ON ROUGH (m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS


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1 U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN ON ROUGH m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this paper, we introduced the concept of m, n) biγhyperideals and rough m, n) biγhyperideals in Γsemihypergroups and some properties of m, n) biγhyperideals in Γsemihypergroups are presented. Keywords: Γsemihypergroups, Rough sets, Rough m, n) biγhyperideals. MSC2010: 20N20, 20M Introduction The notion of m, n)ideals of semigroups was introduced by Lajos [13, 14]. Later m, n) quasiideals and m, n) biideals and generalized m, n) biideals were studied in various algebraic structures. The notion of a rough set was originally proposed by Pawlak [16] as a formal tool for modeling and processing incomplete information in information systems. Some authors have studied the algebraic properties of rough sets. Kuroki, in [12], introduced the notion of a rough ideal in a semigroup. Anvariyeh et al. [3], introduced Pawlak s approximations in Γsemihypergroups. Abdullah et al. [1], introduced the notion of Mhypersystem and Nhypersystem in Γsemihypergroups and Aslam et al. [6], studied rough Mhypersystems and fuzzy Mhypersystems in Γsemihypergroups, also see [4, 5, 19]. Yaqoob et al. [18], Applied rough set theory to Γhyperideals in left almost Γsemihypergroups. The algebraic hyperstructure notion was introduced in 1934 by a French mathematician Marty [15], at the 8th Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, non commutative groups. In 1986, Sen and Saha [17], defined the notion of a Γsemigroup as a generalization of a semigroup. One can see that Γsemigroups are generalizations of semigroups. Many classical notions of semigroups have been extended to Γsemigroups and a lot of results on Γsemigroups are published by a lot of mathematicians, for instance, Chattopadhyay [7], Chinram and Jirojkul [8], Chinram and Siammai [9], Hila [11]. Then, in [2, 10], Davvaz et al. introduced the notion of Γsemihypergroup 1 Corresponding author: Department of Mathematics, QuaidiAzam University, Islamabad, Pakistan., 2 Department of Mathematics, QuaidiAzam University, Islamabad, Pakistan. 3 Department of Mathematics, Yazd University, Yazd, Iran. 4 Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran 119
2 120 Naveed Yaqoob, Muhammad Aslam, Bijan Davvaz, Arsham Borumand Saeid as a generalization of a semigroup, a generalization of a semihypergroup and a generalization of a Γsemigroup. They presented many interesting examples and obtained a several characterizations of Γsemihypergroups. In this paper, we have introduced the notion of m, n) biγhyperideals and we have applied the concept of rough set theory to m, n) biγhyperideals, which is a generalization of m, n) biγhyperideals of Γsemihypergroups. 2. Preliminaries In this section, we recall certain definitions and results needed for our purpose. Definition 2.1. A map : S S P S) is called hyperoperation or join operation on the set S, where S is a nonempty set and P S) denotes the set of all nonempty subsets of S. A hypergroupoid is a set S with together a binary) hyperoperation. A hypergroupoid S, ), which is associative, that is x y z) = x y) z, x, y, z S, is called a semihypergroup. Let A and B be two nonempty subsets of S. Then, we define AΓB = γ Γ AγB = {aγb a A, b B and γ Γ}. Let S, ) be a semihypergroup and let Γ = { }. Then, S is a Γsemihypergroup. So, every semihypergroup is Γsemihypergroup. Let S be a Γsemihypergroup and γ Γ. A nonempty subset A of S is called a sub Γsemihypergroup of S if xγy A for every x, y A. A Γsemihypergroup S is called commutative if for all x, y S and γ Γ, we have xγy = yγx. Example 2.1. [2] Let S = [0, 1] and[ Γ = ] N. For every x, y S and γ Γ, we define γ : S S S) by xγy = 0, xy γ. Then, γ is hyperoperation. For every [ ] x, y, z S and α, β Γ, we have xαy)βz = 0, xyz αβ = xαyβz). This means that S is Γsemihypergroup. Example 2.2. [2] Let S, ) be a semihypergroup and Γ be a nonempty subset of S. We define xγy = x y for every x, y S and γ Γ. Then, S is a Γsemihypergroup. Definition 2.2. [2] A nonempty subset A of a Γsemihypergroup S is a right left) Γhyperideal of S if AΓS A SΓA A), and is a Γhyperideal of S if it is both a right and a left Γhyperideal. Definition 2.3. [2] A sub Γsemihypergroup B of a Γsemihypergroup S is called a biγhyperideal of S if BΓSΓB B. A biγhyperideal B of a Γsemihypergroup S is proper if B S. Lemma 2.1. In a Γsemihypergroup S, AΓB) m = A m ΓB m holds if AΓB = BΓA for all A, B S and m is a positive integer. Proof. We prove the result AΓB) m = A m ΓB m by induction on m. For m = 1, AΓB = AΓB, which is true. For m = 2, AΓB) 2 = AΓB)ΓAΓB) = AΓBΓA)ΓB =
3 On Rough m, n) BiΓhyperideals in Γsemihypergroups 121 A 2 ΓB 2. Suppose that the result is true for m = k. That is, AΓB) k = A k ΓB k. Now for m = k + 1, we have AΓB) k+1 = AΓB) k ΓAΓB) = A k ΓB k )ΓAΓB) = A k ΓB k ΓA)ΓB = A k ΓA)ΓB k ΓB) = A k+1 ΓB k+1. Thus, the result is true for m = k +1. By induction hypothesis the result AΓB) m = A m ΓB m is true for all positive integers m. 3. m, n) BiΓhyperideals in Γsemihypergroups From [14], a subsemigroup A of a semigroup S is called an m, n)ideal of S if A m SA n A. A subset A of a Γsemihypergroup S is called an m, 0) Γhyperideal 0, n) Γhyperideal) if A m ΓS A SΓA n A). A sub Γsemihypergroup A of a Γ semihypergroup S is called m, n) biγhyperideal of S, if A satisfies the condition A m ΓSΓA n A, where m, n are nonnegative integers A m is suppressed if m = 0). Here if m = n = 1 then A is called biγhyperideal of S. By a proper m, n) biγhyperideal we mean an m, n) biγhyperideal, which is a proper subset of S. Example 3.1. Let S, ) be a semihypergroup and Γ be a nonempty subset of S. Define a mapping S Γ S P S) by xγy = x y for every x, y S and γ Γ. By Example 2.2, we know that S is a Γsemihypergroup. Let B be an m, n) bihyperideal of the semihypergroup S. Then, B m S B n B. So, B m ΓSΓB n = B m S B n B. Hence, B is an m, n) biγhyperideal of S. Example [ 3.2. ] Let S = [0, 1] and Γ = N. Then, S together with the hyperoperation xγy = is a Γsemihypergroup. Let t [0, 1] and set T = [0, t]. Then, clearly it 0, xy γ can be seen that T is a sub Γsemihypergroup of S. Since T m ΓS = [0, t m ] [0, t] = T SΓT n = [0, t n ] [0, t] = T ), so T is an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Since T m ΓSΓT n = [0, t m+n ] [0, t] = T, then T is an m, n) biγhyperideal of Γsemihypergroup S. Example 3.3. [ Let ] S = [ 1, 0] and Γ = { 1, 2, 3, }. Define the hyperoperation xγy = xy γ, 0 for all x, y S and γ Γ. Then, clearly S is a Γsemihypergroup. Let λ [ 1, 0] and the set B = [λ, 0]. Then, clearly B is a sub Γsemihypergroup of S. Since B m ΓS = [λ 2m+1, 0] [λ, 0] = B SΓB n = [λ 2n+1, 0] [λ, 0] = B), so B is an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Since B m ΓSΓB n = [λ 2m+n)+1, 0] [λ, 0] = B, then B is an m, n) biγhyperideal of Γsemihypergroup S. Proposition 3.1. Let S be a Γsemihypergroup, B be a sub Γsemihypergroup of S and let A be an m, n) biγhyperideal of S. Then, the intersection A B is an m, n) biγhyperideal of Γsemihypergroup B. Proof. The intersection A B evidently is a sub Γsemihypergroup of S. We show that A B is an m, n) biγhyperideal of B, for this A B) m ΓBΓA B) n A m ΓSΓA n A, 1)
4 122 Naveed Yaqoob, Muhammad Aslam, Bijan Davvaz, Arsham Borumand Saeid because of A is an m, n) biγhyperideal of S. Secondly A B) m ΓBΓA B) n B m ΓBΓB n B. 2) Therefore, 1) and 2) imply that A B) m ΓBΓA B) n A B, that is, the intersection A B is an m, n) biγhyperideal of B. Theorem 3.1. Suppose that {A i : i I} be a family of m, n) biγhyperideals of a Γsemihypergroup S. Then, the intersection A i is an m, n) biγhyperideal i I of S. Proof. Let {A i : i I} be a family of m, n) biγhyperideals in a Γsemihypergroup S. We know that the intersection of sub Γsemihypergroups is a sub Γsemihypergroup. Let B = A i. Now we have to show that B = A i is an m, n) biγhyperideal i I i I of S. Here we need only to show that B m ΓSΓB n B. Let x B m ΓSΓB n. Then, x = a m 1 αsβan 2 for some am 1, an 2 B, s S and α, β Γ. Thus, for any arbitrary i I as a m 1, an 2 B i. So, x Bi mγsγbn i. Since B i is an m, n) biγhyperideal so Bi mγsγbn i B i and therefore x B i. Since i was chosen arbitrarily so x B i for all i I and hence x B. So, B m ΓSΓB n B and hence B = A i is an m, n) i I biγhyperideal of S. It is obvious that the intersection of two or more m, 0) Γhyperideals 0, n) Γhyperideals) is an m, 0) Γhyperideal 0, n) Γhyperideal). Similarly, the union of two or more m, 0) Γhyperideals 0, n) Γhyperideals) is an m, 0) Γhyperideal 0, n) Γhyperideal). Theorem 3.2. Let S be a Γsemihypergroup. If A is an m, 0) Γhyperideal and also 0, n) Γhyperideal of S, then A is an m, n) biγhyperideal of S. Proof. Suppose that A is an m, 0) Γhyperideal and also 0, n) Γhyperideal of S. Then, A m ΓSΓA n AΓA n SΓA n A, which implies that A is an m, n) biγhyperideal of S. Theorem 3.3. Let m, n be arbitrary positive integers. Let S be a Γsemihypergroup, B be an m, n) biγhyperideal of S and A be a sub Γsemihypergroup of S. Suppose that AΓB = BΓA. Then, 1) BΓA is an m, n) biγhyperideal of S. 2) AΓB is an m, n) biγhyperideal of S. Proof. 1) The suppositions of the theorem imply that BΓA)ΓBΓA) = BΓAΓB)ΓA = BΓA. This shows that BΓA is a sub Γsemihypergroup of S. On the other hand, as B is an m, n) biγhyperideal of S, so BΓA) m ΓSΓBΓA) n = B m ΓA m ΓSΓB n )ΓA n BΓA n BΓA. Hence, the product BΓA is an m, n) biγhyperideal of S. 2) The proof is similar to 1).
5 On Rough m, n) BiΓhyperideals in Γsemihypergroups 123 Theorem 3.4. Let S be a Γsemihypergroup and for a positive integer n, B 1, B 2,, B n be m, n) biγhyperideals of S. Then, B 1 ΓB 2 Γ ΓB n is an m, n) biγhyperideal of S. Proof. We prove the theorem by induction. By Theorem 3.3, B 1 ΓB 2 is an m, n) biγhyperideal of S. Next, for k n, suppose that B 1 ΓB 2 Γ...ΓB k is an m, n) biγhyperideal of S. Then, B 1 ΓB 2 Γ...ΓB k ΓB k+1 = B 1 ΓB 2 Γ...ΓB k )ΓB k+1 is an m, n) biγhyperideal of S by Theorem 3.3. Theorem 3.5. Let S be a Γsemihypergroup, A be an m, n) biγhyperideal of S, and B be an m, n) biγhyperideal of the Γsemihypergroup A such that B 2 = BΓB = B. Then, B is an m, n) biγhyperideal of S. Proof. It is trivial that B is a sub Γsemihypergroup of S. Secondly, since A m ΓSΓA n A and B m ΓAΓB n B, we have B m ΓSΓB n = B m ΓB m ΓSΓB n )ΓB n B m ΓA m ΓSΓA n )ΓB n B m ΓAΓB n B. Therefore, B is an m, n) biγhyperideal of S. 4. Lower and Upper Approximations in Γsemihypergroups In what follows, let S denote a Γsemihypergroup unless otherwise specified. Definition 4.1. Let S be a Γsemihypergroup. An equivalence relation on S is called regular on S if for all x S and γ Γ. a, b) implies aγx, bγx) and xγa, xγb), If is a regular relation on S, then, for every x S, [x] stands for the class of x with the represent. A regular relation on S is called complete if [a] γ[b] = [aγb] for all a, b S and γ Γ. In addition, on S is called congruence if, for every a, b) S and γ Γ, we have c [a] γ[b] = [c] [a] γ[b]. Let A be a nonempty subset of a Γsemihypergroup S and be a regular relation on S. Then, the sets { } { } Apr A) = x S : [x] A and Apr A) = x S : [x] A are called lower and upper approximations of A, respectively. For a nonempty subset A of S, Apr A) = Apr A), Apr A)) is called a rough set with respect to if Apr A) Apr A). Theorem 4.1. [3] Let be a regular relation on a Γsemihypergroup S and let A and B be nonempty subsets of S. Then, 1) Apr A) ΓApr B) Apr AΓB) ; 2) If is complete, then Apr A) ΓApr B) Apr AΓB). Theorem 4.2. [3] Let be a regular relation on a Γsemihypergroup S. Then, 1) Every sub Γsemihypergroup of S is a upper rough sub Γsemihypergroup of S. 2) Every right left) Γhyperideal of S is a upper rough right left) Γ hyperideal of S.
6 124 Naveed Yaqoob, Muhammad Aslam, Bijan Davvaz, Arsham Borumand Saeid Theorem 4.3. [3] Let = A S and let be a complete regular relation on S such that the lower approximation of A is nonempty. Then, 1) If A is a sub Γsemihypergroup of S, then A is a lower rough sub Γ semihypergroup of S. 2) If A is a right left) Γhyperideal of S, then A is a lower rough right left) Γhyperideal of S. A subset A of a Γsemihypergroup S is called a upper [lower] rough biγhyperideal of S if Apr A)[Apr A)] is a biγhyperideal of S. Theorem 4.4. [3] Let be a regular relation on S and A be a biγhyperideal of S. Then, 1) A is a upper rough biγhyperideal of S. 2) If is complete such that the lower approximation of A is nonempty, then A is a lower rough biγhyperideal of S. Lemma 4.1. Let be a regular relation on a Γsemihypergroup S. Then, for a nonempty subset A of S 1) Apr A) ) n Apr A n ) for all n N. n 2) If is complete, then Apr A)) Apr A n ) for all n N. Proof. 1) Let A be a nonempty subset of S, then for n = 2, and by Theorem 4.11), we get Apr A) ) 2 = Apr A)ΓApr A) Apr AΓA) = Apr A 2 ). Now for n = 3, we get Apr A) ) 3 = Apr A)Γ Apr A) ) 2 Apr A)ΓApr A 2 ) Apr AΓA 2 ) = Apr A 3 ). Suppose that the result is true for n = k 1, such that Apr A)) k 1 Apr A k 1 ), then for n = k, we get Apr A) ) k = Apr A)Γ Apr A) ) k 1 Apr A)ΓApr A k 1 ) Apr AΓA k 1 ) = Apr A k ). Hence, this shows that Apr A) ) k Apr A k ). This implies that Apr A) ) n Apr A n ) is true for all n N. By using Theorem 4.12), the proof of 2) can be seen in a similar way. This completes the proof. 5. Rough m, n) BiΓhyperideals in Γsemihypergroups Let be a regular relation on a Γsemihypergroup S. A subset A of S is called a upper rough m, 0) Γhyperideal 0, n) Γhyperideal) of S if Apr A) is an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Similarly, a subset A of a Γsemihypergroup S is called a lower rough m, 0) Γhyperideal 0, n) Γhyperideal) of S if Apr A) is an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Theorem 5.1. Let be a regular relation on a Γsemihypergroup S and A be an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Then, 1) Apr A) is an m, 0) Γhyperideal 0, n) Γhyperideal) of S.
7 On Rough m, n) BiΓhyperideals in Γsemihypergroups 125 2) If is complete, then Apr A) is, if it is nonempty, an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Proof. 1) Let A be an m, 0) Γhyperideal of S, that is, A m ΓS A. Note that Apr S) = S. Then, by Theorem 4.11) and Lemma 4.11), we have Apr A) ) m ΓS = Apr A) ) m ΓApr S) Apr A m )ΓApr S) Apr A m ΓS) Apr A). This shows that Apr A) is an m, 0) Γhyperideal of S, that is, A is a upper rough m, 0) Γhyperideal of S. Similarly, we can show that the upper approximation of a 0, n) Γhyperideal is a 0, n) Γhyperideal of S. 2) Let A be an m, 0) Γhyperideal of S, that is, A m ΓS A. Note that Apr S) = S. Then, by Theorem 4.12) and Lemma 4.12), we have ) m m Apr A) ΓS = Apr A)) ΓApr S) Apr A m )ΓApr S) Apr A m ΓS) Apr A). This shows that Apr A) is an m, 0) Γhyperideal of S, that is, A is a lower rough m, 0) Γhyperideal of S. Similarly, we can show that the lower approximation of a 0, n) Γhyperideal is a 0, n) Γhyperideal of S. This completes the proof. A subset A of a Γsemihypergroup S is called a upper [lower] rough m, n) biγhyperideal of S if Apr A) [Apr A)] is an m, n) biγhyperideal of S. Theorem 5.2. Let be a regular relation on a Γsemihypergroup S. If A is an m, n) biγhyperideal of S, then it is a upper rough m, n) biγhyperideal of S. Proof. Let A be an m, n) biγhyperideal of S. Lemma 4.11), we have Then, by Theorem 4.11) and Apr A)) m ΓSΓApr A)) n = Apr A)) m ΓApr S)ΓApr A)) n Apr A m )ΓApr S)ΓApr A n ) Apr A m ΓS)ΓApr A n ) Apr A m ΓSΓA n ) Apr A). From this and Theorem 4.21), we obtain that Apr A) is an m, n) biγhyperideal of S, that is, A is a upper rough m, n) biγhyperideal of S. This completes the proof. Theorem 5.3. Let be a complete regular relation on a Γsemihypergroup S. If A is an m, n) biγhyperideal of S, then Apr A) is, if it is nonempty, an m, n) biγhyperideal of S.
8 126 Naveed Yaqoob, Muhammad Aslam, Bijan Davvaz, Arsham Borumand Saeid Proof. Let A be an m, n) biγhyperideal of S. Then, by Theorem 4.12) and Lemma 4.12), we have Apr A)) m ΓSΓApr A)) n = Apr A)) m ΓApr S)ΓApr A)) n Apr A m )ΓApr S)ΓApr A n ) Apr A m ΓS)ΓApr A n ) Apr A m ΓSΓA n ) Apr A). From this and Theorem 4.31), we obtain that Apr A) is, if it is nonempty, an m, n) biγhyperideal of S. This completes the proof. The following example shows that the converse of Theorem 5.2 and Theorem 5.3 does not hold. Example 5.1. Let S = {x, y, z} and Γ = {β, γ} be the sets of binary hyperoperations defined below: β x y z x x {x, y} z y {x, y} {x, y} z z z z z γ x y z x {x, y} {x, y} z y {x, y} y z z z z z Clearly S is a Γsemihypergroup. Let be a complete regular relation on S such that the regular classes are the subsets {x, y}, {z}. Now for A = {x, z} S, Apr A) = {x, y, z} and Apr A) = {z}. It is clear that Apr A) and Apr A) are m, n) biγhyperideals of S, but A is not an m, n) biγhyperideal of S. Because A m ΓSΓA n = S A. 6. Rough m, n) BiΓhyperideals in the Quotient Γsemihypergroups Let be a regular relation on a Γsemihypergroup S. We put Γ = { γ : γ Γ}. For every [a], [b] S/, we define [a] γ[b] = {[z] : z aγb}. Theorem 6.1. [3, Theorem 4.1]) If S is a Γsemihypergroup, then S/ is a Γsemihypergroup. Definition 6.1. Let be a regular relation on a Γsemihypergroup S. The lower approximation and upper approximation of a nonempty subset A of S can be presented in an equivalent form as shown below: Apr A) = respectively. { [x] S/ : [x] A } and Apr A) = { } [x] S/ : [x] A, Theorem 6.2. [3, Theorems 4.3, 4.4]) Let be a regular relation on a Γsemihypergroup S. If A is a sub Γsemihypergroup of S. Then, 1) Apr A) is a sub Γsemihypergroup of S/. 2) Apr A) is, if it is nonempty, a sub Γsemihypergroup of S/.
9 On Rough m, n) BiΓhyperideals in Γsemihypergroups 127 Theorem 6.3. Let be a regular relation on a Γsemihypergroup S. If A is an m, 0) Γhyperideal 0, n) Γhyperideal) of S. Then, 1) Apr A) is an m, 0) Γhyperideal 0, n) Γhyperideal) of S/. of S/. 2) Apr A) is, if it is nonempty, an m, 0) Γhyperideal 0, n) Γhyperideal) Proof. 1) Assume that A is a 0, n) Γhyperideal of S. Let [x] and [s] be any elements of Apr A) and S/, respectively. Then, [x] A. Hence, x Apr A). Since A is a 0, n) Γhyperideal of S, by Theorem 101), Apr A) is a 0, n) Γ hyperideal of S. So, for γ Γ, we have sγx n Apr A). Now, for every t sγx n, we have [t] A. On the other hand, from t sγx n, we obtain [t] [s] γ[x] n. Therefore, [s] γ[x] n Apr A). This means that Apr A) is a 0, n) Γhyperideal of S/. 2) Let A be a 0, n) Γhyperideal of S. Let [x] and [s] be any elements of Apr A) and S/, respectively. Then, [x] A, which implies x Apr A). Since A is a 0, n) Γhyperideal of S, by Theorem 102), Apr A) is a 0, n) Γhyperideal of S. Thus, for every γ Γ, we have sγx n Apr A). Now, for every t sγx n, we have t Apr A), which implies that [t] A. Hence, [t] Apr A). On the other hand, from t sγx n, we have [t] [s] γ[x] n. Therefore, [s] γ[x] n Apr A). This means that Apr A) is, if it is nonempty, a 0, n) Γhyperideal of S/. The other cases can be seen in a similar way. This completes the proof. Theorem 6.4. Let be a regular relation on a Γsemihypergroup S. If A is an m, n) biγhyperideal of S. Then, 1) Apr A) is an m, n) bi Γhyperideal of S/. 2) Apr A) is, if it is nonempty, an m, n) bi Γhyperideal of S/. Proof. 1) Let [x] and [y] be any elements of Apr A) and [s] be any element of S/. Then, [x] A and [y] A. Hence, x Apr A) and y Apr A). By Theorem 11, Apr A) is an m, n) bi Γhyperideal of S. So, for every α, β Γ, we have x m αsβy n Apr A). Now, for every t x m αsβy n, we obtain [t] [x] m αs β[y] n. On the other hand, since t Apr A), we have [t] A. Thus, [x] m αs β[y] n Apr A). Therefore, Apr A) is an m, n) bi Γhyperideal of S/. 2) Let [x] and [y] be any elements of Apr A) and [s] be any element of S/. Then, [x] A and [y] A. Hence, x Apr A) and y Apr A). By Theorem 12, Apr A) is an m, n) bi Γhyperideal of S. So, for every α, β Γ, we have x m αsβy n Apr A). Then,
10 128 Naveed Yaqoob, Muhammad Aslam, Bijan Davvaz, Arsham Borumand Saeid for every t x m αsβy n, we obtain [t] [x] m αa β[y] n. On the other hand, since t Apr A), we have [t] A. So, [x] m αa β[y] n Apr A). Therefore, Apr A) is, if it is nonempty, an m, n) bi Γhyperideal of S/. This completes the proof. 7. Conclusion The relations between rough sets and algebraic systems have been already considered by many mathematicians. In this paper, the properties of m, n) biγhyperideal in Γsemihypergroup are investigated and hence the concept of rough set theory is applied to m, n) biγhyperideals. Acknowledgements. The authors are highly grateful to referees for their valuable comments and suggestions which were helpful in improving this paper. R E F E R E N C E S [1] S. Abdullah, M. Aslam and T. Anwar, A note on Mhypersystems and Nhypersystems in Γsemihypergroups, Quasigroups Relat. Syst. 192) 2011) [2] S.M. Anvariyeh, S. Mirvakili and B. Davvaz, On Γhyperideals in Γsemihypergroups, Carpathian J. Math. 261) 2010) [3] S.M. Anvariyeh, S. Mirvakili and B. Davvaz, Pawlak s approximations in Γsemihypergroups, Comput. Math. Applic ) [4] M. Aslam, M. Shabir and N. Yaqoob, Roughness in left almost semigroups, J. Adv. Res. Pure Math. 33) 2011) [5] M. Aslam, M. Shabir, N. Yaqoob and A. Shabir, On rough m, n)biideals and generalized rough m, n)biideals in semigroups, Ann. Fuzzy Math. Inform. 22) 2011) [6] M. Aslam, S. Abdullah, B. Davvaz and N. Yaqoob, Rough Mhypersystems and fuzzy Mhypersystems in Γsemihypergroups, Accepted in Neural Comput. Applic., doi: /s x. [7] S. Chattopadhyay, Right inverse Γsemigroup, Bull. Cal. Math. Soc ) [8] R. Chinram and C. Jirojkul, On biγideals in Γsemigroups, Songklanakarin J. Sci. Technol ) [9] R. Chinram and P. Siammai, On Green s relations for Γsemigroups and reductive Γ semigroups, Int. J. Algebra ) [10] D. Heidari, S.O. Dehkordi and B. Davvaz, Γsemihypergroups and their properties, U.P.B. Sci. Bull., Series A ) [11] K. Hila, On regular, semiprime and quasireflexive Γsemigroup and minimal quasiideals, Lobachevskian J. Math ) [12] N. Kuroki, Rough ideals in semigroups, Inform. Sci ) [13] S. Lajos, Generalized ideals in semigroups, Acta Sci. Math ) [14] S. Lajos, Notes on m, n)ideals I, Proc. Japan Acad ) [15] F. Marty, Sur une generalization de la notion de groupe, 8iem congres Math. Scandinaves, Stockholm, 1934, [16] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci ) [17] M.K. Sen and N.K. Saha, On Γsemigroup I, Bull. Cal. Math. Soc ) [18] N. Yaqoob, Applications of rough sets to Γhyperideals in left almost Γsemihypergroups, Accepted in Neural Comp. Applic., doi: /s [19] N. Yaqoob, M. Aslam and R. Chinram, Rough prime biideals and rough fuzzy prime biideals in semigroups, Ann. Fuzzy Math. Inform. 32) 2012)
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