International Journal of Analysis and Applications Volume 17, Number 3 (2019), 329-341 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-329 m-POLAR FUZZY HYPERIDEALS IN LA-SEMIHYPERGROUPS AHMED ELMOASRY1,2,∗, NAVEED YAQOOB1 1Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia 2Department of Mathematics, Faculty Of science, Aswan University, Aswan, Egypt ∗Corresponding author: a.elmoasry@mu.edu.sa Abstract. In this paper, we initiate a study of m-polar fuzzy sets in hyperstructure theory, particularly in left almost semihypergroups. We define an m-polar fuzzy left (right, two sided) hyperideal in a left almost semihypergroup and provided some results on these m-polar fuzzy left (right, two sided) hyperideals. 1. Introduction In 1934, hyperstructure theory was introduced by a French mathematician Marty [1] and then, several authors continued their researches in this direction. Hyperstructures are now widely studied from theoretical point of view and for their applications in many subjects of pure and applied mathematics. Some preliminary results and definitions about hyperstructure theory can be found in the books written by Corsini [2] and Vougiouklis [3]. The concept of a semihypergroup is a generalization of the concept of a semigroup. Many authors studied different aspects of semihypergroups. Some principal notions about semihypergroup theory can be found in [4]. In 2011, Hila and Dine [5] introduced the concept of non-associative semihypergroups (LA-semihypergroups) which is a generalization of semigroups, semihypergroups, and LA-semigroups. Later, Yaqoob et al. [6] explored this concept further and studied intra-regular left almost semihypergroups by their hyperideals using pure left identity. Other results on LA-semihypergroups can be found in [7–9]. Received 2019-03-07; accepted 2019-04-01; published 2019-05-01. 1991 Mathematics Subject Classification. 20N20, 03E72. Key words and phrases. LA-semihypergroups; hyperideals; m-polar fuzzy sets; m-polar fuzzy hyperideals. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 329 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-329 Int. J. Anal. Appl. 17 (3) (2019) 330 In 1965, Zadeh [10] introduced the notion of a fuzzy subset of a non-empty set X, as a function from X to [0, 1]. After the introduction of fuzzy sets, several researchers conducted the researches on generalizations of fuzzy sets with huge applications in computer, logics, automata and many branches of pure and applied mathematics. In 1971, Rosenfeld [11] defined the concept of a fuzzy subgroup of a group, and provided a tool and a novel approach to establish the fuzzy analogs of many results in group theory. However, in 1979, Anthony and Sherwood [12] generalized the Rosenfeld’s definition of fuzzy group. Sherwood [13] defined products of fuzzy subgroups using t-norms and gave some properties of these products. Asaad [14] proposed the idea about the relation between structure of a group and the structure of a fuzzy subgroup. Mukherjee and Bhattacharya, introduced the concept of a fuzzy normal subgroup and a fuzzy coset in [15]. In 1981, Das [16] introduced the concept of level subsets and level subgroups. Fuzzy set theory has been developed in the context of hyperalgebraic structure theory. In [17], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup. More on fuzzy hyperstructures one can find in [18]. In 1994, Zhang [19] generalized the idea of a fuzzy set and gave the concept of a bipolar fuzzy set on a given set X as a map, which associates each element of X, to a real number, in the interval [−1, 1]. In 2014, Chen et al. [20] introduced the idea of m-polar fuzzy sets, as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one [20]. Recently, Akram with others applied the concept of m-polar fuzzy sets in different directions, like, Lie algebras [21, 22], graph theory [23, 24], groups [25], lattices [26] and matroids [27]. Akrem published a book on m-polar fuzzy graphs [28], in which he provided several types of m-polar fuzzy graphs with applications. Al-Masarwah and Ahmad [29] studied m-polar (α,β)-fuzzy ideals in BCK/BCI-Algebra. 2. Preliminaries Definition 2.1. A map ∗ : H × H → P∗(H) is called a hyperoperation or a join operation on a set H, where H is a non-empty set and P∗(H) = P(H)\{∅} denotes the set of all non-empty subsets of H. Definition 2.2. A hypergroupoid is a pair (H,∗), where ∗ is a hyperoperation on H. Definition 2.3. [5, 6] A hypergroupoid (H,∗) is called an LA-semihypergroup if for all x,y,z ∈ H, (x∗y) ∗z = (z ∗y) ∗x. The law (x∗y) ∗ z = (z ∗y) ∗x is called a left invertive law. In an LA-semihypergroup, the medial law (x∗y) ∗ (z ∗w) = (x∗z) ∗ (y ∗w) holds for all x,y,z,w ∈ H. Let A and B be two non-empty subsets of H. Then we define Int. J. Anal. Appl. 17 (3) (2019) 331 A∗B = ⋃ a∈A,b∈B a∗ b, a∗A = {a}∗A and a∗B = {a}∗B. Definition 2.4. An element e ∈ H is called a left identity (resp., pure left identity) if for all x ∈ H, x ∈ e∗x (resp., x = e∗x). An LA-semihypergroup may or may not contains a left identity and a pure left identity. Lemma 2.1. [6] Let H be an LA-semihypergroup with a pure left identity e. Then x∗ (y ∗z) = y ∗ (x∗z) holds for all x,y,z ∈ H. Lemma 2.2. [6] Let H be an LA-semihypergroup with a pure left identity e. Then the paramedial law (x∗y) ∗ (z ∗w) = (w ∗z) ∗ (y ∗x) holds for all x,y,z,w ∈ H. Definition 2.5. [5] Let H be an LA-semihypergroup. A non-empty subset A of H is called a sub LA- semihypergroup of H if x∗y ⊆ A for every x,y ∈ A. Definition 2.6. [5] A subset I of an LA-semihypergroup H is called a right (resp., left) hyperideal of H if I ∗H ⊆ I (resp., H ∗ I ⊆ I) and is called a hyperideal if it is two-sided hyperideal. Definition 2.7. [6] By a bi-hyperideal (resp., generalized bi-hyperideal) of an LA-semihypergroup H, we mean a sub LA-semihypergroup (resp., non-empty subset) B of H such that (B ∗H) ∗B ⊆ B. Definition 2.8. [6] By an interior hyperideal of an LA-semihypergroup H, we mean a subset A of H such that (H ∗A) ∗H ⊆ A. Definition 2.9. An LA-semihypergroup H is called a regular LA-semihypergroup if for every x ∈ H, x ∈ (x∗y) ∗x, for some y ∈ H. An element a of H is called a left regular element of H if there exists x ∈ H such that a ∈ x∗ (a∗a) and H is called left regular if every element of H is left regular. Definition 2.10. [20] An m-polar fuzzy set (or a [0, 1]m-set) on X is a mapping M̂ : X → [0, 1]m. The membership value of every element x ∈ X is denoted by M̂(x) = ( p1 ◦M̂(x),p2 ◦M̂(x), ...,pm ◦M̂(x) ) , where pi : [0, 1] m → [0, 1] is the i-th projection mapping (i ∈ m). Note that [0, 1]m (m-power of [0, 1]) is considered a poset with the point-wise order �, where m is an arbitrary ordinal number (we make an appointment that m = {n|n < m} when m > 0), � is defined by u � v ⇔ pi(u) � pi(v) for each i ∈ m ( u,v ∈ [0, 1]m). Also, 0̂ = (0, 0, · · · , 0) is the smallest element in [0, 1]m and 1̂ = (1, 1, · · · , 1) is the largest element in [0, 1]m. Int. J. Anal. Appl. 17 (3) (2019) 332 3. m-Polar fuzzy hyperideals In this section, we will provide some results on m-polar fuzzy hyperideals. Let M̂1 and M̂2 be two m-polar fuzzy sets in an LA-semihypergroup H. Then for any x ∈ H, we define the following: (M̂1 ∩M̂2)(x) = min { M̂1(x),M̂2(x) } , (M̂1 ∪M̂2)(x) = max { M̂1(x),M̂2(x) } , and (M̂1 }M̂2)(x) =   sup x∈y∗z {min{M̂1(y),M̂2(z)} if x ∈ y ∗z (0, 0, ..., 0) if x /∈ y ∗z. We denote the set of all m-polar fuzzy subsets in an LA-semihypergroup H by Fm(H). For any M̂1, M̂2 ∈ Fm(H), M̂1 ⊆ M̂2 ⇔ M̂1(x) � M̂2(x), for each x ∈ H. Proposition 3.1. Let H be an LA-semihypergroup. Then the set (Fm(H),}) is an LA-semihypergroup. Proof. The proof is straightforward. � Proposition 3.2. If H is an LA-semihypergroup, then the medial law (M̂1 }M̂2) } (M̂3 }M̂4) = (M̂1 }M̂3) } (M̂2 }M̂4) holds in Fm(H), for all M̂1, M̂2, M̂3 and M̂4 in Fm(H). Proof. The proof is straightforward. � Proposition 3.3. Let H be an LA-semihypergroup with a pure left identity. Then the following properties hold in Fm(H) : (i): M̂1 } (M̂2 }M̂3) = M̂2 } (M̂1 }M̂3); (ii): (M̂1 }M̂2) } (M̂3 }M̂4) = (M̂4 }M̂3) } (M̂2 }M̂1); for all M̂1,M̂2,M̂3 and M̂4 in Fm(H). Int. J. Anal. Appl. 17 (3) (2019) 333 Proof. (i) Let M̂1, M̂2 and M̂3 be in Fm(H). There exist x and y in H such that z ∈ x∗y. Then (M̂1 } (M̂2 }M̂3))(z) = sup z∈x∗y { min { M̂1(x), (M̂2 }M̂3)(y) }} = sup z∈x∗y { min { M̂1(x), sup y∈t∗w { min { M̂2(t),M̂3(w) }}}} = sup z∈x∗y sup y∈t∗w { min { M̂1(x),M̂2(t),M̂3(w) }} = sup z∈x∗(t∗w) { min { M̂1(x),M̂2(t),M̂3(w) }} = sup z∈t∗(x∗w) { min { M̂2(t),M̂1(x),M̂3(w) }} , = sup z∈t∗m sup m∈x∗w { min { M̂2(t),M̂1(x),M̂3(w) }} = sup z∈t∗m { min { M̂2(t), sup m∈x∗w { min { M̂1(x),M̂3(w) }}}} = sup z∈t∗m { min { M̂2(t), (M̂1 }M̂3)(m) }} = (M̂2 } (M̂1 }M̂3))(z). Let z be an element of H such that z /∈ x∗y, for some x,y ∈ H. Then we have (M̂1 } (M̂2 }M̂3))(z) = (0, 0, ..., 0) = (M̂2 } (M̂1 }M̂3))(z). Hence this shows that M̂1 } (M̂2 }M̂3) = M̂2 } (M̂1 }M̂3) holds in Fm(H). (2) The proof is similar to (1). � Lemma 3.1. Let H be an LA-semihypergroup with a pure left identity. Then for all x ∈ H, H}H = H, where H(x) = (1, 1, ..., 1). Proof. The proof is straightforward. � Definition 3.1. An m-polar fuzzy set M̂ of an LA-semihypergroup H is called an m-polar fuzzy sub LA- semihypergroup of H if for all x,y ∈ H, the following condition hold: inf z∈x∗y { M̂(z) } � min { M̂(x),M̂(y) } . That is inf z∈x∗y { pi ◦M̂(z) } � min { pi ◦M̂(x),pi ◦M̂(y) } , for each i = 1, 2, 3, · · · ,m. Int. J. Anal. Appl. 17 (3) (2019) 334 Example 3.1. Consider an LA-semihypergroup H = {e,a,b,c,d}, with the hyperoperation ” ∗ ” defined as follow: ∗ e a b c d e e e e e e a e a a a a b e a a {a,d} a c e a {a,b} {a,c} {a,d} d e a a {a,b} a We define a 5-polar fuzzy set M̂ : H → [0, 1]5 as follows: M̂(x) =   (0.6, 0.7, 0.8, 0.8, 0.9) if x = e (0.5, 0.6, 0.8, 0.8, 0.8) if x = a (0.4, 0.4, 0.7, 0.7, 0.7) if x ∈{b,d} (0.1, 0.3, 0.2, 0.4, 0.1) if x = c. By routine calculations, it is easy to verify that M̂ is a 5-polar fuzzy sub LA-semihypergroup of H. Definition 3.2. An m-polar fuzzy set M̂ of an LA-semihypergroup H is called an m-polar fuzzy left (resp., right) hyperideal of H if for all x,y ∈ H, the following condition hold: inf z∈x∗y { M̂(z) } � M̂(y) (resp., inf z∈x∗y { M̂(z) } � M̂(x)). That is inf z∈x∗y { pi ◦M̂(z) } � pi ◦M̂(y) (resp., inf z∈x∗y { pi ◦M̂(z) } � pi ◦M̂(x)), for each i = 1, 2, 3, · · · ,m. Definition 3.3. An m-polar fuzzy set M̂ of an LA-semihypergroup H is called an m-polar fuzzy hyperideal of H if for all x,y ∈ H, the following condition hold: inf z∈x∗y { M̂(z) } � max { M̂(x),M̂(y) } . That is inf z∈x∗y { pi ◦M̂(z) } � max { pi ◦M̂(x),pi ◦M̂(y) } , for each i = 1, 2, 3, · · · ,m. Int. J. Anal. Appl. 17 (3) (2019) 335 Example 3.2. Consider an LA-semihypergroup H = {e,a,b,c,d}, with the hyperoperation ” ∗ ” defined as follow: ∗ e a b c d e e e e e e a e a {e,b} a a b e {e,b} b {e,b} {e,b} c e a {e,b} {a,c,d} d d e a {e,b} {a,d} {a,d} We define a 3-polar fuzzy set M̂ : H → [0, 1]3 as follows: M̂(x) =   (0.8, 0.9, 0.8) if x = e (0.5, 0.5, 0.5) if x = a (0.7, 0.7, 0.6) if x = b (0.1, 0.2, 0.2) if x = c (0.3, 0.4, 0.3) if x = d. By routine calculations, it is easy to verify that M̂ is a 3-polar fuzzy hyperideal of H. Definition 3.4. Let H be an LA-semihypergroup. Let t̂ = (t1, t2, ..., tm) ∈ [0, 1]m and M̂ be an m-polar fuzzy set in H. Then, the set (i): M̂t̂ = {x ∈ H |M̂(x) � t̂} is called a t̂-level cut of M̂. (ii): M̂s t̂ = {x ∈ H |M̂(x) � t̂} is called a strong t̂-level cut of M̂. (iii): Im(M̂) is called the image of M̂. Theorem 3.1. Let M̂ be an m-polar fuzzy set in H, such that the least upper bound t̂0 of Im(M̂) exist. Then the following conditions are equivalent: (i): M̂ is an m-polar fuzzy sub LA-semihypergroup of H. (ii): For all t̂ ∈ Im(M̂), the non-empty t̂-level subset M̂t̂ of M̂ is a sub LA-semihypergroup of H. (iii): For all t̂ ∈ Im(M̂)\ t̂0, the non-empty strong t̂-level subset M̂st̂ of M̂ is a sub LA-semihypergroup of H. (iv): For all t̂ ∈ [0, 1]m, the non-empty strong t̂-level subset M̂s t̂ of M̂ is a sub LA-semihypergroup of H. (v): For all t̂ ∈ [0, 1]m, the non-empty t̂-level subset M̂t̂ of M̂ is a sub LA-semihypergroup of H. Proof. (i) → (iv). Let M̂ be a m-polar fuzzy sub LA-semihypergroup of H, t̂ ∈ [0, 1]m, and x,y ∈ M̂s t̂ . Then we have M̂(x),M̂(y) � t̂. Thus, min{M̂(x),M̂(y)} � t̂. Since M̂ is a sub LA-semihypergroup of H, so inf z∈x∗y M̂(z) � t̂. Thus z ∈ M̂s t̂ for each z ∈ x∗y. Hence M̂s t̂ is a sub LA-semihypergroup of H. Int. J. Anal. Appl. 17 (3) (2019) 336 (iv) → (iii). It is clear. (iii) → (ii). Let t̂ ∈ Im(M̂). Then M̂t̂ is non-empty. Since M̂t̂ = H⋂̂ t≺α̂ M̂α̂, where α̂ ∈ Im(M̂) \ t̂0. Then by (iii) we get that M̂t̂ is a sub LA-semihypergroup of H. (ii) → (v). Let t̂ ∈ [0, 1]m and M̂t̂ be non-empty. Suppose that x,y ∈ M̂t̂. Then, we have M̂(x),M̂(y) � t̂. Let α̂ = min{M̂(x),M̂(y)}. It is clear that α̂ � t̂. Thus x,y ∈ M̂t̂ and α̂ ∈ Im(M̂), by (ii) M̂α̂ is a sub LA-semihypergroup of H, hence z ∈ M̂α̂ for each z ∈ x∗ y. Then we have inf z∈x∗y M̂(z) � α̂ � t̂. Therefore x∗y ⊆ M̂t̂. Then M̂t̂ is a sub LA-semihypergroup of H. (v) → (i). Assume that the non-empty set M̂t̂ is a sub LA-semihypergroup of H, for any t̂ ∈ [0, 1] m. Let x,y ∈ H. Let us take t̂ = min{M̂(x),M̂(y)}. Then M̂(x),M̂(y) � t̂. Thus x,y ∈ M̂t̂. Since M̂t̂ is a sub LA-semihypergroup of H, so for each z ∈ x∗y, z ∈ M̂t̂. Thus, inf z∈x∗y M̂(z) � t̂ = min{M̂(x),M̂(y)}. This shows that M̂ is an m-polar fuzzy sub LA-semihypergroup of H. This completes the proof. � Theorem 3.2. Let M̂ be an m-polar fuzzy set in H, such that the least upper bound t̂0 of Im(M̂) exist. Then the following conditions are equivalent: (i): M̂ is an m-polar fuzzy left (resp., right) hyperideal of H. (ii): For all t̂ ∈ Im(M̂), the non-empty t̂-level subset M̂t̂ of M̂ is a left (resp., right) hyperideal of H. (iii): For all t̂ ∈ Im(M̂) \ t̂0, the non-empty strong t̂-level subset M̂st̂ of M̂ is a left (resp., right) hyperideal of H. (iv): For all t̂ ∈ [0, 1]m, the non-empty strong t̂-level subset M̂s t̂ of M̂ is a left (resp., right) hyperideal of H. (v): For all t̂ ∈ [0, 1]m, the non-empty t̂-level subset M̂t̂ of M̂ is a left (resp., right) hyperideal of H. Proof. The proof is similar to the proof of Theorem 3.1. � Theorem 3.3. An m-polar fuzzy subset M̂ of an LA-semihypergroup H is an m-polar fuzzy (1) sub LA-semihypergroup of H if and only if M̂ }M̂ ⊆ M̂, (2) left hyperideal of H if and only if H}M̂ ⊆ M̂, (3) right hyperideal of H if and only if M̂ }H⊆ M̂, (4) hyperideal of H if and only if H}M̂ ⊆ M̂ and M̂ }H⊆ M̂. Int. J. Anal. Appl. 17 (3) (2019) 337 Proof. (1) Let M̂ be an m-polar fuzzy sub LA-semihypergroup of H and z ∈ H. Let us suppose that z ∈ x∗y for x,y ∈ H. Then (M̂ }M̂) (z) = sup z∈x∗y {min{M̂(x),M̂(y)} � sup z∈x∗y { inf z∈x∗y M̂ (z) } � sup z∈x∗y { M̂ (x∗y) } = M̂ (z) . Therefore (M̂ }M̂) ⊆ M̂. If there do not exist any x,y ∈ H such that z ∈ x∗y, then (M̂ }M̂) (z) = 0 � M̂ (z) . Hence for all cases M̂ }M̂ ⊆ M̂. Conversely, let us assume that M̂}M̂ ⊆ M̂ holds for all m-polar fuzzy subsets of H. Let x,y ∈ H. Then, we have inf z∈x∗y M̂ (z) � inf z∈x∗y { (M̂ }M̂) (z) } = inf z∈x∗y { sup z∈x∗y {min{M̂(x),M̂(y)} } = min{M̂(x),M̂(y)}. This means that M̂ is an m-polar fuzzy sub LA-semihypergroup of H. The other cases can be seen in a similar way. � Theorem 3.4. If { M̂i } i∈Λ is a family of m-polar fuzzy left (resp., right) hyperideals of an LA-semihypergroup H, then ⋂ i∈Λ M̂i is also a fuzzy left (resp., right) hyperideal of H. Proof. Let { M̂i } i∈Λ be a family of m-polar fuzzy left hyperideals of an LA-semihypergroup H and let x,y ∈ H. Then we have ⋂ i∈Λ M̂i (y) = ∧ i∈Λ { M̂i (y) } � ∧ i∈Λ { inf z∈x∗y { M̂i (z) }} = inf z∈x∗y { ∧ i∈Λ { M̂i (z) }} = inf z∈x∗y { ⋂ i∈Λ M̂i (z) } . Hence ⋂ i∈Λ M̂i is an m-polar fuzzy left hyperideal of LA-semihypergroup H. Similarly we can prove it for right hyperideals. This completes the proof. � Int. J. Anal. Appl. 17 (3) (2019) 338 Let H be an LA-semihypergroup and let ∅ 6= A ⊆ H. Then the m-polar fuzzy characteristic function χA M̂ of A is defined by χA M̂ (x) =   (1, 1, ..., 1) if x ∈ A(0, 0, ..., 0) if x /∈ A . Theorem 3.5. Let A be a nonempty subset of an LA-semihypergroup H. Then A is a sub LA-semihypergroup (resp., left hyperideal, right hyperideal) of H if and only if χA M̂ is an m-polar fuzzy sub LA-semihypergroup (resp., left hyperideal, right hyperideal) of H. Proof. Let A be a left hyperideal of H. For any x,y ∈ H, we have the following cases: Case (1) : If x,y ∈ A, then x∗y ⊆ A. Then inf z∈x∗y χA M̂ (z) = 1̂ and χA M̂ (y) = 1̂. Therefore inf z∈x∗y χA M̂ (z) = χA M̂ (y) . Case (2) : If x,y /∈ A, then χA M̂ (y) = 0̂. So inf z∈x∗y χA M̂ (z) � χA M̂ (y) . Case (3) : If x ∈ A and y /∈ A, then χA M̂ (y) = 0̂. So inf z∈x∗y χA M̂ (z) � χA M̂ (y) . Case (4) : If x /∈ A and y ∈ A, then x∗y ⊆ A. Then inf z∈x∗y χA M̂ (z) = 1̂ and χA M̂ (y) = 1̂. Therefore inf z∈x∗y χA M̂ (z) = χA M̂ (y) . Hence χA M̂ is an m-polar fuzzy left hyperideal of H. Conversely, suppose χA M̂ is an m-polar fuzzy left hyperideal of H and let x ∈ H and y ∈ A. Then we have inf z∈x∗y χA M̂ (z) � χA M̂ (y) = 1̂. But, we know that inf z∈x∗y χA M̂ (z) � 1̂. Therefore inf z∈x∗y χA M̂ (z) = 1̂. This implies that z ∈ A for each z ∈ x∗y. Hence A is a left hyperideal of H. The other cases can be seen in a similar way. � Proposition 3.4. Let M̂1 be an m-polar fuzzy right hyperideal of H and M̂2 be an m-polar fuzzy left hyperideal of H. Then M̂1 }M̂2 ⊆ M̂1 ∩M̂2. Int. J. Anal. Appl. 17 (3) (2019) 339 Proof. Let M̂1 be an m-polar fuzzy right hyperideal of H and M̂2 be an m-polar fuzzy left hyperideal of H. Let z ∈ H and suppose that there exist x,y ∈ H such that z ∈ x∗y. Then (M̂1 }M̂2)(z) = sup z∈x∗y { min{M̂1(x),M̂2(y)} } � sup z∈x∗y { min { inf z∈x∗y { M̂1(z) } , inf z∈x∗y { M̂2(z) }}} = min{M̂1(z),M̂2(z)} = (M̂1 ∩M̂2)(z). Let us suppose that there do not exist x,y ∈ H such that z ∈ x∗y. Then, (M̂1}M̂2)(z) = 0̂ � (M̂1∩M̂2)(z). This completes the proof. � Lemma 3.2. Let H be an LA-semihypergroup with a pure left identity. Then every m-polar fuzzy right hyperideal is an m-polar fuzzy hyperideal. Proof. The proof is straightforward. � Theorem 3.6. If M̂ is an m-polar fuzzy left hyperideal of H with a pure left identity, then M̂ ∪ (M̂ }H) and M̂ ∪ (M̂ }M̂) are m-polar fuzzy hyperideals of H. Proof. Let M̂ be an m-polar fuzzy left hyperideal of H. We have( M̂ ∪ (M̂ }H) ) }H = (M̂ }H) ∪ ((M̂ }H) }H) = (M̂ }H) ∪ ((H}H) }M̂) = (M̂ }H) ∪ (H}M̂) (by Lemma 3.1) ⊆ (M̂ }H) ∪M̂. Hence, M̂ ∪(M̂ }H) is an m-polar fuzzy right hyperideal of H. By Lemma 3.2, M̂ ∪(M̂ }H) is an m-polar fuzzy hyperideal of H. In a similar way we can prove that M̂ ∪(M̂ }M̂) is an m-polar fuzzy hyperideal. � Proposition 3.5. Let H be a regular LA-semihypergroup with a pure left identity e. If M̂ is an m-polar fuzzy right hyperideal of H, then M̂(x∗y) = M̂(y ∗x) holds for all x,y ∈ H. Proof. Let M̂ be an m-polar fuzzy right hyperideal of a regular LA-semihypergroup H with a pure left identity e. Let x,y ∈ H. Since H is regular, x ∈ (x ∗ a) ∗ x and y ∈ (y ∗ b) ∗ y for some a,b ∈ H. Now by using the medial and paramedial laws, we get x∗y ⊆ ((x∗a) ∗x) ∗ ((y ∗ b) ∗y) = (y ∗x) ∗ ((y ∗ b) ∗ (x∗a)). Int. J. Anal. Appl. 17 (3) (2019) 340 Since M̂ is an m-polar fuzzy right hyperideal, then for every w ∈ x∗y ⊆ (y∗x) ∗ ((y∗b) ∗ (x∗a)), we have inf w∈x∗y {M̂(w)} � inf w1∈(y∗x)∗((y∗b)∗(x∗a)) {M̂(w1)} � inf s∈y∗x {M̂(s)}, Again by using the medial and paramedial laws, we get y ∗x ⊆ ((y ∗ b) ∗y) ∗ ((x∗a) ∗x) = (x∗y) ∗ ((x∗a) ∗ (y ∗ b)). Since M̂ is an m-polar fuzzy right hyperideal, for every t ∈ y ∗x ⊆ (x∗y) ∗ ((x∗a) ∗ (y ∗ b)), we have inf t∈y∗x {M̂(t)} � inf t1∈(x∗y)∗((x∗a)∗(y∗b)) {M̂(t1)} � inf p∈x∗y {M̂(p)}, This shows that M̂(x∗y) = M̂(y ∗x) holds for all x,y ∈ H. � Lemma 3.3. If H is a left regular LA-semihypergroup, then every m-polar fuzzy left (resp., right) hyperideal M̂ of H is an m-polar fuzzy idempotent. Proof. Let M̂ be any m-polar fuzzy left hyperideal of a left regular LA-semihypergroup H with a pure left identity. Then M̂ }M̂ ⊆H∗M̂ ⊆ M̂. Since H is left regular, for every a ∈ H, there exists x ∈ H such that a ∈ x∗ (a∗a). By Lemma 2.1, we have a ∈ x∗ (a∗a) = a∗ (x∗a). Therefore (M̂ }M̂)(a) = sup a∈a∗(x∗a) { min { M̂(a), inf s∈x∗a M̂(s) }} � min{M̂(a),M̂(a)} = M̂(a). Thus we get M̂ ⊆ M̂ } M̂. This implies that M̂ } M̂ = M̂. Hence this shows that M̂ is m-polar fuzzy idempotent. The other case can be proved in a similar way. � 4. Conclusions In this paper, we studied the hyperideal-structure of m-polar fuzzy sets in left almost semihypergroups. We defined m-polar fuzzy sub LA-semihypergroups and m-polar fuzzy left (right, two sided) hyperideal in a left almost semihypergroup and used these m-polar fuzzy left (right, two sided) hyperideals to characterize some classes of left almost semihypergroups. Acknowledgements. The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number 35/37. Int. J. Anal. Appl. 17 (3) (2019) 341 References [1] F. 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