Bulletin of
Mathematics
c
SEAMS. 2018
Some Properties of Quasi-Γ-Hyperideals and
Hyperfilters in Ordered Γ-Semihypergroups
Saber Omidi
and
Bijan Davvaz
Department of Mathematics, Yazd University, Yazd Iran Email: omidi.saber@yahoo.com; davvaz@yazd.ac.ir
Cihat Abdioˆ
glu
Department of Primary Education, Karamanoˆglu Mehmetbey University, Yunus Emre Campus, 70100, Karaman, Turkey
Email: cabdioglu@kmu.edu.tr
Dedicated to the 75th birthday of Prof. K.P. Shum
Received 22 March 2016 Accepted 20 August 2016
Communicated by Nguyen Van Sabh
AMS Mathematics Subject Classification(2000): 16Y99
Abstract. In this paper, we give some characterizations of the regularity of an ordered Γ-semihypergroup in terms of left, right and quasi-Γ-hyperideals. Also, we define and use the notion of hyperfilters in ordered Γ-semihypergroups to examine some classical results and properties in ordered Γ-semihypergroups. Moreover, by using the notion of pseudoorder, we obtain an ordered Γ-semigroup from an ordered Γ-semihypergroup. Keywords: Ordered Γ-semihypergroup; Quasi-Γ-hyperideal; Hyperfilter; Pseudoorder.
1. Introduction
Recently, Davvaz et al. [3, 14, 21, 19, 23] introduced the notion of Γ-semihyper-group as a generalization of a semiΓ-semihyper-group, a generalization of a semihyperΓ-semihyper-group and a generalization of a Γ-semigroup. They proved some results in this respect and presented many exmaples of Γ-semihypergroups. Many classical notions of semigroups and semihypergroups have been extended to Γ-semihypergroups. The notion of a Γ-hyperideal of a Γ-semihypergroup was introduced in [3].
Davvaz et al. [4] introduced the notion of Pawlak’s approximations in Γ-semi-hypergroups. Abdullah et al. [2] studied M -hypersystems and N -hypersystems in a Γ-semihypergroup.
In 1986, Sen and Saha [36] introduced the notion of a Γ-semigroup as a gener-alization of a semigroup as well as of a ternary semigroup. Since then, hundreds of papers have been written on this topic, see [7, 16]. Chinram and Tinpun stud-ied the isomorphism theorems of Γ-semigroups in [7]. Dutta and Adhikari [16] introduced the concept of m-systems in Γ-semigroups. Many classical notions of semigroups have been extended to Γ-semigroups. Let S = {a, b, c, · · · } and Γ = {α, β, γ, · · · } be two non-empty sets. Then, S is called a Γ-semigroup if there exists a mapping from S × Γ × S to S, written as (a, γ, b) → aγb, satisfying the identity (aαb)βc = aα(bβc) for all a, b, c in S and α, β in Γ. In this case by (S, Γ) we mean S is a Γ-semigroup. The notions of an ordered Γ-groupoid and an ordered Γ-semigroup were defined by Sen and Seth in [37]. Several au-thors, especially, Abbasi and Basar [1], Chinram and Tinpun [7, 8], Dutta and Adhikari [16, 17], Hila [22], Kehayopulu [25] and Kwon and Lee [31, 32, 33], stud-ied different aspects of ordered Γ-semigroups. Recall from [37], that an ordered Γ-semigroup (S, Γ, ≤) is a Γ-semigroup (S, Γ) together with an order relation ≤ such that a ≤ b implies that aγc ≤ bγc and cγa ≤ cγb for all a, b, c ∈ S and γ ∈ Γ.
A semigroup is an algebraic structure consisting of a non-empty set S to-gether with an associative binary operation [24]. An ordered semigroup is a structure (S, ·, ≤) satisfying the following axioms:
(1) S is a semigroup with respect to the multiplication ·; (2) S is a partial ordered set by ≤;
(3) If a and b are elements of S such that a ≤ b, then a·x ≤ b ·x and x·a ≤ x·b for all x ∈ S.
Ordered semigroups have been studied extensively by Kehayopulu and Tsin-gelis, for example, see [26, 27, 28, 29]. In [28, 29], Kehayopulu and Tsingelis introduced the concept of pseudoorder of ordered semigroups. In 2004, Ke-hayopulu [26] introduced the concept of m-systems and n-systems in ordered semigroups.
The connection between ordered semihypergroups and ordered semigroups have been analyzed by Davvaz et al. [12]. The concept of ordered semihyper-groups is a generalization of the concept of ordered semisemihyper-groups. The concept of ordering hypergroups investigated by Chvalina [9] as a special class of hyper-groups. Many authors studied different aspects of ordered semihypergroups, for instance, Changphas and Davvaz [6], Davvaz et al. [12], Gu and Tang [18], Hei-dari and Davvaz [20], Tang et al. [39], and many others. Recall from [20], that an ordered semihypergroup (S, ◦, ≤) is a semihypergroup (S, ◦) together with a partial order ≤ that is compatible with the hyperoperation ◦, meaning that for any x, y, z in S,
Here, if A and B are two non-empty subsets of S, then we say that A ≤ B if for every a ∈ A there exists b ∈ B such that a ≤ b.
Algebraic hyperstructures are a suitable generalization of classical algebraic structures. The concept of hyperstructure was first introduced by Marty [34] at the eighth Congress of Scandinavian Mathematicians in 1934. A comprehensive review of the theory of hyperstructures can be found in [10, 11, 13, 40]. Let S be a non-empty set and P∗(S) be the family of all non-empty subsets of S. A
mapping ◦ : S × S → P∗(S) is called a hyperoperation on S. A hypergroupoid is
a set S together with a (binary) hyperoperation. In the above definition, if A and B are two non-empty subsets of S and x ∈ S, then we denote
A ◦ B = S
a∈A b∈B
a ◦ b, x ◦ A = {x} ◦ A and B ◦ x = B ◦ {x}.
A hypergroupoid (S, ◦) is called a semihypergroup if for every x, y, z in S, x ◦ (y ◦ z) = (x ◦ y) ◦ z. That is, S u∈y◦z x ◦ u = S v∈x◦y v ◦ z.
A non-empty subset K of a semihypergroup S is called a subsemihypergroup of S if K ◦ K ⊆ K. A hypergroupoid (S, ◦) is called a quasihypergroup if for every x ∈ S, x ◦ S = S = S ◦ x. This condition is called the reproduction axiom. The couple (S, ◦) is called a hypergroup if it is a semihypergroup and a quasihypergroup. A hypergroup is an algebraic structure similar to a group except that ◦ is multivalued. A hypergroup (S, ◦) is called commutative if x◦y = y ◦ x, for every x, y ∈ S.
The purpose of this paper is twofold: On the one hand, we intend to study the nonion of quasi-Γ-hyperideals in ordered Γ-semihypergroups. On the other hand, we will focus on some properties of hyperfilters, M -hypersystems and pseudoorders in ordered Γ-semihypergroups.
2. Preliminaries
In 2015, Davvaz and Omidi initiated the study of ordered semihyperring [15] as a generalization of ordered semiring. In [20], Heidari and Davvaz introduced the concept of ordered semihypergroups, which is a generalization of ordered semigroups. The concept of Γ-semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of Γ-semigroups. The concept of a Γ-semihypergroup was introduced and studied by Davvaz et al. [3, 21]. In this section, we recall the notion of an ordered Γ-semihypergroup and then we present certain definitions and notations which we will need in this paper. Throughout this paper, unless otherwise specified, S is always an ordered Γ-semihypergroup (S, Γ, ≤).
Definition 2.1. [3, 4] Let S and Γ be two non-empty sets. Then, S is called a
Γ-semihypergroup if every γ ∈ Γ is a hyperoperation on S, i.e., xγy ⊆ S for every
x, y ∈ S, and for every α, β ∈ Γ and x, y, z ∈ S, we have xα(yβz) = (xαy)βz.
If every γ ∈ Γ is an operation, then S is a Γ-semigroup.
Let A and B be two non-empty subsets of S. We define AγB = ∪{aγb | a ∈ A, b ∈ B}. Also
AΓB = ∪{aγb | a ∈ A, b ∈ B and γ ∈ Γ} = S
γ∈Γ
AγB.
A non-empty subset A of S is called a sub Γ-semihypergroup of S if aγb ⊆ A for every a, b ∈ A and γ ∈ Γ. A Γ-semihypergroup S is called commutative if for all x, y ∈ S and γ ∈ Γ, we have xγy = yγx. A Γ-semihypergroup S is called a Γ-hypergroup if for every γ ∈ Γ, (S, γ) is a hypergroup.
In the following, we present the definition of an ordered Γ-semihypergroup and give some examples.
Definition 2.2. [30] An algebraic hyperstructure (S, Γ, ≤) is called an ordered
Γ-semihypergroup if (S, Γ) is a Γ-Γ-semihypergroup and (S, ≤) is a partially ordered set such that for any x, y, z ∈ S, x ≤ y implies zγx ≤ zγy and xγz ≤ yγz. Here,
A ≤ B means that for any a ∈ A, there exists b ∈ B such that a ≤ b, for all
non-empty subsets A and B of S.
A non-empty subset A of an ordered Γ-semihypergroup (S, Γ, ≤) is called a
sub Γ-semihypergroup of S if AΓA ⊆ A.
Example 2.3. [30] Let (S, ◦, ≤) be an ordered semihypergroup and Γ a non-empty
set. We define xγy = x ◦ y for every x, y ∈ S and γ ∈ Γ. Then (S, Γ, ≤) is an ordered Γ-semihypergroup.
Example 2.4. Every Γ-semihypergroup induces an ordered Γ-semihypergroup.
Indeed: Let (S, Γ) be a Γ-semihypergroup. Define the order on S by ≤:= {(x, y) | x = y}. Then (S, Γ, ≤) is an ordered Γ-semihypergroup.
Example 2.5. Let S = {a, b, c} and Γ = {γ, β} be the sets of binary
hyperoper-ations defined as follows:
γ a b c a {a, b} {a, b} c b {a, b} b c c c c c β a b c a a {a, b} c b {a, b} {a, b} c c c c c
Then S is a Γ-semihypergroup. We have (S, Γ, ≤) is an ordered Γ-semihyper-group where the order relation ≤ is defined by:
The covering relation and the figure of S are given by: ≺= {(a, b)}. b a b b b c
3. Quasi-Γ-hyperideals
The class of quasi-Γ-hyperideals in Γ-semihypergroups is a generalization of quasi-hyperideals in semihypergroups. The notion of quasi-ideal was first intro-duced by Steinfeld [38] for rings and semigroups as a generalization of the one-sided ideal. In this section, we investigate some properties of quasi-Γ-hyperideals in an ordered Γ-semihypergroup. Also, we give some characterizations of the regularity of an ordered Γ-semihypergroup in terms of left, right and quasi-Γ-hyperideals.
Definition 3.1. Let (S, Γ, ≤) be an ordered Γ-semihypergroup. A non-empty
sub-set I of S is called a left Γ-hyperideal of S if it satisfies the following conditions:
(1) SΓI ⊆ I;
(2) For every x ∈ I and y ∈ S, y ≤ x implies that y is in I.
A right Γ-hyperideal of an ordered Γ-semihypergroup S is defined in a similar way. By two-sided Γ-hyperideal or simply Γ-hyperideal, we mean a non-empty subset of S which both left and right Γ-hyperideal of S. A Γ-hyperideal I of S is said to be proper if I 6= S.
Let K be a non-empty subset of an ordered Γ-semihypergroup (S, Γ, ≤). If H is a non-empty subset of K, then we define (H]K := {k ∈ K | k ≤ h for some h ∈
H}. Note that if K = S, then we define (H] := {x ∈ S | x ≤ h for some h ∈ H}. For H = {h}, we write (h] instead of ({h}]. Note that the condition (2) in Definition 3.1 is equivalent to (I] ⊆ I. If A and B are non-empty subsets of S, then we have (1) A ⊆ (A]; (2) ((A]] = (A]; (3) If A ⊆ B, then (A] ⊆ (B]; (4) (A]Γ(B] ⊆ (AΓB]; (5) ((A]Γ(B]] = (AΓB].
Lemma 3.2. Let a be an element of an ordered Γ-semihypergroup (S, Γ, ≤). We
denote by LS(a) (resp. RS(a)) the left (resp. right) Γ-hyperideal of S generated
(1) LS(a) = (a ∪ SΓa];
(2) RS(a) = (a ∪ aΓS].
Proof. The proof is straightforward.
Definition 3.3. [30] A non-empty subset Q of an ordered Γ-semihypergroup (S, Γ, ≤) is called a quasi-Γ-hyperideal of S if the following conditions hold:
(1) (QΓS] ∩ (SΓQ] ⊆ Q;
(2) When x ∈ Q and y ∈ S such that y ≤ x, imply that y ∈ Q.
Example 3.4. Let S = {x, y, z, r, s, t} and Γ = {γ, β} be the sets of binary
hyperoperations defined as follows:
γ x y z r s t x {x, y} {x, y} z {r, s} {r, s} t y {x, y} y z {r, s} s t z z z z t t t r {r, s} {r, s} t {x, y} {x, y} z s {r, s} s t {x, y} y z t t t t z z z β x y z r s t x {x, r} {x, y, r, s} {z, t} {x, r} {x, y, r, s} {z, t} y {x, y, r, s} {x, y, r, s} {z, t} {x, y, r, s} {x, y, r, s} {z, t} z {z, t} {z, t} {z, t} {z, t} {z, t} {z, t} r {x, r} {x, y, r, s} {z, t} {x, r} {x, y, r, s} {z, t} s {x, y, r, s} {x, y, r, s} {z, t} {x, y, r, s} {x, y, r, s} {z, t} t {z, t} {z, t} {z, t} {z, t} {z, t} {z, t}
Then S is a Γ-semihypergroup. We have (S, Γ, ≤) is an ordered Γ-semihyper-group where the order relation ≤ is defined by:
≤:= {(x, x), (x, y), (y, y), (z, z), (r, r), (r, s), (s, s), (t, t)}.
The covering relation and the figure of S are given by: ≺= {(x, y), (r, s)}. b z b r b x b s b y b t
Every left, right and two-sided Γ-hyperideal of an ordered Γ-semihypergroup S is a quasi-Γ-hyperideal of S. The converse is not true, in general, that is, a quasi-Γ-hyperideal may not be a left, right or a two-sided Γ-hyperideal of S.
Example 3.5. Let S = {e, a, b, c, d} and Γ = {γ, β} be the sets of binary
hyper-operations defined as follows:
γ e a b c d e e e e e e a e {a, b} b b b b e b b b b c e c c c c d e d d d d β e a b c d e e e e e e a e a a a a b e a {a, b} a a c e c c c c d e d d d d
Then S is a Γ-semihypergroup [41]. We have (S, Γ, ≤) is an ordered Γ-semihyper-group where the order relation ≤ is defined by:
≤:= {(a, a), (a, b), (b, b), (c, c), (c, d), (e, e)}.
The covering relation and the figure of S are given by: ≺= {(a, b), (c, d)}. b e b c b a b d b b
Now, it is easy to see that Q = {e, a, b} is a quasi-Γ-hyperideal of S, but it is not a left Γ-hyperideal of S.
Lemma 3.6. Let (S, Γ, ≤) be an ordered Γ-semihypergroup. Then, the following
statements are hold:
(1) Every quasi-Γ-hyperideal of S is a sub Γ-semihypergroup of S.
(2) If L is a left Γ-hyperideal and R a right Γ-hyperideal of S, then Q = L ∩ R
Proof. (1): Let Q be a quasi-Γ-hyperideal of an ordered Γ-semihypergroup
(S, Γ, ≤). Then, QΓQ ⊆ SΓQ and QΓQ ⊆ QΓS. Hence QΓQ ⊆ SΓQ ∩ QΓS ⊆ (SΓQ] ∩ (QΓS] ⊆ Q. Therefore, Q is a sub Γ-semihypergroup of S.
(2): Since L is a left Γ-hyperideal and R a right Γ-hyperideal of S, we have RΓL ⊆ SΓL ⊆ L and RΓL ⊆ RΓS ⊆ R.
This implies that RΓL ⊆ L ∩ R. Thus, L ∩ R 6= ∅. We have (SΓQ] ∩ (QΓS] = (SΓ(L ∩ R)] ∩ ((L ∩ R)ΓS]
⊆ (SΓL] ∩ (RΓS] ⊆ (L] ∩ (R] ⊆ L ∩ R = Q.
Now, let x ∈ Q and y ∈ S such that y ≤ x. Then, we have x ∈ L and x ∈ R. So, y ∈ L and y ∈ R. Hence, y ∈ L ∩ R = Q. Therefore, Q is a quasi-Γ-hyperideal of S.
Theorem 3.7. Let {Qk | k ∈ Λ} be a family of quasi-Γ-hyperideals of an ordered
Γ-semihypergroup (S, Γ, ≤). If Tk∈ΛQk 6= ∅, then it is a quasi-Γ-hyperideal of
S.
Proof. Let {Qk | k ∈ Λ} be a family of quasi-Γ-hyperideals of S. If Q =
T
k∈ΛQk 6= ∅, then for each k ∈ Λ, (SΓQ] ∩ (QΓS] ⊆ (SΓQk] ∩ (QkΓS] ⊆ Qk.
We have (SΓQ] ∩ (QΓS] ⊆ Q.
Now, let x ∈ Q and y ∈ S such that y ≤ x. Then for every k ∈ Λ, y ∈ Qk.
Thus, y ∈ Q. Hence, Q =Tk∈ΛQk is a quasi-Γ-hyperideal of S.
Let A be a non-empty subset of an ordered Γ-semihypergroup (S, Γ, ≤). We define Θ = {Q | Q is a quasi-Γ-hyperideal of S containing A}. Since S ⊆ Θ, it follows that Θ is a non-empty set. We denote by QS(A) the quasi-Γ-hyperideal
of S generated by A. It is clear that TQ∈ΘQ is non-empty as A ⊆TQ∈ΘQ = QS(A). By Theorem 3.7, QS(A) is a quasi-Γ-hyperideal of S. Moreover, QS(A)
is the smallest quasi-Γ-hyperideal of S containing A. The quasi-Γ-hyperideal QS(A) is called the quasi-Γ-hyperideal of S generated by A.
Theorem 3.8. Let A be a non-empty subset of an ordered Γ-semihypergroup (S, Γ, ≤). Then, QS(A) = (A ∪ ((AΓS] ∩ (SΓA])].
Proof. Set Q = (A ∪ ((AΓS] ∩ (SΓA])]. It is evident that Q 6= ∅. We have
(SΓQ] ∩ (QΓS] = (SΓ(A ∪ ((AΓS] ∩ (SΓA])]] ∩ ((A ∪ ((AΓS] ∩ (SΓA])]ΓS] ⊆ (SΓ(A ∪ (SΓA])] ∩ ((A ∪ (AΓS])ΓS]
⊆ (SΓA ∪ SΓ(SΓA]] ∩ (AΓS ∪ (AΓS]ΓS] ⊆ (SΓA ∪ (SΓA]] ∩ (AΓS ∪ (AΓS]]
⊆ (SΓA] ∩ (AΓS] ⊆ ((SΓA] ∩ (AΓS]] ⊆ (A ∪ ((AΓS] ∩ (SΓA])] = Q.
Therefore, Q is a quasi-Γ-hyperideal of S, and hence QS(A) ⊆ (A ∪ ((AΓS] ∩
(SΓA])]. Let C be a quasi-Γ-hyperideal of S containing A. Then, (AΓS] ∩ (SΓA] ⊆ (CΓS] ∩ (SΓC] ⊆ C.
Thus, we have Q = (A ∪ ((AΓS] ∩ (SΓA])] ⊆ (C] = C. Hence, Q is the smallest quasi-Γ-hyperideal of S containing A. Therefore, QS(A) = Q = (A ∪ ((AΓS] ∩
(SΓA])].
Corollary 3.9. Let a be an element of an ordered Γ-semihypergroup (S, Γ, ≤).
Then, QS(a) = (a ∪ ((aΓS] ∩ (SΓa])].
In the following, we characterize regular ordered Γ-semihypergroups in terms of left, right and quasi-Γ-hyperideals.
Definition 3.10. An ordered Γ-semihypergroup (S, Γ, ≤) is called regular if for
every a ∈ S there exist x ∈ S, α, β ∈ Γ such that a ≤ aαxβa. This is equivalent to saying that a ∈ (aΓSΓa], for every a ∈ S or A ⊆ (AΓSΓA], for every A ⊆ S.
Theorem 3.11. An ordered Γ-semihypergroup (S, Γ, ≤) is regular if and only
if for every right Γ-hyperideal R, every left Γ-hyperideal L and every quasi-Γ-hyperideal Q of S, we have R ∩ Q ∩ L ⊆ (RΓQΓL].
Proof. Let R be right Γ-hyperideal, L a left Γ-hyperideal and Q a
quasi-Γ-hyperideal of S. Since S is regular, we have R ∩ Q ∩ L ⊆ ((R ∩ Q ∩ L)ΓSΓ(R ∩ Q ∩ L)]
⊆ ((R ∩ Q ∩ L)ΓSΓ(R ∩ Q ∩ L)ΓSΓ(R ∩ Q ∩ L)ΓSΓ(R ∩ Q ∩ L)] ⊆ (RΓSΓQΓSΓLΓSΓL]
= ((RΓS)ΓQΓ(SΓLΓSΓL)] ⊆ (RΓQΓL].
Conversely, suppose that R ∩ Q ∩ L ⊆ (RΓQΓL] for every right Γ-hyperideal R, every left Γ-hyperideal L and every quasi-Γ-hyperideal Q of S. Since S is a quasi-Γ-hyperideal of S, we have
R ∩ L = R ∩ S ∩ L ⊆ (RΓSΓL] ⊆ (RΓL].
As RΓL ⊆ SΓL ⊆ L and RΓL ⊆ RΓS ⊆ R, we have RΓL ⊆ R ∩ L. So, (RΓL] ⊆ (R ∩ L] ⊆ (R] ∩ (L] ⊆ R ∩ L. Hence, R ∩ L = (RΓL] for any right
Γ-hyperideal R and any left Γ-hyperideal L of S. Let a ∈ S. Since a ∈ RS(a)
and a ∈ LS(a), it follows that a ∈ RS(a) ∩ LS(a). So, we have
a ∈ (RS(a)ΓLS(a)] = ((a ∪ aΓS]Γ(a ∪ SΓa]]
⊆ (aΓa ∪ aΓSΓa ∪ aΓSΓSΓa] ⊆ (aΓa ∪ aΓSΓa].
Hence, a ≤ u for some u ∈ aΓa ∪ aΓSΓa. If u ∈ aΓSΓa, then a ≤ aαxβa for some x ∈ S, α, β ∈ Γ. Thus, a ∈ (aΓSΓa]. Therefore, S is a regular ordered Γ-semihypergroup. If u ∈ aΓa, then a ≤ aαa ≤ aα(aβa) ⊆ aΓSΓa. So, we have a ∈ (aΓSΓa]. Therefore, S is regular.
Theorem 3.12. Let (S, Γ, ≤) be a regular ordered Γ-semihypergroup. Then, for
every right Γ-hyperideal R and left Γ-hyperideal L of S, Q = (RΓL] is a
quasi-Γ-hyperideal of S.
Proof. Let R be a right Γ-hyperideal and L a left Γ-hyperideal of S. As RΓL ⊆
SΓL ⊆ L and RΓL ⊆ RΓS ⊆ R, we have RΓL ⊆ R ∩ L. So, (RΓL] ⊆ (R ∩ L] ⊆ (R] ∩ (L] ⊆ R ∩ L. Let S be regular ordered Γ-semihypergroup. We prove that R ∩ L ⊆ (RΓL]. Since S is regular, we have
R ∩ L ⊆ ((R ∩ L)ΓSΓ(R ∩ L)] ⊆ (RΓSΓ(R ∩ L)] ⊆ (RΓSΓL] ⊆ (RΓL]. Hence, R ∩ L = (RΓL]. By Lemma 3.6, Q = (RΓL] is a quasi-Γ-hyperideal of S.
4. Some Notions in Ordered Γ-Semihypergroups
Let (S, ·, ≤) be an ordered semigroup. As in [27], a subsemigroup F of S is called a filter of S if (1) a, b ∈ S, a · b ∈ F implies a ∈ F and b ∈ F ; (2) If a ∈ F , c ∈ S and a ≤ c, then c ∈ F . In [31], Kwon introduced the concept of filters in ordered Γ-semigroups. Now, we state some lemmas and theorems on hyperfilter of an ordered Γ-semihypergroup. Tang et al. studied the hyperfilters of ordered semihypergroups in [39]. Most of the results in this section are analogous to the results on hyperfilters in ordered semihypergroups.
Definition 4.1. Let S be an ordered Γ-semihypergroup and F a sub
Γ-semi-hypergroup. Then F is called a left (resp. right) hyperfilter of S if
(1) a, b ∈ S, γ ∈ Γ, aγb ∩ F 6= ∅ ⇒ b ∈ F (resp. a ∈ F ); (2) a ∈ F, c ∈ S, a ≤ c ⇒ c ∈ F .
If F is both a left hyperfilter and a right hyperfilter of S, then F is called a
hyperfilter of S. A hyperfilter F of S is said to be proper if F 6= S. Clearly, if
(S, Γ, ≤) is an ordered Γ-semigroup, then the hyperfilters of S coincide with the filters of S defined in [31].
Example 4.2. Let S = {a, b, c, d, e} and Γ = {γ, β} be the sets of binary
hyper-operations defined as follows:
γ a b c d e a {a, b} {b, c} c {d, e} e b {b, c} c c {d, e} e c c c c {d, e} e d {d, e} {d, e} {d, e} d e e e e e e e β a b c d e a {b, c} c c {d, e} e b c c c {d, e} e c c c c {d, e} e d {d, e} {d, e} {d, e} d e e e e e e e
Then S is a Γ-semihypergroup [41]. We have (S, Γ, ≤) is an ordered Γ-semihyper-group where the order relation ≤ is defined by:
≤:= {(a, a), (a, b), (a, c), (b, b), (b, c), (c, c), (d, d), (e, e)}. The covering relation and the figure of S are given by:
≺= {(a, b), (b, c)}. b a b b b c b d b e
Then by routine calculations, F = {a, b, c} is a hyperfilter of S.
Let K be a sub Γ-semihypergroup of an ordered Γ-semihypergroup (S, Γ, ≤). If T ⊆ K, then we define
[T )K := {k ∈ K | k ≥ t for some t ∈ T }.
Note that if K = S, then we define
[T ) := {x ∈ S | x ≥ t for some t ∈ T }.
For T = {t}, we write [t) instead of [{t}). Note that the condition (2) in Definition 4.1 is equivalent to [F ) ⊆ F .
Lemma 4.3. Let {Fk | k ∈ Λ} be a family of hyperfilters of an ordered
Proof. Let Fk be a hyperfilter of S for all k ∈ Λ and Tk∈ΛFk 6= ∅. Let a, b ∈
T
k∈ΛFk. Then a, b ∈ Fk for all k ∈ Λ. Since Fk is a hyperfilter of S for all
k ∈ Λ, it follows that aγb ⊆Tk∈ΛFk for all γ ∈ Γ. Hence,Tk∈ΛFk is a sub
Γ-semihypergroup of S. Now, let x, y ∈ S, γ ∈ Γ and xγy ∩ (Tk∈ΛFk) 6= ∅. Then,
there exists z ∈Tk∈ΛFk for some z ∈ xγy. Since z ∈Tk∈ΛFk, it follows that
z ∈ Fk for all k ∈ Λ. Since Fk is a hyperfilter of S for all k ∈ Λ, it follows that
a, b ∈ Fk for all k ∈ Λ. This implies that a, b ∈Tk∈ΛFk. So, the first condition
of the definition of hyperfilter is verified. Now, suppose that a ∈Tk∈ΛFk, c ∈ S
and a ≤ c. Then, a ∈ Fk for all k ∈ Λ. Since Fk is a hyperfilter of S for all
k ∈ Λ, it follows that c ∈ Fk for all k ∈ Λ. This means that c ∈
T
k∈ΛFk, and so
the second condition of the definition of hyperfilter is verified. Hence, Tk∈ΛFk
is a hyperfilter of S.
The following is a natural question to ask:
Question 4.4. Suppose {Fk | k ∈ Λ} is a family of hyperfilters of an ordered
Γ-semihypergroup (S, Γ, ≤). Is it true thatSk∈ΛFk is a hyperfilter of S?
The following example shows that it is not true in general case.
Example 4.5. Let S = {a, b, c, d, e} and Γ = {γ}. We define
γ a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c
Then S is a Γ-semihypergroup. We have (S, Γ, ≤) is an ordered Γ-semihyper-group where the order relation ≤ is defined by:
≤ := {(a, a), (a, b), (b, b), (c, c), (c, e), (d, d), (e, e)}. The covering relation and the figure of S are given by:
≺= {(a, b), (c, e)}. b c b a b e b b b d
Here, F1 = {a, b} and F2 = {c, e} are both hyperfilters of S, but F1∪ F2 =
{a, b, c, e} is not a hyperfilters of S. Indeed: Since a, c ∈ F1∪ F2and aγc = d /∈
Lemma 4.6. Let F1 and F2 be hyperfilters of an ordered Γ-semihypergroup
(S, Γ, ≤). Then, F1∪ F2 is a hyperfilter of S if and only if F1⊆ F2 or F2⊆ F1.
Proof. If F1 ⊆ F2 or F2 ⊆ F1, then it is clear that F1∪ F2 is a hyperfilter of
S. Now, suppose that F1∪ F2 is a hyperfilter of S and F1 * F2 and F2 * F1.
Then, there exist a, b ∈ F1∪ F2 such that a ∈ F1, a /∈ F2, b ∈ F2, b /∈ F1. Since
F1∪ F2is a hyperfilter of S, it follows that aγb ⊆ F1∪ F2 for each γ ∈ Γ. Now,
one of two following cases happens:
Case 1. aγb ∩ F16= ∅. Since F1 is a hyperfilter of S, we have b ∈ F1 and this is
a contradiction.
Case 2. aγb ∩ F26= ∅. Since F2 is a hyperfilter of S, we have a ∈ F2 and this is
a contradiction. This completes the proof.
Theorem 4.7. Suppose {Fk | k ∈ Λ} is a family of hyperfilters of an ordered
Γ-semihypergroup (S, Γ, ≤) such that Fs⊆ Ftor Ft⊆ Fs, for all s, t ∈ Λ. Then,
S
k∈ΛFk is a hyperfilter of S, where | Λ |≥ 2.
Proof. We shall prove the theorem by induction on k, where k ∈ Λ and k ≥ 2. If
k = 2, then it is true by Lemma 4.6. Suppose that the assertion is true for less than k. We claim that Skn=1Fn is a hyperfilter of S. Consider F =Sk−1n=1Fn.
Then Skn=1Fn = F ∪ Fk. By hypothesis, F is a hyperfilter of S. We show that
F ⊆ Fk or Fk ⊆ F . By hypothesis, we can consider the following two cases:
Case 1. Let Fs⊆ Fkfor any s ∈ {1, 2, · · · , k −1}. Then we have F =Sk−1n=1Fn ⊆
Fk.
Case 2. Let Fk ⊆ Fs for some s ∈ {1, 2, · · · , k − 1}. Then we have Fk ⊆ Fs ⊆
Sk−1
n=1Fn = F . By Lemma 4.6,
Sk
n=1Fn = F ∪ Fk is a hyperfilter of S. Thus,
by induction, we complete the proof.
Davvaz et al. [21] defined the concept of prime Γ-hyperideal of a Γ-semihyper-group. A proper Γ-hyperideal P of a Γ-semihypergroup (S, Γ) is called a prime Γ-hyperideal, if for every Γ-hyperideals I and J of S, IΓJ ⊆ P implies I ⊆ P or J ⊆ P . If Γ-semihypergroup S is commutative, then a proper Γ-hyperideal P is prime if and only if aΓb ⊆ P implies a ∈ P or b ∈ P for any a, b ∈ S. A Γ-hyperideal I of an ordered Γ-semihypergroup S is called completely prime if for any a, b of S and γ ∈ Γ such that aγb ∩ I 6= ∅, then a ∈ I or b ∈ I. A left Γ-hyperideal P of a Γ-semihypergroup S is called quasi-prime, if for any left Γ-hyperideals I and J of S, IΓJ ⊆ P implies I ⊆ P or J ⊆ P . P is called
quasi-semiprime if for every left Γ-hyperideal I of S, IΓI ⊆ P implies I ⊆ P .
Let (S, Γ) be a Γ-semihypergroup and P a left Γ-hyperideal of S. Then, P is quasi-prime if and only if for all x, y ∈ S, xΓSΓy ⊆ P implies that x ∈ P or y ∈ P .
Theorem 4.8. Let (S, Γ, ≤) be an ordered Γ-semihypergroup and F a non-empty
(1) F is a hyperfilter of S.
(2) S \ F = ∅ or S \ F is a completely prime Γ-hyperideal of S.
Proof. (1) ⇒ (2): Assume that (1) holds. Let S \ F 6= ∅. First, we show that
S \ F is a Γ-hyperideal of S. Let x ∈ S, y ∈ S \ F , γ ∈ Γ and xγy ∩ F 6= ∅. Since F is a hyperfilter of S, it follows that y ∈ F , a contradiction. Thus xγy ⊆ S \ F , i.e., SΓ(S \ F ) ⊆ S \ F . Similarly, we have (S \ F )ΓS ⊆ S \ F . Suppose that a ∈ S \ F and b ≤ a, where b ∈ S. If b ∈ F . Since F is a hyperfilter of S, and we get a ∈ F , a contradiction. This means that b ∈ S \ F , and so (S \ F ] ⊆ S \ F . Hence, S\F is a Γ-hyperideal of S. Next, let a, b ∈ S, γ ∈ Γ and aγb∩(S\F ) 6= ∅. Then, there exists c ∈ aγb such that c ∈ S \ F . If a ∈ F and b ∈ F , then, since F is a sub Γ-semihypergroup of S, we get c ∈ F , a contradiction. Hence, a ∈ S \ F or b ∈ S \ F . Therefore, S \ F is a completely prime Γ-hyperideal of S and the proof is completed.
(2) ⇒ (1): If S \ F = ∅, then F = S, and so F is a hyperfilter of S. Suppose that S \ F is a completely prime Γ-hyperideal of S. Now, let a, b ∈ F and γ ∈ Γ. If aγb* F , then we have aγb ∩ (S \ F ) 6= ∅. Since S \ F is completely prime, it follows that a ∈ S \ F or b ∈ S \ F , which is a contradiction. Thus, aγb ⊆ F . Hence, F is a sub Γ-semihypergroup of S. Let a, b ∈ S, γ ∈ Γ and aγb ∩ F 6= ∅. If a ∈ S \ F , then, since S \ F is a Γ-hyperideal, we have aγb ⊆ S \ F . This means that (S \ F ) ∩ F 6= ∅, which is a contradiction. Thus, a ∈ F . Similarly, we have b ∈ F . Suppose that a ∈ F and a ≤ c, where c ∈ S. If c ∈ S \ F , then, since S \ F is a Γ-hyperideal of S, we get a ∈ S \ F , a contradiction. This means that c ∈ F , and so F is a hyperfilter of S.
Example 4.9. Let (S, Γ, ≤) be the ordered Γ-semihypergroup defined as in
Ex-ample 4.2. We have shown that F = {a, b, c} is a hyperfilter of S. Thus, by Theorem 4.8, I = S \ F = {d, e} is a completely prime Γ-hyperideal of S.
In 2004, Kehayopulu [26] introduced the concept of m-systems and n-systems in ordered semigroups. In [16], Dutta and Adhikari introduced the concept of m-systems in Γ-semigroups. Abdullah et al. [2, 5] studied M -hypersystems and N -hypersystems in a Γ-semihypergroup. A subset M of a Γ-semihypergroup S is called M -hypersystem if for all a, b ∈ M , there exist x ∈ S and α, β ∈ Γ such that aαxβb ⊆ M . A subset N of Γ-semihypergroup S is called N -hypersystem if for all a ∈ N , there exist x ∈ S and α, β ∈ Γ such that aαxβa ⊆ N . In the following, we introduce the concepts of M -hypersystems and N -hypersystems in ordered Γ-semihypergroup. The results generalizes the results on Γ-semihypergroups proved by Abdullah et al. [2].
Definition 4.10. Let (S, Γ, ≤) be an ordered Γ-semihypergroup. A non-empty
subset M of S is called an M -hypersystem of S if for each a, b ∈ M , there exist
x ∈ S, c ∈ M and α, β ∈ Γ such that c ≤ aα(xβb).
Definition 4.11. Let (S, Γ, ≤) be an ordered Γ-semihypergroup. A non-empty
subset N of S is called an N -hypersystem of S if for each a ∈ N , there exist
x ∈ S, c ∈ N and α, β ∈ Γ such that c ≤ aα(xβa). Equivalent Definition: ∀a ∈ N, ∃x ∈ S, (aΓxΓa] ∩ N 6= ∅.
Note that every M -hypersystem of S is an N -hypersystem of S, but the converse is not true in general, that is, an N hypersystem may not be an M -hypersystem.
Example 4.12. Let S = {a, b, c, d, e} and Γ = {γ, β} be the sets of binary
hyper-operations defined as follows: γ a b c d e a a e c d e b a e c d e c a e c d e d a e c d e e a e c d e β a b c d e a a e c d e b a b c d e c a e c d e d a e c d e e a e c d e
Then, (S, Γ) is a Γ-semihypergroup. We have (S, Γ, ≤) is an ordered Γ-semi-hypergroup where the order relation ≤ is defined by:
≤:= {(a, a), (b, b), (c, a), (c, c), (d, a), (d, d), (e, a), (e, e)}. The covering relation and the figure of S are given by:
≺= {(c, a), (d, a), (e, a)}.
b c b d b e b a b b @ @ @
Now, it is easy to see that the set {a, b} is an N -hypersystem of S, but it is not an M -hypersystem of S, Since if we consider the elements a, b, then for all x ∈ S and γ, β ∈ Γ, we have aγxβb = {e} and (aΓxΓb] ∩ {a, b} 6= ∅ is false.
Theorem 4.13. A proper left Γ-hyperideal P of an ordered Γ-semihypergroup (S, Γ, ≤) is quasi-prime if and only if S \ P is an M -hypersystem.
Proof. Suppose that P is quasi-prime and a, b ∈ S \ P . Take any x ∈ S and
α, β ∈ Γ such that (aαxβb] ∩ (S \ P ) 6= ∅; then aαxβb ⊆ (aαxβb] ⊆ P . Hence, a ∈ P or b ∈ P . This implies that a ∈ S \P or b ∈ S \P , which is a contradiction. Therefore, S \ P is an M -hypersystem.
Conversely, assume that S \ P is an M -hypersystem and aΓSΓb ⊆ P with a, b ∈ S \ P . By hypothesis, there exist c ∈ S \ P , x ∈ S and α, β ∈ Γ such that c ≤ aαxβb ⊆ aΓSΓb ⊆ P . Since P is a left Γ-hyperideal of S, we get c ∈ P , which is a contradiction. Therefore, P is quasi-prime.
Analogous to the proof of Theorem 4.13, we have the following result.
Theorem 4.14. A proper left Γ-hyperideal P of an ordered Γ-semihypergroup (S, Γ, ≤) is quasi-semiprime if and only if S \ P is an N -hypersystem.
Regular and strongly regular relations are important in order to study the quotient structures. Let σ be an equivalence relation on a Γ-semihypergroup (S, Γ). If A and B are non-empty subsets of S, then AσB means that for all a ∈ A, there exists b ∈ B such that aσb and for all b′ ∈ B, there exists a′ ∈ A
such that a′σb′. Also, AσB means that for all a ∈ A and for all b ∈ B, we have
aσb. An equivalence relation σ on S is said to be regular if for all a, b, x ∈ S and γ ∈ Γ, we have
aσb implies (aγx)σ(bγx) and (xγa)σ(xγb).
In addition, σ on S is said to be strongly regular if for all a, b, x ∈ S and γ ∈ Γ, we have
aσb implies (aγx)σ(bγx) and (xγa)σ(xγb).
Theorem 4.15. [35] Let σ be a regular equivalence relation on a Γ-semihypergroup S and γ ∈ Γ. If we define the following hyperoperation on the set of all
equiva-lence classes with respect to σ, that is, S/σ = {[x]σ | x ∈ S}:
[a]σγσ[b]σ= {[z]σ | z ∈ aγb},
then (S/σ, Γσ) is a Γσ-semihypergroup.
Theorem 4.16. [35] Let σ be a strongly regular equivalence relation on a
Γ-semihypergroup S. Then (S/σ, Γσ) is a Γσ-semigroup.
Definition 4.17. Let (S, Γ, ≤S) and (T, Γ, ≤T) be two ordered Γ-semihypergroups.
The map ϕ : R → T is called a Γ-homomorphism if for all a, b ∈ S and γ ∈ Γ, we have
(1) ϕ(aγb) ⊆ ϕ(a)γϕ(b);
(2) ϕ is isotone, that is, for any a, b ∈ S, a ≤Sb implies ϕ(a) ≤T ϕ(b).
If the condition (1) holds for the equality instead of the inclusion, ϕ is said to be a strong homomorphism.
Let ϕ be a Γ-homomorphism from a Γ-semihypergroup (S, Γ) into (T, Γ). The relation ϕ−1◦ ϕ is an equivalence relation σ on S (aσb ⇔ ϕ(a) = ϕ(b)) is
known as the kernel of ϕ. One can see that kerϕ is a regular relation (see [4, Prop. 2.3]). The following is a natural question to ask:
Question 4.18. Is there a strongly regular relation σ on S for which S/σ is an ordered Γσ-semigroup?
Similar to ordered Γ-semigroups [7], a probable order on S/σ could be the relation on S/σ defined by means of the order ≤ on S, that is
:= {(σ(a), σ(b)) ∈ S/σ × S/σ | ∃x ∈ σ(a), ∃y ∈ σ(b) such that (x, y) ∈≤}. But this relation is not an order, in general. It is enough we consider the or-dered semigroup defined in the example of page 235 in [7] as an oror-dered Γ-semihypergroup.
In case of ordered Γ-semigroups, pseudoorders play the role of congruences of Γ-semigroups. In [7], Chinram and Tinpun introduced the concept of pseu-doorder of ordered semigroups. Now, we extend this notion for ordered Γ-semihypergroups.
Definition 4.19. Let (S, Γ, ≤) be an ordered Γ-semihypergroup. A relation σ on S is called a pseudoorder on S if
(1) ≤⊆ σ;
(2) aσb and bσc imply aσc for all a, b, c ∈ S;
(3) aσb implies aγcσbγc and cγaσcγb for all a, b, c ∈ S and γ ∈ Γ.
Lemma 4.20. Let {σi | i ∈ Ω} be a set of pseudoorders on an ordered
Γ-semihypergroup (S, Γ, ≤). Then, σ =Ti∈Ωσi is a pseudoorder on (S, Γ, ≤).
Proof. Obviously, ≤⊆ σ and σ is transitive. Now, let aσb, c ∈ S and γ ∈ Γ.
Then aσib for all i ∈ Ω. Since each σi is a pseudoorder on S, by condition (3)
of Definition 4.19, we conclude that
aγcσibγc and cγaσicγb.
Hence, for every u ∈ aγc and v ∈ bγc, we have uσiv for all i ∈ Ω. So, we have
uσv. This implies that aγcσbγc. Similarly, we obtain cγaσcγb. Hence, σ is a pseudoorder on S.
In the following, Theorem 4.21 is a generalization of Proposition 3.3 in [7]. Indeed, in Theorem 4.21, if our ordered semihypergroup is an ordered Γ-semigroup, i.e., our hyperoperation is a binary operation, we obtain Prop. 3.3 in [7].
Theorem 4.21. Let (S, Γ, ≤) be an ordered Γ-semihypergroup and σ a
pseudo-order on S. Then, there exists a strongly regular equivalence relation σ∗ =
{(a, b) ∈ S × S | aσb and bσa} on S such that (S/σ∗, Γ
σ∗, σ∗) is an ordered
Γσ∗-semigroup, where σ∗:= {(σ∗(x), σ∗(y)) ∈ S/σ∗× S/σ∗ | ∃a ∈ σ∗(x), ∃b ∈
σ∗(y) such that (a, b) ∈ σ}.
Proof. Suppose that σ∗is the relation on S defined as follows:
σ∗= {(a, b) ∈ S × S | aσb and bσa}.
First, we show that σ∗ is a strongly regular relation on S. Since (a, a) ∈≤ and
≤⊆ σ, we have aσa. So, aσ∗a. If (a, b) ∈ σ∗, then aσb and bσa. This means
that (b, a) ∈ σ∗. Next, let a, b, c ∈ S such that (a, b) ∈ σ∗ and (b, c) ∈ σ∗. Then,
aσb, bσa, bσc and cσb. Hence, aσc and cσa, which imply that (a, c) ∈ σ∗. Now,
let a, b ∈ S such that aσ∗b. Then, aσb and bσa. Since σ is a pseudoorder on S,
by condition (3) of Definition 4.19, we conclude that aγcσbγc, cγaσcγb, bγcσaγc, cγbσcγa,
for all c ∈ S and γ ∈ Γ. Hence, for every x ∈ aγc and y ∈ bγc, we have xσy and yσx which imply that xσ∗y. So, aγcσ∗bγc. Similarly, we have cγaσ∗cγb.
Therefore, σ∗is a strongly regular relation on S. Hence, by Theorem 4.16, S/σ∗
with the following operation is a Γσ∗-semigroup:
[x]σ∗γσ∗[y]σ∗ = {[z]σ∗ | z ∈ xγy}.
Now, for each σ∗(x), σ∗(y) ∈ S/σ∗, define the order relation
σ∗ on S/σ∗ by:
σ∗:= {(σ∗(x), σ∗(y)) ∈ S/σ∗× S/σ∗ | ∃a ∈ σ∗(x), ∃b ∈ σ∗(y) such that
(a, b) ∈ σ}. We show that
σ∗(x)
σ∗ σ∗(y) ⇔ xσy.
Let σ∗(x)
σ∗ σ∗(y). We show that for every a ∈ σ∗(x) and b ∈ σ∗(y), aσb.
Since σ∗(x)
σ∗ σ∗(y), there exist x′ ∈ σ∗(x) and y′ ∈ σ∗(y) such that x′σy′.
Since a ∈ σ∗(x) and x′ ∈ σ∗(x), we obtain aσ∗x′, and so aσx′ and x′σa. Since
b ∈ σ∗(y) and y′∈ σ∗(y), we obtain bσ∗y′, and so bσy′ and y′σb. Now, we have
aσx′, x′σy′ and y′σb, which imply that aσb. Since x ∈ σ∗(x) and y ∈ σ∗(y),
we conclude that xσy. Conversely, let xσy. Since x ∈ σ∗(x) and y ∈ σ∗(y), we
obtain σ∗(x)
σ∗ σ∗(y).
Now, we prove that σ∗ is an order on S/σ∗. Let a, b, c ∈ S. Since (a, a) ∈≤⊆
σ, we have σ∗(a)
σ∗ σ∗(a). If σ∗(a) σ∗ σ∗(b) and σ∗(b) σ∗ σ∗(a), then
(a, b) ∈ σ and (b, a) ∈ σ. This means that (a, b) ∈ σ∗, and so σ∗(a) = σ∗(b).
Let σ∗(a)
σ∗ σ∗(b) and σ∗(b) σ∗ σ∗(c). Then, (a, b) ∈ σ and (b, c) ∈ σ. This
means that (a, c) ∈ σ, and so σ∗(a)
σ∗ σ∗(c). Therefore, σ∗ is an order on
Finally, we prove that (S/σ∗, Γ
σ∗, σ∗) is an ordered Γσ∗-semigroup. Let
σ∗(x)
σ∗ σ∗(y) and σ∗(z) ∈ S/σ∗. Then, xσy and z ∈ S. By condition (3) of
Definition 4.19, we have xγzσyγz and zγxσzγy. So, for all a ∈ xγz and b ∈ yγz, we have aσb. This implies that σ∗(a)
σ∗ σ∗(b). Hence, σ∗(x)γσ∗ρ∗(z) σ∗
σ∗(y)γ
σ∗σ∗(z). Similarly, we get σ∗(z)γσ∗σ∗(x) σ∗ σ∗(z)γσ∗σ∗(y). Hence, the
theorem is proved.
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