1 Introduction
The formal study of semigroups began in the early 20th century. As a generalization of semigroups, the notion of a Γ-semigroup was first introduced by Sen and Saha [1] in 1986. The notion of an ordered Γ-semigroup was first introduced by Sen and Seth [2] in 1993. Many authors studied various aspects of ordered Γ-semigroups, for instance Chinram and Tinpun [3], Dutta and Adhikari [4], Hila [5], Iampan [6], Kehayopulu [7], Kwon [8], Sen and Seth [2], Tang and Xie [9] and many others. In 2010, Hila [5] investigated some properties of quasi-prime and weakly quasi-prime left ideals in ordered Γsemigroups. In 2013, Tang and Xie [9] studied on ܽ-maximal ideals and Cideals of ordered Γ-semigroups. Recall from [2] that an ordered Γ-semigroup ሺܵ, Γ, ሻ is a Γ-semigroup ሺܵ, Γሻ together with an order relation such that ܾܽ implies that ܽߛܿ ܾߛܿ and ܿߛܽ ܿߛܾ for all ܽ, ܾ, ܿ ∈ ܵ and ߛ∋Γ. In [10-12], Fabrici introduced the concept of the covered ideal, which is a proper ideal ܫ of ܵ satisfying ܫ⊇ܵሺܵ∖ܫሻܵ, and obtained some properties in terms of maximal ideals of ܵ. In [13], Xie introduced covered ideals in ordered semigroups. In 2016, Changphas and Summaprab [14] discussed the structure of ordered semigroups containing covered ideals.
Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The concept of hyperstructure was first introduced by Marty [15] at the eighth Congress of Scandinavian Mathematicians in 1934. Since then, several books have been published on this topic, for example see [16-19]. Ameri and Hoskova [20] studied the fuzzy continuous polygroup as a fuzzy
polygroup with a continuous membership function. In [21], some properties of hyperlattices are studied and the relationship between prime ideals and prime filters in hyperlattices is discussed. Motivation of compatibility of orderings with hyperoperations can be found in [22,23]. In 2010, Davvaz, et al. [24-26] introduced the notion of the Γ-semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup and a generalization of a Γsemigroup. They proved some results in this respect and presented many examples of Γ- semihypergroups. Since then, properties of Γ- semihypergroups were studied by some mathematicians, for example, see [24,27-29]. Anvariyeh, et al., studied the Γ-hyperideals of Γ-semihypergroups in [24]. Many classical notions of semigroups and semihypergroups have been extended to Γsemihypergroups.
Davvaz and Omidi [30] investigated several notions, for example, hyperideals, quasi-hyperideals and bi-hyperideals of an ordered semihyperring. The concept of ordered semihypergroups is a generalization of the concept of ordered semigroups. The concept of ordering hypergroups was introduced by Chvalina [31] as a special class of hypergroups. Many authors studied various aspects of ordered semihypergroups, for instance, Davvaz, et al., [32], Gu and Tang [33], Heidari and Davvaz [23], Tang, et al. [34], and many others. Explicit study of ordered semihypergroups seems to have begun with Heidari and Davvaz [23] in 2011. We have noticed that the relationships between ordered semigroups and ordered semihypergroups have been already considered by Davvaz, et al. [32]. In [33], the authors answered to the open problem given by Davvaz, et al. [32].
In this paper, we give some properties of Γ-hyperideals and C-Γ-hyperideals of ordered Γ-semihypergroups. We make a connection between proper Γhyperideals and C-Γ-hyperideals. Moreover, we prove that for a Γ-hyperideal ܫ of a regular ordered Γ-semihypergroup ሺܵ, Γ, ሻ, every C-Γ-hyperideal ܬ of ܫ is also a C-Γ-hyperideal of ܵ.
2 Basic Definitions
In the following, we recall the notion of an ordered Γ-semihypergroup and then we present some definitions and properties that we will need in this paper. Throughout this paper, unless otherwise stated, ܵ is always an ordered Γsemihypergroup ሺܵ, Γ, ሻ. One can see that ordered Γ-semihypergroups are generalizations of ordered semihypergroups.
Definition 2.1 [35] An algebraic hyperstructure ሺܵ, Γ, ሻ is said to be an ordered Γ-semihypergroup if ሺܵ, Γሻ is a Γ-semihypergroup and ሺܵ, ሻ is a partially ordered set such that : for any ݔ, ݕ, ݖ ∋ ܵ and ߛ∋Γ, ݔݕ implies
exists there ܣ∋ܽ any for means ܤ≽ܣ ,Here .ݖߛݕ ≽ ݖߛݔ and ݕߛݖ ≽ ݔߛݖ ܾ∈ܤ such that ܾܽ , for all non-empty subsets ܣ and ܤ of ܵ.
Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. By a sub Γ-semihypergroup of ܵ we mean a non-empty subset ܣ of ܵ such that ܽߛܾ ⊇ ܣ for all ܽ, ܾ ∈ ܣ and ߛ∋Γ. Let ܭ be a non-empty subset of ܵ. If ܪ is a non-empty subset of ܭ, then we define:
\[(H)_K := \{k \in K \mid k \le h \text{ for some } h \in H\}.\]
Note that if ܭൌܵ, then we define:
\[(H] = \{x \in S \mid x \le h \text{ for some } h \in H\}.\]
Example 1. [35] Let ሺܵ,∘, ሻ be an ordered semihypergroup and Γ a non-empty set. We define ݕߛݔ ൌ ݔ ∘ ݕ for every ݔ, ݕ ∋ ܵ and ߛ∋Γ. Then ሺܵ, Γ, ሻ is an ordered Γ-semihypergroup.
Definition 2.2 [35] Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. A non-empty subset ܫ of ܵ is called a left (resp. right) Γ-hyperideal of ܵ if it satisfies the following conditions:
- ;ሻܫ ⊇ ܵΓܫ .ݏ݁ݎሺ ܫ ⊇ ܫΓܵ 1.
- . ܫ ∋ ݕ that imply , ݔݕ that such ܵ∈ݕ and ܫ∋ݔ When 2.
Equivalent Definition:
- ;(ܫ ⊇ ܵΓܫ resp. ) ܫ ⊇ ܫΓܵ 1.
- .ܫ ⊇ ሿܫሺ 2.
By two-sided Γ-hyperideal or simply Γ-hyperideal, we mean a non-empty subset ܫ of ܵ that is both a right and a left Γ-hyperideal of ܵ. A Γ-hyperideal ܫ of ܵ is said to be proper if ܫ്ܵ.
3 Main Results
An ordered Γ-semihypergroup ሺܵ, Γ, ሻ is said to be simple if it has no proper Γhyperideals, i.e., for any Γ-hyperideal ܫ് ∅of ܵ, we have ܫൌܵ. A Γhyperideal ܫ്ܵ of ܵ is said to be maximal if for any proper Γ-hyperideal ܬ of ܵ, ܫ ⊇ ܬ implies that ܫൌܬ. A Γ-hyperideal ܫ of ܵ is called a minimal Γhyperideal of ܵ if there is no Γ-hyperideal ܬ of ܵ such that ܬ⊃ܫ. The concept of C-ideals in ordered Γ-semigroups was introduced by Tang and Xie [9]. As a continuation of Tang and Xie's works on ordered Γ-semigroups, the aim of this section is to study and characterize Γ-hyperideals and C-Γ-hyperideals of an ordered Γ-semihypergroup.
Lemma 3.1 Let \((S, \Gamma, \leq)\) be an ordered Γ-semihypergroup. If A and B are non-empty subsets of S, then we have:
- 1. \(A \subseteq (A]\);
- 2. If \(A \subseteq B \subseteq S\), then \((A] \subseteq (B]\);
- 3. ((A)] = (A);
- 4. \((A]\Gamma(B] \subseteq (A\Gamma B]\);
- 5. \(((A|\Gamma(B)] = (A\Gamma B)\).
Proof. The proof is straightforward.
Let a be an element of an ordered \(\Gamma\)-semihypergroup \((S, \Gamma, \leq)\). We denote by \(L_S(a)\) (resp. \(R_S(a), I_S(a)\)) the left (resp. right, two-sided) \(\Gamma\)-hyperideal of S generated by a.
Lemma 3.2 Let a be an element of an ordered \(\Gamma\)-semihypergroup \((S, \Gamma, \leq)\). Then,
- 1. \(L_{S}(a) = (a \cup S\Gamma a);\)
- 2. \(R_{S}(a) = (a \cup a \Gamma S);\)
- 3. \(I_S(a) = (a \cup S\Gamma a \cup a\Gamma S \cup S\Gamma a\Gamma S)\).
Proof. The proof is straightforward.
We continue this section with the following definition.
Definition 3.3 An element a of an ordered \(\Gamma\)-semihypergroup \((S, \Gamma, \leq)\) is regular if there exist \(x \in S\) and \(\alpha, \beta \in \Gamma\) such that \(\alpha \leq a\alpha x\beta a\). An ordered \(\Gamma\)-semihypergroup S is said to be regular if every element of S is regular.
Lemma 3.4 Let \((S, \Gamma, \leq)\) be an ordered Γ-semihypergroup. Then, the following statements are equivalent:
- 1. S is regular.
- 2. \(a \in (a\Gamma S\Gamma a)\) for every \(a \in S\).
- 3. \(A \subseteq (A \Gamma S \Gamma A)\) for every \(A \subseteq S\).
Proof. The proof is straightforward.
Theorem 3.5 Let \((S, \Gamma, \leq)\) be a regular ordered \(\Gamma\)-semihypergroup. If I is a one-sided \(\Gamma\)-hyperideal of S, then \((I\Gamma I] = I\).
Proof. Let I be a left \(\Gamma\)-hyperideal of S. Then, \((I\Gamma I] \subseteq (S\Gamma I) \subseteq (I] = I\). Now, let \(a \in I\). Since S is regular, there exist \(x \in S\) and \(\alpha, \beta \in \Gamma\) such that \(a \leq I\).
ܽߚݔߙܽ. As ܫ is a left Γ-hyperideal of ܵ, we have ߚݔܽ ⊇ ܫ. Hence, ܽ ܽߚݔߙܽ ⊇ ܫΓܫ. This means that ܽ ∈ ሺܫΓܫሿ, and so ܫ ⊇ ሺܫΓܫሿ. Hence, the theorem is proved.
Let ܵ be a Γ-semihypergroup and ߛ∋Γ. We define ݔ ∘ ݕ ൌ ݕߛݔ for every ݔ, ݕ ∋ ܵ. Then ሺܵ,∘ሻ is a semihypergroup and we denote it by ܵఊ. A Γsemihypergroup ܵ is called a ߁-hypergroup if ሺܵఊ, ߛሻ is a hypergroup for every ߛ∋Γ. An ordered Γ-semihypergroup ሺܵ, Γ, ሻ is called an ordered Γhypergroup if for every ߛ∋Γ, ሺܵ, ߛሻ is a hypergroup. In fact, for all ܽ∈ܵ and ߛ∋Γ, we have ܽߛܵ ൌ ܵߛܽ ൌ ܵ.
Theorem 3.6 Every ordered Γ-hypergroup is a regular ordered Γsemihypergroup.
Proof. Let ሺܵ, Γ, ሻ be an ordered Γ-hypergroup and ܽ∈ܵ. Then ሺܽߛܵሿ ൌ ሺܵߛܽሿ ൌ ܵ for all ߛ∋Γ. Hence, ܽ ܽݔߛ for some ݔ∋ܵ. Again, ݔ ߛݕܽ for some ݕ∋ܵ. Then, ܽ ܽݔߛ ܽߛሺߛݕܽሻ which implies that ܽ ∈ ሺܽΓܵΓܽሿ for every ܽ∈ܵ. Therefore, ܵ is a regular ordered Γ-semihypergroup.
Definition 3.7 A proper Γ-hyperideal ܫ of an ordered Γ-semihypergroup ሺܵ, Γ, ሻ is called a covered Γ-hyperideal (simply C-Γ-hyperideal) of ܵ if it satisfies ܫ ⊇ ሺܵΓሺܵ∖ܫሻΓܵሿ.
Note that ܵ itself is not a C-Γ-hyperideal.
Example 2. Let ܵ ൌ ሼܽ, ܾ, ܿ, ݀ሽ and Γ ൌ ሼߛ, ߚሽ be the sets of binary hyperoperations defined as follows:
| γ | ܽ | ܾ | ܿ | ݀ | β | ܽ | ܾ | ܿ | ݀ |
|---|---|---|---|---|---|---|---|---|---|
| ܽ | ܽ | ሼܾ, ݀ሽ | ܿ | ݀ | ܽ | ሼܽ, ܿሽ | ሼܾ, ݀ሽ | ሼܽ, ܿሽ | ݀ |
| ܾ | ሼܾ, ݀ሽ | ܾ | ሼܾ, ݀ሽ | ݀ | ܾ | ሼܾ, ݀ሽ | ܾ | ሼܾ, ݀ሽ | ݀ |
| ܿ | ܿ | ሼb,dሽ | ܽ | ݀ | ܿ | ሼܽ, ܿሽ | ሼܾ, ݀ሽ | ሼܽ, ܿሽ | ݀ |
| ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ |
| γ | ܽ | ܾ | ܿ | ݀ | β | ܽ | ܾ | ܿ | ݀ |
|---|---|---|---|---|---|---|---|---|---|
| ܽ | ܽ | ሼܾ, ݀ሽ | ܿ | ݀ | ܽ | ሼܽ, ܿሽ | ሼܾ, ݀ሽ | ሼܽ, ܿሽ | ݀ |
| ܾ | ሼܾ, ݀ሽ | ܾ | ሼܾ, ݀ሽ | ݀ | ܾ | ሼܾ, ݀ሽ | ܾ | ሼܾ, ݀ሽ | ݀ |
| ܿ | ܿ | ሼb,dሽ | ܽ | ݀ | ܿ | ሼܽ, ܿሽ | ሼܾ, ݀ሽ | ሼܽ, ܿሽ | ݀ |
| ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ | ݀ |
Then ܵ is a Γ-semihypergroup [29]. We have ሺܵ, Γ, ሻ is an ordered Γ– semihypergroup where the order relation is defined by:
\[\leq := \{(a, a), (b, b), (b, d), (c, c), (d, d)\}.\]
The covering relation and the figure of ܵ are given by :
\[\prec = \{(b,d)\}.\]
It is a routine matter to verify that ܫ ൌ ሼܾ, ݀ሽ is a C-Γ-hyperideal of ܵ.
Lemma 3.8 If ܫ, ܬ are two C-Γ-hyperideals of an ordered Γ-semihypergroup ሺܵ, Γ, ሻ, then ܫ∩ܬ is also a C-Γ-hyperideal of ܵ.
⊇ ݕߛݔ and ܫ ⊇ ܵΓܫ ⊇ ܬΓܫ ⊇ ݕߛݔ ,Then .Γ∈ߛ and ܬ∋ݕ ,ܫ∋ݔ Let .Proof :have We ܵ.⊆ܬ∩ܫ്∅ hence and ܬ ∩ ܫ ⊇ ݕߛݔ ,So .ܬ ⊇ ܬΓܵ ⊆ ܬΓܫ
\[(I \cap J)\Gamma S \subseteq I\Gamma S \subseteq I\] and \(S\Gamma(I \cap J) \subseteq S\Gamma J \subseteq J\).
Similarly, ሺܫ ∩ ܬሻΓܵ ⊆ ܬ and ܵΓሺܫ ∩ ܬሻ ⊆ ܫ. So, we have ሺܫ ∩ ܬሻΓܵ ⊆ ܫ ∩ ܬ and a is ܬ∩ܫ ,Hence .ܬ∩ܫ∋ݕ ,So .ܬ∋ݕ and ܫ∋ݕ obtain we ܵ, of hyperideals-Γ are ܬ and ܫ Since .ݔݕ and ܵ∈ݕ ,ܬ∩ܫ∋ݔ let ,Now .ܬ ∩ ܫ ⊇ ሻܬ ∩ ܫΓሺܵ proper Γ-hyperideal of ܵ. By hypothesis, we have:
\[I \cap J \subseteq I \subseteq (S\Gamma(S \setminus I)\Gamma S] \subseteq (S\Gamma(S \setminus (I \cap J))\Gamma S].\]
Hence, ܫ∩ܬ is a C-Γ-hyperideal of ܵ.
Theorem 3.9 Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. If ܵ has a C-Γhyperideal, then every C-Γ-hyperideal of ܵ is minimal if and only if the intersection of any two different C-Γ-hyperideals is empty.
Proof. Suppose ܫ and ܬ are two different C-Γ-hyperideals of ܵ. Then ܫ and ܬ are minimal. If ܫ∩ܬ്∅, then, by Lemma 3.8, ܫ∩ܬ is a C-Γ-hyperideal of ܵ. As ܫ and ܬ are minimal, we obtain ܫൌܬ. This is a contradiction. Hence, ܫ∩ܬൌ∅. The converse is clear.
Theorem 3.10 Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. If ܵ is not simple, then ܵ contains at least one C-Γ-hyperideal of ܵ.
Proof. Let ܣ be any proper Γ-hyperideal of ܵ. We have:
\[S\Gamma(S\Gamma(S \setminus A)\Gamma S] = (S]\Gamma(S\Gamma(S \setminus A)\Gamma S]\] \[\subseteq (S\Gamma(S\Gamma(S \setminus A)\Gamma S)]\] \[= ((S\Gamma S)\Gamma(S \setminus A)\Gamma S]\] \[\subseteq (S\Gamma(S \setminus A)\Gamma S].\]
Similarly, we have ሺܵΓሺܵ∖ܣሻΓܵሿΓܵ ⊆ ሺܵΓሺܵ∖ܣሻΓܵሿ. Now, suppose that ݔ ∋ ሺܵΓሺܵ∖ܣሻΓܵሿ and ݕ∋ܵ such that ݕݔ. Since ݔ ∋ ሺܵΓሺܵ∖ܣሻΓܵሿ, it follows that ݔݑ for some ݑ ∋ ܵΓሺܵ∖ܣሻΓܵ. Since ݕݔ and ݔݑ, we get ݕݑ. So, we have ݕ ∋ ሺܵΓሺܵ∖ܣሻΓܵሿ. Hence, ሺܵΓሺܵ∖ܣሻΓܵሿ is a Γhyperideal of ܵ. Consider ܫ ൌ ܣ ∩ ሺܵΓሺܵ∖ܣሻΓܵሿ. By the proof of Lemma 3.8, ܫ is a Γ-hyperideal of ܵ. Next, we show that ܫ is a C-Γ-hyperideal of ܵ. We have:
\[I \subseteq (S\Gamma(S \setminus A)\Gamma S] \subseteq (S\Gamma(S \setminus I)\Gamma S].\]
Therefore, we conclude that ܫ is a C-Γ-hyperideal of ܵ.
Example 3. Let ܵ ൌ ሼܽ, ܾ, ܿሽ and Γ ൌ ሼߛ, ߚሽ be the sets of binary hyperoperations defined as follows:
| γ | ܽ | ܾ | ܿ | β | ܽ | ܾ | ܿ |
|---|---|---|---|---|---|---|---|
| ܽ | ሼܽ, ܾሽ | ሼܾ, ݀ሽ | ܿ | ܽ | ܽ | ሼܽ, ܾሽ | ܿ |
| ܾ | ሼܽ, ܾሽ | ܾ | ܿ | ܾ | ሼܽ, ܾሽ | ሼܽ, ܾሽ | ܿ |
| ܿ | ܿ | c | ܿ | ܿ | ܿ | ܿ | ܿ |
| β | а | b | С |
|---|---|---|---|
| а | а | {a, b} | С |
| b | {a, b} | \(\{a,b\}\) | С |
| С | С | С | С |
Then ܵ is a Γ-semihypergroup. We have ሺܵ, Γ, ሻ is an ordered Γ– semihypergroup, where the order relation is defined by:
\[\leq := \{(a,a),(a,b),(b,b),(c,c)\}.\]
The covering relation and the figure of ܵ are given by :
\[\prec = \{(a, b)\}.\]
Since ܫ ൌ ሼܿሽ is a Γ-hyperideal of ܵ, it follows that ܵ is not simple. Since ܿ ൌ ܿߛܾߚܿ, ܾ ∋ ܫ, we have ܫ ⊇ ሺܵΓሺܵ∖ܫሻΓܵሿ. Hence, ܫ is a C-Γ-hyperideal of ܵ.
Theorem 3.11 Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. If ܵ is not simple, then every C-Γ-hyperideal of ܵ is minimal if and only if the intersection of any two different C-Γ-hyperideals is empty.
Proof. It follows immediately from Theorems 3.9 and 3.10.
Theorem 3.12 Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. If ܵ contains two different maximal Γ-hyperideals ܫ, ܬ, then none of them is a C-Γ-hyperideal of ܵ.
Proof. Let ܫ and ܬ be two different maximal Γ-hyperideals of ܵ. Then, ܫ∪ܬൌ ܵ. Hence, ܵ∖ܫ⊇ܬ and ܵ∖ܬ⊇ܫ. If ܫ is a C-Γ-hyperideal of ܵ, then ܫ⊇ ሺܵΓሺܵ∖ܫሻΓܵሿ ⊆ ሺܵΓܬΓܵሿ ⊆ ሺܬሿ ൌ ܬ. Since ܫ∪ܬൌܵ, it follows that ܬൌܵ, which is a contradiction. Therefore, ܫ is not a C-Γ-hyperideal of ܵ. If ܬ is a C-Γhyperideal of ܵ, then ܬ ⊇ ሺܵΓሺܵ∖ܬሻΓܵሿ ⊆ ሺܵΓܫΓܵሿ ⊆ ሺܫሿ ൌ ܫ. Since ܫ∪ܬൌܵ, it follows that ܫൌܵ, which is a contradiction. Therefore, ܬ is not a C-Γhyperideal of ܵ.
Theorem 3.13 Let ሺܵ, Γ, ሻ be an ordered Γ-semihypergroup. If ܫ, ܬ are two C-Γ-hyperideals of ܵ, then ܫ∪ܬ is a C-Γ-hyperideal of ܵ.
Proof. Clearly, ܫ∪ܬ്∅. We have ሺܫ∪ܬሻΓܵ ൌ ܫΓܵ ∪ ܬΓܵ ⊆ ܫ ∪ ܬ. So, for each let ,Now .ܬ ∪ ܫ ⊇ ܽߛݔ ,Similarly .ܬ ∪ ܫ ⊇ ݔߛܽ ,Γ∈ߛ and ܵ∈ݔ ,ܬ∪ܫ∋ܽ ܽ∈ܫ∪ܬ, ݕ∋ܵ and ݕܽ. Since ܫ, ܬ are Γ-hyperideals of ܵ, we obtain ݕ∋ܫ∪ܬ. Hence, ܫ∪ܬ is a Γ-hyperideal of ܵ. Let ܫ, ܬ be two C-Γ-hyperideals of ܵ; we need to prove that ܫ∪ܬ ⊇ሺܵΓሺܵ ∖ ሺܫ ∪ ܬሻሻΓܵሿ. By hypothesis, we have:
\[I \subseteq (S\Gamma(S \setminus I)\Gamma S]\] and \(J \subseteq (S\Gamma(S \setminus J)\Gamma S]\).
If ܽ∈ܫ, then there exists ݔ∋ܵ∖ܫ such that ܽ ∈ ሺܵΓݔΓܵሿ. Now, one of two following cases happens:
Case 1. If ݔ ∋ ܵ ∖ ሺܫ ∪ ܬሻ, then ܽ ∈ ሺܵΓሺܵ ∖ ሺܫ ∪ ܬሻሻΓܵሿ.
Case 2. If ݔ ∋ ሺܵ ∖ ܫሻ ∩ ܬ, then ݔ∋ܬ. Thus, there exists ܾ∈ܵ∖ܬ such that ݔ ∋ ሺܵΓܾΓܵሿ. If ܾ∈ܫ, then ݔ ∋ ሺܵΓܾΓܵሿ ⊆ ሺܵΓܫΓܵሿ ⊆ ሺܫሿ ൌ ܫ. This is a contradiction. This means that ܾ∈ܵ∖ܫ, and so ܾ ∈ ሺܵ∖ܫሻ ∩ ሺܵ∖ܬሻ ൌܵ∖ ሺܫ ∪ ܬሻ. We have:
\[a \in (S\Gamma x\Gamma S] \subseteq (S\Gamma (S\Gamma b\Gamma S)\Gamma S]\]\[\subseteq (S\Gamma b\Gamma S)\]\[\subseteq (S\Gamma (S \setminus (I \cup J))\Gamma S].\]
Therefore, ܫ ⊇ ൫SΓ൫ܵ ∖ ሺܫ∪ܬሻ൯ΓS൧. Similarly, we have ܬ ⊇ ൫SΓ൫ܵ ∖ ሺܫ∪ܬሻ൯ΓS൧. Hence the proof is completed.
We now describe the relationship between the proper Γ-hyperideal and the C-Γhyperideal.
Theorem 3.14 Let ሺܵ, Γ, ሻ be a regular ordered Γ-semihypergroup. If for any proper Γ-hyperideal ܫ and for every ܫௌሺܽሻ ⊆ ܫ, there exists ܾ∈ܵ∖ܫ such that ܫௌሺܽሻ ⊆ ܫௌሺܾሻ, then every proper Γ-hyperideal of ܵ is a C-Γ-hyperideal of ܵ.
Proof. Let ܵ be regular. Clearly, ሺܵΓܵሿ ⊆ ܵ. Since ܵ is regular, we have ܵ ⊆ ሺܵΓܵΓܵሿ ⊆ ሺܵΓܵሿ. Thus, we have ܵ ൌ ሺܵΓܵሿ. Let ܫ be any proper Γhyperideal of ܵ and ܽ∈ܫ. Then ܫௌሺܽሻ ⊆ ܫ. So, there exists ܾ∈ܵ∖ܫ such that ܫௌሺܽሻ ⊆ ܫௌሺܾሻ ⊆ ܵ. Since ܵ ൌ ሺܵΓܵሿ, it follows that ܵ ൌ ሺܵΓܵΓܵሿ. Thus, ⊇ ݕߚݑߙݔ ܾ then ,ܫ∋ݑ If .Γ ∈ ߚ ,ߙ and ܵ ∈ ݑ ,ݕ ,ݔ some for ݕߚݑߙݔ ܾ ܵΓܫΓܵ, which is a contradiction. This leads to ݑ∋ܫ. We have:
\[a \in I_S(a) \subseteq I_S(b) \subseteq (S\Gamma u\Gamma S] \subseteq (S\Gamma (S \setminus I)\Gamma S].\]
Hence, ܫ ⊇ ሺܵΓሺܵ ∖ ܫሻΓܵሿ. This completes the proof.
We now give the main result of this paper as below.
Proposition 3.15 Let ܫ be a ߁-hyperideal of a regular ordered Γsemihypergroup ሺܵ, Γ, ሻ. Then, any C-Γ-hyperideal ܬ of ܫ is a C-Γ-hyperideal of ܵ.
Proof. Since ܫ is a ߁-hyperideal of ܵ, it follows that ܫ is a sub Γsemihypergroup of ܵ. Let ܽ∈ܫ⊇ܵ. Then, there exist ݔ∋ܵ and ߙ, ߚ ∋ Γ such that ܽ ܽߚݔߙܽ ܽߚݔߙሺܽߚݔߙܽሻ ൌ ܽߙሺߚݔܽݔߙሻߚܽ. Since ܫ is a Γ-hyperideal of ܵ, it follows that ߚݔܽݔߙ ⊇ ܵΓܫΓܵ ⊆ ܫ. Hence, ܽݑ for some ݑ∋ ܽߙሺߚݔܽݔߙሻߚܽ ⊇ ܽΓܫΓܽ. This means that ܽ ∈ ሺܽΓܫΓܽሿூ. Hence, ܫ is a regular sub Γ-semihypergroup of ܵ.
Let ܽ∈ܬ⊇ܫ and ݏ∋ܵ. Then, ܽݏߛ ⊇ ܫ where ߛ∋Γ. Now, suppose that ⊇ ݒߤݕߣݒ ݒ that such Γ ∈ ߤ ,ߣ and ܫ∋ݕ exist there ,Then .ܫ ⊇ ݏߛܽ ∋ ݒ ሺܽݏߛሻߤݕߣሺܽݏߛሻ ⊆ ܬΓሺܵΓܫΓܫሻΓܵ ⊆ ܬΓܫΓܵ ⊆ ܬΓܫ ⊇ ܬ. Since ܬ is a Γ-hyperideal of ܫ, it follows that ܽݏߛ ⊇ ܬ. Similarly, we have ߛݏܽ ⊇ ܬ. If ݁∈ܬ⊇ܫ and ݂∈ܵ such that ݂݁, then ݂∈ܫ. Since ܬ is a Γ-hyperideal of ܵ, it follows that ݂∈ܬ. Hence, ܬ is a Γ-hyperideal of ܵ. By hypothesis, we have ܬ ⊇ ሺܵΓሺܫ ∖ ܬሻΓܵሿ. As :obtain we ,ܬ∖ܵ⊇ܬ∖ܫ്∅
\[J \subseteq (S\Gamma(I \setminus J)\Gamma S] \subseteq (S\Gamma(S \setminus J)\Gamma S].\]
Therefore, ܬ is a C-Γ-hyperideal of ܵ.
4 Conclusion
In this paper, we introduced and studied some properties of C-Γ-hyperideals of ordered Γ-semihypergroups. In particular, we described a connection between
proper Γ-hyperideals and C-Γ-hyperideals. Furthermore, we proved that in regular ordered Γ-semihypergroups, every C-Γ-hyperideal of a Γ-hyperideal is also a C-Γ-hyperideal. The results of this paper can also be used on ordered semihypergroups by some moderate modifications. We hope that this work will offer the foundation for further study of the theory on ordered Γsemihypergroups.
Acknowledgements
The authors wish to thank the anonymous reviewers for their specific and useful comments.