International Journal of Analysis and Applications Volume 16, Number 4 (2018), 528-541 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-528 ON LEFT ALMOST SEMIHYPERRINGS SHAH NAWAZ1, INAYATUR REHMAN2, MUHAMMAD GULISTAN1,∗ 1Department of Mathematics, Hazara University, Mansehra, Pakistan 2Department of Mathematics and Science, College of Arts and Applied Sciences, Dhofar University, Salalah, Oman ∗Corresponding author: gulistanmath@hu.edu.pk Abstract. The purpose of this article is to introduce the notion of left almost semihyperrings which is a generalization of left almost semirings. We investigate the basic properties of left almost semihyperrings. By using the concept of hyperideal and regular relations we prove some useful results on it. 1. Introduction Kazim and Naseeudin [10] studied left almost semigroup (abbreviated as LA-semigroup). They gener- alized some handy sequal of semigroup theroy. Mushtaq and others [14–16] added many useful result of theory of LA-semigroups, also see [2, 8, 9]. LA-semigroup is the midway structure between a commutative semigroup and a groupoid. On the other hand it posses many interesting properties which we usully find in commutativ and associative algebric structure. Hyperstructures were introduced in 1934, when Marty [13] defined hypergroups, began to study their properties, and applied them to groups. A number of papers and several book have been written on hyper- structure theory; see [3,19]. Currently a book published on hyperstructure [4] points out on its applications in rough set theory, cryptography, automata, codes automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Hila and Dine [7] introduced the notion of left almost semihypergroups. Yaqoob Received 2018-02-18; accepted 2018-04-27; published 2018-07-02. 2010 Mathematics Subject Classification. 20N25. Key words and phrases. LA-semihypergroups; LA-semihyperrings; regular relations. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 528 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-528 Int. J. Anal. Appl. 16 (4) (2018) 529 et al. [20] extended the work of Hila and Dine and characterized intra-regular left almost semihypergroups by their hyperideals using pure left identity. Gulistan et al. [20] introduced the notion of Hv-LA-semigroups. Rehman et al. [20] studied hyperideals and hypersystems in LA-hyperrings. Yaqoob and Gulistan [21] intro- duced the concept of partial ordering on left almost semihypergroups. Yaqoob et al. [22] introduced the idea of left almost polygroups. Fuzzy set theory and rough set theory are also applied to LA-semihypergroups, see [1, 6, 12, 23, 24, 26]. In [25], Yusuf extended the concept of LA-groups to a non-associative structure with respect to the binary operations ‘+’ and ‘×’ namely left almost ring (LA-ring). Further M. Shah and T. Shah studied some basic properties of LA-ring in [18]. Kellil [11] studied left almost semirings. The purpose of this article is to introduce a new and more gerenal class of left almost semihyperrings which is of cource a generalization of left almost semiring. We have investigated the basic properties of left almost semihyperrings. By using the concept of hyperideal and regular relations we have proved some useful results on it. 2. LA-semihypergroups In this section we recalled some basic ideas from litrature about an LA-semihypergroup which helped in further development of this article. Definition 2.1. A map ◦ : X ×X −→ P∗(X) is called an hyperoperation or join operation on the set X, where X is a non-empty set and P∗(X) = P(X)\{∅} shows the all non empty subset of X. A hypergroupoid is a set X with the binary operation is a hyperoperation. If A and B be two non-empty subsets of X then we write product as fallow A◦B = ⋃ x1∈A,x2∈B x1 ◦x2, x1 ◦A = {x1}◦A and x1 ◦B = {x1}◦B. Definition 2.2. [7, 20] A hypergroupoid (X,◦) is called an LA-semihypergroup if for all x1,x2,x3 ∈ X, (x1 ◦x2) ◦x3 = (x3 ◦x2) ◦x1, the law (x1 ◦x2) ◦x3 = (x3 ◦x2) ◦x1 is known as left invertive law. Every LA-semihypergroup satisfies the law (x1 ◦x2) ◦ (x3 ◦x4) = (x1 ◦x3) ◦ (x2 ◦x4) for all x1,x2,x3,x4 ∈ X. This law is known as medial law (cf. [7]). Lemma 2.1. [20] Let X be an LA-semihypergroup with pure left idetity e, then x1◦(x2◦x3) = x2◦(x1◦x3) holds for all x1,x2,x3 ∈ X. Int. J. Anal. Appl. 16 (4) (2018) 530 Lemma 2.2. [20] Let X be an LA-semihypergroup with pure left identity e, then (x1 ◦ x2) ◦ (x3 ◦ x4) = (x4 ◦x2) ◦ (x3 ◦x1) holds for all x1,x2,x3,x4 ∈ X. The law (x1 ◦x2) ◦ (x3 ◦x4) = (x4 ◦x2) ◦ (x3 ◦x1) is called a paramedial law. 3. Left Almost Semihyperrings In this section we define the notion of left almost semihyperrings and provided some examples with some basic properties. Definition 3.1. An algebraic hyperstructure (R,⊕,⊗) is said to be an LA-semihyperring if it satisfies the following axioms: (1) (R,⊕) is an LA-semihypergroup; (2) (R,⊗) is an LA-semihypergroup, with an absorbing element 0 ∈ R, that is, 0 ⊗x = x⊗ 0 = 0 for all x ∈ R. (3) The operation ⊗ is distributive with respect to the hyperoperation ⊕, that is x1 ⊗ (x2 ⊕x3) = (x1 ⊗x2) ⊕ (x1 ⊗x3), (x1 ⊕x2) ⊗x3 = (x1 ⊗x3) ⊕ (x2 ⊗x3), for all x1,x2,x3 ∈ R. Example 3.1. Let R = {0,a,b} be a set with the hyperoperations ⊕ and ⊗ defined as follow: ⊕ 0 a b 0 0 R R a {a,b} {a,b} {a,b} b R R R ⊗ 0 a b 0 0 0 0 a 0 R b b 0 R R Then (R,⊕,⊗) is an LA-semihyperring. Definition 3.2. Let R be an LA-semihyperring. An element e ∈ R is called (i) Left identity (resp., pure left identity) if for all x ∈ R, x ∈ e⊗x (resp., x = e⊗x), (ii) Right identity (resp., pure right identity) if for all x ∈ R, x ∈ x⊗e (resp., x = x⊗e), (iii) Identity (resp., pure identity) if for all x ∈ R, x ∈ e⊗x∩x⊗e (resp., x = e⊗x∩x⊗e). Lemma 3.1. If an LA-semihypering R has a pure left identity e, then it is unique. Int. J. Anal. Appl. 16 (4) (2018) 531 Proof. Let us consider that there exist another pure left identity fL such that e⊗fL = fL and fL ⊗e = e. Consider e = e⊗e = (fL ⊗e) ⊗e = (e⊗e) ⊗fL = e⊗fL = fL. � Definition 3.3. An element x ∈ (R,⊕,⊗) is called additively idempotent if x ∈ x⊕x. Set of all additively idempotent element is denoted by I⊕ (R) . If every element of R is additively idem- potent then R is said to be additively idempotent. Definition 3.4. An element x ∈ (R,⊕,⊗) is called multiplicatively idempotent if x ∈ x⊗x. Set of all multiplicatively idempotent element is denoted by I⊗ (R) . Lemma 3.2. In an LA-semihyperring the following laws hold . (i) (x⊗y) ⊗ (z ⊗w) = (x⊗z) ⊗ (y ⊗w) ,∀x,y,z,w ∈ R, known as medial law. (ii) With left identity of an LA-semihyperring R, (x⊗y) ⊗ (z ⊗w) = (w ⊗y) ⊗ (z ⊗x) ,∀x,y,z,w ∈ R, known as paramedial law. Proof. (i) Consider (x⊗y) ⊗ (z ⊗w) = ((z ⊗w) ⊗y) ⊗x = ((y ⊗w) ⊗z) ⊗x = (x⊗z) ⊗ (y ⊗w) . Int. J. Anal. Appl. 16 (4) (2018) 532 (ii) Let e ∈ R be the left identity. Consider (x⊗y) ⊗ (z ⊗w) = (e⊗ (x⊗y)) ⊗ (z ⊗w) = ((z ⊗w) ⊗ (x⊗y)) ⊗e = ((z ⊗x) ⊗ (w ⊗y)) ⊗e = (e⊗ (w ⊗y)) ⊗ (z ⊗x) = (w ⊗y) ⊗ (z ⊗x) . � Remark 3.1. By medial law we have (x⊗y) ⊗ (z ⊗w) = (w ⊗y) ⊗ (z ⊗x) = (w ⊗z) ⊗ (y ⊗x) . Theorem 3.1. Let R is an LA-semihyperring with pure left identity e then x⊗(y ⊗z) = y⊗(x⊗z) for all x,y,z ∈ R. Proof. Consider x⊗ (y ⊗z) = (e⊗x) ⊗ (y ⊗z) = (e⊗y) ⊗ (x⊗z) by medial law = y ⊗ (x⊗z) , for all x,y,z,w ∈ R. � Theorem 3.2. An LA-semihyperring R is said to be semihyperring if and only if x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ R. Proof. Let R is semihyperring then (x⊗y) ⊗z = x⊗ (y ⊗z) , but (x⊗y) ⊗z = (z ⊗y) ⊗x, so x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ R. Conversely let x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ R. Int. J. Anal. Appl. 16 (4) (2018) 533 Since R is LA-semihyperring so (x⊗y) ⊗z = (z ⊗y) ⊗x = x⊗ (y ⊗z) . � Lemma 3.3. In an LA-semihyperring R, with pure left identity e, x⊗y = z ⊗w ⇒ y ⊗x = w ⊗ z for all x,y,z,w ∈ R. Proof. Consider y ⊗x = (e⊗y) ⊗x = (x⊗y) ⊗e = (z ⊗w) ⊗e since x⊗y = z ⊗w = (e⊗w) ⊗z by left invertive law = w ⊗z for all x,y,z,w ∈ R. � Definition 3.5. Let ∅ 6= K ⊆ R, then K is called sub LA-semihyperring if K itself form the LA-semihyperring. Proposition 3.1. Let ∅ 6= K ⊆ R, then K is called sub LA-semihyperring if ∀ x,y ∈ K we get x⊕y ∈ K and x⊗y ∈ K. Proof. Straightforward. � Definition 3.6. Let ∅ 6= A ⊆ R, then A is called a left hyperideal (resp., right hyperideal) of an LA- semihyperring R if A⊕A ⊆ A and R⊗A ⊆ A (resp., A⊗R ⊆ A) . If A is both a left and a right hyperideal of R, then it is called a hyperideal of R. Definition 3.7. Let ∅ 6= B ⊆ R, then B is called a bi-hyperideal of an LA-semihyperring R if B ⊕B ⊆ B, B ⊗B ⊆ B and (B ⊗R) ⊗B ⊆ B. Definition 3.8. Let ∅ 6= I ⊆ R, then I is called an interior hyperideal of an LA-semihyperring R if I⊕I ⊆ I and (R⊗ I) ⊗R ⊆ I. Definition 3.9. Let ∅ 6= Q ⊆ R, then Q is called a quasi-hyperideal of an LA-semihyperring R if Q⊕Q ⊆ Q and Q⊗R∩R⊗Q ⊆ Q. Int. J. Anal. Appl. 16 (4) (2018) 534 Definition 3.10. A hyperideal A of an LA-semihyperring R is called prime hyperideal of R, if for hyperideals I and J of R satisfying, I ⊗J ⊆ A, implies, either I ⊆ A, or J ⊆ A. Definition 3.11. For any non-empty subsets X,Y of an LA-semihyperring (R,⊕,⊗) we define X ⊕Y = ⋃ l1∈X l2∈Y (l1 ⊕ l2) and X ⊗Y = ⋃ l1i∈X l2i∈Y ( n∑ i=1 l1i ⊗ l2i ) . Proposition 3.2. Let X and Y be any two hyperideals of an LA-semihyperring (R,⊕,⊗) then X⊗Y ⊆ X∩Y. Proof. Let x ∈ X ⊗ Y = ⋃ l1∈X l2∈Y ( n∑ =1 l1 ⊗ l2 ) ⇒ x ∈ ⋃ l1∈X l2∈Y l3, where l3 = n∑ =1 l1 ⊗ l2 for each l1 ∈ X and l2 ∈ Y. Now since X is an hyperideal so l1 ⊗ l2 ∈ X for l2 ∈ Y ⊆ R. This implies that n∑ =1 l1 ⊗ l2 ⊆ X. we x ∈ X. Similarly by using the fact that Y is also hyperideal we get x ∈ Y, and thus x ∈ X ∩Y. Hence X ⊗Y ⊆ X ∩Y. � Proposition 3.3. Any left hyperideal of an LA-semihyperring R is a sub LA-semihyperring. Proof. Let I is a left hyperideal of an LA-semihyperring R. Then obviously ∀ x,y ∈ I we get x⊕y ∈ I. Also for any l1,x ∈ I and since l1 ∈ I ⊆ R so we have l1 ⊗x ∈ I. Thus I is a sub LA-simihyperring. � In particular every right hyperideal becomes sub LA-semihyperring and so does the hyperideal. Theorem 3.3. Intersection of any family of hyperideal of an LA-semihyperring R is an hyperideal of R. Proof. Let {Ii}i∈∧ be a family of hyperideals of an LA-semihyperring R and we have to show that ⋂ i∈∧ Ii is also an hyperideal of R. Let x,y ∈ ⋂ i∈∧ Ii, then x,y ∈ Ii. Now since each Ii is an hyperideal so x⊕y ∈ Ii for all i ∈∧. Thus x⊕y ∈ ⋂ i∈∧ Ii. Again let x ∈ ⋂ i∈∧ Ii and l1 ∈ R. From x ∈ ⋂ i∈∧ Ii we have x ∈ Ii for all i ∈∧. Since each Ii is an hyperideal so l1 ⊕x ∈ Ii for all i ∈∧. Which implies that l1 ⊕x ∈ ⋂ i∈∧ Ii. Thus ⋂ i∈∧ Ii is an left hyperideal of R. Similarly it can easily be proved for right hyperideals and hence ⋂ i∈∧ Ii is an hyperideal of R. � Corollary 3.1. Intersection of any family of sub LA-semihyperring of an LA-semihyperring R is again sub LA-semihyperring of R. Proof. Straightforward. � Theorem 3.4. If I and J are hyperideals of an LA-semihyperring R, then I ⊕ J are hyperideals of R. Moreover it is the smallest hyperideal of R containing both I and J. Int. J. Anal. Appl. 16 (4) (2018) 535 Proof. Let us define I ⊕ J = ⋃ l1∈I l2∈J (l1 ⊕ l2) . Let x,y ∈ I ⊕ J then ∃ l11, l12 ∈ x and l21, l22 ∈ J such that x ∈ l11⊕l21 and y ∈ l12⊕l22. Consider x⊕y ⊆ (l11 ⊕ l21)⊕(l12 ⊕ l22) = (l11 ⊕ l12)⊕(l21 ⊕ l22) by medial law. Since I and J are hyperideals so (l11 ⊕ l12) ⊆ I and (l21 ⊕ l22) ⊆ J. Thus x⊕y ⊆ (l11 ⊕ l12) ⊕ (l21 ⊕ l22) ⊆ I ⊕J. Hence x⊕y ⊆ I ⊕J for all x ∈ I, and y ∈ J. Again consider x ∈ I ⊕ J and r ∈ R. For x ∈ I ⊕ J there exist some l1 ∈ I and l2 ∈ J such that x ∈ l1 ⊕ l2. Now r⊗x ∈ r⊗ (l1 ⊕ l2) = (r ⊗ l1) ⊕ (r ⊗ l2) by distributive law. Since I and J are hyperideals so (r ⊗ l1) ⊆ I and (r ⊗ l2) ⊆ J for any r ∈ R, l1 ∈ I and l2 ∈ J. Thus (r ⊗ l1) ⊕ (r ⊗ l2) ⊆ I ⊕J and hence r⊗x ∈ I⊕J for x ∈ I⊕J and r ∈ R. Which shows that I⊕J is left hyperideal of R. Similarly it can easily be proved for right ideals and thus I ⊕J is an hyperideal of R. Now we will show that I ⊕J contains both I and j i.e. I ∪J ⊆ I ⊕J. Let x ∈ I ∪J ⇒ x ∈ I or x ∈ J. Since I and J are hyperideals so 0 ∈ I and 0 ∈ J. Now if l1 ∈ I, since x = x⊕0 ⊆ I ⊕J. And if x ∈ J, since x = 0 ⊕x ⊆ I ⊕J. Hence x ∈ I ⊕J and thus I ∪J ⊆ I ⊕J. Next we will show that I ⊕J is the smallest hyperideal. Let M be any other hyperideal containing both I and J and we have to show that I ⊕J ⊆ M. For this let x ∈ I ⊕J then there exist l1 ∈ I and l2 ∈ J such that x ∈ l1 ⊕ l2. Since l1 ∈ I ⊆ I ∪J ⊆ M and l2 ∈ J ⊆ I ∪J ⊆ M. Which implies that l1, l2 ∈ M, but M is the hyperideal so l1 ⊕ l2 ⊆ M. Thus x ∈ M, and so I ⊕J ⊆ M. Hence I ⊕J is the smallest hyperideal of R containing both I and J. � Proposition 3.4. Let ∅ 6= K ⊆ R and ∅ 6= I ⊆ R such that K is sub LA-semihyperring and I is an hyperideal then (i) K ⊕ I is a sub LA-semihyperring of R. (ii) K ∩ I is a hyperideal of R. Proof. (i) let us define K ⊕ I = ⋃ l1∈K l2∈I (l1 ⊕ l2) . Since 0 ∈ K and 0 ∈ I. Therefore we have {0} = 0 ⊕ 0 ⊆ K ⊕ I. thus K ⊕ I is non-empty. Let x,y ∈ K ⊕ I then there exist l11, l12 ∈ K and l21, l22 ∈ I such that x ∈ l11 ⊕ l21 and y ∈ l12 ⊕ l22. Consider x⊕y ⊆ (l11 ⊕ l21) ⊕ (l12 ⊕ l22) = (l11 ⊕ l12) ⊕ (l21 ⊕ l22) by medial law. Since K is sub LA-semihyperring and I is an hyperideal so (l11 ⊕ l12) ⊆ K and (l21 ⊕ l22) ⊆ I. Thus x⊕y ⊆ (l11 ⊕ l12)⊕(l21 ⊕ l22) ⊆ K⊕I. Hence x⊕y ⊆ K⊕I for all x,y ∈ K⊕I. And again consider x⊗y ⊆ (l11 ⊕ l21)⊗(l12 ⊕ l22) = (l11 ⊗ l12)⊕(l11 ⊗ l22)⊕(l21 ⊗ l12)⊕(l21 ⊗ l22) by distributive law. Now since K is sub LA-semihyperring so (l11 ⊗ l12) ∈ K, and I is an hyperideal so ((l11 ⊗ l22) ⊕ (l21 ⊗ l12) ⊕ (l21 ⊗ l22)) ∈ I. Eventually x⊗y ⊆ K ⊕ I. Hence K ⊕ I is a sub LA-semihyperring of R. (ii) Let x,y ∈ K ∩ I ⇒ x,y ∈ K and x,y ∈ I. Since K is sub LA-semihyperring and I is an hyperideal so x⊕y ∈ K and x⊕y ∈ I. Which implies that x⊕y ∈ K ∩ I. Now again let x ∈ K ∩ I and l1 ∈ R. From x ∈ K∩I we have x ∈ K and x ∈ I. Since K is sub LA-semihypergroup so l1⊗x ∈ K and I is an hyperideal Int. J. Anal. Appl. 16 (4) (2018) 536 so l1 ⊕ x ∈ I. Which implies that l1 ⊕ x ∈ K ∩ I. Thus K ∩ I is an left hyperideal of R. Similarly it can easily be proved for right hyperideals and hence K ∩ I is an hyperideal of R. � Theorem 3.5. Let R be an LA-semihyperring with pure left identity then right distributive implies left distributive. Proof. Let R is right distributive, then (l1 ⊕ l2) ⊗ l3 = (l1 ⊗ l3) ⊕ (l2 ⊗ l3) . Consider ((l1 ⊗ l3) ⊕ (l2 ⊗ l3)) ⊗e = ((l1 ⊗ l3)⊗) e⊕ ((l2 ⊗ l3) ⊗e) by right distributive law = ((e⊗ l3) ⊗ l1) ⊕ ((e⊗ l3) ⊗ l2) by left invertive law = (l3 ⊗ l1) ⊕ (l3 ⊗ l2) as e is the left identity = l3 ⊗ (l1 ⊕ l2) which shows that it is left distributive. � Theorem 3.6. Let e ∈ R be an LA-semihyperring with pure left identity then every right hyperideal is also a left hyperideal. Proof. Let l1 be any right hyperideal of R, then it is a sub LA-semihyperring of R. Now let l1 ∈ X and h ∈ R then h⊗ l1 = (e⊗h) ⊗ l1 = (l1 ⊗h) ⊗e ∈ X. which shows that X is left hyperideal of R and hence hyperideal of R. � Lemma 3.4. Let X is an right hyperideal of R with pure left identity e then X ⊗X is an hyperideal of R. Proof. Let x ∈ X ⊗X then l1 = l2 ⊗ l3 where l2, l3 ∈ X. Consider l1 ⊗h = (l2 ⊗ l3) ⊗h = (h⊗ l3) ⊗ l2 ∈ X ⊗X, for all h ∈ R. Hence X ⊗ X is an right hyperideal of R. As e is the left identity of R so by theorem 3.6 ,X ⊗ X is left hyperideal of R and hence hyperideal of R. � Lemma 3.5. Let R is an LA-semihyperring with pure left identity e. If X is a proper ideal of R then e /∈ X. Int. J. Anal. Appl. 16 (4) (2018) 537 Proof. Suppose that e ∈ X. Let h ∈ R and consider h = e⊗h ∈ R⊗X ⊆ X for all h ∈ R. which implies that R ⊆ X. But X ⊆ R is obvious and thus X = R. Which is contradiction to the fact that X is a proper ideal of R. Hence e /∈ X. � 4. Homomorphisms on LA-semihyperrings Definition 4.1. A map γ : R1 → R2 where both R1 and R2 are LA-semihyperring is called inclusion homorphism if (i) γ(x⊕y) ⊆ γ(x) ⊕γ(y) (ii) γ(x⊗y) ⊆ γ(x) ⊗γ(y) for all x,y ∈ R1. Definition 4.2. A map γ : R1 → R2 where both R1 and R2 are LA-semihyperring is called strong homor- phism if (i) γ(x⊕y) = γ(x) ⊕γ(y) (ii) γ(x⊗y) = γ(x) ⊗γ(y) for all x,y ∈ R1. Definition 4.3. Let σ be an equivalence relation on R, then σ is said to be left regular if for x,y,z ∈ R such that (x,y) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ z ⊕ x,l6 ∈ z ⊕ y and (z ⊗ x,z ⊗ y) ∈ σ. σ is said to be right regular if for x,y,z ∈ R such that (x,y) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ x ⊕ z,l6 ∈ y ⊕ z and (x ⊗ z ∈ y ⊗ z) ∈ σ. σ is said to be regular if for x,y,z,w ∈ R such that (x,y) ∈ σ and (z,w) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ x⊕z,l6 ∈ y ⊕z and (x⊗z,y ⊗z) ∈ σ. Proposition 4.1. Let (R,⊕,⊗) an LA-semihyperring and σ is an equivalence relation on R. Then σ is a regular relation on R if and only if σ is a left and right regular respectively. Proof. Suppose that σ is a regular relation on R. Let x,y,z ∈ R such that (x,y) ∈ σ and (z,z) ∈ σ then (l5, l6) ∈ σ for l5 ∈ z⊕x,l6 ∈ z⊕y and (z⊗x,z⊗y) ∈ σ. Hence σ is a left regular relation on R. Similarly σ is a right regular relation on R. Conversely let σ is a left and right regular respectively, and let for x,y,z,w ∈ R such that (x,y) ∈ σ and (z,w) ∈ σ. Then by left regularity we have (l5, l6) ∈ σ for l5 ∈ x⊕z,l6 ∈ x⊕w and (x⊗z,x⊗w) ∈ σ. Now by right regularity we have (l6, l4) ∈ σ for l6 ∈ x⊕w,l4 ∈ y⊕w and (x⊗w,y⊗w) ∈ σ. Now by using transitivity of σ we have (l5, l4) ∈ σ for l5 ∈ x⊕z,l4 ∈ y ⊕w and (x⊗z,y ⊗w) ∈ σ. Thus σ is a regular relation on R. � Proposition 4.2. Let γ : R1 → R2 where both R1 and R2 are LA-semihyperring is called inclusion hom- morphism, then this inclusion hommorphism defines a regular relation σ on R1 given by (h1,h2) ∈ σ if and only if γ(h1) = γ(h2) for all h1,h2 ∈ R1. Int. J. Anal. Appl. 16 (4) (2018) 538 Proof. Straightforward. � Definition 4.4. Let x be an hyperideal of LA-semihyperring. Define a relation % on R as (x,y) ∈ % if and only if x = y or x ∈ A and y ∈ A, Then % is a regular relation on R and it is known as Rees regular relation. Lemma 4.1. % is a regular relation on R. Proof. It is obviously an equivalence relation. Now let (x,y) ∈ % and z ∈ R. Case(i) If x = y then z⊕x = z⊕y and z⊗x = z⊗y so (l5, l5) ∈ % for l5 ∈ z⊕x and (z⊗x,z⊗y) ∈ %. So % is left regular relation on R. Similarly % is right regular relation on R. Hence % is a regular relation on R. Case(ii) If both x and y ∈ A then for z ∈ R,l5 ∈ z⊕x ⊆ I and l6 ∈ z⊕y ⊆ I and z⊗x ∈ I and z⊗y ∈ I so we have (l5, l6) ∈ % for l5 ∈ z ⊕ x, l6 ∈ z ⊕ y and (z ⊗ x,z ⊗ y) ∈ %. So % is left regular relation on R. Similarly % is right regular relation on R. Hence % is a regular relation on R. � Lemma 4.2. Let σ be a regular relation on LA-semihyperring R, then {σ(l3) : l3 ∈ σ(l1)⊕σ(l2) ∈ σ(l1⊕l2)} and σ(l1) ⊗σ(l2) = σ(l1 ⊗ l2) for all l1, l2 ∈ R. Proof. Straightforward. � Theorem 4.1. Let σ be a regular relation on LA-semihyperring R, then (R/σ,⊕,⊗) is an LA-semihyperring with the mapping ⊕ : R/σ ×R/σ → P∗(R/σ) and ⊗ : R/σ ×R/σ → R/σ by σ(l1) ⊕σ(l2) = {σ(l3) : l3 ∈ σ(l1) ⊕σ(l2) ∈ σ(l1 ⊕ l2)} and σ(l1) ⊗σ(l2) = σ(l1 ⊗ l2) for all σ(l1),σ(l2) ∈ R/σ. Proof. Indeed by Proposition 4.2, the hyperoperation ⊕ and binary operation ⊗ are well defined. Now let σ(x),σ(y),σ(z) ∈ R/σ then (1) (σ(x) ⊕σ(y)) ⊕σ(z) = ({σ(w) : w ∈ σ(x) ⊕σ(y) ∈ σ(x⊕y)}) ⊕σ(z) = {σ(e) : e ∈ (σ(x) ⊕σ(y)) ⊕σ(z) ∈ σ((x⊕y) ⊕z)} = {σ(z51) : e ∈ (σ(z) ⊕σ(y)) ⊕σ(x) ∈ σ((z ⊕y) ⊕x)} = (σ(z) ⊕σ(y)) ⊕σ(x). Int. J. Anal. Appl. 16 (4) (2018) 539 (2) (σ(x) ⊗σ(y)) ⊗σ(z) = σ(x⊗y) ⊗σ(z) = σ((x⊗y) ⊗z) = σ((z ⊗y) ⊗x) = σ(z ⊗y) ⊗σ(x) = (σ(z) ⊗σ(y)) ⊗σ(x). (3) σ(x) ⊗ (σ(y) ⊕σ(z)) = σ(x) ⊗ ({σ(w) : w ∈ σ(y) ⊕σ(z) ∈ σ(y ⊕z)}) = σ(x⊗w) = σ(x⊗ (y ⊕z)) = σ((x⊗y) ⊕ (x⊗z)) = {σ(e) : e ∈ σ(x⊗y) ⊕σ(x⊗z) ∈ σ((x⊗y) ⊕ (x⊗z))} = {σ(e) : e ∈ ((σ(x) ⊗σ(y)) ⊕ (σ(x) ⊗σ(z)))} = (σ(x) ⊗σ(y)) ⊕ (σ(x) ⊗σ(z))), which shows that R/σ is left distributive and similarly it is right distributive. Hence (R/σ,⊕,⊗) is an LA-semihyperring. � Theorem 4.2. (R/%,⊕,⊗) is an LA-semihyperring. Proof. It follows from the proof of the Theorem, 4.1. � Proposition 4.3. Let (R,⊕,⊗) be an LA-semihyperring and ∅ 6= N ⊆ R. If we define a well defined hyperoperation � and binary operation � on R/N = {N(x)|x ∈ R} as (N(x))�(N(y)) = {N(n)|n ∈ x⊕y} , and (N(x)) � (N(y)) = N(x⊗y) ∀ x,y ∈ R. Then (R/N,�,�) is an LA-semihyperring. Proof. Let (N(x)) , (N(y)) , (N(z)) ∈ R/N, ∀ x,y,z ∈ R. (1)Consider Int. J. Anal. Appl. 16 (4) (2018) 540 (N(x) � N(y)) � N(z) = ({N(n)|n ∈ x⊕y}) � (N(z)) = {N(z)|z ∈ n⊕z} = {N(z)|z ∈ (x⊕y) ⊕z} = {N(z)|z ∈ (z ⊕y) ⊕x} = {N(z)|z ∈ n⊕x} = ({N(n)|n ∈ z ⊕y}) � (N(x)) = ((N(z)) � (N(y))) � (N(x)) . Hence (R/N,�) is an LA-semihypergroup. (2) Consider for (N(x)) , (N(y)) , (N(z)) ∈ R/N, ∀ x,y,z ∈ R, we have (N(x) � N(y)) � N(z) = (N(x⊗y)) � (N(z)) = N((x⊗y) ⊗z) = N((z ⊗y) ⊗x) = (N(z ⊗y)) � (N(x)) = (N(z) � N(y)) � N(x). Hence (R/N,�) is an LA-semigroup. (3) Now let N(x),N(y),N(z) ∈ R/N, ∀ x,y,z ∈ R, then consider N(x) � (N(y) � N(z)) = N(x) � ({N(n)|n ∈ y ⊕z}) = N(x⊗n) = N(x⊗ (y ⊕z)) = N((x⊗y) ⊕ (x⊗z)) = N(x⊗y) � N(x⊗z) = (N(x) � N(y)) � (N(x) � N(z)), and similarly (N(x) � N(y)) � N(z) = (N(x) � N(z)) � (N(y) � N(z)). Thus the operation � is distributive with respect to the hyperoperation � for all N(x),N(y),N(z) ∈ R/N. Hence (R/N,�,�) is an LA-semihyperring. � Int. J. Anal. Appl. 16 (4) (2018) 541 References [1] M. Azhar, M. Gulistan, N. Yaqoob and S. 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