International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 (2017), 155-171 DOI: 10.28924/2291-8639-15-2017-155 FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS ASGHAR KHAN1, MUHAMMAD FAROOQ1, MUHAMMAD IZHAR1,∗ AND BIJAN DAVVAZ2 Abstract. This paper explores the foundations of fuzzy left (resp. right) hyperideals of left almost semihypergroups (briefly, LA-semihypergroups). We investigate the properties of fuzzy left hyperide- als and fuzzy right hyperideals in regular and intra-regular LA-semihypergroups. We also characterize regular and intra-regular LA-semihypergroups in terms of fuzzy hyperideals. 1. Introduction The idea of generalization of a commutative semigroup, (known as left almost semigroup) was introduced by Kazim and Naseeruddin in 1972 (see [1]). A groupoid (S, ·) is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a for all a,b,c ∈ S. This structure is closely related with a commutative semigroup because if an AG-groupoid contains right identity then it becomes a commutative monoid. An AG-groupoid may or may not contain a left identity. Some other names have also been used in literature for left almost semigroups. Cho et al. [2] studied this structure under the name of right modular groupoid. Holgate [3] studied it as left invertive groupoid. Similarly, Stevanovic and Protic [4] called this structure an Abel-Grassmann groupoid (or simply LA-semigroup), which is the primary name under which this structure is known nowadays. There are many important applications of AG-groupoids in the theory of flocks [5]. The concept of a fuzzy set was introduced by Zadeh [9], in 1965. Since its inception, the theory has developed in many directions and found applications in a wide variety of fields. Many researchers published high-quality research articles on fuzzy sets in a variety of international journals. The study of fuzzy set in algebraic structure has been started in the definitive paper of Rosenfeld 1971 [15], in which he defined fuzzy subgroup and gave its important properties. In 1981, Kuroki introduced the concept of fuzzy ideals and fuzzy bi-ideals in semigroups in his paper [16]. The theory of hyperstructures was introduced by Marty in 1934 during the 8th Congress of the Scandinavian Mathematicians [20]. Marty introduced hypergroups as a generalization of groups. He published some papers on hypergroups, using them in different contexts as algebraic functions, rational fractions, non commutative groups. In the following decades and nowadays, a number of different hy- perstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. In [17] Corsini and Leoreanu- Fotea collected numerous applications of algebraic hyperstructures such as: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities. Especially, semihypergroups are the sim- plest algebraic hyperstructures which possess the properties of closure and associativity. Nowadays many scholars have studied different aspects of semihypergroups see [18, 19, 21, 22]. Recently, Hila and Dine [12] introduced the notion of LA-semihypergroups. They investigated several properties of hy- perideals of LA-semihypergroup and defined the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. In [13], Yaqoob, Corsini and Yousafzai have charac- terized intra-regular LA-semihypergroups by using the properties of their left and right hyperideals, Received 26th July, 2017; accepted 29th September, 2017; published 1st November, 2017. 2010 Mathematics Subject Classification. 08A72, 20N20. Key words and phrases. Fuzzy set; LA-semihypergroup; fuzzy LA-semihypergroup; fuzzy left (resp. right) hyperideal; regular and intra-regular LA-semihypergroup. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 155 156 KHAN, FAROOQ, IZHAR AND DAVVAZ and investigated some useful conditions for an LA-semihypergroup to become an intra-regular LA- semihypergroup. This non-associative hyper structure has been further explored in [14], by Yousafzai and Corsini. In this paper, we introduce the notion of fuzzy left (resp. right) hyperideals in LA-semihypergroups and present some related examples of these concepts. We characterize regular and intra-regular LA- semihypergroups in terms of fuzzy hyperideals. 2. Preliminaries A hypergroupoid is a nonempty set S equipped with a hyperoperation ◦, that is a map ◦ : S ×S −→ P∗ (S), where P∗ (S) denotes the set of all nonempty subsets of S (see [20]). We shall denote by x ◦ y, the hyperproduct of elements x,y of S. Let A, B be two nonempty subsets of S. Then the hyperproduct of A and B is defined as A◦B = ⋃ a∈A,b∈B a◦ b. We shall write A◦x instead of A◦{x} and x◦A for {x}◦A. A hypergroupoid (S,◦) is called an LA-semihypergroup [12], if it satisfies the left invertive law: (a◦ b) ◦ c = (c◦ b) ◦a for all a,b,c ∈ S. Every LA-semihypergroup satisfies the medial law [12]. That is, (x◦y) ◦ (z ◦w) = (x◦z) ◦ (y ◦w) for all w,x,y,z ∈ S. Definition 2.1. (see [14]). Let (S,◦) be an LA-semihypergroup then an element e ∈ S is called (i) left identity (resp. pure left identity) if ∀ a ∈ S, a ∈ e◦a (resp. a = e◦a); (ii) right identity (resp. pure right identity) if ∀ a ∈ S, a ∈ a◦e (resp. a = a◦e); (iii) identity (resp. pure identity) if ∀ a ∈ S, a ∈ e◦a∩a◦e (resp. a = e◦a∩a◦e). An LA-semihypergroup (S,◦) with pure left identity e, paramedial law holds. That is (x◦y) ◦ (z ◦w) = (w ◦z) ◦ (y ◦x) for all w,x,y,z ∈ S. An LA-semihypergroup (S,◦) with pure left identity e, satisfies the following law x◦ (y ◦z) = y ◦ (x◦z) (1) . A nonempty subset A of an LA-semihypergroup (S,◦) is called an LA-subsemihypergroup of S if A◦A ⊆ A. A nonempty subset A of an LA-semihypergroup (S,◦) is a called left ( resp. right ) hyperideal of S if S ◦A ⊆ A (resp. A◦S ⊆ A). If A is both a left hyperideal and a right hyperideal of S then it is called a two-sided hyperideal or simply a hyperideal of S. An LA-semihypergroup S is called [13]; (i) regular if for all a ∈ S, there exist x ∈ S such that a ∈ (a◦x) ◦a. (ii) intra-regular if for all a ∈ S, there exist x,y ∈ S such that a ∈ (x◦a2) ◦y. 3. Fuzzy concepts in LA-semihypergroups Let S be an LA-semihypergroup. A function f from a nonempty set X to the unit interval [0, 1] is called a fuzzy subset of S. Let S be an LA-semihypergroup and f be a fuzzy subset of S. Then for every t ∈ (0, 1] the set U (f; t) = {x | x ∈ S, f (x) ≥ t} , is called the level set of f. For x ∈ S, define Ax = {(y,z) ∈ S ×S : x ∈ y ◦z or x = y ◦z} . We denote by F (S) the set of all fuzzy subsets of S. FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 157 Let S be an LA-semihypergroup and f,g are any two fuzzy subsets of S. We define the product f ∗g of f and g as follows: (f ∗g) (x) = ∨ (y,z)∈Ax {f (y) ∧g (z)} . The fuzzy subsets defined by S : S −→ [0, 1],x −→ S (x) = 1 and 0 : S −→ [0, 1] ,x −→ 0 (x) = 0 for all x ∈ S are the greatest and least elements of F (S) . Definition 3.1. Let S be an LA-semihypergroup and ∅ 6= A ⊆ S. Then the characteristic function χA of A is defined as: χA : S −→ [0, 1] ,−→ χA (x) = { 1 if x ∈ A 0 if x /∈ A Definition 3.2. Let S be an LA-semihypergroup and f be a fuzzy subset of S. Then f is called a fuzzy LA-subsemihypergroup of S if: (∀x,y ∈ S) ∧ α∈x◦y f (α) ≥ f (x) ∧f (y) . Definition 3.3. Let S be an LA-semihypergroup and f be a fuzzy subset of S. Then f is called a fuzzy left (resp. right) hyperideal of S if: (∀x,y ∈ S) ∧ α∈x◦y f (α) ≥ f (y) (resp. ∧ α∈x◦y f (α) ≥ f (x) ). Definition 3.4. A fuzzy hyperideal f of an LA-semihypergroup S is called idempotent if f ∗f = f. Example 3.1. Let us consider an LA-semihypergroup S = {a,b,c} in the following cayley’s table ◦ a b c a {a} {a} {a} b {a} {a} {a,c} c {a} {a} {a} Let us define a fuzzy subset f : S −→ [0, 1] as follows f (x) =   0.9 if x = a 0.7 if x = b 0.5 if x = c Then it is easy to observe that f is a fuzzy LA-subsemihypergroup of S. Example 3.2. Let us consider an LA-semihypergroup S = {e1,e2,e3} in the following cayley’s table ◦ e1 e2 e3 e1 {e1} {e1} {e1} e2 {e1} {e1} {e1,e3} e3 {e1} {e1} {e1} Let us define a fuzzy subset f : S −→ [0, 1] as follows f (x) =   0.8 if x = e1 0.4 if x = e2 0.6 if x = e3 Then it is easy to see that f is a fuzzy hyperideal of LA-semihypergroup S. Example 3.3. Let us consider an LA-semihypergroup S = {e1,e2,e3} in the following cayley’s table ◦ e1 e2 e3 e1 {e1,e3} {e2} {e2,e3} e2 {e2,e3} {e2,e3} {e2,e3} e3 {e2,e3} {e2,e3} {e2,e3} 158 KHAN, FAROOQ, IZHAR AND DAVVAZ Let us define a fuzzy subset f : S −→ [0, 1] as follows f (x) =   0.5 if x = e1 0.7 if x = e2 0.7 if x = e3 Then it is easy to see that f is a fuzzy hyperideal of LA-semihypergroup S. Proposition 3.1. The set (F (S) ,∗) is an LA-semihypergroup. Proof. Clearly F (S) is closed. Let f,g and h be in F (S) . If Ax = ∅ for any x ∈ S. Then ((f ∗g) ∗h) (x) = 0 = ((h∗g) ∗f) (x) . Let Ax 6= ∅, then there exist y and z in S such that (y,z) ∈ Ax. Therefore by using left invertive law, we have ((f ∗g) ∗h) (x) = ∨ (y,z)∈Ax {(f ∗g) (y) ∧h (z)} = ∨ (y,z)∈Ax   ∨ (p,q)∈Ay {f (p) ∧g (q)}∧h (z)   = ∨ x∈((p◦q)◦z) {f (p) ∧g (q) ∧h (z)} = ∨ x∈((z◦q)◦p) {h (z) ∧g (q) ∧f (p)} = ∨ (w,p)∈Ax   ∨ (z,q)∈Aw (h (z) ∧g (q)) ∧f (p)   = ∨ (w,p)∈Ax {(h∗g) (w) ∧f (p)} = ((h∗g) ∗f) (x) . Hence (F (S) ,∗) is an LA-semihypergroup. � Lemma 3.1. Let S be an LA-semihypergroup. Then the medial law holds in F (S) . Proof. Let f,g,h and k be the arbitrary elements of F (S) . By successive use of left invertive law, (f ∗g) ∗ (h∗k) = ((h∗k) ∗g) ∗f = ((g ∗k) ∗h) ∗f = (f ∗h) ∗ (g ∗k) . � Proposition 3.2. An LA-semihypergroup with F (S) = (F (S)) 2 is a commutative semihypergroup if and only if (f ∗g) ∗h = f ∗ (h∗g) holds for all fuzzy subsets f,g,h ∈ F (S) . Proof. Let S be a commutative semihypergroup. For any fuzzy subsets f,g,h ∈ F (S) . If Ax = ∅ then ((f ∗g) ∗h) (x) = 0 = (f ∗ (h∗g)) (x) . Let Ax 6= ∅ then (s,t) ∈ Ax, therefore by the use of left FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 159 invertive law and commutative law, we get ((f ∗g) ∗h) (x) = ∨ (s,t)∈Ax {(f ∗g) (s) ∧h (t)} = ∨ (s,t)∈Ax   ∨ (m,n)∈As (f (m) ∧g (n)) ∧h (t)   = ∨ x∈((m◦n)◦t) {f (m) ∧h (t) ∧g (n)} = ∨ x∈((t◦n)◦m) {f (m) ∧h (t) ∧g (n)} = ∨ x∈(m◦(t◦n)) {f (m) ∧h (t) ∧g (n)} = ∨ (m,p)∈Ax  f (m) ∧ ∨ (t,n)∈Ap (h (t) ∧g (n))   = ∨ (m,p)∈Ax {f (m) ∧ (h∗g) (p)} = (f ∗ (h∗g)) (x) . Conversely, let (f ∗g) ∗h = f ∗ (h∗g) holds for all fuzzy subsets f,g,h ∈ F (S) . We have to show that F (S) is a commutative semihypergroup. Let f and g be any fuzzy subsets of S. If Ax = ∅ for any x ∈ S, then (f ∗g) (x) = 0 = (g ∗f) (x) . Let Ax 6= ∅. Then (s,t) ∈ Ax. Since F (S) = (F (S)) 2 . So f = (h∗k) where h and k are any fuzzy subsets of S. Now by using left invertive law, we have (f ∗g) (x) = ((h∗k) ∗g) (x) = ∨ (s,t)∈Ax {(h∗k) (s) ∧g (t)} = ∨ (s,t)∈Ax   ∨ (m,n)∈As (h (m) ∧k (n)) ∧g (t)   = ∨ x∈((m◦n)◦t) {h (m) ∧k (n) ∧g (t)} = ∨ x∈((t◦n)◦m) {g (t) ∧k (n) ∧h (m)} = ∨ (p,m)∈Ax   ∨ (t,n)∈Ap (g (t) ∧k (n)) ∧h (m)   = ∨ (p,m)∈Ax {(g ∗k) (p) ∧h (m)} = ((g ∗k) ∗h) (x) = (g ∗ (h∗k)) (x) . This shows that f ∗g = g ∗ (h∗k) = g ∗f. Thus commutative law holds in F (S) . 160 KHAN, FAROOQ, IZHAR AND DAVVAZ Now if Ax = ∅. Then ((f ∗g) ∗h) (x) = 0 = (f ∗ (h∗g)) (x). Let Ax 6= ∅. Then (s,t) ∈ Ax. Therefore by the use of commutative law and left invertive law we get ((f ∗g) ∗h) (x) = ∨ (s,t)∈Ax {(f ∗g) (s) ∧h (t)} = ∨ (s,t)∈Ax   ∨ (m,n)∈As (f (m) ∧g (n)) ∧h (t)   = ∨ x∈((m◦n)◦t) {f (m) ∧g (n) ∧h (t)} = ∨ x∈((t◦n)◦m) {f (m) ∧g (n) ∧h (t)} = ∨ x∈(m◦(t◦n)) {f (m) ∧g (n) ∧h (t)} = ∨ x∈(m◦(n◦t)) {f (m) ∧g (n) ∧h (t)} = ∨ (m,p)∈Ax  f (m) ∧ ∨ (n,t)∈Ap (g (n) ∧h (t))   = ∨ (m,p)∈Ax {f (m) ∧ (g ∗h) (p)} = (f ∗ (g ∗h)) (x) . � Theorem 3.1. If S has a pure left identity then the following properties holds in F (S) . (1) f ∗ (g ∗h) = g ∗ (f ∗h) for all f,g and h ∈ F (S) . (2) (f ∗g) ∗ (h∗k) = (k ∗h) ∗ (g ∗f) for all f,g,h and k ∈ F (S) . Proof. (1). Let x ∈ S. If Ax = ∅. Then (f ∗ (g ∗h)) (x) = 0 = (g ∗ (f ∗h)) (x) . Let Ax 6= ∅. Then (y,z) ∈ Ax. Now by using medial law with pure left identity, we have (f ∗ (g ∗h)) (x) = ∨ (y,z)∈Ax {f (y) ∧ (g ∗h) (z)} = ∨ (y,z)∈Ax  f (y) ∧ ∨ (p,q)∈Az (g (p) ∧h (q))   = ∨ x∈(y◦(p◦q)) {f (y) ∧g (p) ∧h (q)} = ∨ x∈(p◦(y◦q)) {g (p) ∧f (y) ∧h (q)} = ∨ (p,w)∈Ax  g (p) ∧ ∨ (y,q)∈Aw (f (y) ∧h (q))   = ∨ (p,w)∈Ax {g (p) ∧ (f ∗h) (w)} = (g ∗ (f ∗h)) (x) . Thus (f ∗ (g ∗h)) (x) = (g ∗ (f ∗h)) (x) for all x ∈ S. FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 161 (2) . If Ax = ∅ for x ∈ S, then ((f ∗g) ∗ (h∗k)) (x) = 0 = ((k ∗h) ∗ (g ∗f)) (x) . Let Ax 6= ∅ then there exist y and z in S such that (y,z) ∈ Ax. Therefore by using paramedial law, we have ((f ∗g) ∗ (h∗k)) (x) = ∨ (y,z)∈Ax {(f ∗g) (y) ∧ (h∗k) (z)} = ∨ (y,z)∈Ax   ∨ (p,q)∈Ay {f (p) ∧g (q)}∧ ∨ (u,v)∈Az {(h (u) ∧k (v))}   = ∨ x∈((p◦q)◦(u◦v)) {f (p) ∧g (q) ∧h (u) ∧k (v)} = ∨ x∈((v◦u)◦(q◦p)) {k (v) ∧h (u) ∧g (q) ∧f (p)} = ∨ (m,n)∈Ax   ∨ (v,u)∈Am {k (v) ∧h (u)} ∨ (q,p)∈An {g (q) ∧f (p)}   = ∨ (m,n)∈Ax {(k ∗h) (m) ∧ (g ∗f) (n)} = ((k ∗h) ∗ (g ∗f)) (x) . Thus (f ∗g) ∗ (h∗k) = (k ∗h) ∗ (g ∗f) for all x ∈ S. � Theorem 3.2. Let S be an LA-semihypergroup. Then L = {f | f ∈ F (S) , f ∗h = f where h = h∗h} is a commutative monoid in S. Proof. The fuzzy subset L of S is nonempty since h ∗ h = h, which implies that h is in L. Let f and g be the fuzzy subsets of S in L, then f ∗ h = f and g ∗ h = g. If Ax = ∅ for x ∈ S, then (f ∗g) (x) = 0 = ((f ∗g) ∗h) (x) . Let Ax 6= ∅. Then by using medial law, we have (f ∗g) (x) = ∨ (y,z)∈Ax {(f ∗h) (y) ∧ (g ∗h) (z)} = ∨ (y,z)∈Ax   ∨ (p,q)∈Ay {f (p) ∧h (q)}∧ ∨ (u,v)∈Az {g (u) ∧h (v)}   = ∨ x∈((p◦q)◦(u◦v)) {f (p) ∧h (q) ∧g (u) ∧h (v)} = ∨ x∈((p◦u)◦(q◦v)) {f (p) ∧g (u) ∧h (q) ∧h (v)} = ∨ (m,n)∈Ax   ∨ (p,u)∈Am {f (p) ∧g (u)} ∨ (q,v)∈An {h (q) ∧h (v)}   = ∨ (m,n)∈Ax {(f ∗g) (m) ∧ (h∗h) (n)} = ((f ∗g) ∗ (h∗h)) (x) . Thus f ∗g = (f ∗g) ∗ (h∗h) = (f ∗g) ∗h which implies that L is closed. 162 KHAN, FAROOQ, IZHAR AND DAVVAZ Now if Ax = ∅. Then (f ∗g) (x) = 0 = (g ∗f) (x) . Let Ax 6= ∅ then (y,z) ∈ Ax. Therefore by using left invertive law, we have (f ∗g) (x) = ∨ (y,z)∈Ax {(f ∗h) (y) ∧g (z)} = ∨ (y,z)∈Ax   ∨ (p,q)∈Ay (f (p) ∧h (q)) ∧g (z)   = ∨ x∈((p◦q)◦z) {f (p) ∧h (q) ∧g (z)} = ∨ x∈((z◦q)◦p) {g (z) ∧h (q) ∧f (p)} = ∨ (t,p)∈Ax   ∨ (z,q)∈At (g (z) ∧h (q)) ∧f (p)   = ∨ (t,p)∈Ax {(g ∗h) (t) ∧f (p)} = ((g ∗h) ∗f) (x) . Thus f ∗g = (g ∗h)∗f = g∗f, which implies that commutative law holds in L and associative law holds in L due to commutativity. Since for any fuzzy subset f in L, we have f ∗ h = f (where h is fixed) implies that h is pure right identity in F (S) and hence an identity. � Lemma 3.2. Let S be an LA-semihypergroup. If S has a pure left identity then S ∗S = S. Proof. Every x in S can be written as x = e◦x, where e is the pure left identity in S. Therefore (S ∗S) (x) = ∨ (y,z)∈Ax {S (y) ∧S (z)} ≥{S (e) ∧S (x)} = 1 = S (x) . Hence S ∗S = S. � Theorem 3.3. Let χA and χB be fuzzy subsets of an LA-semihypergroup S, where A and B are nonempty subsets of S. Then the following properties hold: (1) If A ⊆ B then χA ⊆ χB. (2) χA ∩χB = χA∩B. (3) χA ∗χB = χA◦B. Proof. (1) . It is obvious. (2) . Let x ∈ S. If x ∈ A∩B, then x ∈ A and x ∈ B. So χA (x) = 1 and χB (x) = 1. Thus we have (χA ∩χB) (x) = χA (x)∧χB (x) = 1 = χA∩B. If x /∈ A∩B, then x /∈ A and x /∈ B. So χA (x) = 0 and χB (x) = 0. Thus we have (χA ∩χB) (x) = χA (x) ∧χB (x) = 0 = χA∩B. Thus χA ∩χB = χA∩B. (3) . For any x ∈ S. If x /∈ A◦B, then χA◦B(x) = 0 (i) This means that there does not exist y ∈ A and z ∈ B such that x ∈ y ◦z. If Ax = ∅ then (χA ∗χB)(x) = 0 (ii) If Ax 6= ∅ and (y,z) ∈ Ax then x ∈ y◦z. Then y /∈ A or z /∈ B. Thus either χA(y) = 0 or χB(z) = 0. So we have, χA(y) ∧χB(z) = 0. Hence (χA ∗λB)(x) = 0. FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 163 Let x ∈ A◦B, then χA◦B(x) = 1. Thus x ∈ a◦ b, for some a ∈ A and b ∈ B, so (a,b) ∈ Ax. Since Ax 6= ∅, we have (χA ∗χB)(x) = ∨ (y,z)∈Ax {χA(y) ∧χB(z)} ≥ χA(a) ∧χB(b) = 1. Thus (χA ∗χB)(x) = 1. Hence χA ∗χB = χA◦B. � Theorem 3.4. A fuzzy subset f of an LA-semihypergroup S is a fuzzy LA-subsemihypergroup of S if and only if f ∗f ⊆ f. Proof. Assume that f is a fuzzy LA-subsemihypergroup of S. If Aa = ∅. Then (f ∗f) (a) = 0 = f (a) . If Aa 6= ∅, then there exist x and y in S such that (x,y) ∈ Aa. Then for any α ∈ x◦y, we have a ∈ α. Since f is a fuzzy LA-subsemihypergroup of S, we have (f ∗f) (a) = ∨ (x,y)∈Aa {f (x) ∧f (y)} ≤ ∨ (x,y)∈Aa f (α) ≤ ∨ (x,y)∈Aa f (a) = f (a) . Thus f ∗f ⊆ f. Conversely, assume that f ∗f ⊆ f. Let x,y ∈ S and α ∈ x◦y. We have, f (α) ≥ (f ∗f) (α) = ∨ (x,y)∈Aα {f (x) ∧f (y)} ≥{f (x) ∧f (y)} f (α) ≥{f (x) ∧f (y)} . Thus ∧ α∈x◦y f (α) ≥{f (x) ∧f (y)} . Thus f is a fuzzy LA-subsemihypergroup of S. � Theorem 3.5. A nonempty subset A of an LA-semihypergroup S is an LA-subsemihypergroup if and only if the characteristic fuzzy set χA is a fuzzy LA-subsemihypergroup. Proof. Let A be a nonempty subset of an LA-semihypergroup S, x and y be arbitrary elements of S. Let A be an LA-subsemihypergroup of S. Let x,y ∈ A, then x◦ y ⊆ A. For any α ∈ x◦ y, we have, χA (x) = 1 and χA (y) = 1. Hence ∧ α∈x◦y χA (α) = 1 = χA (x) ∧χA (y) . Now let x ∈ A and y /∈ A, then χA (x) = 1 and χA (y) = 0, so we have ∧ α∈x◦y χA (α) ≥ 0 = χA (x) ∧χA (y) . Now let both x and y are not in A, then χA (x) = 0 and χA (y) = 0, so we have ∧ α∈x◦y χA (α) ≥ 0 = χA (x)∧χA (y) . Thus for all x,y ∈ S, we have ∧ α∈x◦y χA (α) ≥ χA (x) ∧χA (y) . Thus χA is a fuzzy LA-subsemihypergroup of S. Conversely, Let χA be a fuzzy LA-subsemihypergroup of S. If the elements x and y are in A, then χA (x) = 1 = χA (y) . But ∧ α∈x◦y χA (α) ≥ χA (x) ∧χA (y) = 1, which implies that χA (α) ≥ 1 for any α ∈ x◦y. Hence for any α ∈ x◦y, χA (α) = 1, i.e., α ∈ A. It thus follows that x◦y ⊆ A. Hence A is an LA-subsemihypergroup of S. � Theorem 3.6. Let S be an LA-semihypergroup and for a nonempty subset A of S the following statements are equivalent: 164 KHAN, FAROOQ, IZHAR AND DAVVAZ (1) A is left (resp. right) hyperideal of S. (2) The characteristic fuzzy set χA is a fuzzy left (resp. right) hyperideal of S. Proof. (1) =⇒ (2) . Assume that A is a left hyperideal of S. Let x,y ∈ S be such that both x and y are in A. Then since A is left hyperideal of S, x ◦ y ⊆ A. For any α ∈ x ◦ y, we have, χA (x) = 1 and χA (y) = 0. Hence ∧ α∈x◦y χA (α) = 1 = χA (y) . Now let x ∈ A and y /∈ A, then χA (x) = 1 and χA (y) = 0, so we have ∧ α∈x◦y χA (α) ≥ 0 = χA (y) . Now let both x and y are not in A, then χA (x) = 0 and χA (y) = 0, so we have ∧ α∈x◦y χA (α) ≥ 0 = χA (y) . Thus for all x,y ∈ S, we have∧ α∈x◦y χA (α) ≥ χA (y) . Thus χA is a fuzzy left hyperideal of S. (2) =⇒ (1) . Let χA be a fuzzy left hyperideal of S. If the elements x and y are in A, then χA (x) = 1 = χA (y) . But 1 = χA (y) ≤ ∧ α∈x◦y χA (α) , which implies that χA (α) ≥ 1 for any α ∈ x◦y. Hence for any α ∈ x◦y, χA (α) = 1, i.e., α ∈ A. It thus follows that S◦A ⊆ A. Therefore A is left hyperideal of S. Similarly we can prove that χA is a fuzzy right hyperideal of S when A is right hyperideal of S. � Theorem 3.7. A fuzzy subset f of an LA-semihypergroup S is a fuzzy left (resp. right) hyperideal of S if and only if for each t ∈ (0, 1], U(f; t) 6= φ is a left (resp. right) hyperideal of S. Proof. Suppose f be a fuzzy left hyperideal of S and x ∈ U(f; t) and y ∈ S. Then f(x) ≥ t. Since f is a fuzzy left hyperideal of S, so f(x) ≤ ∧ α∈y◦x f (α). Hence f(α) ≥ t for all α ∈ y ◦x, this implies α ∈ U(f; t) that is y ◦x ⊆ U(f; t). Hence U(f; t) is a fuzzy left hyperideal of S. Conversely, assume that U(f; t) 6= ∅ is a left hyperideal of S. Let x ∈ S such that f(x) > ∧ α∈y◦x f (α) for all y ∈ S. Select t ∈ (0, 1] such that f(x) = t > ∧ α∈y◦x f (α). Then x ∈ U(f; t) but y ◦x * U(f; t), a contradiction. Hence f(x) ≤ ∧ α∈y◦x f (α), that is f is a fuzzy left hyperideal of S. � Proposition 3.3. Let S be an LA-semihypergroup then the following properties hold. (1) Let f and g be two fuzzy LA-subsemihypergroups of S. Then f∩g is also fuzzy LA-subsemihypergroup of S. (2) The intersection of any family of fuzzy left (resp. right, two sided) hyperideals of S is a fuzzy left (resp. right, two sided) hyperideal of S. Proof. (1) . Let f and g be two fuzzy LA-subsemihypergroups of S. Let x,y ∈ S. Then for any α ∈ x◦y, we have ∧ α∈x◦y f (α) ≥ f (x) ∧ f (y) and ∧ α∈x◦y g (α) ≥ g (x) ∧ g (y) . Hence f (α) ≥ f (x) ∧ f (y) and g (α) ≥ g (x) ∧g (y) . Thus (f ∩g) (α) = f (α) ∧g (α) ≥ f (x) ∧f (y) ∧g (x) ∧g (y) = f (x) ∧g (x) ∧f (y) ∧g (y) = (f ∩g) (x) ∧ (f ∩g) (y) . Hence ∧ α∈x◦y (f ∩g) (α) ≥ (f ∩g) (x)∧(f ∩g) (y) . Therefore f∩g is a fuzzy LA-subsemihypergroup of S. (2) . Let g = ⋂ i∈I gi be a family of fuzzy left hyperideals of S. Let x,y ∈ S. Then, since each gi (i ∈ I) is a fuzzy left hyperideals of S, so ∧ α∈x◦y gi (α) ≥ gi (y) . Thus for any α ∈ x◦y, gi (α) ≥ gi (y) , and we FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 165 have g (α) = (⋂ i∈I gi ) (α) = ∧ i∈I (gi (α)) ≥ ∧ i∈I gi (y) = (⋂ i∈I gi ) (y) = g (y) . Thus ∧ i∈I g (α) ≥ g (y) . Therefore g = ⋂ i∈I gi is a fuzzy left hyperideal of S. � Proposition 3.4. Let S is an LA-semihypergroup. If f is fuzzy left (resp. right or two-sided) hyper- ideal of S. Then f is a fuzzy LA-subsemihypergroup. Proof. Let f be a fuzzy left hyperideal of S. Let x,y ∈ S. Then ∧ α∈x◦y f (α) ≥ f (y) ≥ f (x) ∧ f (y) . Thus ∧ α∈x◦y f (α) ≥ f (x) ∧f (y) . Therefore f is a fuzzy LA-subsemihypergroup of S. � Proposition 3.5. A fuzzy subset f of an LA-semihypergroup S is a fuzzy left (resp. right) hyperideal of S if and only if S ∗f ⊆ f (resp. f ∗S ⊆ f). Proof. Let f be a fuzzy left hyperideal of S and x ∈ S. Then (S ∗f) (x) = ∨ x∈y◦z {S (y) ∧f (z)} = ∨ x∈y◦z {f (z)} (∵S (y) = 1) ≤ ∨ x∈y◦z f (x) , because f (z) ≤ ∧ α∈y◦z {f (α)}≤ f (α) for each α ∈ y ◦z. = f (x) . Hence, (S ∗f)(x) ≤ f(x). Thus S ∗f ⊆ f. Conversely, suppose that S∗f ⊆ f. We show that f is a fuzzy left hyperideal of S. Let x ∈ S. Then f (x) ≥ (S ∗f)) (x) = ∨ x∈y◦z {S (y) ∧f (z)} = ∨ x∈y◦z {f (z)} , (because S (y) = 1) ≥ f (z) , for each z such that x ∈ y ◦z. Thus ∧ x∈y◦z f (x) ≥ f (z) . Hence f is a fuzzy left hyperideal of S. Similarly we can prove the case of fuzzy right hyperideal of S. � Theorem 3.8. If S is an LA-semihypergroup with pure left identity. Then every fuzzy right hyperideal is a fuzzy left hyperideal of S. Proof. Let S be an LA-semihypergroup with pure left identity e, and f be a fuzzy right hyperideal of S. Since f is a fuzzy right hyperideal of S, so f ∗S ⊆f. Thus by Lemma 3.2, and left invertive law, 166 KHAN, FAROOQ, IZHAR AND DAVVAZ we have S∗f = (S ∗S) ∗f = (f∗S)∗S ⊆ f∗S ⊆ f. Thus, S∗f ⊆ f. Thus, f is a fuzzy left hyperideal of S. � Proposition 3.6. The product of two fuzzy left (resp. right) hyperideals of an LA-semihypergroup S with pure left identity is a fuzzy left (resp. right) hyperideal of S. Proof. Let f and g be any two fuzzy left hyperideals of S. Then by using (1) , we have, S∗(f ∗g) = f ∗ (S∗g) ⊆ f ∗g. Let f and g be any two fuzzy right hyperideals of S. Then by using medial law and 3.2, we have (f ∗g) ∗S = (f ∗g) ∗ (S ∗S) = (f ∗S) ∗ (g ∗S) ⊆ f ∗g. Therefore f ∗g is a fuzzy hyperideal of S. � Proposition 3.7. In LA-semihypergroup with pure left identity for every fuzzy left hyperideal f of S, we have S∗f = f. Proof. It suffices to show that f ⊆S∗f. Since every element x ∈ S can be written as x = e◦x, where e is the pure left identity in S, (S∗f) (x) = ∨ (y,z)∈Ax {S (y) ∧f (z)} ≥{S (e) ∧f (x)} = f (x) . Hence S∗f = f. � Proposition 3.8. In an LA-semihypergroup S with pure left identity for every fuzzy right hyperideal h of S, we have h∗S = h. Proof. It suffices to show that h ⊆ h ∗S. Since every element a ∈ S can be written as a = e ◦ a = (e◦e) ◦a = (a◦e) ◦e, then there exist u ∈ a◦e such that a ∈ u◦e. Then (u,e) ∈ Aa, where e is the pure left identity in S, (h∗S) (a) = ∨ (x,y)∈Aa {h (x) ∧S (y)} ≥{h (u) ∧S (e)} . Since h is a fuzzy right hyperideal of S. Then ∧ u∈a◦e h (u) ≥ h (a) . Hence h (u) ≥ h (a) . Thus (h∗S) (a) ≥{h (u) ∧S (e)} ≥ h (a) ∧ 1 = h (a) . Hence h∗S = h. � Proposition 3.9. Let S be an LA-semihypergroup with pure left identity, f be a fuzzy subset and k be a fuzzy left hyperideal of S. Then for any fuzzy subset h and fuzzy left hyperideal g of S, f ∗g = h∗k implies that g ∗f = k ∗h. Proof. Since g and k are fuzzy left hyperideals of S, by Proposition 3.7, S∗g = g and S∗k = k. Then, g ∗f = (S∗g) ∗f = (f ∗g) ∗S = (h∗k) ∗S = (S∗k) ∗h = k ∗h. � FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 167 Proposition 3.10. Every idempotent fuzzy left hyperideal of an LA-semihypergroup S is a fuzzy hyperideal of S. Proof. Let f be a fuzzy left hyperideal of S which is idempotent. Then f ∗S = (f ∗f) ∗S = (S∗f) ∗f⊆f ∗f = f. Hence f is a fuzzy right hyperideal of S and so f is a fuzzy hyperideal of S. � Proposition 3.11. If f is an idempotent element in an LA-semihypergroup S with pure left identity. Then S∗f is an idempotent element. Proof. Let f be an idempotent element in an LA-semihypergroup S with pure left identity. Then by using medial law, (S∗f) ∗ (S∗f) = (S ∗S) ∗ (f ∗f) = S∗f. � Proposition 3.12. If f is an idempotent element in an LA-semihypergroup S with pure left identity. Then every fuzzy left hyperideal g of S commutes with f. Proof. Let f be an idempotent element in an LA-semihypergroup S with pure left identity. Then f ∗g = (f ∗f) ∗g = (g ∗f) ∗f ⊆ (g ∗S) ∗f ⊆ g ∗f. Also, g ∗f = g ∗ (f ∗f) = f ∗ (g ∗f) ⊆ f ∗ (g ∗S)⊆f ∗g. � Proposition 3.13. If f is a fuzzy left hyperideal of an LA-semihypergroup S with pure left identity, then f ∪ (f ∗S) is a fuzzy hyperideal of S. Proof. Assume that f is a fuzzy left hyperideal of S. Then (f ∪ (f ∗S)) ∗S= (f ∗S) ∪ ((f ∗S) ∗S) = (f ∗S) ∪ ((S ∗S) ∗f) = (f ∗S) ∪ (S∗f) = (f ∗S) ∪f = f ∪ (f ∗S) . Hence f ∪ (f ∗S) is a fuzzy right hyperideal of S. and by Theorem 3.8, it is a fuzzy hyperideal of S. � Proposition 3.14. If f is a fuzzy right hyperideal of an LA-semihypergroup S with pure left identity, then f ∪ (S∗f) . is a fuzzy hyperideal of S. Proof. Assume that f is a fuzzy right hyperideal of S. Then (f ∪ (S∗f)) ∗S= ((f ∗S) ∪ (S∗f) ∗S) ⊆f ∪ (S∗f) ∗ (S ∗S) = f ∪ (S ∗S) ∗ (f ∗S) = f ∪ (S∗(f ∗S)) = f ∪ (f ∗ (S ∗S)) = f ∪ (f ∗S) = f ⊆ f ∪ (S∗f) . 168 KHAN, FAROOQ, IZHAR AND DAVVAZ Also, S∗(f ∪ (S∗f)) = (S∗f) ∪ (S∗(S∗f)) = (S∗f) ∪ ((S ∗S)∗(S∗f)) = (S∗f) ∪ ((f ∗S) ∗ (S ∗S)) ⊆ (S∗f) ∪ (f ∗ (S ∗S)) = (S∗f) ∪ (f∗S) ⊆ (S∗f) ∪f = f ∪ (S∗f) Hence f ∪ (S∗f) is a fuzzy hyperideal of S. � 4. Characterizations of regular and intra-regular LA-semihypergroups in terms of fuzzy hyperideals In this section, we characterize regular as well as intra-regular LA-semihypergroups in terms of fuzzy hyperideals. Theorem 4.1. Let S be a regular LA-semihypergroup. Then for every fuzzy right hyperideal f and every fuzzy left hyperideal g of S, we have f ∗g = f ∩g. Proof. Let S be a regular LA-semihypergroup and f is a fuzzy right hyperideal and g a fuzzy left hyperideal of S. Then f ∗g ⊆ f ∗S ⊆f and f ∗g ⊆S∗g ⊆ g. This implies that f ∗g ⊆ f ∩g. Now let a be any element of S, then, since S is a regular LA-semihypergroup, so there exist an element x ∈ S such that a ∈ (a◦x) ◦a. Then there exist u ∈ a◦x such that a ∈ u◦a. Then (u,a) ∈ Aa. Thus we have (f ∗g) (a) = ∨ (y,z)∈Ax {f (y) ∧g (z)} ≥{f (u) ∧g (a)} . Since f is fuzzy right hyperideal of S, ∧ u∈a◦x f (u) ≥ f (a) . Hence f (u) ≥ f (a) . Thus (f ∗g) (a) ≥{f (u) ∧g (a)} ≥{f (a) ∧g (a)} = (f ∩g) (a) . Thus f ∗g ⊇ f ∩g. Therefore f ∗g = f ∩g. � Corollary 4.1. Let S be a regular LA-semihypergroup. Then for every fuzzy hyperideal f and every fuzzy hyperideal g of S, we have f ∗g = f ∩g. Proposition 4.1. Let S be a regular LA-semihypergroup. Then for every fuzzy right hyperideal f of S is idempotent. Proof. Let S be a regular LA-semihypergroup and f is a fuzzy right hyperideal of S. Then f ∗ f ⊆ f∗S ⊆f. Next since S is regular so for any a ∈ S, there exist an element x ∈ S such that a ∈ (a◦x)◦a. Then there exist α ∈ a◦x such that a ∈ α◦a. Then (α,a) ∈ Aa. Thus we have (f ∗f) (a) = ∨ (y,z)∈Ax {f (y) ∧f (z)} ≥{f (α) ∧f (a)} . FUZZY HYPERIDEALS OF LEFT ALMOST SEMIHYPERGROUPS 169 Since f is fuzzy right hyperideal of S, ∧ α∈a◦x f (α) ≥ f (a) . Hence f (α) ≥ f (a) . Thus (f ∗f) (a) ≥{f (α) ∧f (a)} ≥ f (a) ∧f (a) = f (a) . Hence f ⊆ f ∗f. Therefore f ∗f = f. � Corollary 4.2. Let S be a regular LA-semihypergroup. Then for every fuzzy hyperideal f of S is idempotent. Proposition 4.2. If S is a regular LA-semihypergroup. Then every fuzzy right hyperideal is a fuzzy left hyperideal of S. Proof. Let S be a regular LA-semihypergroup and f be a fuzzy right hyperideal of S. Let x,y ∈ S. Since S is regular and x ∈ S, so there exist an element a ∈ S such that x ∈ (x◦a) ◦ x. Thus we have ∧ α∈x◦y f (α) = ∧ α∈(((x◦a)◦x)◦y) f (α) = ∧ α∈((y◦x)◦(x◦a)) f (α) = ∧ α∈u◦v u∈y◦x,v∈x◦a f (α) ≥ f (u) ≥ ∧ u∈y◦x f (u) ≥ f (y) . Hence ∧ α∈x◦y f (α) ≥ f (y) . Therefore f is a fuzzy left hyperideal of S. � Proposition 4.3. A fuzzy set of an intra-regular LA-semihypergroup S is a fuzzy right hyperideal if and only if it is a fuzzy left hyperideal of S. Proof. Let f be a fuzzy right hyperideal of S. Let a,b ∈ S. Since a ∈ S and S is intra-regular LA- semihypergroup, so there exist x,y ∈ S such that a ∈ ( x◦a2 ) ◦ y. Thus for any α ∈ a◦ b, we have,∧ α∈a◦b f (α) = ∧ α∈(((x◦a2)◦y)◦b) f (α) = ∧ α∈((b◦y)◦(x◦a2)) f (α) = ∧ α∈u◦v u∈b◦y,v∈x◦a2 f (α) ≥ f (u) ≥ ∧ u∈b◦y f (u) ≥ f (b) . Thus f is a fuzzy left hyperideal of S. Conversely, assume that f is a fuzzy left hyperideal of S. Now for any α ∈ a ◦ b, we have,∧ α∈a◦b f (α) = ∧ α∈(((x◦a2)◦y)◦b) f (α) = ∧ α∈((b◦y)◦(x◦a2)) f (α) = ∧ α∈u◦v u∈b◦y,v∈x◦a2 f (α) ≥ f (v) ≥ ∧ v∈x◦a2 f (v) ≥ f ( a2 ) ≥ ∧ β∈a◦a f (β) ≥ f (a) . Thus ∧ α∈a◦b f (α) ≥ f (a) . Hence f is a fuzzy right hyperideal of S. � Proposition 4.4. Every fuzzy two-sided hyperideal of an intra-regular LA-semihypergroup S with pure left identity is idempotent. Proof. Assume that f is a fuzzy two-sided hyperideal of S. Then clearly f ∗f ⊆ f ∗S ⊆f. Since S is intra-regular, so for each a ∈ S, there exist x,y ∈ S such that a ∈ ( x◦a2 ) ◦y. So by using (1) and left invertive law, we have a ∈ ( x◦a2 ) ◦y = (x◦ (a◦a)) ◦y = (a◦ (x◦a)) ◦y = (y ◦ (x◦a)) ◦a. Then there exist u ∈ (y ◦ (x◦a)) such that a ∈ u◦a. Then (u,a) ∈ Aa. Thus we have, (f ∗f) (a) = ∨ (p,q)∈Aa {f (p) ∧f (q)} ≥{f (u) ∧f (a)} . 170 KHAN, FAROOQ, IZHAR AND DAVVAZ Since f is a fuzzy two-sided hyperideal of S, so we have ∧ u∈(y◦(x◦a)) f (u) = ∧ u∈y◦v v∈x◦a f (u) ≥ f (v) ≥ ∧ v∈x◦a f (v) ≥ f (a) . Hence f (u) ≥ f (a) . Thus (f ∗f) (a) ≥{f (u) ∧f (a)} ≥ f (a) ∧f (a) = f (a) . Hence f ∗f = f. � Proposition 4.5. If S is an intra-regular LA-semihypergroup with pure left identity. Then f = (S∗f)2 for all fuzzy left hyperideal f of S. Proof. Let S be an intra-regular LA-semihypergroup with pure left identity and f be a fuzzy left hyperideal of S. Then S∗f ⊆ f. 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Davvaz, An investigation on fuzzy hyperideals of ordered semihypergroups, Quasigroups and related systems 23 (2015), 297-308. [20] F .Marty, Sur Une generalization de la notion de group,” 8iemcongress, Math Scandinaves Stockholm (1934), 45-49. [21] M. Farooq, A. Khan and B. Davvaz, Characterizations of ordered semihypergroups by the properties of their intersectional-soft generalized bi-hyperideals, Soft Comput DOI 10.1007/s00500-017-2550-6. [22] A. Khan, M. Farooq and B. Davvaz, Int-soft interior-hyperideals of ordered semihypergroups. Intl. J. Anal. Appl. 14(2) (2017), 193-202. 1Department of Mathematics, Abdul Wali Khan University Mardan, 23200, KP, Pakistan 2Department of Mathematics, Yazd Univesity, Yazd, Iran ∗Corresponding author: mizharmath@gmail.com 1. Introduction 2. Preliminaries 3. Fuzzy concepts in LA-semihypergroups 4. Characterizations of regular and intra-regular LA-semihypergroups in terms of fuzzy hyperideals 5. Conclusion References