ON FUZZY ORDERED LA-SEMIHYPERGROUPS

We introduce the notion of fuzzy ordered LA-semihypergroups and provide different examples. We also discuss some results related with fuzzy left and right hyperideals.

Another non-associative algebraic hyperstructure known as LA-semihypergroup which is a useful generalization of semigroup, semihypergroups and LA-semigroups was introduced by Hilla and Dine [14] in 2011 based on left invertive law given by Kazim and Naseerudin [15] in 1972.Yaqoob et al. [16] extended the work of Hila and Dine and characterized intra-regular left almost semihypergroups by their hyperideals using pure left identity.The ordering in LA-semihypergroups was introduced by Yaqoob and Gulistan [17].
The concept of fuzzy set was introduced by Zadeh in 1965 [18].Rosenfeld [19] introduced fuzzy sets in the context of group theory and formulated the concept of fuzzy subgroup of a group in 1971.Later many researcher are engaged in extending the concept of abstract algebra to the frame work of fuzzy setting.
As a further study of ordered LA-semihypergroups, we attempt in the present paper to study the fuzzy ordered LA-semihypergroups in detail.

Preliminaries
Let H be a non-empty set.Then the map • : H × H → P * (H) is called hyperoperation or join operation on the set H, where P * (H) = P(H)\{∅} denotes the set of all non-empty subsets of H.A hypergroupoid is a set H together with a (binary) hyperoperation.For any non-empty subsets A, B of H, we denote Recently, in [14,16] authors introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semigroups.A hypergroupoid (H, •) is called an LA-semihypergroup if for every x, y, z ∈ H, we have (x x, y, z, w ∈ H.An LA-semihypergroup may or may not contains a left identity and pure left identity.In an LA-semihypergroup H with pure left identity, the paramedial law (x • y) for all x, y, z, w ∈ H.If an LA-semihypergroup contains a pure left identity, then by using medial law, we  Definition 2.2.[17] If (H, •, ≤) is an ordered LA-semihypergroup and A ⊆ H, then (A] is the subset of H defined as follows: (A] = {t ∈ H : t ≤ a, for some a ∈ A}. If A is both right hyperideal and left hyperideal of H, then A is called a hyperideal (or two sided hyperideal) of H.

Fuzzy ordered LA-semihypergroups
Let x ∈ H, then A x = {(y, z) ∈ H • H : x ≤ y • z} .Let f and g be two fuzzy subsets of an ordered LA-semihypergroup H, then f * g is defined as Let F (H) denote the set of all fuzzy subsets of an ordered LA-semihypergroup.
Proof.Clearly F (H) is closed.Let f, g and h be in F (H) and let x be any element of H such that it is not expressible as product of two elements in H. Then we have, Let A x = ∅.Then there exist y and z in H such that (y, z) ∈ A x .Therefore by using left invertive law, we have Similarly we can show that f * h ⊇ g * h.Let A x = ∅.Then there exist y and z in H such that (y, z) Similarly we can show that f * h ⊇ g * h.It is easy to see that F (H) is a poset.Thus (F (H), * , ⊆) is an ordered LA-semihypergroup.
Theorem 3.2.Let H be an ordered LA-semihypergroup.Then the property holds in F (H), for all f, g, h and k in F (H). Proof.Straightforward.
Theorem 3.3.If an ordered LA-semihypergroup H has a pure left identity, then the following properties hold in F (H).
for all f, g, h and k in F (H). Proof.Straightforward.
Proof.Let an ordered LA-semihypergroup F (H) be a commutative ordered semihypergroup.For any fuzzy then there exist s and t in H such that (s, t) ∈ A x , therefore by use of left invertive law and commutative law, we get We have to show that H is a commutative ordered semihypergroup.Let f and g be any arbitrary fuzzy where h and k are any fuzzy subsets of H.
Now by left invertive law, we have This shows that f * g = g * (h * k) = g * f.Therefore commutative law holds in F (H).
Let A x = ∅, then there exist s and t in H such that (s, t) ∈ A x , therefore by using left invertive law and commutative law, we get Therefore associative law holds in F (H). Thus F (H) is commutative ordered semihypergroup.
Proof.The fuzzy subset C M of H is non-empty since α * α = α, which implies that α is in C M .Let f and γ be fuzzy subsets of Let A x = ∅, then there exist y and z in H such that (y, z) ∈ A x .Therefore by using medial law, we have .
Let A x = ∅, then there exist y and z in H such that (y, z) ∈ A x .Therefore by using left invertive law, we have Thus f * γ = (γ * α) * f = γ * f, which implies that commutative law holds in C M and associative law holds in C M due to commutativity.Since for any fuzzy subset f in C M , we have f * α = f (where α is fixed) implies that α is a right identity in H and hence an identity.
For an ordered LA-semihypergroup H, the fuzzy subset H of H is defined as follows: for every a, b ∈ H.
Clearly f is a fuzzy LA-subsemihypergroup of H.
We give the covering relation " ≺ " and the figure of H as follows: ≺= {(x, y), (x, z), (w, x)} Then (H, •, ≤) is an ordered LA-semihypergroup.Now let f be a fuzzy subset of H such that Then f is a fuzzy two sided hyperideal of H.
Proof.The proof is similar to the proof of the Theorem 4.1.
Definition 4.4.Let H be an ordered LA-semihypergroup and f be a fuzzy subset of H. Then for every t ∈ [0, 1] the set Definition 4.5.Let H be an ordered LA-semihypergroup and ∅ = A ⊆ H. Then the characteristic function χ A of A is defined as: Theorem 4.3.Let H be an ordered LA-semihypergroup and f be a fuzzy subset of H. Then f is a fuzzy LA-subsemihypergroup (resp., right hyperideal, left hyperideal) of H if and only if for every t ∈ [0, 1], the non-empty level subset f t is a fuzzy LA-subsemihypergroup (resp., right hyperideal, left hyperideal) of H.
Conversely, we assume that for every t ∈ [0, 1] , f t is a right hyperideal of H.We show that Then, for every c ∈ a • b, we obtain t • ≤ f (c) and hence, f (a Proof.Straightforward.Proof.Let f 1 and f 2 be two fuzzy right hyperideals of an ordered LA-semihypergroup If A x = ∅.Since f 1 and f 2 are a fuzzy right hyperideals of H, then Thus f 1 * f 2 is a fuzzy right hyperideal of H.
Theorem 4.5.Let f 1 be a fuzzy right hyperideal and f 2 a fuzzy left hyperideal of H. Then Lemma 4.1.Let H be an ordered LA-semihypergroup with left identity.Then every fuzzy right hyperideal of H is fuzzy left hyperideal of H.
Proof.Let H be an LA-semihypergroup with pure left identity e, and f be a fuzzy right hyperideal of H.
Since f is a fuzzy right hyperideal of H, so f * H ⊆ f.Thus by Lemma 3.1, and left invertive law, we have Thus H * f ⊆ f.Hence f is a fuzzy left hyperideal of H. Proof.Let f be a fuzzy left hyperideal of H which is idempotent.Then Hence f is a fuzzy right hyperideal of H and so f is a fuzzy hyperideal of H. Thus (H * f ) * (H * f ) = H * f.The case for f * H can be seen in a similar way.

Conclusion
Fuzzy set theory is a mathematical tools for dealing with uncertainties.This paper is devoted to the discussion of the combinations of fuzzy set in ordered LA-semihypergroup.We combined these concepts to introduce fuzzy left (resp., right) hyperideals and discussed some interesting results.
Instead of {a} • A and B • {a} , we write a • A and B • a, respectively.

Definition 2 . 1 .
[17] Let H be non-empty set and ≤ be an ordered relation on H.The triplet (H, •, ≤) is called an ordered LA-semihypergroup if the following conditions are satisfied.

( 1 )
(H, •) is an LA-semihypergroup, (2) (H, ≤) is a partially ordered set, (3) for every a, b, c ∈ H, a ≤ b implies a • c ≤ b • c and c • a ≤ c • b, where a • c ≤ b • c means that for x ∈ a • c there exist y ∈ b • c such that x ≤ y.

Theorem 4 . 1 .
A fuzzy subset f of an ordered LA-semihypergroup H is a fuzzy LA-subsemihypergroup of H if and only if

Definition 4 . 3 .
2) a ≤ b implies f (a) ≥ f (b) , for every a, b ∈ H.If f is both fuzzy right hyperideal and fuzzy left hyperideal of H, then f is called a fuzzy hyperideal of H.We consider a set H = {x, y, z} with the following hyperoperation "•" and the order " ≤ " : w} {x, w} w y {x, w} {y, z} {y,

Theorem 4 . 2 .
A fuzzy subset f of an ordered LA-semihypergroup H is a fuzzy left (resp., right) hyperideal of H if and only if

Proposition 4 . 1 .
The fuzzy product of two fuzzy right (resp., left) hyperideals of an ordered LA-semihypergroup H is again a fuzzy right (resp., left) hyperideal of H.

Theorem 4 . 6 .Definition 4 . 6 .Proposition 4 . 2 .
If f is a fuzzy left hyperideal of H with left identity, then f ∪ (f * H) is a fuzzy hyperideal of H.Proof.Let f be a fuzzy left hyperideal of H.We have to show that f ∪ (f * H) is fuzzy hyperideal.Let(f ∪ (f * H)) * H = (f * H) ∪ (f * H) * H = (f * H) ∪ (H * H) * f = (f * H) ∪ (H * f ) ⊆ (f * H) ∪ f = f ∪ (f * H) .Hence f ∪ (f * H) is fuzzy right hyperideal of H. Since every fuzzy right hyperideal of an ordered LAsemihypergroup with left identity is a fuzzy left hyperideal of H, so f ∪ (f * H) is a fuzzy hyperideal of H.A fuzzy hyperideal f of an ordered LA-semihypergroup H is called idempotent if f * f = f.Every idempotent fuzzy left hyperideal of an ordered LA-semihypergroup H is a fuzzy hyperideal of H.

Proposition 4 . 3 .
If f is an idempotent fuzzy set in an ordered LA-semihypergroup H with left identity.Then H * f and f * H are idempotents.Proof.Let f be an idempotent element in an ordered LA-semihypergroup H with left identity.Then by using medial law, we have(H * f ) * (H * f ) = (H * H) * (f * f ) = H * f.
Therefore f is a fuzzy right hyperideal of H. Corollary 4.1.Let H be an ordered LA-semihypergroup and χ I be the characteristic function of I.Then, then I is an LA-subsemihypergroup (resp., right hyperideal, left hyperideal) of H if and only if χ I is a fuzzy LA-subsemihypergroup (resp., right hyperideal, left hyperideal) of H. Theorem 4.4.If {f i } i∈J is a family of fuzzy left hyperideals (resp., right hyperideals) of an ordered LAsemihypergroup H, then i∈J f i is a fuzzy left hyperideal (resp., right hyperideal) of H, where