Int. J. Anal. Appl. (2023), 21:77 On (Fuzzy) Weakly Almost Interior Γ-Hyperideals in Ordered Γ-Semihypergroups Warud Nakkhasen1,∗, Ronnason Chinram2, Aiyared Iampan3 1Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand 2Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 3Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand ∗Corresponding author: warud.n@msu.ac.th Abstract. In this paper, we concentrate on studying the generalization of almost interior Γ-hyperideals in ordered Γ-semihypergroups. The notion of weakly almost interior Γ-hyperideals of ordered Γ- semihypergroups is introduced. This concept generalizes the notion of almost interior Γ-hyperideals in ordered Γ-semihypergroups. Then, the characterization of ordered Γ-semihypergroups having no proper weakly almost interior Γ-hyperideals is provided. Next, we introduce the concept of fuzzy weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups. Also, some properties of fuzzy weakly almost interior Γ-hyperideals are considered. Moreover, the concepts of weakly almost interior Γ-hyperideals and fuzzy weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups are char- acterized. The connections between strongly prime (resp., prime, semiprime) weakly almost interior Γ-hyperideals and fuzzy strongly prime (resp., prime, semiprime) weakly almost interior Γ-hyperideals in ordered Γ-semihypergroups are presented. 1. Introduction When it comes to studying in semigroups, ideal theory is essential. Grošek and Satko [5] extended the concept of ideals in semigroups to the concept of almost ideals in 1980, characterizing the semi- groups that have proper almost ideals. Afterwards, Bogdanović [2] introduced the concept almost bi-ideals in semigroups, as a generalization of bi-ideals, by using the concepts of almost ideals and Received: May 20, 2023. 2020 Mathematics Subject Classification. 03E72, 20M12. Key words and phrases. weakly almost interior Γ-hyperideals; fuzzy weakly almost interior Γ-hyperideals; ordered Γ-semihypergroups. https://doi.org/10.28924/2291-8639-21-2023-77 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-77 2 Int. J. Anal. Appl. (2023), 21:77 bi-ideals of semigroups. Zadeh [22] introduced the concept of fuzzy subsets as a function from a nonempty set X to the unit interval [0, 1]. Wattanatripop et al. [21] applied the concept of fuzzy subsets to define the notion of fuzzy almost bi-ideals of semigroups in 2018, they examined at some of the connections between almost bi-ideals and fuzzy almost bi-ideals in semigroups. The concepts of (resp., weakly) almost interior ideals and fuzzy (resp., weakly) almost interior ideals in semigroups were introduced and discussed by Kaopusek et al. [8] and Krailoet et al. [9], respectively. In 2022, Chinram and Nakkhasen [3] introduced the concept of almost bi-quasi-interior ideals of semigroups and considered some relationships between almost bi-quasi-interior ideals and their fuzzification in semigroups. The notion of Γ-semigroups generalized from the classical semigroups, was first introduced by Sen and Saha [15]. Then, Simuen et al. [16] defined the concepts of almost quasi-Γ-ideals and fuzzy almost quasi-Γ-ideals of Γ-semigroups. Later, Jantanan et al. [7] studied the concepts of almost interior Γ-ideals and fuzzy almost interior Γ-ideals in Γ-semigroups. The notion of ordered semigroups is another generalization of the semigroups. In 2022, Suebsung et al. [17] introduced the concepts of (resp., fuzzy) almost bi-ideals and (resp., fuzzy) almost quasi-ideals of ordered semigroups, and they have investigated the characterizations of these concepts. Since 1934, the research of Marty [10], who developed the notion of hyperstructures, has been studied by many mathematicians. The concept of almost hyperideals in semihypergroups, which is a generalization of hyperideals, was introduced and presented some properties by Suebsung et al. [18]. Then, they have defined the concept of almost quasi-hyperideals in semihypergroups and gave some interesting properties, see [19]. Next, Muangdoo et al. [11] introduced the notions of (resp., fuzzy) almost bi-hyperideals of semihypergroups and discussed some connections between almost bi- hyperideals and their fuzzification in semihypergroups. In 2022, Nakkhasen et al. [12] surveyed some properties of fuzzy almost interior hyperideals in semihypergroups and considered some links between almost interior hyperideals and fuzzy almost interior hyperideals in semihypergroups. It is known that ordered Γ-semihypergroups are a generalization of semihypergroups. Recently, Rao et al. [14] defined the concept of almost interior Γ-hyperideals of ordered Γ-semihypergroups and provided the relationships between interior Γ-hyperideals and almost interior Γ-hyperideals in ordered Γ-semihypergroups. This article presents the notions of weakly almost interior Γ-hyperideals in ordered Γ-semihypergroups, which extend the idea of almost interior Γ-hyperideals, and provides certain char- acteristics of these hyperideals. Furthermore, we define the concept of fuzzy weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups, and consider some connections between weakly almost interior Γ-hyperideals and fuzzy weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups. 2. Preliminaries Firstly, we recall some of the basis definitions and properties, which are necessary for this paper. Int. J. Anal. Appl. (2023), 21:77 3 A hypergroupoid (H,◦) is a nonempty set H together with a mapping ◦ : H×H →P∗(H) called a hyperoperation, where P∗(H) denotes the set of all nonempty set of H (see [4,10]). We denote by a◦b the image of the pair (a,b) in H ×H. If x ∈ H and A,B ∈P∗(H), then we denote A◦B := ⋃ a∈A,b∈B a◦b,A◦x := A◦{x} and x ◦B := {x}◦B. Definition 2.1. (see [6]) A hypergroupoid (S,◦) is called a semihypergroup if (x ◦y) ◦z = x ◦ (y ◦z) for all x,y,z ∈ S. In 2010, Anvariyeh et al. [1] introduced the notion of Γ-semihypergroups, which is a generalization of semihypergroups. Definition 2.2. (see [1]) Let S and Γ be two nonempty sets. Then, (S, Γ) is called a Γ-semihypergroup if for each γ ∈ Γ is a hyperoperation on S, i.e., xγy ⊆ S for all x,y ∈ S, and for any α,β ∈ Γ and x,y,z ∈ S, (xαy)βz = xα(yβz). Let A and B be two nonempty subsets of a Γ-semihypergroup (S, Γ). We define AΓB := ⋃ γ∈Γ AγB = ⋃ γ∈Γ {aγb | a ∈ A,b ∈ B}. Particularly, if A = {a} and B = {b}, then we define aΓb := {a}Γ{b}. Definition 2.3. (see [20]) Let S and Γ be two nonempty sets and ≤ be an order relation on S. An algebraic hyperstructure (S, Γ,≤) is called an ordered Γ-semihypergroup if the following conditions are satisfied: (i) (S, Γ) is a Γ-semihypergroup; (ii) (S,≤) is a partially ordered set; (iii) for every x,y,z ∈ S and γ ∈ Γ, x ≤ y implies xγz ≤ yγz and zγx ≤ zγy. Here, A ≤ B means that for each a ∈ A, there exists b ∈ B such that a ≤ b, for all nonempty subsets A and B of S. Throughout this paper, we say an ordered Γ-semihypergroup S instead of an ordered Γ- semihypergroup (S, Γ,≤), unless otherwise mentioned. For any nonempty subset A of an ordered Γ-semihypergroup S, we denote (A] := {t ∈ S | t ≤ a for some a ∈ A}. For A = {a}, we write (a] instead of ({a}]. Lemma 2.1. [20] Let A and B be nonempty subsets of an ordered Γ-semihypergroup S. Then, the following statements holds: (i) A ⊆ (A]; 4 Int. J. Anal. Appl. (2023), 21:77 (ii) if A ⊆ B, then (A] ⊆ (B]; (iii) (A]Γ(B] ⊆ (AΓB] and ((A]Γ(B]] = (AΓB]; (iv) ((A]] = (A]. The notion of almost interior Γ-hyperideals in ordered Γ-semihypergroups, as a generalization of interior Γ-hyperideals, has been introduced by Rao et al. [14] in 2021 as follows. Definition 2.4. [14] Let S be an ordered Γ-semihypergroup. A nonempty subset K of S is called an almost interior Γ-hyperideal of S if (i) (xΓKΓy] ∩K 6= ∅ for every x,y ∈ S, (ii) (K] ⊆ K. Now, we review the concept of fuzzy subsets, was defined by Zadeh [22]. We say that µ is a fuzzy subset [22] of a nonempty set X if µ : X → [0, 1]. For any two fuzzy subsets µ and λ of a nonempty set X, we denote (i) µ ⊆ λ if and only if µ(x) ≤ λ(x) for all x ∈ X, (ii) (µ∩λ)(x) := min{f (x),g(x)} for all x ∈ X, (iii) (µ∪λ)(x) := max{f (x),g(x)} for all x ∈ X. For any fuzzy subset µ of a nonempty set X, the support of µ is defined by supp(µ) := {x ∈ X | µ(x) 6= 0}. The characteristic mapping CA of A, where A is a subset of a nonempty set X, is a fuzzy subset of X defined by for every x ∈ X, CA(x) :=  1 if x ∈ A, 0 otherwise. Lemma 2.2. [11] Let A and B be nonempty subsets of a nonempty set X and let µ and λ be fuzzy subsets of X. Then, the following statements hold: (i) CA∩B = CA ∩CB; (ii) A ⊆ B if and only if CA ⊆ CB; (iii) supp(CA) = A; (iv) if µ ⊆ λ, then supp(µ) ⊆ supp(λ). For any element s of X and α ∈ (0, 1], a fuzzy point sα [13] of X is a fuzzy subset of X defined by for every x ∈ X, sα(x) :=  α if x = s, 0 otherwise. Int. J. Anal. Appl. (2023), 21:77 5 Let S be an ordered Γ-semihypergroup. For each x ∈ S, we define Hx := {(y,z) ∈ S×S | x ≤ yΓz}. Then, for any two fuzzy subsets µ and λ of S, the product µ◦λ [20] of µ and λ is defined by (µ◦λ)(x) =   sup (y,z)∈Hx [min{µ(y),λ(z)}] if Hx 6= ∅, 0 if Hx = ∅, for all x ∈ S. Let µ be a fuzzy subset of an ordered Γ-semihypergroup S. Then, we define (µ] : S → [0, 1] by (µ](x) = sup x≤y µ(y) for all x ∈ S (see [20]). The following results can be verified straightforward. Lemma 2.3. Let A and B be subsets of an ordered Γ-semihypergroup S. Then CA ◦CB = C(AΓB]. Proposition 2.1. Let µ,λ and ν be fuzzy subsets of an ordered Γ-semihypergroup S. Then, the following conditions hold: (i) µ ⊆ (µ]; (ii) if µ ⊆ λ, then (µ] ⊆ (λ]; (iii) if µ ⊆ λ, then (µ◦ν] ⊆ (λ◦ν] and (ν ◦µ] ⊆ (ν ◦λ]. Proposition 2.2. Let µ be a fuzzy subset of an ordered Γ-semihypergroup S. Then, the following statements are equivalent: (i) if x ≤ y, then µ(x) ≥ µ(y) for all x,y ∈ S; (ii) (µ] = µ. 3. Weakly almost interior Γ-hyperideals In this section, we present and study the notion of weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups as a generalization of almost interior Γ-hyperideals. Definition 3.1. Let S be an ordered Γ-semihypergroup. A nonempty subset I of S is called a weakly almost interior Γ-hyperideal of S if it satisfies the following conditions: (i) (xΓIΓx] ∩ I 6= ∅ for all x ∈ S; (ii) (I] ⊆ I. The following proposition obtains direct from the definition of almost interior Γ-hyperideals and weakly almost interior Γ-hyperideals in ordered Γ-semihypergroups. Proposition 3.1. Every almost interior Γ-hyperideal of an ordered Γ-semihypergroup S is also a weakly almost interior Γ-hyperideal of S. The converse of Proposition 3.1 is not true in general, as shown by the following example below. 6 Int. J. Anal. Appl. (2023), 21:77 Example 3.1. Let S = {a,b,c,d,e,f} and Γ = {γ} with the hyperoperation on S defined by γ a b c d e f a {a} {b} {c} {d} {e} {f} b {b} {c} {a} {f} {d} {e} c {c} {a} {b} {e} {f} {d} d {d} {f} {e} {a,b} {a,c} {b,c} e {e} {d} {f} {a,c} {b,c} {a,b} f {f} {e} {d} {b,c} {a,b} {a,c} Then, (S, Γ,≤) is an ordered Γ-semihypergroup, where the order relation ≤ on S defined by ≤:= {(x,y) | x = y}. Let I = {a,b}. Hence, by routine calculation, we have that I is a weakly almost interior Γ-hyperideal of S. But I is not an almost interior Γ-hyperideal of S, because (dΓIΓa] ∩ I = ∅. Theorem 3.1. Let I be a weakly almost interior Γ-hyperideal of ordered Γ-semihypergroup S. If A is any subset of S containing I, then A is also a weakly almost interior Γ-hyperideal of S. Proof. Assume that A is a subset of S such that I ⊆ A. Let x ∈ S. Then, (xΓIΓx] ∩ I 6= ∅. Thus, ∅ 6= (xΓIΓx]∩I ⊆ (xΓAΓx]∩A. It follows that (xΓAΓx]∩A 6= ∅. Hence, A is a weakly almost interior Γ-hyperideal of S. � Corollary 3.1. Let S be an ordered Γ-semihypergroup. If I1 and I2 are weakly almost interior Γ- hyperideals of S, then I1 ∪ I2 is a weakly almost interior Γ-hyperideal of S. Example 3.2. Let S = {a,b,c,d,e} and Γ = {α} be the nonempty sets. Define the hyperoperation as: α a b c d e a {d} {a,b,d} {a,b,d} {d} {a,b,d,e} b {a,b,d} {a,b,d} {a,b,d} {a,b,d} {a,b,d,e} c {a,b,d} {a,b,d} {a,b,d} {a,b,d} {a,b,d,e} d {d} {a,b,d} {a,b,d} {d} {a,b,d,e} e {a,b,d} {a,b,d} {a,b,d,e} {a,b,d} {a,b,d,e} Next, we define an order relation ≤ on S as: ≤:={(a,a), (b,b), (c,c), (d,d), (e,e), (a,b), (a,c), (a,e), (b,c), (b,e), (d,b), (d,c), (d,e)}. Then, (S, Γ,≤) is an ordered Γ-semihypergroup. Let I1 = {a,b} and I2 = {d}. Verifying that I1 and I2 are weakly almost interior Γ-hyperideals of S is a routine process. However, I1 ∩ I2 is not a weakly almost interior Γ-hyperideal of S. The intersection of any two weakly almost interior Γ-hyperideals of an ordered Γ-semihypergroup S does not necessarily have to be a weakly almost interior Γ-hyperideal of S, as shown by Example 3.2. Int. J. Anal. Appl. (2023), 21:77 7 Theorem 3.2. Let S be an ordered Γ-semihypergroup and |S| > 1. Then, the following statements are equivalent: (i) S has no proper weakly almost interior Γ-hyperideal; (ii) for every x ∈ S, there exists ax ∈ S such that (ax Γ(S \{x})Γax ] = {x}. Proof. (i) ⇒ (ii) Assume that (i) holds. For any x ∈ S, we have that S\{x} is not a weakly almost interior Γ-hyperideal of S. So, there exists ax ∈ S such that (ax Γ(S \{x})Γax ] ∩ (S \{x}) = ∅. We obtain that (ax Γ(S \{x}Γax )] ⊆ S \ (S \{x}) = {x}. It turns out that (ax Γ(S \{x})Γax ] = {x}. (ii) ⇒ (i) Assume that (ii) holds. Let A be any a proper subset of S. Then, A ⊆ S\{x} for some x ∈ S. By assumption, there exists ax ∈ S such that (ax Γ(S \{x})Γax ] = {x}. Thus, (ax ΓAΓax ] ∩A ⊆ (ax Γ(S \{x})Γax ] ∩ (S \{x}) = {x}∩ (S \{x}) = ∅. Hence, A is not a weakly almost interior Γ-hyperideal of S. This shows that S has no proper weakly almost interior Γ-hyperideal of S. � 4. Fuzzy weakly almost interior Γ-hyperideals The concept of fuzzy weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups and some of the relationships between them are discussed in this section. Definition 4.1. Let µ be a nonzero fuzzy subset of an ordered Γ-semihypergroup S. Then, µ is called a fuzzy weakly almost interior Γ-hyperideal of S if for every fuzzy point sα of S, (sα◦µ◦sα] ∩µ 6= 0. From the Definition 4.1, we obtain that the following remark holds. Remark 4.1. Let sα be any fuzzy point of an ordered Γ-semihypergroup S. Then, (sα◦µ◦sα]∩µ 6= 0 if and only if there exist x,a ∈ S such that x ≤ sΓaΓs and µ(x),µ(a) 6= 0. Theorem 4.1. Let µ be a fuzzy weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup S. If λ is a fuzzy subset of S such that µ ⊆ λ, then λ is also a fuzzy weakly almost interior Γ-hyperideal of S. Proof. Assume that λ is a fuzzy subset of S such that µ ⊆ λ. Let sα be a fuzzy point of S. Then, (sα◦µ◦sα]∩µ 6= 0. Since µ ⊆ λ, 0 6= (sα◦µ◦sα]∩µ ⊆ (sα◦λ◦sα]∩λ. Also, (sα◦λ◦sα]∩λ 6= 0. Hence, λ is a fuzzy weakly almost interior Γ-hyperideal of S. � Corollary 4.1. Let µ and λ be fuzzy weakly almost interior Γ-hyperideals of an ordered Γ- semihypergroup S. Then, µ∪λ is a fuzzy weakly almost interior Γ-hyperideal of S. 8 Int. J. Anal. Appl. (2023), 21:77 Example 4.1. Consider the ordered Γ-semihypergroup (S, Γ,≤) in Example 3.2, we define two fuzzy subsets µ and λ of S by for every x ∈ S, µ(x) =  0.8 if x ∈{a,b}, 0 otherwise and λ(x) =  0.5 if x = d, 0 otherwise. By routine computations, we find out that µ and λ are fuzzy weakly almost interior Γ-hyperideals of S. However, µ∩λ is not a fuzzy weakly almost interior Γ-hyperideal of S, because µ∩λ = 0. From Example 4.1, we know that the intersection of two fuzzy weakly almost interior Γ-hyperideals of an ordered Γ-semihypergroup S need not be a fuzzy weakly almost interior Γ-hyperideal of S. Theorem 4.2. Let I be a nonempty subset of an ordered Γ-semihypergroup S. Then, I is a weakly almost interior Γ-hyperideal of S if and only if CI is a fuzzy weakly almost interior Γ-hyperideal of S. Proof. Assume that I is a weakly almost interior Γ-hyperideal of S. Let sα be any fuzzy point of S. Then, (sΓIΓs] ∩ I 6= ∅. Thus, there exists a ∈ S such that a ∈ (sΓIΓs] and a ∈ I. So, CI(a) = 1 and a ≤ sΓxΓs for some x ∈ I. Since x ∈ I, CI(x) = 1. It follows that (sα ◦CI ◦ sα](a) ≥ min{sα(s),CI(x),sα(s)} 6= 0. We obtain that [(sα◦CI ◦sα]∩CI](a) 6= 0. Thus, CI is a fuzzy weakly almost interior Γ-hyperideal of S. Conversely, assume that CI is a fuzzy weakly almost interior Γ-hyperideal of S. Let s ∈ S. Choose t = 1. Then, (s1◦CI◦s1]∩CI 6= 0. So, there exist x,a ∈ S such that x ≤ sΓaΓs and CI(x),CI(a) 6= 0. This implies that x,a ∈ I. Also, x ∈ (sΓIΓs]. Thus, x ∈ (sΓIΓs] ∩ I, and then (sΓIΓs] ∩ I 6= ∅. Therefore, I is a weakly almost interior Γ-hyperideal of S. � Theorem 4.3. Let µ be a fuzzy subset of an ordered Γ-semihypergroup S. Then, µ is a fuzzy weakly almost interior Γ-hyperideal of S if and only if supp(µ) is a weakly almost interior Γ-hyperideal of S. Proof. Assume that µ is a fuzzy weakly almost interior Γ-hyperideal of S. Let s ∈ S. Choose t = 1. So, (s1◦µ◦s1]∩µ 6= 0. So, there exist x,a ∈ S such that x ≤ sΓaΓs and µ(x),µ(a) 6= 0. Also, x,a ∈ supp(µ). Since x ≤ sΓaΓs, x ∈ (sΓ(supp(µ))Γs]. It turns out that x ∈ (sΓ(supp(µ))Γs] ∩supp(µ), that is, (sΓ(supp(µ))Γs] ∩ supp(µ) 6= ∅. Hence, supp(µ) is a weakly almost interior Γ-hyperideal of S. Conversely, assume that supp(µ) is a weakly almost interior Γ-hyperideal of S. Let sα be any fuzzy point of S. Then, (sΓ(supp(µ))Γs] ∩ supp(µ) 6= ∅. Thus, there exists x ∈ S such that x ∈ (sΓ(supp(µ))Γs] and x ∈ supp(µ). So, x ≤ sΓaΓs for some a ∈ supp(µ). This means that µ(x),µ(a) 6= 0. We have that (sα ◦µ◦ sα] ∩µ 6= 0. Therefore, µ is a fuzzy weakly almost interior Γ-hyperideal of S. � Int. J. Anal. Appl. (2023), 21:77 9 Let S be an ordered Γ-semihypergroup. A weakly almost interior Γ-hyperideal I of S is called minimal if for any weakly almost interior Γ-hyperideal A of S such that A ⊆ I implies that A = I. Definition 4.2. Let S be an ordered Γ-semihypergroup. A fuzzy weakly almost interior Γ-hyperideal µ of S is called minimal if for any fuzzy weakly almost interior Γ-hyperideal λ of S such that λ ⊆ µ implies that supp(λ) = supp(µ). Now, the relationship between minimal weakly almost interior Γ-hyperideals and minimal fuzzy weakly almost interior Γ-hyperideals in ordered Γ-semihypergroups is then briefly examined. Theorem 4.4. Let S be an ordered Γ-semihypergroup, and I be a nonempty subset of S. Then, I is a minimal weakly almost almost interior Γ-hyperideal of S if and only if CI is a minimal fuzzy weakly almost interior Γ-hyperideal of S. Proof. Assume that I is a minimal weakly almost interior Γ-hyperideal of S. By Theorem 4.2, CI is a fuzzy weakly almost interior Γ-hyperideal of S. Let λ be any fuzzy weakly almost interior Γ- hyperideal of S such that λ ⊆ CI. By Lemma 2.2 and Theorem 4.3, we have that supp(λ) is a weakly almost interior Γ-hyperideal of S such that supp(λ) ⊆ supp(CI). Since I is minimal, supp(λ) = I = supp(CI). Hence, CI is a minimal fuzzy weakly almost interior Γ-hyperideal of S. Conversely, assume that CI is a minimal fuzzy weakly almost interior Γ-hyperideal of S. Thus, I is a weakly almost Γ-hyperideal of S by Theorem 4.2. Now, let A be any weakly almost interior Γ-hyperideal of S such that A ⊆ I. Then, CA is a fuzzy weakly almost interior Γ-hyperideal of S such that CA ⊆ CI. Since CI is minimal and by Lemma 2.2, we have that A = supp(CA) = supp(CI) = I. Therefore, I is a minimal weakly almost interior Γ-hyperideal of S. � The following corollary can be achieved by Theorem 4.2 and Theorem 4.3. Corollary 4.2. Let S be an ordered Γ-semihypergroup. Then, S has no proper weakly almost interior Γ-hyperideal if and only if for every fuzzy weakly almost interior Γ-hyperideal µ of S, supp(µ) = S. Let S be an ordered Γ-semihypergroup and P be a weakly almost interior Γ-hyperideal of S. Then: (i) P is said to be prime if for any weakly almost interior Γ-hyperideals A and B of S such that (AΓB] ⊆ P implies that A ⊆ P or B ⊆ P; (ii) P is said to be semiprime if for any weakly almost interior Γ-hyperideal A of S such that (AΓA] ⊆ P implies that A ⊆ P ; (iii) P is said to be strongly prime if for any weakly almost interior Γ-hyperideals A and B of S such that (AΓB] ∩ (BΓA] ⊆ P implies that A ⊆ P or B ⊆ P. Definition 4.3. Let µ be a fuzzy weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup S. Then, µ is said to be a fuzzy prime weakly almost interior Γ-hyperideal of S if for any fuzzy weakly almost interior Γ-hyperideals λ and ν of S such that λ◦ν ⊆ µ implies that λ ⊆ µ or ν ⊆ µ. 10 Int. J. Anal. Appl. (2023), 21:77 Definition 4.4. Let µ be a fuzzy weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup S. Then, µ is said to be a fuzzy semiprime weakly almost interior Γ-hyperideal of S if for any fuzzy weakly almost interior Γ-hyperideal λ of S such that λ◦λ ⊆ µ implies that λ ⊆ µ. Definition 4.5. Let µ be a fuzzy weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup S. Then, µ is said to be a fuzzy strongly prime weakly almost interior Γ-hyperideal of S if for any fuzzy weakly almost interior Γ-hyperideals λ and ν of S such that (λ◦ν) ∩ (ν ◦λ) ⊆ µ implies that λ ⊆ µ or ν ⊆ µ. It is obvious that every fuzzy strongly prime weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup is a fuzzy prime weakly almost interior Γ-hyperideal, and every fuzzy prime weakly almost interior Γ-hyperideal of an ordered Γ-semihypergroup is a fuzzy semiprime weakly almost interior Γ-hyperideal. Finally, we consider the connections between strongly prime (resp., prime, semiprime) weakly almost interior Γ-hyperideals and their fuzzifications in ordered Γ-semihypergroups. Theorem 4.5. Let S be an ordered Γ-semihypergroup and P be a nonempty subset of S. Then, P is a strongly prime weakly almost interior Γ-hyperideal of S if and only if CP is a fuzzy strongly prime weakly almost interior Γ-hyperideal of S. Proof. Assume that P is a strongly prime weakly almost interior Γ-hyperideal of S. Also, CP is a fuzzy weakly almost interior Γ-hyperideal of S by Theorem 4.2. Let λ and ν be any two fuzzy weakly almost interior Γ-hyperideals of S such that (λ ◦ ν) ∩ (ν ◦ λ) ⊆ CP . Suppose that λ 6⊆ CP and ν 6⊆ CP . Thus, there exist x,y ∈ S such that λ(x) 6= 0 and ν(y) 6= 0, but CP (x) = 0 and CP (y) = 0. So, x,y 6∈ P. By using Theorem 4.3, we have that supp(λ) and supp(ν) are weakly almost interior Γ-hyperideals of S such that x ∈ supp(λ) and y ∈ supp(ν). We obtain that, supp(λ) 6⊆ P and supp(ν) 6⊆ P. By assumption, ((supp(λ))Γ(supp(ν))] ∩ ((supp(ν))Γ(supp(λ))] 6⊆ P. Then, there exists t ∈ ((supp(λ))Γ(supp(ν))] ∩ ((supp(ν))Γ(supp(λ))], but t 6∈ P. It follows that CP (t) = 0, and then [(λ◦ν)∩(ν◦λ)](t) = 0. Since t ∈ ((supp(λ))Γ(supp(ν))] and t ∈ ((supp(ν))Γ(supp(λ))], we have that t ≤ a1Γb1 and t ≤ b2Γa2 for some a1,a2 ∈ supp(λ) and b1,b2 ∈ supp(ν). It turns out that (λ◦ν)(t) = sup t≤a1Γb1 [min{λ(a1),ν(b1)}] 6= 0 and (ν ◦λ)(t) = sup t≤b2Γa2 [min{ν(b2),λ(a2)}] 6= 0. This implies that [(λ◦ν) ∩ (ν ◦λ)](t) 6= 0, as a contradiction. So, λ ⊆ CP or ν ⊆ CP . This shows that CP is a fuzzy strongly prime weakly almost interior Γ-hyperideal of S. Conversely, assume that CP is a fuzzy strongly prime weakly almost interior Γ-hyperideal of S. Then, P is a weakly almost interior Γ-hyperideal of S by Theorem 4.2. Let A and B be any two weakly almost interior Γ-hyperideals of S such that (AΓB] ∩ (BΓA] ⊆ P. By using Lemma 2.2 and Int. J. Anal. Appl. (2023), 21:77 11 Lemma 2.3, it follows that (CA ◦CB) ∩ (CB ◦CA) = C(AΓB] ∩C(BΓA] = C(AΓB]∩(BΓA] ⊆ CP . By the hypothesis, CA ⊆ CP or CB ⊆ CP . It follows that A ⊆ P or B ⊆ P. Therefore, P is a strongly prime weakly almost interior Γ-hyperideal of S. � Theorem 4.6. Let P be a nonempty subset of an ordered Γ-semihypergroup S. Then, P is a prime weakly almost interior Γ-hyperideal of S if and only if CP is a fuzzy prime weakly almost interior Γ-hyperideal of S. Proof. Assume that P is a prime weakly almost interior Γ-hyperideal of S. By using Theorem 4.2, we obtain that CP is a fuzzy weakly almost interior Γ-hyperideal of S. Let λ and ν be any two fuzzy weakly almost interior Γ-hyperideals of S such that λ◦ν ⊆ CP . Suppose that λ 6⊆ CP and ν 6⊆ CP . Then, there exist x,y ∈ S such that λ(x) 6= 0 and ν(y) 6= 0, while CP (x) = 0 and CP (y) = 0. So, x ∈ supp(λ), y ∈ supp(ν) with x,y 6∈ P. By Theorem 4.3, we have that supp(λ) and supp(ν) are weakly almost interior Γ-hyperideals of S. This implies that supp(λ) 6⊆ P and supp(ν) 6⊆ P. By assumption, it follows that ((supp(λ)Γ(supp(ν)))] 6⊆ P. Also, there exists t ∈ ((supp(λ)Γ(supp(ν)))] such that t 6∈ P . This means that CP (t) = 0. It turns out that (λ ◦ ν)(t) = 0, because λ ◦ ν ⊆ CP . Since t ∈ ((supp(λ)Γ(supp(ν)))], t ≤ aΓb for some a ∈ supp(λ) and b ∈ supp(ν). Thus, (λ◦ν)(t) = sup t≤aΓb [min{λ(a),ν(b)}] 6= 0. This is a contradiction to the fact that (λ◦ν)(t) = 0. This shows that λ ⊆ CP or ν ⊆ CP . Hence, CP is a fuzzy prime weakly almost interior Γ-hyperideal of S. Conversely, assume that CP is a fuzzy prime weakly almost interior Γ-hyperideal of S. By Theorem 4.2, P is a weakly almost Γ-hyperideal of S. Let A and B be any weakly almost interior Γ-hyperideals of S such that (AΓB] ⊆ P. By Lemma 2.2 and Lemma 2.3, it follows that CA ◦CB = C(AΓB] ⊆ CP . By the given assumption, CA ⊆ CP or CB ⊆ CP . This implies that, A ⊆ P or B ⊆ P. Therefore, P is a prime weakly almost interior Γ-hyperideal of S. � Theorem 4.7. Let S be an ordered Γ-semihypergroup and P be a nonempty subset of S. Then, P is a semiprime weakly almost interior Γ-hyperideal of S if and only if CP is a fuzzy semiprime weakly almost interior Γ-hyperideal of S. Proof. Assume that P is a semiprime weakly almost interior Γ-hyperideal of S. By Theorem 4.2, we obtain that CP is a fuzzy weakly almost Γ-hyperideal of S. Let λ be any fuzzy weakly almost interior Γ-hyperideal of S such that λ ◦ λ ⊆ CP . Suppose that λ 6⊆ CP . So, there exists x ∈ S such that λ(x) 6= 0 and CP (x) = 0. Also, x ∈ supp(λ) and x 6∈ P. By Theorem 4.3, supp(λ) is a weakly almost interior Γ-hyperideal of S where supp(λ) 6⊆ P. By assumption, ((supp(λ)Γ(supp(λ)))] 6⊆ P. Thus, there exists t ∈ S such that t ∈ ((supp(λ)Γ(supp(λ)))], but t 6∈ P. This implies that CP (t) = 0. It 12 Int. J. Anal. Appl. (2023), 21:77 follows that (λ◦λ)(t) = 0, because λ◦λ ⊆ CP . Since t ∈ ((supp(λ)Γ(supp(λ)))], t ≤ aΓb for some a,b ∈ supp(λ). It turns out that (λ◦λ)(t) = sup t≤aΓb [min{λ(a),λ(b)}] 6= 0, which is a contradiction. Hence, λ ⊆ CP . Therefore, CP is a fuzzy semiprime weakly almost Γ-hyperideal of S. Conversely, assume that CP is a fuzzy semiprime weakly almost Γ-hyperideal of S. It follows that P is a weakly almost interior Γ-hyperideal of S by Theorem 4.2. Let A be a weakly almost interior Γ-hyperideal of S such that (AΓA] ⊆ P. By using Lemma 2.2 and Lemma 2.3, we have that CA ◦CA = C(AΓA] ⊆ CP . Since CP is semiprime, CA ⊆ CP . It follows that A ⊆ P. This shows that P is a semiprime weakly interior Γ-hyperideal of S. � 5. Conclusions In 2021, Rao et al. [14] introduced the concept of almost interior Γ-hyperideals as a generalization of interior Γ-hyperideals of ordered Γ-semihypergroups. In this paper, we introduced the notion of weakly almost interior Γ-hyperideals of ordered Γ-semihypergroups which is a generalization of almost interior Γ-hyperideals. Next, we shown that the union of (fuzzy) weakly almost interior Γ-hyperideals is also a (fuzzy) weakly almost interior Γ-hyperideal, but the intersection of them need not to be a (fuzzy) weakly almost interior Γ-hyperideal in ordered Γ-semihypergroups. Then, we characterized the ordered Γ-semihypergroups having no proper weakly almost interior Γ-hyperideal. Finally, we discussed the connections between weakly almost interior Γ-hyperideals and their fuzzification in ordered Γ- semihypergroups. 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Fuzzy Syst. 32 (2017), 3821-3838. https://doi.org/10.3233/ifs-162117. [21] K. Wattanatripop, R. Chinram, T. Changphas, Fuzzy Almost Bi-Ideals in Semigroups, Int. J. Math. Computer Sci. 13 (2018), 51-58. [22] L.A. Zadeh, Fuzzy Sets, Inform. Control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.1080/09720529.2019.1696917 https://doi.org/10.28919/jmcs/5609 https://doi.org/10.1016/0022-247x(80)90048-7 https://doi.org/10.1155/2021/6683910 https://doi.org/10.1155/2021/6683910 https://doi.org/10.1080/09720529.2020.1826167 https://doi.org/10.3233/ifs-162117 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. Introduction 2. Preliminaries 3. Weakly almost interior -hyperideals 4. Fuzzy weakly almost interior -hyperideals 5. Conclusions References