ALMOST BI-HYPERIDEALS AND THEIR FUZZIFICATION OF SEMIHYPERGROUPS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we introduce the concept of almost bi-hyperideals of semihypergroups which is a generalization of bi-hyperideals, and we give some properties of them. Moreover, we consider the connections between almost bi-hyperideals and their fuzzification of semihypergroups.


INTRODUCTION
The concepts of left, right, two-sided almost ideals of semigroups were introduced by Grosek and Satko [7] in 1980. They studied the characterization of these ideals when a semigroup contains no proper left, right, two-sided ideals. Later in 1981, Bogdanovic [1] introduced the notion of almost bi-ideals in semigroups as a generalization of bi-ideals. The concept of fuzzy subsets was first introduced by Zadeh [18] as a function from a nonempty set X to the unit interval [0, 1]. The fuzzy subset theory is a generalization of traditional mathematics set theory.
In 2018, Wattanatripop, Chinram and Changphas [17] introduced the notion of fuzzy almost bi-ideals in semigroups and discussed some relationships between almost bi-ideals and fuzzy almost bi-ideals of semigroups. Then, Simuen, et al. [14] investigated some properties of fuzzy almost bi-Γ-ideals of Γ-semigroups.
In this paper, we introduce the concept of almost bi-hyperideals of semihypergroups as a generalization of bi-hyperideals and investigate some properties of them. Then, we discuss the connections between almost bi-hyperideals and fuzzy almost bi-hyperideals of semihypergroups.

PRELIMINARIES
Let H be a nonempty set. A hyperoperation on H is a mapping • : H × H → P * (H), where P * (H) denotes the set of all nonempty subsets of H. Then, the structure (H, •) is called a hypergroupoid. If A, B ∈ P * (H) and x ∈ H, then we denote For more convenient, we write S instead of a semihypergroup (S, •) and AB instead of A • B, for any nonempty subsets A and B of S.

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A fuzzy subset [18] of a nonempty set X is a mapping f : X → [0, 1]. Let f and g be any two fuzzy subsets of a nonempty set X. Then, f ⊆ g if and only if f (x) ≤ g(x) for all x ∈ X. The intersection and the union of two fuzzy subsets f and g of a nonempty set X, denoted by f ∩ g and f ∪ g, respectively, are defined by letting x ∈ X, Let X be a nonempty set. For a fuzzy subset f of X, the support of f is defined by For any element s of X and t ∈ (0, 1], a fuzzy point s t of X defined by for all x ∈ X. Lemma 2.1. Let A and B be nonempty subsets of a nonempty set X and let f and g be fuzzy subsets of X. Then the following statements hold: Proof. The proof is straightforward.
Let f and g be fuzzy subsets of a semihypergroup S. A product f • g is defined by for all x ∈ S.
Lemma 2.2. If A and B are subsets of a semihypergroup S, then χ A • χ B = χ AB .
Proof. Let x ∈ S. If χ AB (x) = 0, then x ∈ AB. This means that x ∈ ab for all a ∈ A and b ∈ B.

ALMOST BI-HYPERIDEALS
In this section, we introduce the concept of almost bi-hyperideals of semihypergroups and give some of its properties.
Therefore, B is an almost bi-hyperideal of S. In general, an almost bi-hyperideal of a semihypergroup need not to be a bi-hyperideal as the following example.
Next, we discuss some properties of almost bi-hyperideals of semihypergroups.
Theorem 3.5. Let B be an almost bi-hyperideal of a semihypergroup S. If A is any subset of S containing B, then A is also an almost bi-hyperideal of S.
Corollary 3.6. The union of any two almost bi-hyperideals of a semihypergroup S is also an almost bi-hyperideal of S.
Proof. Let A and B be any two almost bi-hyperideals of a semihypergroup S. Since A ⊆ A ∪ B and by Theorem 3.5, we get that A ∪ B is an almost bi-hyperideal of S.  This is a contradiction. Therefore, S has no proper almost bi-hyperideals.

FUZZY ALMOST BI-HYPERIDEALS
In this section, we introduce the concept of fuzzy almost bi-hyperideals of semihypergroups, and we study the connections between almost bi-hyperideals and their fuzzification of semihypergroups. of S such that f ⊆ g, then g is a fuzzy almost bi-hyperideal of S.
Proof. Assume that g is a fuzzy subset of S such that f ⊆ g. By assumption, for all fuzzy points s t of S. Hence, g is a fuzzy almost bi-hyperideal of S.   Proof. Assume that B is an almost bi-hyperideal of S. Then, BsB ∩ B = / 0 for all s ∈ S. So, there exists x ∈ S such that x ∈ BsB and x ∈ B. Thus, x ∈ b 1 sb 2 for some b 1 , b 2 ∈ B. It follows that This implies that (χ B • s t • χ B ) ∩ χ B = 0 for all fuzzy points s t of S. Hence, χ B is a fuzzy almost bi-hyperideal of S.
Conversely, assume that χ B is a fuzzy almost bi-hyperideal of S. Let s ∈ S. Then, such that x ∈ b 1 sb 2 and x ∈ B. This means that x ∈ BsB and x ∈ B. That is, BsB ∩ B = / 0. Therefore, B is an almost bi-hyperideal of S. Proof. Assume that f is a fuzzy almost bi-hyperideal of S. Let s ∈ A and s t be a fuzzy point of We obtain that there exist y 1 , y 2 ∈ S such that x ∈ y 1 sy 2 , So, f (y 1 ) = 0 and f (y 2 ) = 0. Hence, x, y 1 , y 2 ∈ supp( f ). It follows that This means that χ supp( f ) is a fuzzy almost bi-hyperideal of S. By Theorem 4.5, supp( f ) is an almost bi-hyperideal of S.
Conversely, assume that supp( f ) is an almost bi-hyperideal of S. By Theorem 4.5, χ supp( f ) is a fuzzy almost bi-hyperideal of S. Let s t be any fuzzy point of S. Then,  minimal if for any fuzzy almost bi-hyperideal g of S such that g ⊆ f , we get supp( f ) = supp(g).
Next, we investigate the minimality of fuzzy almost bi-hyperideals of semihypergroups. Conversely, assume that χ B is a minmal fuzzy almost bi-hyperideal of S. Then, B is an almost bi-hyperideal of S. Let A be any almost bi-hyperideal of S such that A ⊆ B. By Theorem 4.5, χ A is a fuzzy almost bi-hyperideal of S such that χ A ⊆ χ B . Since χ B is minimal, we get that supp(χ A ) = supp(χ B ). We obtain that A = supp(χ A ) = supp(χ B ) = B by Lemma 2.1.
Consequently, B is a minimal almost bi-hyperideal of S. Let S be a semihypergroup. An almost bi-hyperideal P of S is prime if for any almost bihyperideals A and B of S such that AB ⊆ P implies that A ⊆ P or B ⊆ P. An almost bi-hyperideal P of S is semiprime if for any almost bi-hyperideal A of S such that AA ⊆ P implies that A ⊆ P.
An almost bi-hyperideal P of S is strongly prime if for any almost bi-hyperideals A and B of S such that AB ∩ BA ⊆ P implies that A ⊆ P or B ⊆ P.
Definition 4.11. A fuzzy almost bi-hyperideal h of a semihypergroup S is called a fuzzy semiprime almost bi-hyperideal of S if for any fuzzy almost bi-hyperideal f of S, We note that every fuzzy strongly prime almost bi-hyperideal of a semihypergroup is a fuzzy prime almost bi-hyperideal, and every fuzzy prime almost bi-hyperideal of a semihypergroup is a fuzzy semiprime almost bi-hyperideal, but the converse is not true in general.
Finally, we study the relationships between prime (resp., semiprime, strongly prime) almost bi-hyperideals and their fuzzification of semihypergroups.
Theorem 4.13. Let P be a nonempty subset of a semihypergroup S. Then P is a prime almost bi-hyperideal of S if and only if χ P is a fuzzy prime almost bi-hyperideal of S.
Proof. Assume that P is a prime almost bi-hyperideal of S. By Theorem 4.5, χ P is a fuzzy almost hyperideal of S. Let f and g be fuzzy almost bi-hyperideals of S such that f • g ⊆ χ P .
Suppose that f χ P and g χ P . Then, there exist x, y ∈ S such that f (x) = 0 and g(y) = 0, but χ P (x) = 0 and χ P (y) = 0. It follows that x ∈ P and y ∈ P. Since f (x) = 0 and g(y) = 0, we get that x ∈ supp( f ) and y ∈ supp(g). Thus, supp( f ) P and supp(g) P. By Theorem 4.6, we have that supp( f ) and supp(g) are almost bi-hyperideals of S. Since P is prime, Then, there exists t ∈ ab for some a ∈ supp( f ) and b ∈ supp(g) such that t ∈ P. So, χ P (t) = 0, and then ( f • g)(t) = 0 because f • g ⊆ χ P . Since a ∈ supp( f ) and b ∈ supp(g), we have that f (a) = 0 and g(b) = 0. Hence, min{ f (a), g(b)} = 0, which implies that ( f •g)(t) = sup t∈ab {min{ f (a), g(b)}} = 0. This is a contradiction to the fact that ( f •g)(t) = 0.
Therefore, f ⊆ χ P or g ⊆ χ P . Consequently, χ P is a fuzzy prime almost bi-hyperideal of S.
Conversely, assume that χ P is a fuzzy prime almost bi-hyperideal of S. By Theorem 4.5, P is an almost bi-hyperideal of S. Let A and B be any two almost bi-hyperideals of S such that AB ⊆ P. By Lemma 2.1 and Lemma 2.2, we get that χ A • χ B = χ AB ⊆ χ P . Again by Theorem 4.5, χ A and χ B are fuzzy almost bi-hyperideals of S. By hypothesis, χ A ⊆ χ P or χ B ⊆ χ P . That is, A ⊆ P or B ⊆ P. Hence, P is a prime almost bi-hyperideal of S.
The proof of the following theorem is similar to Theorem 4.13.
Theorem 4.14. Let P be a nonempty subset of a semihypergroup S. Then P is a semiprime almost bi-hyperideal of S if and only if χ P is a fuzzy semiprime almost bi-hyperideal of S.
Theorem 4.15. Let P be a nonempty subset of a semihypergroup S. Then P is a strongly prime almost bi-hyperideal of S if and only if χ P is a fuzzy strongly prime almost bi-hyperideal of S.
Proof. Assume that P is a strongly prime almost bi-hyperideal of S. By Theorem 4.5, χ P is a fuzzy almost bi-hyperideal of S. Let f and g be any two fuzzy almost bi-hyperideals of S such that ( f • g) ∩ (g • f ) ⊆ χ P . Suppose that f χ P and g χ P . Then, there exist x, y ∈ S such that f (x) = 0 and g(y) = 0, but χ P (x) = 0 and χ P (y) = 0. So, x ∈ supp( f ) and y ∈ supp(g) such that x ∈ P and y ∈ P. It follows that supp( f ) P and supp(g) P. By Theorem 4.6 and the hypothesis, we have that [supp( f )supp(g)] ∩ [supp(g)supp( f )] P. Hence, there exists t ∈ [supp( f )supp(g)] ∩ [supp(g)supp( f )] such that t ∈ P. Also, χ P (t) = 0, and then Since t ∈ supp( f )supp(g) and t ∈ supp(g)supp( f ), we have that t ∈ a 1 b 1 and t ∈ b 2 a 2 for some a 1 , a 2 ∈ supp( f ) and b 1 , b 2 ∈ supp(g). It turns out that This implies that min{( f • g)(t), (g • f )(t)} = 0, that is, [( f • g) ∩ (g • f )](t) = 0. This is a contradiction with the fact that [( f • g) ∩ (g • f )](t) = 0. Hence, f ⊆ χ P or g ⊆ χ P . Therefore, χ P is a fuzzy strongly prime almost bi-hyperideal of S.
Conversely, assume that χ P is a fuzzy strongly prime almost bi-hyperideal of S. Then, P is an almost bi-hyperideal of S by Theorem 4.5. Let A and B be any two almost bi-hyperideals of S such that (AB) ∩ (BA) ⊆ P. By Theorem 4.5, χ A and χ B are fuzzy almost bi-hyperideals of S. χ A ) = χ AB ∩ χ BA = χ (AB)∩(BA) ⊆ χ P . By assumption, we get that χ A ⊆ χ P or χ B ⊆ χ P implies that A ⊆ P or B ⊆ P. Consequently, P is a strongly prime almost bi-hyperideal of S.