Impact Factor 2.638 | CiteScore 2.3
More on impact ›

Brief Research Report ARTICLE

Front. Phys., 05 November 2020 | https://doi.org/10.3389/fphy.2020.555573

Some Properties of Relative Bi-(Int-)Γ-Hyperideals in Ordered Γ-Semihypergroups

Yongsheng Rao1, Peng Xu1,2, Zehui Shao1, Saeed Kosari1* and Saber Omidi3
  • 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
  • 2School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun, China
  • 3Department of Education in Tehran, Tehran, Iran

In this article, we study the concept of relative bi-Γ-hyperideals (J-bi-Γ-hyperideals) in ordered Γ-semihypergroups and present some related examples of this concept. Especially, characterization of J-simple ordered Γ-semihypergroups in terms of J-bi-Γ-hyperideals is given. Furthermore, we define the notion of J-(bi-)int-Γ-hyperideals in ordered Γ-semihypergroups and investigate its related properties. We show that J-int-Γ-hyperideals and J-Γ-hyperideals coincide in a J-regular ordered Γ-semihypergroup.

1. Introduction

One of the motivations for the study of hyperstructures comes from biological inheritance and physical phenomenon as the nuclear fission. Another motivation for the study of hyperstructures comes from chemical reactions and redox reactions. Dehghan Nezhad et al. [1] provided a physical example of hyperstructures associated with the elementary particle physics: leptons. As we know, the Higgs boson is an elementary particle in the standard model of particle physics. In [2], it is shown that the leptons and gauge bosons along with the interactions between their members construct an Hν-structure. Yaqoob et al. [3] studied some properties of (fuzzy) Γ-hyperideals in involution Γ-semihypergroups. In [4], the concepts of uni-soft Γ-hyperideals and uni-soft interior Γ-hyperideals of ordered Γ-semihypergroups are investigated.

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 hyperstructures was first introduced by Marty [5] at the eighth congress of Scandinavian Mathematicians in 1934. Nowadays, we can easily find well-written books for the introduction to hyperstructures, which include Corsini [6], Corsini and Leoreanu [7], Davvaz [8], Davvaz and Leoreanu-Fotea [9], Davvaz and Vougiouklis [10], and Vougiouklis [11]. For the information about hyper rings, we refer the reader to Ref. 9.

The study of ordered semihypergroups began with the work of Heidari and Davvaz [12]. In 2015, Davvaz et al. [13] discussed the notion of a pseudo-order in an ordered semihypergroup. The focus of the study was to find out if there is a relationship between ordered semihypergroups and ordered semigroups by using pseudo-orders. In 2016, Gu and Tang [14] answered to the open problem given by Davvaz et al. [13]. In [15], Tipachot and Pibaljommee introduced the concept of fuzzy interior hyperideals on ordered semihypergroups. Recently, Mahboob et al. [16] studied the concept of (m,n)-hyperideals on ordered semihypergroups. Recall, from Ref. 12, that an ordered semihypergroup (S,,) is a semihypergroup (S) together with a (partial) order relation that is compatible with the hyperoperation , meaning that, for any x,y,zS,

xyzxzyandxzyz.

Here, AB means that for any aA, there exists bB such that ab, for all nonempty subsets A and B of S.

Good and Hughes [17] introduced the notion of bi-ideals of a semigroup as early as 1952. In 1962, Wallace [18] introduced the notion of relative ideals (J-ideals) on semigroups. In 1967, Hrmová [19] generalized the notion of J-ideal by introducing the notion of a (J1,J2)-ideal of a semigroup S. Recently, Khan and Ali [20] introduced the concept of relative bi-ideals in ordered semigroups. The notion of Γ-semigroup was introduced by Sen and Saha [21] in 1986, which is a generalization of semigroups. In 2011, Anvariyeh et al. [22] introduced the notion of Γ-hyperideal of a Γ-semihypergroup. Later on, Yaqoob and Aslam [23] studied prime (m,n) bi-Γ-hyperideals of Γ-semihypergroups. Omidi et al. [24, 25] discussed some important properties of bi-Γ-hyperideals of an ordered Γ-semihypergroup. Bi-Γ-hyperideal is a special case of (m,n)-Γ-hyperideal [26]. In 2017, Tang et al. [27] considered and proved some theorems on fuzzy interior Γ-hyperideals in ordered Γ-semihypergroups. Pseudo-orders are the bridge between ordered Γ-semigroups and ordered Γ-semihypergroups, see Ref. 28.

After an introduction, in Section 2, we introduce some notation and terminologies. Section 2 aims for summarizing the fundamental materials on ordered Γ-semihypergroups. Section 3 is devoted to the study of relative Γ-hyperideals (J-Γ-hyperideals) and relative bi-Γ-hyperideals (J-bi-Γ-hyperideals) of an ordered Γ-semihypergroup. In this section, our main results are stated and proved. J-simple ordered Γ-semihypergroups are characterized by using the notions of J-Γ-hyperideals and J-bi-Γ-hyperideals. Finally, in Section 4, the notion of J-(bi-)int-Γ-hyperideals (J-(bi-)interior-Γ-hyperideals) are studied and their related properties are discussed. It is shown that, in J-regular ordered Γ-semihypergroups the J-Γ-hyperideals and the J-int-Γ-hyperideals coincide.

2. Preliminaries

Let S and Γ be two nonempty sets. Then, S is called a Γ-semihypergroup [22] if every γΓ is a hyperoperation on S, i.e., xγyS, for every x,yS, α,βΓ, and x,y,zS, we have

xα(yβz)=(xαy)βz.

If every γΓ is an operation, then S is a Γ-semigroup. Let A and B be two nonempty subsets of S. We define

AγB={aγb|aA,bB}.

Also,

AΓB={aγb|aA,bBandγΓ}=γΓAγB.

In the following, we recall the notion of an ordered Γ-semihypergroup, and then we present basic definitions and notations, which we will need in this article. Throughout this article, unless otherwise specified, S is always an ordered Γ-semihypergroup (S,Γ,).

Definition 2.1 (see [29]). 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,zS and γΓ, xy implies zγxzγy and xγzyγz. Here, if A and B are two nonempty subsets of S, then we say that AB if, for every aA, there exists bB such that ab.

Let S be an ordered Γ-semihypergroup. By a sub Γ-semihypergroup of S, we mean a nonempty subset A of S such that aγbA for all a,bA and γΓ. A nonempty subset A of S is called idempotent if A=(AΓA].

Example 1. (See Ref. 25.) Let (S,,) be an ordered semihypergroup and Γ a nonempty set. We define aγb=ab for every a,bS and γΓ. Then, (S,Γ,) is an ordered Γ-semihypergroup.

Let J be a nonempty subset of an ordered Γ-semihypergroup (S,Γ,). If H is a nonempty subset of J, then we define

(H]J:={jJ|jhforsomehH}.

Note that if J=S, then we define

(H]:={xS|xhforsomehH}.

If A and B are nonempty subsets of S, then we have

(1) A(A]J for all AJ

(2) ((A]J]J=(A]J

(1) If ABJ, then (A]J(B]J

(2) (A]JΓ(B]J(AΓB]J

An element a of an ordered Γ-semihypergroup (S,Γ,) is regular [25] if there exist xS and α,βΓ such that aaαxβa. This is equivalent to saying that a(aΓSΓa], for each aS. An ordered Γ-semihypergroup S is said to be regular if every element of S is a regular element.

Definition 2.2. Let (S,Γ,) be an ordered Γ-semihypergroup. A nonempty subset A of S is called a left (resp. right) Γ-hyperideal [24] of S if it satisfies the following conditions:

(1) SΓAA (resp. AΓSA)

(2) If xA, yS, and yx, then yA

If A is both a left Γ-hyperideal and a right Γ-hyperideal of S, then it is called a Γ-hyperideal (or two-sided Γ-hyperideal) of S.

3. Basic Properties of Relative Bi-Γ-Hyperideals (J-Bi-Γ-Hyperideals)

Let (S,Γ,) be an ordered Γ-semihypergroup and and J be the nonempty subsets of S. Then, is called a left J-Γ-hyperideal of S if it satisfies the following conditions:

(1) JΓ

(2) When xJ and y such that xy, it implies that x

A right J-Γ-hyperideal of an ordered Γ-semihypergroup S is defined in a similar way. By two-sided J-Γ-hyperideal or simply J-Γ-hyperideal, we mean a nonempty subset of S which is both left and right J-Γ-hyperideal of S.

Definition 3.1. Let J1 and J2 be the nonempty subsets of S. A nonempty subset of S is said to be an (J1,J2)-Γ-hyperideal of S if it satisfies the following conditions:

(1) J1Γ and ΓJ2

(2) When xJ1J2 and y such that xy, it implies that x

Example 2. Let S={a,b,c,d,e,f} and Γ={γ,β} be the sets of binary hyperoperations defined as follows:

www.frontiersin.org

Then, S is a Γ-semihypergroup [23]. We have (S,Γ,) as an ordered Γ-semihypergroup, where the order relation is defined by

:={(a,a),(b,a),(b,b),(b,c),(b,f),(c,c),(d,d),(e,e),(f,f)}.

The covering relation and the figure of S are given by

={(b,a),(b,c),(b,f)}.

www.frontiersin.org

Let J1={a,b,c},J2={b,d,f}, and ={a,b,f}. One can check that is a (J1,J2)-Γ-hyperideal of S.Here are some elementary properties of these concepts.

Lemma 3.2. Let (S,Γ,) be an ordered Γ-semihypergroup. If J is a sub Γ-semihypergroup of S, then (JΓaΓJ]J is a J-Γ-hyperideal of S for each aS.

Proof. Let aS. We show that (JΓaΓJ]J is a J-Γ-hyperideal of S. We have

JΓ(JΓaΓJ]J=(J]JΓ(JΓaΓJ]J(JΓ(JΓaΓJ)]J=(JΓJ(ΓaΓJ)]J(JΓaΓJ]J.

Similarly, we have (JΓaΓJ]JΓJ(JΓaΓJ]J. Now, let xJ and y(JΓaΓJ]J such that xy. Then, xyz for some zJΓaΓJ. Hence, xz and so x(JΓaΓJ]J. Therefore, (JΓaΓJ]J is a J-Γ-hyperideal of S.

Theorem 3.3. Let I be a left Γ-hyperideal of an ordered Γ-semihypergroup (S,Γ,) and JS such that IJ. If M is an idempotent left J-Γ-hyperideal of I, then M is a left Γ-hyperideal of S.

Proof. Clearly, I is an ordered sub-Γ-semihypergroup of S. We have

SΓM=SΓ(MΓM]=(S]Γ(MΓM](SΓ(MΓM)]=((SΓM)ΓM]((SΓI)ΓM](IΓM](JΓM](M]J=M.

If mMI and xS such that xm, then we have xIJ. Since M is a left J-Γ-hyperideal of I, it follows that xM. Hence, M is a left Γ-hyperideal of S. ∎

Theorem 3.4. Let I be a Γ-hyperideal of a regular ordered Γ-semihypergroup (S,Γ,) and J1, J2S such that IJ1J2. Then, any (J1,J2)-Γ-hyperideal of I is a Γ-hyperideal of S.

Proof. Let I be a Γ-hyperideal of S. Then, IΓISΓII. So, I is an ordered sub-Γ-semihypergroup of S. Let A(AI) be a (J1,J2)-Γ-hyperideal of I. We prove that A is a Γ-hyperideal of S. Let iIS. Then, there exist xS and α,βΓ such that

iiαxβiiαxβ(iαxβi)=iα(xβiαx)βi.

Since I is a Γ-hyperideal of S, it follows that

xβiαx(SΓI)ΓSIΓSI.

Hence, it, for some tiα(xβiαx)βiiΓIΓi. This means that i(iΓIΓi]I. Therefore, I is a regular ordered sub-Γ-semihypergroup of S.

Let aA(AI) and sS. Then, aγsI where γΓ. Now, suppose that vaγsI. Then, there exist yI and λ,μΓ such that

vvλyμv(aγs)λyμ(aγs)AΓ(SΓIΓI)ΓSAΓIΓSAΓIAΓ(J1J2)AΓJ2A.

Similarly, we have sγaA.

If uAI and vS such that vu, then we have uIJ1J2J1J1J2. Since A is a (J1,J2)-Γ-hyperideal of I, it follows that vA. Hence, A is a Γ-hyperideal of S.

Let a be an element of an ordered Γ-semihypergroup (S,Γ,). We denote by L1(a) (resp. R2(a), IJ(a)) the left (resp. right, two-sided) relative Γ-hyperideal of S generated by a. The intersection of all (J1,J2)-Γ-hyperideals of S containing the element a is denoted by IJ(a), where J=J1J2.

Lemma 3.5. Let a be an element of an ordered Γ-semihypergroup (S,Γ,) and J1 and J2 be two sub-Γ-semihypergroups of S. Then,

(1) L1(a)=(aJ1Γa]J1

(2) R2(a)=(aaΓJ2]J2

(3) IJ(a)=(aJ1ΓaaΓJ2J1ΓaΓJ2]J, where J=J1J2

Proof. Since aL1(a) and J1ΓaL1(a), it follows that (aJ1Γa]J1L1(a). We have

J1Γ(aJ1Γa]J1(J1]J1Γ(aJ1Γa]J1(J1Γ(aJ1Γa)]J1(J1Γa]J1(aJ1Γa]J1.

On the contrary, we have (L1(a)]J1=L1(a). Thus, L1(a)=(aJ1Γa]J1 is a left J1-Γ-hyperideal of S containing a. This means that L1(a)(aJ1Γa]J1.

Now, we show that L1(a) is the smallest left J1-Γ-hyperideal of S containing a. Suppose that A is a left J1-Γ-hyperideal of S containing a. We have

L1(a)=(aJ1Γa]J1(AJ1ΓA]J1(A]J1A.

This proves that (1) holds. Conditions (2) and (3) are proved similarly.

Let (S,Γ,) be an ordered Γ-semihypergroup and J1 and J2 be nonempty subsets of S. Then, S is said to be left J1-simple if it has no proper left J1-Γ-hyperideal. In the same way, we can define a right J2-simple ordered Γ-semihypergroup. If S is a left J1-simple and right J2-simple) ordered Γ-semihypergroup, then S is a (J1, J2)-simple ordered Γ-semihypergroup.

Lemma 3.6. Let (S,Γ,) be an ordered Γ-semihypergroup and J1 and J2 be sub-Γ-semihypergroups of S such that J=J1J2. Then, the following assertions hold:

(1) S is left J1-simple if and only if (J1Γa]J=S for each aS

(2) S is right J2-simple if and only if (aΓJ2]J=S for each aS

(3) S is (J1, J2)-simple if and only if (J1ΓaΓJ2]J=S for each aS

Proof. The proof is straightforward.

We continue this section with the following definition.

Definition 3.7. Let (S,Γ,) be an ordered Γ-semihypergroup and J a nonempty subset of S. A sub-Γ-semihypergroup of S is called a relative bi-Γ-hyperideal (J-bi-Γ-hyperideal) of S if the following conditions hold:

(1) ΓJΓ

(2) When aJ and b such that ab, it implies that a

Example 3. We come back to Example 2 and consider ordered Γ-semihypergroup (S,Γ,). Let ={a,b,c} and J={b,f}. It is a routine matter to verify that is a J-bi-Γ-hyperideal of S.

Lemma 3.8. The intersection of any family of J-bi-Γ-hyperideals of an ordered Γ-semihypergroup (S,Γ,) is a J-bi-Γ-hyperideal of S.

Proof. This proof is straightforward.

Let (S,Γ,) be an ordered Γ-semihypergroup and J be any nonempty subset of S. Then, S is said to be J-regular if, for each jJ, there exist xJ and α,βΓ such that jjαxβj.

Theorem 3.9. Let (S,Γ,) be an ordered Γ-semihypergroup and J a sub-Γ-semihypergroup of S. Then, the following assertions are equivalent:

(1) S is J-regular

(2) =(ΓJΓ]J for every J-bi-Γ-hyperideal (J) of S

Proof. (1)(2) Assume that (1) holds. Since is a J-bi-Γ-hyperideal of S, we get ΓJΓ. Thus, (ΓJΓ]J(]J=. Now, let b(J). Since S is J-regular, there exist xJ and α,βΓ such that

bbαxβbΓJΓ.

Hence, b(ΓJΓ]J and so (ΓJΓ]J. Therefore, =(ΓJΓ]J.(2)(1) Let be a right J-Γ-hyperideal and a left J-Γ-hyperideal of S. By routine checking, we can easily verify that and are J-bi-Γ-hyperideals of S. By Lemma 3.8, is a J-bi-Γ-hyperideal of S. By hypothesis, we have

=(()ΓJΓ()]J(ΓJΓ]J(Γ]J.

Let aJ. Since aR2(a) and aL1(a), it follows that aR2(a)L1(a). By (1) and (2) of Lemma 3.5, we have

a(R2(a)ΓL1(a)]J=((aaΓJ]JΓ(aJΓa]J]J=((aaΓJ)Γ(aJΓa)]J(aΓaaΓJΓa]J.

Then, aw, for some waΓaaΓJΓa. If waΓa, then aaγaaγ(aγa). So, a(aΓJΓa]J. Therefore, S is J-regular. If waΓJΓa, then aaδxλa, for some xJ and δ,λΓ. Thus, a(aΓJΓa]J. Therefore, S is J-regular.

Definition 3.10. Let (S,Γ,) be an ordered Γ-semihypergroup and J1 and J2 be the nonempty subsets of S. A sub-Γ-semihypergroup of S is said to be a (J1,J2)-bi-Γ-hyperideal of S if it satisfies the following conditions:

(1) Γ(J1J2)Γ

(2) When aJ1J2 and b such that ab, it implies that a

Example 4. Let S={e,a,b,c,d} and Γ={γ,β} be the sets of binary hyperoperations defined as follows:

www.frontiersin.org

Then, S is a Γ-semihypergroup [23]. We have (S,Γ,) as an ordered Γ-semihypergroup, where the order relation is defined by

:={(a,a),(b,a),(b,b),(c,c),(d,c),(d,d),(e,e)}.

The covering relation and the figure of S are given by

={(b,a),(d,c)}.

www.frontiersin.org

Let ={e,a,b}, J1={c}, and J2={d}. It is easy to see that is a (J1,J2)-bi-Γ-hyperideal of S.

The concept of J-bi-Γ-hyperideals of an ordered Γ-semihypergroup is a generalization of the concept of J-Γ-hyperideals (left J-Γ-hyperideals and right J-Γ-hyperideals) of an ordered Γ-semihypergroup. Obviously, every left (right) J-Γ-hyperideal of an ordered Γ-semihypergroup S is a J-bi-Γ-hyperideal of S, but the following example shows that the converse is not true in the general case.

Example 5. Consider the ordered Γ-semihypergroup (S,Γ,) given in Example 2. It is easy to check that ={a,b,c} is a J-bi-Γ-hyperideal on S, where J={b,f}, but it is not a right J-Γ-hyperideal on S. Since c and fJ, but cγf={a,f}{a,b,c} which implies that ΓJ does not hold.

Theorem 3.11. Let (S,Γ,) be an ordered Γ-semihypergroup and J1 and J2 be two sub-Γ-semihypergroups of S. Then, S is left J1-simple and right J2-simple if and only if S does not contain proper (J1,J2)-bi-Γ-hyperideals.

Proof. Let S be a left J1-simple and right J2-simple ordered Γ-semihypergroup and B a (J1,J2)-bi-Γ-hyperideal of S. It is sufficient to prove that SB. Consider sS and bB. Since S is left J1-simple, we obtain

S=L1(b)=(bJ1Γb]J1,

by Lemma 3.5. We need to consider only two cases:

Case 1. Let sb. As B is (J1,J2)-bi-Γ-hyperideal, we have sB.

Case 2. Let sj1γb, for some j1J1 and γΓ. By assumption, S is a right J2-simple ordered Γ-semihypergroup. Therefore,

S=R2(b)=(bbΓJ2]J2,

by Lemma 3.5. Since j1J1S, we have j1b or j1(bλj2]J2 for some j2J2 and λΓ. By Lemma 3.6, we have

S=(bΓJ2]J=(J1Γb]J,

and so

b(bΓJ2]J(bΓ(J1Γb]J]J(bΓJ1Γb]J(bΓJΓb]J,

where J=J1J2. Hence, S is a J-regular ordered Γ-semihypergroup. So, there exist xJ and α,βΓ such that bbαxβb. We now turn to the case j1b. From this, we conclude that j1γbbγb. So, we obtain

(j1γb]J(bγb]J(bγ(bαxβb)]J(bγbα(xβb)]J(BΓJΓB]J(B]J=B.

Since sj1γb, it follows that sB. This gives SB. If j1(bλj2]J2, then

(j1γb]J((bλj2]J2γb]J(BΓJΓB]J(B]J=B,

and so sB. We thus get SB.

Conversely, suppose that S does not contain proper (J1,J2)-bi-Γ-hyperideals. Let M be a left J1-Γ-hyperideal and right J2-Γ-hyperideal of S. We have

MΓ(J1J2)ΓM=MΓJ1ΓMMΓJ2ΓMMΓMM.

Hence, M is a (J1,J2)-bi-Γ-hyperideal of S. By assumption, we have S=M. Therefore, S is a left J1-simple and right J2-simple ordered Γ-semihypergroup.

4. Relative (Bi)-Int-Γ-Hyperideals (J-(Bi)-Int-Γ-Hyperideals)

Definition 4.1. Let (S,Γ,) be an ordered Γ-semihypergroup and J a nonempty subset of S. A sub-Γ-semihypergroup CI of S is called a J-int-Γ-hyperideal of S if the following conditions hold:

(1) JΓCIΓJCI

(2) If xJ, yCI, and xy, then xCI

It is not difficult to see that every J-Γ-hyperideal of an ordered Γ-semihypergroup S is a J-int-Γ-hyperideal of S. The following example shows that the converse is not true in general.

Example 6. Let S={a,b,c,d} and Γ={γ}. We define

www.frontiersin.org

Then, S is a Γ-semihypergroup. We have (S,Γ,) is an ordered Γ-semihypergroup, where the order relation is defined by

:={(a,a),(a,b),(a,c),(a,d),(b,b),(c,c),(d,d)}.

The covering relation and the figure of S are given by

={(a,b),(a,c),(a,d)}.

www.frontiersin.org

Let CI={a,c} and J={b,d}. Here, CI is a J-int-Γ-hyperideal of S, but not a J-Γ-hyperideal of S. Indeed, since JΓCI={a,b}CI, it follows that CI is not a J-Γ-hyperideal of S.The following example shows that a J-regular ordered Γ-semihypergroup is not regular in general.

Example 7. Consider the Γ-semihypergroup (S,Γ) given in Example 6. Let be the relation on S defined as follows:

:={(a,a),(b,a),(b,b),(b,c),(c,c),(d,d)}.

Then, (S,Γ,) is an ordered Γ-semihypergroup. The covering relation and the figure of S are given by:

={(b,a),(b,c)}.

www.frontiersin.org

Let J={a,b}S. An easy computation shows that S is a J-regular ordered Γ-semihypergroup, but it is clearly not regular.

Theorem 4.2. Let (S,Γ,) be a J-regular ordered Γ-semihypergroup, where J is a nonempty subset of S. Then, every J-int-Γ-hyperideal CI(CIJ) of S is a J-Γ-hyperideal of S.

Proof. Let CI be a J-int-Γ-hyperideal of S. Then, CI is a sub-Γ-semihypergroup of S and (CI]J=CI. Let aCIJ. Since S is J-regular, there exist xJ and α,βΓ such that aaαxβa. Now, let jJ and γΓ. Then,

jγajγ(aαxβa)=(jγaαx)βa(JΓCIΓJ)ΓCICIΓCICI.

Thus, JΓCI(CI]J=CI. By a similar argument, we have CIΓJCI. Hence, the result follows.In the following, we introduce the notion of J-bi-int-Γ-hyperideals as a generalization of J-Γ-hyperideals, J-bi-Γ-hyperideals, and J-int-Γ-hyperideals of ordered Γ-semihypergroups.

Definition 4.3. Let (S,Γ,) be an ordered Γ-semihypergroup and J a nonempty subset of S. A sub-Γ-semihypergroup Dbi of S is called a J-bi-int-Γ-hyperideal of S if the following conditions hold:

(1) JΓDbiΓJDbiΓJΓDbiDbi

(2) If xJ, yDbi, and xy, then xDbi

Lemma 4.4. Let (S,Γ,) be an ordered Γ-semihypergroup and J a nonempty subset of S. Then, the following statements hold:

(1) Every J-Γ-hyperideal of S is a J-bi-int-Γ-hyperideal of S

(2) The intersection of J-bi-int-Γ-hyperideals of S is a J-bi-int-Γ-hyperideal of S

(3) If B is a J-bi-Γ-hyperideal and C a J-int-Γ-hyperideal of S, then Dbi=BC is a J-bi-int-Γ-hyperideal of S

Proof. (1) Let A be a J-Γ-hyperideal of S. Then, JΓAA and AΓJA. We have

JΓAΓJAΓJΓAAΓJΓAAΓAA,

and

JΓAΓJAΓJΓAJΓAΓJAΓJA.

Therefore, A is a J-bi-int-Γ-hyperideal of S.

(2) The proof is similar to the proof of Lemma 3.8.

(3) Clearly, Dbi=BC is a sub-Γ-semihypergroup of S. We have

DbiΓJΓDbi=(BC)ΓJΓ(BC)BΓJΓBB,

and

JΓDbiΓJ=JΓ(BC)ΓJJΓCΓJC.

Hence, JΓDbiΓJDbiΓJΓDbiBC=Dbi. Therefore, Dbi=BC is a J-bi-int-Γ-hyperideal of S.

Example 8. Let S={a,b,c,d,e} and Γ={γ}. We define

www.frontiersin.org

Then, S is a Γ-semihypergroup. We have (S,Γ,) as an ordered Γ-semihypergroup [29], where the order relation is defined by

:={(a,a),(b,b),(c,c),(d,d),(e,e),(a,c),(b,a),(b,c),(d,a),(d,c),(e,a),(e,c)}.

The covering relation and the figure of S are given by

={(a,c),(b,a),(d,a),(e,a)}.

www.frontiersin.org

Let Dbi={a,b,d} and J={c,d}. Then, JΓDbiΓJ={a,c,d} and DbiΓJΓDbi={a,b,d}, so {a,c,d}{a,b,d}={a,d}Dbi. Hence, Dbi is a J-bi-int-Γ-hyperideal of S. It is easy to see that Dbi (DbiΓJDbi) is not a J-Γ-hyperideal of S.

5. Conclusion

In this article, we studied some properties of J-(bi-)Γ-hyperideals of ordered Γ-semihypergroups. In particular, we introduced and studied J-int-Γ-hyperideals and J-bi-int-Γ-hyperideals. Furthermore, we proved that J-int-Γ-hyperideals and J-Γ-hyperideals coincide in J-regular ordered Γ-semihypergroups. When we deal with J-(bi-)Γ-hyperideals of ordered Γ-semihypergroups, it is natural to talk about fuzzy J-(bi-)Γ-hyperideal. According to the research results, it is suggested to define and investigate some properties of fuzzy J-(bi-)Γ-hyperideals, rough prime J-bi-Γ-hyperideals, fuzzy prime J-bi-Γ-hyperideals, and uni-soft J-int-Γ-hyperideals in ordered Γ-semihypergroups. As an application of the results of this article, the corresponding results of ordered semihypergroups can be also obtained by moderate modification.

Data Availability Statement

All datasets presented in this study are included in the article/supplementary material.

Author Contributions

All authors have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This work was supported by the National Key R&D Program of China (Grant no. 2019YFA0706402), the Natural Science Foundation of Guangdong Province (Grant no. 2018A0303130115), and Guangzhou Academician and Expert Workstation (Grant no. 20200115-9).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

References

1. Dehghan Nezhad, A, Moosavi Nejad, SM, Nadjafikhah, M, and Davvaz, B. A physical example of algebraic hyperstructures: leptons. Indian J Phys. (2012). 86(11):1027–32. doi: 10.1007/s12648-012-0151-x

CrossRef Full Text | Google Scholar

2. Davvaz, B, Nezhad, AD, and Nejad, SMM. Algebraic hyperstructure of observable elementary particles including the Higgs boson. Proc Natl Acad Sci, India, Sect A Phys Sci. (2020). 90:169–76. doi: 10.1007/s40010-018-0553-z

CrossRef Full Text | Google Scholar

3. Yaqoob, N, Tang, J, and Chinram, R. Structures of involution Γ-semihypergroups. Honam Mathematical J (2018). 40(1):109–24. doi: 10.5831/HMJ.2018.40.1.109

Google Scholar

4. Khan, A, Farooq, M, and Yaqoob, N. Uni-soft structures applied to ordered Γ-semihypergroups. Proc Natl Acad Sci, India, Sect A Phys Sci. (2020). 90:457. doi: 10.1007/s40010-019-00602-x

CrossRef Full Text | Google Scholar

5. Marty, F. Sur une generalization de la notion de groupe. In: Proceedings of the 8th Congress Math. Scandinaves, Stockholm, Sweden; Stockholm, Sweden (1934). p. 45–49.

Google Scholar

6. Corsini, P. Prolegomena of hypergroup theory. 2nd ed. Udine, Italy: Aviani Editore (1993).

Google Scholar

7. Corsini, P, and Leoreanu, V. Applications of hyperstructure theory. In: Advances in mathematics. Dordrecht, Netherlands: Kluwer Academic Publishers (2003).

Google Scholar

8. Davvaz, B. Semihypergroup theory. New York, NY: Elsevier (2016).

Google Scholar

9. Davvaz, B, and Leoreanu-Fotea, V. Hyperring theory and applications. Palm Harbor, FL: International Academic Press (2007).

Google Scholar

10. Davvaz, B, and Vougiouklis, T. A walk through weak hyperstructures-Hυ-structures. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. (2019).

Google Scholar

11. Vougiouklis, T. Hyperstructures and their representations. Palm Harbor, FL: Hadronic Press Inc. (1994).

Google Scholar

12. Heidari, D, and Davvaz, B. On ordered hyperstructures. Politehn Univ Bucharest Sci Bull Ser A Appl Math Phys. (2011). 73(2):85–96.

Google Scholar

13. Davvaz, B, Corsini, P, and Changphas, T. Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder. Eur J Combinator. (2015). 44:208–17. doi: 10.1016/j.ejc.2014.08.006

CrossRef Full Text | Google Scholar

14. Gu, Z, and Tang, X. Ordered regular equivalence relations on ordered semihypergroups. J Algebra. (2016). 450:384–97. doi: 10.1016/j.jalgebra.2015.11.026

CrossRef Full Text | Google Scholar

15. Tipachot, N, and Pibaljommee, B. Fuzzy interior hyperideals in ordered semihypergroups. Ital J Pure Appl Math. (2016). 36:859–70.

Google Scholar

16. Mahboob, A, Khan, NM, and Davvaz, B. (m, n)-hyperideals in ordered semihypergroups. Categ General Alg Structures Appl. (2020). 12(1):43–67. doi: 10.29252/cgasa.12.1.43

CrossRef Full Text | Google Scholar

17. Good, RA, and Hughes, DR. Associated groups for a semigroup. Bull Am Math Soc. (1952). 58:624–5.

Google Scholar

18. Wallace, A. Relative ideals in semigroups, I (Faucett’s theorem). Colloq Math. (1962). 9:55–61. doi: 10.4064/cm-9-1-55-61.

CrossRef Full Text | Google Scholar

19. Hrmová, R. Relative ideals in semigroups. Matematický časopis (1967). 17(3):206–23.

Google Scholar

20. Khan, NM, and Ali, MF. Relative bi-ideals and relative quasi ideals in ordered semigroups. Hacet J Math Stat (2020). 49(3):950–61. doi: 10.15672/hujms.624046

CrossRef Full Text | Google Scholar

21. Sen, MK, and Saha, NK. On Γ-semigroup I. Bull Calcutta Math Soc. (1986). 78:180–6.

Google Scholar

22. Anvariyeh, SM, Mirvakili, S, and Davvaz, B. On Γ-hyperideals in Γ-semihypergroups. Carpathian J Math. (2010). 26(1):11–23.

Google Scholar

23. Yaqoob, N, and Aslam, M. Prime (m, n) Bi-Γ -hyperideals in Γ-semihypergroups. Appl Math Inf Sci. (2014). 8(5):2243–9. doi: 10.12785/amis/080519

CrossRef Full Text | Google Scholar

24. Omidi, S, and Davvaz, B. Bi-Γ-hyperideals and Green’s relations in ordered Γ-semihypergroups. Eurasian Mathematical J (2017). 8(4):63–73.

Google Scholar

25. Omidi, S, Davvaz, B, and Hila, K. Characterizations of regular and intra-regular ordered Γ-semihypergroups in terms of bi-Γ-hyperideals. Carpathian Math Publ. (2019). 11(1):136–51. doi: 10.15330/cmp.11.1.136-151

CrossRef Full Text | Google Scholar

26. Basar, A, Abbasi, MY, and Bhavanari, S. On generalized Γ-hyperideals in ordered Γ-semihypergroups. Fundam J Math Appl. (2019). 2(1):18–23. doi: 10.33401/fujma.543712

Google Scholar

27. Tang, J, Davvaz, B, Xie, X-Y, and Yaqoob, N. On fuzzy interior Γ-hyperideals in ordered Γ-semihypergroups. J Intell Fuzzy Syst. (2017). 32:2447–60. doi: 10.3233/jifs-16431

CrossRef Full Text | Google Scholar

28. Omidi, S, Davvaz, B, and Abdiogluv, C. Some properties of quasi-Γ-hyperideals and hyperfilters in ordered Γ-semihypergroups. Southeast Asian Bull Math. (2018). 42(2):223–42.

Google Scholar

29. Omidi, S, and Davvaz, B. Convex ordered Gamma-semihypergroups associated to strongly regular relations. Matematika (2017). 33(2):227–40. doi: 10.11113/matematika.v33.n2.838

CrossRef Full Text | Google Scholar

Keywords: ordered -semihypergroup, J -bi- -hyperideal, J -(bi-)int- -hyperideal, J -regular AMS mathematics subject classification: 16Y99, 20N20, 06F99

Citation: Rao Y, Xu P, Shao Z, Kosari S and Omidi S (2020) Some Properties of Relative Bi-(Int-)Γ-Hyperideals in Ordered Γ-Semihypergroups. Front. Phys. 8:555573. doi: 10.3389/fphy.2020.555573

Received: 25 April 2020; Accepted: 24 August 2020;
Published: 05 November 2020.

Edited by:

Muhammad Javaid, University of Management and Technology, Lahore, Pakistan

Reviewed by:

Kazuharu Bamba, Fukushima University, Japan
Naveed Yaqoob, Majmaah University, Saudi Arabia

Copyright © 2020 Rao, Xu, Shao, Kosari and Omidi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: S. Kosari, saeedkosari38@yahoo.com