A study on (fuzzy) quasi-Γ-hyperideals in ordered Γ-semihypergroups

1 Introduction

As we know, the concept of Γ-semigroups was first introduced by Sen in 1981 [41]. Also see [39, 42]. Γ-semigroups generalize semigroups. Many classical notions of semigroups have been extended to Γ-semigroups (see [4, 5]), and some properties of Γ-semigroups were studied by some researchers, for instance, see [6, 40]. In 1993, Sen and Seth [43] introduced the concept of po-Γ-semigroups (some authors called ordered Γ-semigroups). Later on, some properties of ordered Γ-semigroups were studied (see [24, 32]).

Algebraic hyperstructures (or hypersystems) represent a natural extension of classical algebraic structures and they were originally proposed in 1934 by a French mathematician Marty [35], at the 8th Congress of Scandinavian Mathematicians. Since then, hyperstructures are widely investigated from the theoretical point of view and for their applications to many branches of pure and applied mathematics (see [7, 10]. One of the main reason which attracts researches towards hyperstructures is its unique property that in hyperstructures composition of two elements is a set, while in classical algebraic structures the composition of two elements is an element. Thus algebraic hyperstructures are natural extension of classical algebraic structures. Especially, semihypergroups are the simplest algebraic hyperstructures which possess the properties of closure and associativity. Nowadays many scholars have studied different aspects of semihypergroups, for instance, Davvaz [13], Davvaz and Poursalavati [17], Hila et al. [27] and Leoreanu [33], also see [36]. It is worth pointing out that Davvaz et al. [23] introduced the concept of Γ-semihypergroups as a generalization of semigroups, semihypergroups and Γ-semigroups. Recently, some authors have worked on Γ-semihypergroups, for example, see [1, 26].

The theory of fuzzy sets, which was initially introduced in 1965 by Zadeh [52], has been applied to many mathematical branches. The study of fuzzy hyperstructures is an interesting research topic of fuzzy set theory. We noticed that the relationships between the fuzzy sets and algebraic hyperstructures have been already considered by Corsini, Davvaz, Leoreanu, Dudek, Jun, Zhan, Hila and others, for example, the reader can refer to [11, 53]. A recent books [10, 48] contain plenty of applications. Recently, Davvaz and Leoreanu-Fotea [15] and Ersoy et al. [20] studied the structure of fuzzy Γ-hyperideals in Γ-semihypergroups, and provided some interesting results, also see [18, 50]. In [51], Yaqoob et al. defined and studied the bipolar fuzzy Γ-hyperideals in Γ-semihypergroups.

A theory of hyperstructures on ordered semigroups has been recently developed. In [22], Heidari and Davvaz applied the theory of hyperstructures to ordered semigroups and introduced the notion of ordered semihypergroups, which is a generalization of ordered semigroups. Since then, several researchers conducted the researches on the properties of ordered semihypergroups, and obtained some important results. For more details, the reader is referred to [3, 47].

As we know, quasi-ideals play an important role in the study of ring, semigroup and ordered semigroup structures. The concept of a quasi-ideal in rings and semigroups was studied by Stienfeld in [45]. Furthermore, Kehayopulu and Tsingelis introduced the concept of quasi-ideals in ordered semigroups (see [30]). The fuzzy quasi-ideals in ordered semigroups were studied in [29, 44], where the basic properties of ordered semigroups in terms of fuzzy quasi-ideals are given. In [27], Hila et al. extended the concept of quasi-ideals to the theory of semihypergroups, and introduced the concept of quasi-hyperideals in semihypergroups. In [2], Anvariyeh et al. defined and studied the quasi-Γ-hyperideals of Γ-semihypergroups. As a continuation of these papers, in this paper we introduce and study ordered Γ-semihypergroups, which generalize ordered semigroups, ordered Γ-semigroups and particularly ordered semihypergroups. Especially, we attempt to investigate the quasi-Γ-hyperideals and fuzzy quasi-Γ-hyperideals in ordered Γ-semihypergroups in detail. The rest of this paper is organized as follows. After an introduction, in Section 2 we recall some basic definitions and results of ordered Γ-semihypergroups which will be used throughout this paper. In Section 3, we define and study the quasi-Γ-hyperideals of an ordered Γ-semihypergroup, and present some examples in this respect. In Section 4, we introduce the concepts of fuzzy Γ-hyperideals and fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups, and investigate their related properties. In particular, we discuss the relationships between fuzzy Γ-hyperideals and fuzzy quasi-Γ-hyperideals in ordered Γ-semihypergroups. Furthermore, we study the structure of fuzzy quasi-Γ-hyperideal generated by a fuzzy subset in an ordered Γ-semihypergroup. In Section 5, we introduce the concepts of completely prime, weakly completely prime and completely semiprime fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups, and characterize bi-regular ordered Γ-semihypergroups in terms of completely semiprime fuzzy quasi-Γ-hyperideals. In Section 6, some characterizations of regular ordered Γ-semihypergroups and intra-regular ordered Γ-semihypergroups by the properties of fuzzy quasi-Γ-hyperideals are given. Some conclusions are given in the last Section.

2 Preliminaries and some notations

Recall first the basic terms and definitions from the hyperstructure theory.

As we know, a hypergroupoid (S, ∘) is a nonempty set S together with a hyperoperation, that is a map ∘ : S × S → P^∗ (S) , where P^∗ (S) denotes the set of all nonempty subsets of S (see [9]). The image of the pair (x, y) is denoted by x ∘ y . If x ∈ S and A, B are nonempty subsets of S, then A ∘ B is defined by A ∘ B = ⋃ a ∈ A , b ∈ B a ∘ b . The notations A ∘ x and x ∘ A are used for A ∘ {x} and {x} ∘ A, respectively. Generally, the singleton {x} is identified by its element x .

We say that a hypergroupoid (S, ∘) is a semihypergroup if the hyperoperation “ ∘ ″ is associative, that is, (x ∘ y) ∘ z = x ∘ (y ∘ z) for all x, y, z ∈ S (see [10]).

We now recall the notion of ordered semihypergroups from [22].

Definition 2.1. An algebraic hyperstructure (S, ∘ , ≤) is called an ordered semihypergroup (also called po-semihypergroup in [22]) if (S, ∘) is a semihypergroup and (S, ≤) is a partially ordered set such that the monotone condition holds as follows: x ≤ y ⇒ a ∘ x ⪯ a ∘ y and x ∘ a ⪯ y ∘ a for all x, y, a ∈ S, where, if A and B are nonempty subsets of S, then we say that A ⪯ B if for every a ∈ A there exists b ∈ B such that a ≤ b .

Definition 2.2. [23]. Let S = {x, y, z, ⋯} and Γ = {α, β, γ, ⋯} be two nonempty sets. Then, (S, Γ) is called Γ-semihypergroup if every γ ∈ Γ is a hyperoperation on S, that is, xγy ⊆ S for every x, y ∈ S, and for every α, β ∈ Γ and x, y, z ∈ S, (xαy) βz = xα (yβz) .

Now we shall introduce the notion of ordered Γ-semihypergroups.

Definition 2.3. Let S = {x, y, z, ⋯} and Γ = {α, β, γ, ⋯} be two nonempty sets. An algebraic hyperstructure (S, Γ, ≤) is called an ordered Γ-semihypergroup if (S, Γ) is a Γ-semihypergroup and (S, ≤) is a partially ordered set such that: for any x, y, z ∈ S and γ ∈ Γ, x ≤ y implies xγz ⪯ yγz and zγx ⪯ zγy . Here, if A, B ∈ P^∗ (S) , then we say that A ⪯ B if for every a ∈ A there exists b ∈ B such that a ≤ b . In particular, if A = {a} , then we write a ⪯ B instead of {a} ⪯ B .

Let (S, ∘ , ≤) be an ordered semihypergroup and Γ = {∘} . Then, (S, Γ, ≤) is an ordered Γ-semihyper- group. Thus every ordered semihypergroup is an ordered Γ-semihypergroup.

Example 2.4. Let (S, ∘ , ≤) be an ordered semihypergroup and Γ a nonempty subset of S . We define x γ y : = x ∘ γ ∘ y for every x, y ∈ S and γ ∈ Γ . Then, (S, Γ, ≤) is an ordered Γ-semihypergroup.

Example 2.5. Let S = [0, 1] and Γ be the set of all positive integers. For every x, y ∈ S and γ ∈ Γ, we define x γ y : = [ 0 , xy γ ] . Then, γ is a hyperoperation on S . For every x, y, z ∈ S and α, β ∈ Γ we have ( x α y ) β z = [ 0 , xyz α β ] = x α ( y β z ) . This means that (S, Γ) is a Γ-semihypergroup. Furthermore, (S, Γ, ≤) is an ordered Γ-semihypergroup, where “≤” means the usual partial order relation on S .

Throughout this paper, unless otherwise mentioned, S will denote an ordered Γ-semihypergroup.

Let A and B be two nonempty subsets of an ordered Γ-semihypergroup S . We define A Γ B : = ⋃ γ ∈ Γ A γ B = ⋃ γ ∈ Γ { a γ b | a ∈ A , b ∈ B } .

Let S be an ordered Γ-semihypergroup. For ∅ ≠ H ⊆ S, we define ( H ] : = { t ∈ S | t ≤ h for some h ∈ H } . For H = {a} , we write (a] instead of ({a}] .

Lemma 2.6. Let S be an ordered Γ-semihypergroup. Then the following statements hold:

Proof. Straightforward. □

Definition 2.7. Let S be an ordered Γ-semi-hypergroup. A nonempty subset I of S is called a left (resp. right) Γ-hyperideal of S if

SΓI ⊆ I  (resp .  IΓS ⊆ I) , and

If I is both a left and a right Γ-hyperideal of S, then it is called a (two-sided) Γ-hyperideal of S . We denote by L (A) (resp. R (A)) the left (resp. right) Γ-hyperideal of S generated by A  (∅ ≠ A ⊆ S) . One can easily prove that L (A) = (A ∪ S ∘ A] and R (A) = (A ∪ A ∘ S] . In particular, if A = {a} , then we write L (a) and R (a) instead of L ({a}) and R ({a}) , respectively.

For the sake of simplicity, throughout this paper, we denote A ⁿ  = AΓAΓ ⋯ ΓA (n-copies).

Let S be an ordered Γ-semihypergroup. A nonempty subset A of S is called a sub Γ-semihypergroup of S if AΓA ⊆ A . S is called regular if for every a ∈ S, there exists x ∈ S such that a ⪯ aΓxΓa . Equivalent Definitions: (1) A ⊆ (AΓSΓA] , for any nonempty subset A of S . (2) a ∈ (aΓSΓa] , for any a ∈ S . S is called intra-regular if for each a ∈ S, there exist x, y ∈ S such that a ⪯ xΓa²Γy . Equivalent Definitions: (1) A ⊆ (SΓA²ΓS] , for any nonempty subset A of S . (2) a ∈ (SΓa²ΓS] , for any a ∈ S .

We next state some fuzzy logic concepts.

Let S be an ordered Γ-semihypergroup. By a fuzzy subset of S, we mean a function from S into the real closed interval [1], that is, f : S → [0, 1] . For an ordered Γ-semihypergroup S, the fuzzy subset 1 of S is defined as follows: 1 : S → [ 0 , 1 ] , x ↦ 1 ( x ) : = 1 , ∀ x ∈ S . We denote by F (S) the set of all fuzzy subsets of S . Let f, g ∈ F (S) . Then, the inclusion relation f ⊆ g is defined by f (x) ≤ g (x) for all x ∈ S, and f ∩ g, f ∪ g are defined by ( f ∩ g ) ( x ) = min { f ( x ) , g ( x ) } = f ( x ) ∧ g ( x ) , ( f ∪ g ) ( x ) = max { f ( x ) , g ( x ) } = f ( x ) ∨ g ( x ) for all x ∈ S, respectively.

Let (S, Γ, ≤) be an ordered Γ-semihypergroup. For x ∈ S, we define H _x  : = {(y, z) ∈ S × S| x ⪯ yΓz} . For any f, g ∈ F (S) , the product f ∗ g of f and g is defined by ( f ∗ g ) ( x ) = { ⋁ ( y , z ) ∈ H x [ f ( y ) ∧ g ( z ) ] , if H x ≠ ∅ , 0 , if H x = ∅ , for all x ∈ S . Similar to the proof of Lemma 2.5 in [47], we can easily prove that the multiplication “∗” on F (S) is well defined and associative.

Definition 2.8. Let S be an ordered Γ-semihypergroup and f ∈ F (S) . The set f t : = { x ∈ S | f ( x ) ≥ t } , where t ∈ [ 0 , 1 ] is called a level subset of f .

Definition 2.9. Let S be an ordered Γ-semihypergroup, a ∈ S and λ ∈ [0, 1] . An ordered fuzzy point a _λ of S is defined by the rule that ( ∀ x ∈ S ) a λ ( x ) = { λ if x ∈ ( a ] , 0 if x ∉ ( a ] . It is evident that every ordered fuzzy point of S is a fuzzy subset of S . For any fuzzy subset f of S, we also denote a _λ  ⊆ f by a _λ  ∈ f in the sequel.

Definition 2.10. Let f be a fuzzy subset of an ordered Γ-semihypergroup S . We define (f] by the rule that ( f ] ( x ) = ⋁ y ≥ x f ( y ) for all x ∈ S. A fuzzy subset f of S is called strongly convex if f = (f] .

Lemma 2.11. Let f be a fuzzy subset of an ordered Γ-semihypergroup S . Then, f is a strongly convex fuzzy subset of S if and only if x ≤ y implies f (x) ≥ f (y) , for all x, y ∈ S .

Proof. The proof is similar to that of Lemma 3.15 in [47], and hence we omit the details. □

Lemma 2.12. Let a _λ , b _μ (λ > 0, μ > 0) be ordered fuzzy points of S, and f, g ∈ F (S) . Then, the following statements are true:

Proof. (1) Let x ∈ S . If x ∈ (aΓb] , then x ⪯ aΓb . Thus (a, b) ∈ H _x , and we have ( a λ ∗ b μ ) ( x ) = ⋁ ( y , z ) ∈ H x [ a λ ( y ) ∧ b μ ( z ) ] ≥ a λ ( a ) ∧ b μ ( b ) = λ ∧ μ . Since a _λ  (y) ∧ b _μ  (z) ≤ λ ∧ μ for any y, z ∈ S, we have (a _λ  ∗ b _μ ) (x) ≤ λ ∧ μ, and so ( a λ ∗ b μ ) ( x ) = λ ∧ μ = ( ⋃ c ∈ ( a Γ b ] c λ ∧ μ ) ( x ) . If x ∉ (aΓb] , then x ∉ (c] for any c ∈ (aΓb] . Thus we have ( ⋃ c ∈ ( a Γ b ] c λ ∧ μ ) ( x ) = ⋁ c ∈ ( a Γ b ] ( c λ ∧ μ ( x ) ) = 0 . On the other hand, (a _λ  ∗ b _μ ) (x) =0 . In fact, if (a _λ  ∗ b _μ ) (x) ≠0, then ( a λ ∗ b μ ) ( x ) = ⋁ ( y , z ) ∈ H x [ a λ ( y ) ∧ b μ ( z ) ] ≠ 0 . Thus there exist u, v ∈ S such that x ⪯ uΓv and a _λ  (u) ∧ b _μ  (v) ≠0 . Hence u ∈ (a] , v ∈ (b] . It follows from Lemma 2.6 that x ∈ (uΓv] ⊆ ((a] Γ (b]] = (aΓb] , which is impossible. Thus, in this case, ( a λ ∗ b μ ) ( x ) = ( ⋃ c ∈ ( a Γ b ] c λ ∧ μ ) ( x ) .

(2) The proof is similar to that of Lemma 2.5 in [47] with suitable modification.

(3) Suppose that f is a strongly convex fuzzy subset of S . Let a _λ  ∈ f . Then, f (a) ≥ a _λ  (a) , i.e., f (a) ≥ λ . Conversely, if f (a) ≥ λ, then a _λ  ∈ f . Indeed, if x ∉ (a] , then a _λ  (x) =0 ≤ f (x) . If x ∈ (a] , i.e., x ≤ a, then, by Lemma 2.11, f (x) ≥ f (a) ≥ λ = a _λ  (x) . Consequently, a _λ  (x) ≤ f (x) for all x ∈ S . It thus implies that a _λ  ∈ f .

(4) The proof is analogous to that of Lemma 3.16 in [47], we omit it.

Let S be an ordered Γ-semihypergroup and ∅ ≠ A ⊆ S . Then, the characteristic function f _A of A is a fuzzy subset of S defined by f A : S → [ 0 , 1 ] | x ↦ f A ( x ) : = { 1 if x ∈ A , 0 if x ∉ A . □

The reader is referred to [10, 49] for notation and terminology not defined in this paper.

3 Quasi-Γ-hyperideals of ordered Γ-semihyper-groups

In this section, we introduce the concept of quasi-Γ-hyperideals of an ordered Γ-semihypergroup, and investigate its related properties. In particular, characterizations of regular ordered Γ-semihypergroups and intra-regular ordered Γ-semihypergroups in terms of quasi-Γ-hyperideals are given.

Definition 3.1. Let S be an ordered Γ-semihypergroup. A nonempty subset Q of S is called a quasi-Γ-hyperideal of S if

(QΓS] ∩ (SΓQ] ⊆ Q, and

The concept of quasi-Γ-hyperideals defined in the above definition is a generalization of the concept of quasi-Γ-hyperideals of Γ-semihypergroups (without order) to ordered Γ-semihypergroup theory, see [2]. In addition, it is clear that every one-sided Γ-hyperideal of an ordered Γ-semihypergroup S is a quasi-Γ-hyperideal of S . However, the converse is not true, in general, as shown in the following example.

Example 3.2. Let S = {a, b, c, d, e} and Γ = {γ} with the hyperoperation and the relation “≤” on S defined by

   ≤ : =   {(a, a) , (a, b) , (a, c) , (a, d) , (a, e) , (b, b) ,  (b, c) , (b, d) , (b, e) , (c, c) , (c, e) , (d, d) , (d, e) , (e, e)} .We give the covering relation “≺” and the figure of S as follows:≺ = {(a, b) , (b, c) , (b, d) , (c, e) , (d, e)}.

Then, (S, Γ, ≤) is an ordered Γ-semihypergroup (see [38]). With a small amount of effort one can verify that the sets {a, b} , {a, b, c} , {a, b, d} and S are all quasi-Γ-hyperideals of S . But the sets {a, b} , {a, b, c} and {a, b, d} are not Γ-hyperideals of S .

Definition 3.3. Let Q be a quasi-Γ-hyperideal of an ordered Γ-semihypergroup S . Q is called prime if for any two elements a, b of S such that aΓb ⊆ Q, we have a ∈ Q or b ∈ Q .

Definition 3.4. Let Q be a quasi-Γ-hyperideal of an ordered Γ-semihypergroup S . Q is called semiprime if for any nonempty subset A of S such that AΓA ⊆ Q, we have A ⊆ Q . Equivalently, if for any element a of S such that aΓa ⊆ Q, we have a ∈ Q .

One can easily observe that every prime quasi-Γ-hyperideal of an ordered Γ-semi- hypergroup is semiprime. However, the converse is not true, in general, as shown in the following example.

Example 3.5. Let (S, Γ, ≤) be the ordered Γ-semihyper-group in Example 3.2. We can easily verify that the sets {a, b} , {a, b, c} , {a, b, d} and S are all semiprime quasi-Γ-hyperideals of S . But {a, b} is not prime. In fact, since cγd = {a} ⊆ {a, b} , but c ∉ {a, b} and d ∉ {a, b} .

Proposition 3.6. Let S be an ordered Γ-semihypergroup and Q a quasi-Γ-hyperideal of S . Then, Q is a sub Γ-semihypergroup of S .

Proof. Suppose that Q is a quasi-Γ-hyperideal of S . Then, we have Q Γ Q ⊆ Q Γ S ∩ S Γ Q ⊆ ( Q Γ S ] ∩ ( S Γ Q ] ⊆ Q . It thus follows that Q is a sub Γ-semihypergroup of S . □

Lemma 3.7. Let S be an ordered Γ-semihypergroup. Then, for every right Γ-hyperideal R and left Γ-hyperideal L of S, R ∩ L is a quasi-Γ-hyperideal of S .

Proof. Let R and L be a right Γ-hyperideal and a left Γ-hyperideal of S, respectively. Then, RΓL ⊆ SΓL ⊆ L and RΓL ⊆ RΓS ⊆ R, which imply that RΓL ⊆ R ∩ L, so R ∩ L ≠ ∅ . Furthermore, by (R ∩ L] ⊆ (R] ∩ (L] ⊆ R ∩ L and ((R ∩ L) ΓS] ∩ (SΓ (R ∩ L)] ⊆ (RΓS] ∩ (SΓL] ⊆ (R] ∩ (L] = R ∩ L, it follows that R ∩ L is a quasi-Γ-hyperideal of S . □

Lemma 3.8. Let Q be any quasi-Γ-hyperideal of an ordered Γ-semihypergroup S . Then, we have

Proof. The inclusion Q ⊆ (Q ∪ QΓS] ∩ (Q ∪ SΓQ] is evident. To prove the inverse inclusion, let a ∈ (Q ∪ QΓS] ∩ (Q ∪ SΓQ] = (Q] ∪ ((QΓS] ∩ (SΓQ]) . Then, a ∈ (Q] or a ∈ (QΓS] ∩ (SΓQ] . Since Q is a quasi-Γ-hyperideal of S, the first case implies a ∈ (Q] ⊆ Q and the second case implies a ∈ (QΓS] ∩ (SΓQ] ⊆ Q . Thus Q = (Q ∪ QΓS] ∩ (Q ∪ SΓQ] . □

Remark 3.9. The above lemma shows that every quasi-Γ-hyperideal of an ordered Γ-semihypergroup S can be expressed as the intersection of a right Γ-hyperideal and a left Γ-hyperideal of S .

Theorem 3.10. Let S be an ordered Γ-semihypergroup and {Q _i  | i ∈ I} a family of quasi-Γ-hyperideals of S . Then, ⋂ i ∈ I Q i is a quasi-Γ-hyperideal of S if ⋂ i ∈ I Q i ≠ ∅ .

Proof. Let Q _i be a quasi-Γ-hyperideal of S for all i ∈ I . Assume that ⋂ i ∈ I Q i ≠ ∅ . Then, by hypothesis we have ( ( ⋂ i ∈ I Q i ) Γ S ] ∩ ( S Γ ( ⋂ i ∈ I Q i ) ] ⊆ ( Q i Γ S ] ∩ ( S Γ Q i ] ⊆ Q i

for all i ∈ I . It shows that ( ( ⋂ i ∈ I Q i ) Γ S ] ∩ ( S Γ ( ⋂ i ∈ I Q i ) ] ⊆ ⋂ i ∈ I Q i . Furthermore, if a ∈ ⋂ i ∈ I Q i and S ∋ b ≤ a, then b ∈ ⋂ i ∈ I Q i . In fact, since a ∈ ⋂ i ∈ I Q i , we have a ∈ Q _i for all i ∈ I . Since Q _i   (i ∈ I) is a quasi-Γ-hyperideal of S and S ∋ b ≤ a, we have b ∈ Q _i for every i ∈ I, which implies that b ∈ ⋂ i ∈ I Q i . Therefore, ⋂ i ∈ I Q i is a quasi-Γ-hyperideal of S . □

Let now S be an ordered Γ-semihypergroup and∅ ≠ A ⊆ S . We denote Ω = {Q | Q isaquasi - Γ - hyperideal of S containing A}. Clearly, Ω is not empty since S ∈ Ω . Let Q ( A ) = ⋂ Q ∈ Ω Q . It is clear that Q (A)≠ ∅ because A ⊆ Q (A) . By Theorem 3.10, Q (A) is a quasi-Γ-hyperideal of S . Moreover, Q (A) is the smallest quasi-Γ-hyperideal of S containing A . The quasi-Γ-hyperideal Q (A) is called the quasi-Γ-hyperideal of S generated by A . For A = {a} , let Q (a) denote the quasi-Γ-hyperideal of S generated by {a} .

Theorem 3.11. Let S be an ordered Γ-semihypergroup. Then,

Proof. (1) Let a ∈ S . By Lemma 3.7, we see that R (a) ∩ L (a) is a quasi-Γ-hyperideal of S containing a, and thus Q (a) ⊆ R (a) ∩ L (a) . On the other hand, by Lemma 3.8 it follows that R ( a ) ∩ L ( a ) = ( a ∪ a Γ S ] ∩ ( a ∪ S Γ a ] ⊆ ( Q ( a ) ∪ Q ( a ) Γ S ] ∩ ( Q ( a ) ∪ S Γ Q ( a ) ] = Q ( a ) . Consequently, Q (a) = R (a) ∩ L (a) .

(2) It can be proved similarly as (1). □

Remark 3.12. By Theorem 3.11 and Lemma 2.6, we can easily show that Q (A) = (A ∪ ((AΓS] ∩ (SΓA])] for every ∅ ≠ A ⊆ S . We shall see that this equality is simpler to use in the sequel.

Example 3.13. Let S be the ordered Γ-semi-hypergroup of Example 3.2 and A = {b, c} ⊆ S . Then, we apply the result of the above remark, and obtain that Q (A) = {a, b, c} .

Now we give a characterization of regular ordered Γ-semihypergroups by quasi-Γ-hyperideals.

Theorem 3.14. An ordered Γ-semihypergroup S is regular if and only if Q ⊆ (QΓSΓQ] for any quasi-Γ-hyperideal Q of S .

Proof. Let S be a regular ordered Γ-semihypergroup and Q a quasi-Γ-hyperideal of S . Then, clearly, Q ⊆ (QΓSΓQ] .

Conversely, let a ∈ S . Then, by hypothesis and Theorem 3.11, we have a ∈ Q ( a ) ⊆ ( Q ( a ) Γ S Γ Q ( a ) ] = ( ( R ( a ) ∩ L ( a ) ) Γ S Γ ( R ( a ) ∩ L ( a ) ) ] ⊆ ( R ( a ) Γ S Γ L ( a ) ] = ( ( a ∪ a Γ S ] Γ S Γ ( a ∪ S Γ a ] ] ⊆ ( a Γ S Γ a ] . Thus, S is a regular ordered Γ-semihypergroup. □

In a similar way, intra-regular ordered Γ-semihyper- groups can be characterized by quasi-Γ-hyperideals.

Theorem 3.15. An ordered Γ-semihypergroup S is intra-regular if and only if Q ⊆ (SΓQ²ΓS] for any quasi-Γ-hyperideal Q of S .

In the following theorem we give a characterization of an ordered Γ-semihypergroup that is both regular and intra-regular in terms of quasi-Γ-hyperideals.

Theorem 3.16. An ordered Γ-semihypergroup S is regular and intra-regular if and only if Q = (Q²] for every quasi-Γ-hyperideal Q of S .

Proof. Let S be both regular and intra-regular, and Q a quasi-Γ-hyperideal of S . Then, Q ⊆ (QΓSΓQ] , Q ⊆ (SΓQ²ΓS] and (QΓS] ∩ (SΓQ] ⊆ Q . Thus, by Lemma 2.6 we have Q ⊆ ( Q Γ S Γ Q ] ⊆ ( Q Γ S Γ ( Q Γ S Γ Q ] ] ⊆ ( ( Q Γ S ] Γ ( Q Γ S Γ Q ] ] = ( Q Γ S Γ Q Γ S Γ Q ] ⊆ ( ( Q Γ S ] Γ ( S Γ Q 2 Γ S ] Γ ( S Γ Q ] ] = ( ( Q Γ S ) Γ ( S Γ Q 2 Γ S ) Γ ( S Γ Q ) ] ⊆ ( ( Q Γ S Γ Q ) Γ ( Q Γ S Γ Q ) ] = ( ( Q Γ S Γ Q ] Γ ( Q Γ S Γ Q ] ] = ( Q Γ Q ] = ( Q 2 ] ( By Theorem 3.14 ) . On the other hand, since Q is a quasi-Γ-hyperideal of S, by Proposition 3.6, (Q²] ⊆ (Q] ⊆ Q . Hence we obtain that Q = (Q²] .

Conversely, let a ∈ S . By hypothesis, we have a ∈ Q ( a ) = ( Q 2 ( a ) ] = ( ( a ∪ ( ( a Γ S ] ∩ ( S Γ a ] ) ] Γ ( a ∪ ( ( a Γ S ] ∩ ( S Γ a ] ) ] ] = ( ( a ∪ ( ( a Γ S ] ∩ ( S Γ a ] ) ) Γ ( a ∪ ( ( a Γ S ] ∩ ( S Γ a ] ) ) ] ⊆ ( ( a ∪ ( a Γ S ] ) Γ ( a ∪ ( S Γ a ] ) ] ⊆ ( ( a Γ a ) ∪ ( a Γ S Γ a ] ] = ( ( a Γ a ) ∪ ( a Γ S Γ a ) ] . Then, a ≤ t for some t ∈ (aΓa) ∪ (aΓSΓa) . If t ∈ aΓa, then a ⪯ aΓa ⪯ aΓaΓa ⊆ aΓSΓa . If t ∈ aΓSΓa, then a ∈ (aΓSΓa] . Thus S is regular. Furthermore, we also have a ∈ Q ( a ) = ( Q 2 ( a ) ] = ( Q 2 ( a ) Γ Q ( a ) ] ⊆ ( ( ( a Γ a ) ∪ ( a Γ S Γ a ) ] Γ ( a ∪ ( ( a Γ S ] ∩ ( S Γ a ] ) ] ] ⊆ ( ( ( a Γ a ) ∪ ( a Γ S Γ a ) ] Γ ( a ∪ ( a Γ S ] ] ] = ( ( ( a Γ a ) ∪ ( a Γ S Γ a ) ] Γ ( a ∪ ( a Γ S ) ] ] = ( ( ( a Γ a ) ∪ ( a Γ S Γ a ) ) Γ ( a ∪ ( a Γ S ) ) ] ⊆ ( a 3 ∪ ( a Γ S Γ a 2 ) ∪ ( S Γ a 2 Γ S ) ] . Then, a ≤ t for some t ∈ a³ ∪ (aΓSΓa²) ∪ (SΓa²ΓS) . If t ∈ a³, then a ⪯ a³ = aΓa² ⪯ a³Γa² = aΓa²Γa² ⊆ SΓa²ΓS, that is, a ∈ (SΓa²ΓS] . If t ∈ aΓSΓa², then a ⪯ aΓSΓa² ⪯ (aΓSΓa²) ΓSΓa² = (aΓS) Γa² Γ (SΓa²) ⊆ SΓa²ΓS, i.e., a ∈ (SΓa²ΓS] . If t ∈ SΓa²ΓS, then a ∈ (SΓa²ΓS] . Hence S is also intra-regular. □

4 Fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups

In what follows, let Z⁺ denote the set of all positive integers. In the current section we introduce the concepts of fuzzy Γ-hyperideals and fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups, and investigate their related properties. In particular, we study the structure of fuzzy quasi-Γ-hyperideal generated by a fuzzy subset in an ordered Γ-semihypergroup.

Definition 4.1. Let S be an ordered Γ-semihypergroup. A fuzzy subset f of S is called a fuzzy sub Γ-semihypergroup of S if ⋀ z ∈ x Γ y f ( z ) ≥ f ( x ) ∧ f ( y ) for all x, y ∈ S .

Definition 4.2. Let S be an ordered Γ-semi-hypergroup. A fuzzy subset f of S is called a fuzzy left (resp. right) Γ-hyperideal of S if

x ≤ y implies f (x) ≥ f (y) for all x, y ∈ S, and

Example 4.3. Let S = (0, 1) , Γ = {γ _n  | n ∈ Z⁺} and for every n ∈ Z⁺ we define the hyperoperation γ _n on S as follows: ( ∀ x , y ∈ S ) x γ n y : = { xy 2 k | 0 ≤ k ≤ n } . Then, xγ _n y ⊆ S, and for every x, y, z ∈ S, γ _m , γ _n  ∈ Γ we have ( x γ m y ) γ n z = { xyz 2 k | 0 ≤ k ≤ m + n } = x γ m ( y γ n z ) . This means that (S, Γ) is a Γ-semihypergroup. Furthermore, (S, Γ, ≤) is an ordered Γ-semihypergroup, where “≤” means the usual partial order relation on S . Now a fuzzy subset f of S is defined by ( ∀ x ∈ S ) f ( x ) = { 0.8 if 0 < x < 1 2 , 0.4 if 1 2 ≤ x < 1 . Then, by routine calculations, f is a fuzzy Γ-hyperideal of S .

Theorem 4.4. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S . Then, f is a fuzzy left Γ-hyperideal of S if and only if f satisfies the following conditions:

Proof. Suppose that f is a fuzzy left Γ-hyperideal of S . By Definition 4.2, the condition (1) holds. To prove the condition (2) holds, it is enough to prove that (1 ∗ f) (x) ≤ f (x) for any x ∈ S . Indeed, let x ∈ S . If H _x  = ∅ , then (1 ∗ f) (x) =0 . Since f (x) ≥0 for all x ∈ S, we have (1 ∗ f) (x) ≤ f (x) . Let H _x  ≠ ∅ . Then, there exist y, z ∈ S such that x ⪯ yΓz, and there exists w ∈ yΓz such that x ≤ w . Since f is a fuzzy left Γ-hyperideal of S, we have f ( z ) ≤ ⋀ w ∈ y Γ z f ( w ) ≤ ⋀ w ∈ y Γ z x ≤ w f ( w ) ≤ ⋀ w ∈ y Γ z x ≤ w f ( x ) = f ( x ) . Hence we have f (z) ≤ f (x) for any x ⪯ yΓz . Thus ( 1 ∗ f ) ( x ) = ⋁ x ⪯ y Γ z [ 1 ( y ) ∧ f ( z ) ] = ⋁ x ⪯ y Γ z [ 1 ∧ f ( z ) ] = ⋁ x ⪯ y Γ z f ( z ) ≤ f ( x ) . Thus (1 ∗ f) (x) ≤ f (x) for any x ∈ S .

Conversely, assume that the conditions (1) and (2) hold. Let y, z ∈ S . Then, we can prove that f (x) ≥ f (z) for any x ∈ yΓz . In fact, since x ∈ yΓz, x ≤ x, we have x ⪯ yΓz . Thus, by hypothesis, we have f ( x ) ≥ ( 1 ∗ f ) ( x ) = ⋁ x ⪯ u Γ v [ 1 ( u ) ∧ f ( v ) ] ≥ 1 ( y ) ∧ f ( z ) = 1 ∧ f ( z ) = f ( z ) for every y, z ∈ S such that x ∈ yΓz, from which we can conclude that ⋀ x ∈ y Γ z f ( x ) ≥ f ( z ) . By Definition 4.2, f is a fuzzy left Γ-hyperideal of S . □

Similar to Theorem 4.4, we have the following theorem.

Theorem 4.5. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S . Then, f is a fuzzy right Γ-hyperideal of S if and only if f satisfies the following conditions:

Theorem 4.6.Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S . Then, f is a fuzzy Γ-hyperideal of S if and only if f satisfies the following conditions:

Proof. It is obvious by Theorems 4.4 and 4.5. □

In the following we shall introduce the notion of quasi-Γ-hyperideals of an ordered Γ-semihypergroup by the ordered fuzzy points.

Definition 4.7. Let S be an ordered Γ-semi-hypergroup. A fuzzy subset f of S is called a fuzzy quasi-Γ-hyperideal of S, if for all λ, μ ∈ (0, 1] and x, y, z, u, v ∈ S, the following conditions hold:

x ≤ y ⇒ f (x) ≥ f (y) .

Theorem 4.8. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S . Then, f is a fuzzy quasi-Γ-hyperideal of S if and only if f satisfies the following conditions:

Proof. Let f be a fuzzy quasi-Γ-hyperideal of S and x, y ∈ S . Then, by Definition 4.7, the condition (1) holds. To prove the condition (2), it suffices to verify that f (x) ≥ (f ∗ 1) (x) ∧ (1 ∗ f) (x) for all x ∈ S . Indeed, if f (x) < (f ∗ 1) (x) ∧ (1 ∗ f) (x) for some x ∈ S, then there exists λ ∈ (0, 1] such that f (x) < λ ≤ (f ∗ 1) (x) ∧ (1 ∗ f) (x) . Since (f ∗ 1) (x) ≥ λ > 0, (1 ∗ f) (x) ≥ λ > 0, there exist y, z, u, v ∈ S such that x ⪯ yΓu and x ⪯ vΓz, and λ ≤ f (y) ∧1 (u) = f (y) , λ ≤ 1 (v) ∧ f (z) = f (z) . Then, y _λ  ∈ f, z _λ  ∈ f and x _λ  ∉ f, which is a contradiction. Hence f (x) ≥ (f ∗ 1) (x) ∧ (1 ∗ f) (x) for all x ∈ S .

Conversely, assume that the conditions (1) and (2) hold. Let x, y, z, u, v ∈ S and λ, μ ∈ (0, 1] be such that x ⪯ yΓu, x ⪯ vΓz and y _λ , z _μ  ∈ f . Then, f (y) ≥ λ, f (z) ≥ μ, and f ( x ) ≥ ( f ∗ 1 ) ( x ) ∧ ( 1 ∗ f ) ( x ) = ( ⋁ ( s , t ) ∈ H x [ f ( s ) ∧ 1 ( t ) ] ) ∧ ( ⋁ ( p , q ) ∈ H x [ 1 ( p ) ∧ f ( q ) ] ) ≥ ( f ( y ) ∧ 1 ( u ) ) ∧ ( 1 ( v ) ∧ f ( z ) ) = f ( y ) ∧ f ( z ) ≥ λ ∧ μ . By the condition (1) and Lemma 2.11, f is a strongly convex fuzzy subset of S . Since f (x) ≥ λ ∧ μ, by Lemma 2.12(3), x_λ∧μ ∈ f . Therefore, f is a fuzzy quasi-Γ-hyperideal of S . □

As we know, the fuzzy Γ-hyperideal is characterized in terms of the fuzzy subset f itself while the fuzzy quasi-Γ-hyperideal in terms of the multiplications f ∗ 1 and 1 ∗ f . A natural question is if a fuzzy quasi-Γ-hyperideal f can be defined in a similar way using only the fuzzy subset f itself. The theorems below give the answer.

Theorem 4.9. Let S be an ordered Γ-semihypergroup. Then, a fuzzy subset f of S is a fuzzy quasi-Γ-hyperideal of S if and only if the following conditions are satisfied:

Proof. ⇒. Let x, y, z, u, v ∈ S be such that x ⪯ yΓu and x ⪯ vΓz . Since f is a fuzzy quasi-Γ-hyperideal of S and x ∈ S, by Theorem 4.8, f (x) ≥ (f ∗ 1) (x) ∧ (1 ∗ f) (x) . Since x ⪯ yΓu, we have (y, u) ∈ H _x , and ( f ∗ 1 ) ( x ) = ⋁ ( s , t ) ∈ H x [ f ( s ) ∧ 1 ( t ) ] ≥ f ( y ) ∧ 1 ( u ) = f ( y ) . By x ⪯ vΓz in a similar way we can get (1 ∗ f) (x) ≥ f (z) . It thus follows that f (x) ≥ f (y) ∧ f (z) .

⇐. Let x ∈ S . Then, f (x) ≥ (f ∗ 1) (x) ∧ (1 ∗ f) (x) for all x ∈ S . Indeed, if H _x  = ∅ , then (f ∗ 1) (x) ∧ (1 ∗ f) (x) =0 ≤ f (x) . Let H _x  ≠ ∅ . Then, ( f ∗ 1 ) ( x ) = ⋁ ( s , t ) ∈ H x [ f ( s ) ∧ 1 ( t ) ] , ( ∗ ) ( 1 ∗ f ) ( x ) = ⋁ ( p , q ) ∈ H x [ 1 ( p ) ∧ f ( q ) ] . We consider the following two cases:

Case 1. If f (x) ≥ (f ∗ 1) (x) , then f (x) ≥ (f ∗ 1) (x) ≥ (f ∗ 1) (x) ∧ (1 ∗ f) (x) .

Case 2. Let f (x) < (f ∗ 1) (x) . Then, by (∗) , there exists (y, u) ∈ H _x such that f (x) < f (y) ∧1 (u) . Since f (y) ∧1 (u) = f (y) , we have f (x) < f (y) . Hence we can show that f (x) ≥1 (v) ∧ f (z) for any (v, z) ∈ H _x  . In fact, since (y, u) ∈ H _x , we have y, u ∈ S and x ⪯ yΓu . Since (v, z) ∈ H _x , we have v, z ∈ S and x ⪯ vΓz . Thus, by hypothesis, we have f ( x ) ≥ f ( y ) ∧ f ( z ) . If f (y) ∧ f (z) = f (y) , then f (x) ≥ f (y) , which contradicts the fact that f (x) < f (y) . Hence we have f ( x ) ≥ f ( z ) ≥ 1 ( v ) ∧ f ( z ) for any (v, z) ∈ H _x  . Then, we have f ( x ) ≥ ⋁ ( v , z ) ∈ H x [ 1 ( v ) ∧ f ( z ) ] = ( 1 ∗ f ) ( x ) ≥ ( f ∗ 1 ) ( x ) ∧ ( 1 ∗ f ) ( x ) , and the proof is completed by Theorem 4.8. □

Theorem 4.10. Let S be an ordered Γ-semi-hypergroup. Then, a fuzzy subset f of S is a fuzzy quasi-Γ-hyperideal of S if and only if the following conditions are satisfied:

Proof. The proof is straightforward verification by Theorem 4.9, and hence we omit the details. □

Then, (S, Γ, ≤) is an ordered Γ-semihypergroup. Let f be a fuzzy subset of S such that f (a) = f (b) =0.4, f (c) = f (d) =0.6 . Then, by Theorem 4.9, we can easily verify that f is a fuzzy quasi-Γ-hyperideal of S .

Proposition 4.12. Let S be an ordered Γ-semihyper- group and f a fuzzy quasi-Γ-hyperideal of S . Then, f is a fuzzy sub Γ-semihypergroup of S .

Proof. Suppose that f is a fuzzy quasi-Γ-hyperideal of S . Let x, y ∈ S . Since z ⪯ xΓy for any z ∈ xΓy, by Theorem 4.9, f (z) ≥ f (x) ∧ f (y) . Thus ⋀ z ∈ x Γ y f ( z ) ≥ f ( x ) ∧ f ( y ) , and f is a fuzzy sub Γ-semihypergroup of S . □

Proposition 4.13. Let S be an ordered Γ-semihyper- group and f, g be a fuzzy right Γ-hyperideal and a fuzzy left Γ-hyperideal of S, respectively. Then, f ∩ g is a fuzzy quasi-Γ-hyperideal of S .

Proof. Assume that f and g are a fuzzy right Γ-hyperideal and a fuzzy left Γ-hyperideal of S, respectively. Then, by Lemma 2.12(2), ((f ∩ g) ∗1) ∩ (1 ∗ (f ∩ g)) ⊆ (f ∗ 1) ∩ (1 ∗ g) ⊆ f ∩ g . Moreover, let x, y ∈ S be such that x ≤ y . Then, (f ∩ g) (x) ≥ (f ∩ g) (y) . In fact, since x ≤ y, by hypothesis, f (x) ≥ f (y) and g (x) ≥ g (y) , and we have (f ∩ g) (x) = f (x) ∧ g (x) ≥ f (y) ∧ g (y) = (f ∩ g) (y) . Therefore, f ∩ g is a fuzzy quasi-Γ-hyperideal of S by Theorem 4.8. □

By Proposition 4.13, we immediately obtain the following corollary:

Corollary 4.14. Let S be an ordered Γ-semihypergroup and f be a fuzzy Γ-hyperideal of S . Then, f is a fuzzy quasi-Γ-hyperideal of S .

The converse of Corollary 4.14 is not true in general. We can illustrate it by the following example:

Example 4.15. Suppose that S is the ordered Γ-semihypergroup of Example 4.11. Let f be a fuzzy subset of S such that f (a) = f (b) =0.4, f (c) = f (d) =0.6 . We have shown that f is a fuzzy quasi-Γ-hyperideal of S . But we claim that f is not a fuzzy Γ-hyperideal of S . Indeed, since bΓc = bγc = {b, d} , we have ⋀ z ∈ b Γ c f ( z ) = f ( b ) ∧ f ( d ) = 0.4 < 0.6 = f ( c ) , i.e., f is not a fuzzy left Γ-hyperideal of S .

Theorem 4.16. Let S be an ordered Γ-semihypergroup. A nonempty subset Q of S is a quasi-Γ-hyperideal of S if and only if the characteristic function f _Q of Q is a fuzzy quasi-Γ-hyperideal of S .

Proof. ⇒. Let Q be a quasi-Γ-hyperideal of S and x, y, z, u, v ∈ S such that x ⪯ yΓu and x ⪯ vΓz . Then, f _Q  (x) ≥ f _Q  (y) ∧ f _Q  (z) . Indeed, if y, z ∈ Q, then, since Q is a quasi-Γ-hyperideal of S, x ∈ (QΓS] ∩ (SΓQ] ⊆ Q, and we have f _Q  (x) =1 = f _Q  (y) ∧ f _Q  (z) . If x ∉ Q or y ∉ Q, then f _Q  (y) ∧ f _Q  (z) =0 . Since x ∈ S, we have f _Q  (x) ≥0 . Thus f _Q  (x) ≥ f _Q  (y) ∧ f _Q  (z) . Let now x, y ∈ S such that x ≤ y . Then, f _Q  (x) ≥ f _Q  (y) . In fact, if y ∉ Q, then f _Q  (y) =0 . Then, f _Q  (x) ≥ f _Q  (y) . If y ∈ Q, then f _Q  (y) =1 . Since x ∈ S, x ≤ y ∈ Q, by hypothesis we have x ∈ Q . Then, f _Q  (x) =1, and again f _Q  (x) ≥ f _Q  (y) . Therefore, f _Q is a fuzzy quasi-Γ-hyperideal of S by Theorem 4.9.

⇐. Assume that f _Q is a fuzzy quasi-Γ-hyperideal of S . Let x ∈ (QΓS] ∩ (SΓQ] . Then, there exist y, z ∈ Q and u, v ∈ S such that x ⪯ yΓu and x ⪯ vΓz . Thus, by Theorem 4.9, f _Q  (x) ≥ f _Q  (y) ∧ f _Q  (z) =1 ∧ 1 =1 . Also, since f _Q  (x) ≤1 for all x ∈ S, we have f _Q  (x) =1, i.e., x ∈ Q . Therefore, (QΓS] ∩ (SΓQ] ⊆ Q . Furthermore, let y ∈ Q, S ∋ x ≤ y . Then, x ∈ Q . Indeed, it is enough to prove that f _Q  (x) =1 . From y ∈ Q, we have f _Q  (y) =1 . Since f _Q is a fuzzy quasi-Γ-hyperideal of S and x ≤ y, we have f _Q  (x) ≥ f _Q  (y) =1 . Since x ∈ S, we have f _Q  (x) ≤1 . Thus f _Q  (x) =1 . Hence Q is a quasi-Γ-hyperideal of S . □

Theorem 4.17. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S. Then, f is a fuzzy quasi-Γ-hyperideal of S if and only if the level subset f _t   (t ∈ (0, 1]) of f is a quasi-Γ-hyperideal of S for f _t  ≠ ∅ .

Proof. Assume that f is a fuzzy quasi-Γ-hyperideal of S . Let t ∈ (0, 1] be such that f _t  ≠ ∅ . To prove that f _t is a quasi-Γ-hyperideal of S, let x ∈ (f _t ΓS] ∩ (SΓf _t ] . Then, x ∈ (f _t ΓS] and x ∈ (SΓf _t ] , and there exist y, z ∈ f _t and u, v ∈ S such that x ⪯ yΓu, x ⪯ vΓz . Then, f (y) ≥ t and f (z) ≥ t . It follows from Theorem 4.9 that f ( x ) ≥ f ( y ) ∧ f ( z ) ≥ t ∧ t = t . Thus x ∈ f _t  . Furthermore, let y ∈ f _t , S ∋ x ≤ y . Then, x ∈ f _t  . Indeed, since y ∈ f _t , f (y) ≥ t, and f is a fuzzy quasi-Γ-hyperideal of S, we have f (x) ≥ f (y) ≥ t, so x ∈ f _t  . Therefore, f _t is a quasi-Γ-hyperideal of S .

Conversely, let x, y, z, u, v ∈ S be such that x ⪯ yΓu and x ⪯ vΓz . Put t = f (y) ∧ f (z) . Then, y, z ∈ f _t  . Hence x ∈ (f _t ΓS] and x ∈ (SΓf _t ] , and x ∈ (f _t ΓS] ∩ (SΓf _t ] . Since f _t is a quasi-Γ-hyperideal of S, we have x ∈ (f _t ΓS] ∩ (SΓf _t ] ⊆ f _t  . So f (x) ≥ t = f (y) ∧ f (z) . Similarly, if x ≤ y, then it can be easily shown that f (x) ≥ f (y) . Therefore, f is a fuzzy quasi-Γ-hyperideal of S by Theorem 4.9. □

Theorem 4.18. Let {f _i  | i ∈ I} be a family of fuzzy quasi-Γ-hyperideals of an ordered Γ-semihypergroup S . Then, f : = ⋂ i ∈ I f i is a fuzzy quasi-Γ-hyperideal of S, where ( ⋂ i ∈ I f i ) ( x ) = ⋀ i ∈ I ( f i ( x ) ) .

Proof. Let x, y, z, u, v ∈ S be such that x ⪯ yΓu and x ⪯ vΓz . Then, since each f _i   (i ∈ I) is a fuzzy quasi-Γ-hyperideal of S, f _i  (x) ≥ f _i  (y) ∧ f _i  (z) , and wehave f ( x ) = ( ⋂ i ∈ I f i ) ( x ) = ⋀ i ∈ I ( f i ( x ) ) ≥ ⋀ i ∈ I ( f i ( y ) ∧ f i ( z ) ) = ( ⋀ i ∈ I ( f i ( y ) ) ) ∧ ( ⋀ i ∈ I ( f i ( z ) ) ) = ( ⋂ i ∈ I f i ) ( y ) ∧ ( ⋂ i ∈ I f i ) ( z ) = f ( y ) ∧ f ( z ) . Furthermore, if x ≤ y, then f (x) ≥ f (y) . Indeed, since every f _i   (i ∈ I) is a fuzzy quasi-Γ-hyperideal of S, it can be obtained that f _i  (x) ≥ f _i  (y) for all i ∈ I . Thus f ( x ) = ( ⋂ i ∈ I f i ) ( x ) = ⋀ i ∈ I ( f i ( x ) ) ≥ ⋀ i ∈ I ( f i ( y ) ) = ( ⋂ i ∈ I f i ) ( y ) = f ( y ) . By Theorem 4.9, f is a fuzzy quasi-Γ-hyperideal of S . □

Definition 4.19. Let S be an ordered Γ-semi-hypergroup and f ∈ F (S) . The smallest fuzzy quasi-Γ-hyperideal of S containing f is called the fuzzy quasi-Γ-hyperideal of S generated by f, denoted by q (f) .

In order to characterize the fuzzy quasi-Γ-hyperideal generated by a fuzzy subset in an ordered Γ-semihyper- group, we need the following lemma.

Lemma 4.20. Let S be an ordered Γ-semihypergroup and f ∈ F (S) . Then, f (x) = sup {k | x ∈ f _k } for any x ∈ S, where f _k   (k ∈ (0, 1]) is the level subset of f .

Proof. Let α = sup {k | x ∈ f _k } . Then, for any ɛ > 0, we have sup { k | x ∈ f k } > α - ɛ , and so there exists t ∈ {k | x ∈ f _k } such that t > α-ɛ . Since x ∈ f _t , we have f (x) ≥ t . Thus f (x) > α - ɛ . By the arbitrariness of ɛ, it follows that f (x) ≥ α . On the other hand, let t = f (x) . Then, x ∈ f _t , and t ∈ {k | x ∈ f _k } . It thus implies that f (x) = t ≤ sup {k | x ∈ f _k } = α . Therefore, we obtain the requested result. □

Theorem 4.21. Let S be an ordered Γ-semihypergroup and f ∈ F (S) . Then, the fuzzy set f^∗ of S defined by

is the fuzzy quasi-Γ-hyperideal q (f) generated by f in S, where Q (f _k )   (k ∈ (0, 1]) is the quasi-Γ-hyperideal of S generated by f _k  .

Proof. In order to prove that f^∗ is the fuzzy quasi-Γ-hyperideal q (f) of S generated by f, we now consider the following three steps:

Case 1. If w ∈ g _{α n} , then g (w) ≥ α _n  .

Case 2. Let w ∈ (g _{α n} ΓS] ∩ (SΓg _{α n} ] . Then, there exist y, z ∈ g _{α n} , u, v ∈ S such that w ⪯ yΓu and w ⪯ vΓz . Thus g (y) ≥ α _n , g (z) ≥ α _n  . Since g is a fuzzy quasi-Γ-hyperideal of S, by Theorem 4.9, g ( w ) ≥ g ( y ) ∧ g ( z ) ≥ α n ∧ α n = α n . Thus, in both cases, we have g (w) ≥ α _n ,  n ∈ Z⁺ . Since g is a fuzzy quasi-Γ-hyperideal of S and x ≤ w, we have g ( x ) ≥ g ( w ) ≥ α n = t - 1 n , n ∈ Z + . By the arbitrariness of n in Z⁺, g (x) ≥ t = f^∗ (x) . Hence f^∗ ⊆ g, and it is shown that f^∗ is the fuzzy quasi-Γ-hyperideal q (f) of S generated by f . □

5 Some types of fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups

In this section, we define and study the completely prime, weakly completely prime and completely semiprime fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups. In addition, we introduce the concept of bi-regular ordered Γ-semihypergroups, and discuss the characterizations of bi-regular ordered Γ-semihypergroups in terms of completely semiprime fuzzy quasi-Γ-hyperideals.

Definition 5.1. A fuzzy quasi-Γ-hyperideal f of an ordered Γ-semihypergroup S is called completely prime if for any two ordered fuzzy points a _λ , b _μ   (λ, μ ∈ (0, 1]) of S, a _λ  ∗ b _μ  ∈ f implies a _λ  ∈ f or b _μ  ∈ f .

Definition 5.2. A fuzzy quasi-Γ-hyperideal f of an ordered Γ-semihypergroup S is called weakly completely prime if for any two ordered fuzzy points a _λ , b _λ   (λ ∈ (0, 1]) of S, a _λ  ∗ b _λ  ∈ f implies a _λ  ∈ f or b _λ  ∈ f .

Definition 5.3. A fuzzy quasi-Γ-hyperideal f of an ordered Γ-semihypergroup S is called completely semiprime if for any ordered fuzzy point a _λ   (λ ∈ (0, 1]) of S, a _λ  ∗ a _λ  ∈ f implies a _λ  ∈ f .

Clearly, every completely prime fuzzy quasi-Γ-hyperideal of S is a weakly completely prime and completely semiprime fuzzy quasi-Γ-hyperideal of S .

Theorem 5.4. Let S be an ordered Γ-semihypergroup. Then, a fuzzy quasi-Γ-hyperideal f of S is completely prime if and only if for any two strongly convex fuzzy subsets g and h of S, g ∗ h ⊆ f implies g ⊆ f or h ⊆ f .

Proof. Let g, h be strongly convex fuzzy subsets of S such that g ∗ h ⊆ f . If gnotsubseteqf, then, by Lemma 2.12(4), there exists an ordered fuzzy point a _λ  ∈ g such that a _λ  ∉ f . For any b _μ  ∈ h, we have a _λ  ∗ b _μ  ∈ g ∗ h ⊆ f . By hypothesis, b _μ  ∈ f . By Lemma 2.12(4), h = ⋃ b μ ∈ h b μ , and thus h ⊆ f .

Conversely, it is evident that every ordered fuzzy point of S is a strongly convex fuzzy subset of S . Thus, by Definition 5.1, the reverse implication holds. □

Theorem 5.5. Let S be an ordered Γ-semihypergroup and f a fuzzy quasi-Γ-hyperideal of S . Then, f is weakly completely prime if and only if ⋀ z ∈ x Γ y f ( z ) ≤ f ( x ) ∨ f ( y ) for any x, y ∈ S .

Proof. Assume that f is a weakly completely prime fuzzy quasi-Γ-hyperideal of S . Let x, y ∈ S . If ⋀ z ∈ x Γ y f ( z ) > f ( x ) ∨ f ( y ) , then there exists λ ∈ (0, 1] such that ⋀ z ∈ x ∘ y f ( z ) ≥ λ > f ( x ) ∨ f ( y ) . Thus, by Lemma 2.12, x λ ∗ y λ = ⋃ z ∈ ( x Γ y ] z λ ∈ f , but x _λ  ∉ f and y _λ  ∉ f, which is impossible. Therefore, ⋀ z ∈ x Γ y f ( z ) ≤ f ( x ) ∨ f ( y ) for all x, y ∈ S .

Conversely, suppose that x _λ , y _λ   (λ ∈ (0, 1]) are the ordered fuzzy points of S such that x _λ  ∗ y _λ  ∈ f . Then, x _λ  ∈ f or y _λ  ∈ f . Indeed, by hypothesis and Lemma 2.12(1), ⋃ z ∈ ( x Γ y ] z λ = x λ ∗ y λ ∈ f . Then, for any z ∈ xΓy ⊆ (xΓy] , z _λ  ∈ f, and f (z) ≥ λ . It thus follows that ⋀ z ∈ x Γ y f ( z ) ≥ λ . Since ⋀ z ∈ x Γ y f ( z ) ≤ f ( x ) ∨ f ( y ) , we have f (x) ∨ f (y) ≥ λ, which implies that f (x) ≥ λ or f (y) ≥ λ . By hypothesis and Lemma 2.11, f is a strongly convex fuzzy subset of S . Thus, by Lemma 2.12(3), x _λ  ∈ f or y _λ  ∈ f . Hence f is weakly completely prime. □

Theorem 5.6. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S. Then, f is a weakly completely prime fuzzy quasi-Γ-hyperideal of S if and only if the level subset f _t   (t ∈ (0, 1]) of f is a prime quasi-Γ-hyperideal of S for f _t  ≠ ∅ .

Proof. Suppose that f is a weakly completely prime fuzzy quasi-Γ-hyperideal of S . By Theorem 4.17, f _t   (t ∈ (0, 1]) is a quasi-Γ-hyperideal of S for f _t  ≠ ∅ . To prove that f _t is prime, let x, y ∈ S such that xΓy ⊆ f _t  . Then, z ∈ f _t for any z ∈ xΓy . It implies that ⋀ z ∈ x Γ y f ( z ) ≥ t . Then, by Theorem 5.5, we have f ( x ) ∨ f ( y ) ≥ ⋀ z ∈ x ∘ y f ( z ) ≥ t , which means that f (x) ≥ t or f (y) ≥ t, and thus x ∈ f _t or y ∈ f _t  . Hence f _t is prime.

Conversely, assume that f _t   (≠ ∅) is a prime quasi-Γ-hyperideal of S . Then, by Theorem 4.17, f is a fuzzy quasi-Γ-hyperideal of S . Now let x, y ∈ S and t = ⋀ z ∈ x Γ y f ( z ) . Then, for any z ∈ xΓy, f (z) ≥ t, and z ∈ f _t  . It implies that xΓy ⊆ f _t  . Since f _t is a prime quasi-Γ-hyperideal of S, we have x ∈ f _t or y ∈ f _t , and thus f (x) ≥ t or f (y) ≥ t . Consequently, f ( x ) ∨ f ( y ) ≥ t = ⋀ z ∈ x Γ y f ( z ) . Therefore, f is weakly completely prime by Theorem 5.5. □

Theorem 5.7. Let S be an ordered Γ-semihypergroup. Then, a fuzzy quasi-Γ-hyperideal f of S is completely semiprime if and only if for any strongly convex fuzzy subset g of S, g ∗ g ⊆ f implies g ⊆ f .

Proof. The proof is similar to that of Theorem 5.4, and hence we omit the details. □

Theorem 5.8. Let f be a fuzzy quasi-Γ-hyperideal of an ordered Γ-semihypergroup S . Then, the following statements are equivalent:

Proof. (1) ⇒ (2) . Assume that f is a completely semiprime fuzzy quasi-Γ-hyperideal of S . Let a ∈ S . Then, f ( a ) ≥ ⋀ x ∈ a Γ a f ( x ) . Indeed, if f ( a ) < ⋀ x ∈ a Γ a f ( x ) , then there exists λ ∈ (0, 1] such that f ( a ) < λ ≤ ⋀ x ∈ a Γ a f ( x ) . Then, for any x ∈ aΓa, f (x) ≥ λ . Since f is strongly convex, by Lemma 2.11 we can easily deduce that f (y) ≥ λ for any y ∈ (aΓa] , and we have y _λ  ∈ f . Thus, by Theorem 2.12(1), a λ ∗ a λ = ⋃ y ∈ ( a Γ a ] y λ ∈ f . Since f is a completely semiprime fuzzy quasi-Γ-hyperideal of S, we have a _λ  ∈ f, and f (a) ≥ λ, which is a contradiction. Therefore, f ( a ) ≥ ⋀ x ∈ a Γ a f ( x ) for any a ∈ S .

(2) ⇒ (3) .  Let a ∈ S . Since f is a fuzzy quasi-Γ-hyperideal of S, by Proposition 4.12, f is a fuzzy sub Γ-semihypergroup of S . Thus ⋀ x ∈ a Γ a f ( x ) ≥ f ( a ) ∧ f ( a ) = f ( a ) , and by (2), f ( a ) = ⋀ x ∈ a Γ a f ( x ) .

(3) ⇒ (1) .  Let f be a fuzzy quasi-Γ-hyperideal of S and a ∈ S . By (3), f ( a ) = ⋀ x ∈ a Γ a f ( x ) . If a _λ  ∗ a _λ  ∈ f, λ ∈ (0, 1] , then ⋃ x ∈ ( a Γ a ] x λ ∈ f . It follows that x _λ  ∈ f for any x ∈ (aΓa] . Thus we have f ( a ) = ⋀ x ∈ a Γ a f ( x ) ≥ ⋀ x ∈ ( a Γ a ] f ( x ) ≥ λ . Then, since f is strongly convex, a _λ  ∈ f . In other words, f is indeed a completely semiprime fuzzy quasi-Γ-hyperideal of S . □

Theorem 5.9. Let Q be a nonempty subset of an ordered Γ-semihypergroup S . Then, Q is a semiprime quasi-Γ-hyperideal of S if and only if the characteristic function f _Q of Q is a completely semiprime fuzzy quasi-Γ-hyperideal of S .

Proof. Let Q be a semiprime quasi-Γ-hyperideal of S . By Theorem 4.16, f _Q is a fuzzy quasi-Γ-hyperideal of S . To prove that f _Q is completely semiprime, it is enough to show that f Q ( a ) ≥ ⋀ x ∈ a Γ a f Q ( x ) for any a ∈ S . In fact, if aΓa ⊆ Q, then, since Q is semiprime, we have a ∈ Q . Thus f Q ( a ) = 1 = ⋀ x ∈ a Γ a f Q ( x ) . If there exists y ∈ aΓa such that y ∉ Q, then we have f Q ( a ) ≥ 0 = ⋀ x ∈ a Γ a f Q ( x ) . Consequently, f Q ( a ) ≥ ⋀ x ∈ a Γ a f Q ( x ) for any a ∈ S . By Theorem 5.8, f _Q is completely semiprime.

Conversely, assume that f _Q is a completely semiprime fuzzy quasi-Γ-hyperideal of S . Then, by Theorem 4.16, Q is a quasi-Γ-hyperideal of S . We claim that Q is semiprime. To prove our claim, let a ∈ S such that aΓa ⊆ Q . Then, x ∈ Q for any x ∈ aΓa . Since f _Q is a completely semiprime fuzzy quasi-Γ-hyperideal of S, by Theorem 5.8 we have f Q ( a ) ≥ ⋀ x ∈ a Γ a f Q ( x ) = 1 . On the other hand, since f _Q is a fuzzy subset of S, we have f _Q  (a) ≤1 for all a ∈ S . Hence f _Q  (a) =1, which implies that a ∈ Q . It thus follows that Q is semiprime. □

Theorem 5.10. Let S be an ordered Γ-semihypergroup and f a fuzzy subset of S. Then, f is a completely semiprime fuzzy quasi-Γ-hyperideal of S if and only if the level subset f _t   (t ∈ (0, 1]) of f is a semiprime quasi-Γ-hyperideal of S for f _t ≠ ∅ .

Proof. The proof is analogous to that of Theorem 5.6, we omit it. □

Example 5.11. Consider the ordered Γ-semihypergroup S given in Example 3.2, and define a fuzzy subset f of S by f (a) = f (b) =0.8, f (c) =0.6, f (d) = f (e) =0.4 . Then, f t = { S , if t ∈ ( 0 , 0.4 ] , { a , b , c } , if t ∈ ( 0.4 , 0.6 ] , { a , b } , if t ∈ ( 0.6 , 0.8 ] , ∅ , if t ∈ ( 0.8 , 1 ] . Thus all nonempty level subsets f _t   (t ∈ (0, 1]) of f are semiprime quasi-Γ-hyperideals of S, and by Theorem 5.10, f is a completely semiprime fuzzy quasi-hyperideal of S . But f is not weakly completely prime. Indeed, by Example 3.2, {a, b} is not a prime quasi-Γ-hyperideal of S, and by Theorem 5.6, f is not a weakly completely prime fuzzy quasi-Γ-hyperideal of S .

Definition 5.12. Let S be an ordered Γ-semihypergroup. S is called right (resp. left) regular if for each a ∈ S there exists x ∈ S such that a ⪯ a²Γx (resp., a ⪯ xΓa²), i.e., a ∈ (a²ΓS] (resp., a ∈ (SΓa²]) . An ordered Γ-semihypergroup (S, ∘ , ≤) is called bi-regular if it is both right regular and left regular.

Clearly, an ordered Γ-semigroup S is bi-regular if and only if a ∈ (a²ΓS] ∩ (SΓa²] for every a ∈ S .

Example 5.13. Let S be the ordered Γ-semihy-pergroup of Example 4.11. Then, S is bi-regular. Indeed, for any x ∈ S, we have x ∈ xΓx = x²Γx² ⊆ (x²ΓS) ∩ (SΓx²) , which implies that x ∈ (x²ΓS] ∩ (SΓx²] .

Now we shall characterize the bi-regular ordered Γ-semihypergroups in terms of completely semiprime fuzzy quasi-Γ-hyperideals.

Theorem 5.14. Let S be an ordered Γ-semihypergroup. Then, the following statements are equivalent:

Proof. (1) ⇒ (2) . Suppose that S is a bi-regular ordered Γ-semihypergroup and f a fuzzy quasi-Γ-hyperideal of S . Let a ∈ S . Then, a ∈ (a²ΓS] and a ∈ (SΓa²] , and there exist x, y ∈ S, u, v ∈ a² = aΓa such that a ⪯ uΓx and a ⪯ yΓv . Thus, by Theorem 4.9, we have f ( a ) ≥ f ( u ) ∧ f ( v ) ≥ ⋀ z ∈ a Γ a f ( z ) . Therefore, f is a completely semiprime fuzzy quasi-Γ-hyperideal of S by Theorem 5.8.