasigroups Related Sstems ram M. l-shehrie N.O. 2012b. Bipolar fu Lie ideals. Utilitas Mathematica nvarieh S.M. Mirvaili S. Davva B On - hperideals in -semihpergroups. Carpathian Journal of Mathematics. 26(1) slam M. bdullah S. Davva B. Yaqoob N Rough M-hpersstems fu M-hpersstems in -semihpergroups. Neural Computing pplications. 21(1) Corsini P. Leoreanu V pplications of hperstructure theor. Kluwer cademic PublicationsThe Netherls. Corsini P. Shabir M. Mahmood T Semisipmle semihpergroups in terms of hperideals fu hperideals. Iranian Journal of Fu Sstems. 8(1) Coumaressane S Near-rings characteried b their ()-fu quasi-ideals. International Journal of Computational Cognition Davva B surve of fu algebraic hperstructures. lgebra Groups Geometries. 27(1) Davva B Fu hperideals in semihpergroups. Italian Journal of Pure pplied Mathematics Davva B Intuitionistic hperideals of semihpergroups. Bulletin of the Malasian Mathematical Sciences Societ. 29(1) Davva B Extensions of fu hperideals in H v groups. Soft Computing. 11(9) semi- Erso B.. Saricaoglu Y. Yenigun M. Davva B On fu interior -hperideals of -semihpergroups. Utilitas Mathematica Erso B.. Davva B. 2013a. Structure of intuitionistic fu sets in -semihperrings. bstract pplied nalsis pages. Erso B.. Davva B. 2013b. tanassov.s intuitionistic fu -hperideals of -semihpergroups. Journal of Intelligent Fu Sstems. 25(2) Heidari D. Dehordi S.O. Davva B semihpergroups their properties. U.P.B. Scientific Bulletin Series Hila K. Davva B. Dine J Stud on the structure of Û-semihpergroups. Communications in lgebra Hila K. bdullah S stud on intuitionistic fu sets in -semihpergroups. Journal of Intelligent Fu Sstems. 26(4) Jun Y.B. Par C.H Filters of BCH-algebras based on bi-polar-valued fu sets. International Mathematical Forum Kehaopulu N. Tsingelis M Fu ideals in ordered semigroups. Quasigroups Related Sstems Khan F.M. Sarmin N.H. Khan Some new characteriation of ordered semigroups in terms of ( )- fu bi-ideals. International Journal of lgebra Statistics. 1(1) Kuroi N Fu bi-ideals in semigroups. Commentarii Mathematici Universitatis Sancti Pauli Kuroi N On fu ideals fu bi-ideals in semigroups. Fu Sets Sstems. 5(2) Lee K.M Bi-polar-valued fu sets their operations. Proceeding of the International Conference on Intelligent Technologies Bango Thail Lee K.M Comparison of interval-valued fu sets intuitionistic fu sets bi-polar-valued fu sets. Journal of Fu Logic Intelligent Sstems Lee K.J Bi-polar fu subalgebras bi-polar fu ideals of BCK/BCI-algebras. Bulletin of the Malasian Mathematical Sciences Societ. 32(3) Li B. Feng Y Intuitionistic ( )-fu sets in - semigroups. Journal of Inequalities pplications pages. Mart F Sur une generaliation de la notion de group. Proceeding of the 8 th Congres Mathematics Scinaves Stocholm Mirvaili S. nvarieh S.M. Davva B semihpergroups regular relations. Journal of Mathematics. rticle ID pages. Rosenfeld Fu groups. Journal of Mathematical nalsis pplications Saeid.B Bipolar-valued fu BCK/BCI-algebras. World pplied Sciences Journal. 7(11) Shabir M Full fu prime semigroups. International Journal of Mathematics Mathematical Sciences Shabir M. Nawa Y. slam M Semigroups characteried b the properties of their fu ideals with thresholds. World pplied Sciences Journal. 14(12) Shabir M. Mahmood T Semihpergroups Characteried b ( q) -fu Hperideals. Information Sciences Letters. 2(2) Shabir M. Mahmood T Semihpergroups characteried b ( q )-fu hperideals. Journal of Intelligent Fu Sstems. 28(6) Yao B ( )-fu normal sub.elds ( )-fu quotient sub.elds. The Journal of Fu Mathematics. 13(3) Yaqoob N. slam M. Davva B. Ghareeb Structures of bipolar fu -hperideals in -semihpergroups. Journal of Intelligent Fu Sstems. 27(6) Yaqoob N. nsari M Bipolar ( )-fu ideals in ternar semigroups. International Journal of Mathematical nalsis. 7(36) Zadeh L Fu sets. Information Control