AFS Advances in Fuzzy Systems 1687-711X 1687-7101 Hindawi 10.1155/2019/3693926 3693926 Research Article On Fuzzy Ordered Hyperideals in Ordered Semihyperrings http://orcid.org/0000-0001-8243-3050 Kazancı O. 1 http://orcid.org/0000-0002-0282-9483 Yılmaz Ş. 1 Davvaz B. 2 Dvorák Antonin 1 Department of Mathematics Karadeniz Technical University 61080 Trabzon Turkey ktu.edu.tr 2 Department of Mathematics Yazd University Yazd Iran yazd.ac.ir 2019 322019 2019 28 05 2018 12 12 2018 322019 2019 Copyright © 2019 O. Kazancı et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce the concept of fuzzy ordered hyperideals of ordered semihyperrings, which is a generalization of the concept of fuzzy hyperideals of semihyperrings to ordered semihyperring theory, and we investigate its related properties. We show that every fuzzy ordered quasi-hyperideal is a fuzzy ordered bi-hyperideal, and, in a regular ordered semihyperring, fuzzy ordered quasi-hyperideal and fuzzy ordered bi-hyperideal coincide.

1. Introduction

The theory of algebraic hyperstructures is a well-established branch of classical algebraic theory which was initiated by Marty . Since then many researchers have worked on algebraic hyperstructures and developed it [2, 3]. A short review of this theory appears in .

The notion of semiring was introduced by Vandiver  in 1934, which is a generalization of rings. Semirings are very useful for solving problems in graph theory, automata theory, coding theory, analysis of computer programs, and so on. We refer to  for the information we need concerning semiring theory. In , quasi-ideals of semirings are studied and some properties and related results are given. In , Vougiouklis generalized the notion of hyperring and named it as semihyperring, where both the addition and multiplication are hyperoperations. Semihyperrings are a generalization of Krasner hyperrings. Davvaz, in , studies the notion of semihyperring in a general form. Ameri and Hedayati define k-hyperideals in semihyperrings in . In 2011, Heidari and Davvaz  studied a semihypergroup (H,) with a binary relation , where is a partial order so that the monotony condition is satisfied. This structure is called an ordered semihypergroup. Properties of hyperideals in ordered semihypergroups are studied in . Also, the properties of fuzzy hyperideals in an ordered semihypergroup are investigated in [18, 19]. Yaqoop and Gulistan  study the concept of ordered LA-semihypergroup. In , Davvaz and Omidi introduce the basic notions and properties of ordered semihyperrings and prove some results in this respect. In 2018, Omidi and Davvaz  studied on special kinds of hyperideals in ordered semihyperrings. Some properties of hyperideals in ordered Krasner hyperrings can be found in .

After the introduction of fuzzy sets by Zadeh , reconsideration of the concept of classical mathematics began. Because of the importance of group theory in mathematics, as well as its many areas of application, the notion of fuzzy subgroup is defined by Rosenfeld  and its structure is investigated. This subject has been studied further by many others [26, 27]. Fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now used in the world both on the theoretical point of view and for their many applications. There exists a rich bibliography: publications that appeared within 2015 can be found in “Fuzzy Algebraic Hyperstructures - An Introduction” by Davvaz and Cristea . Recently, many researchers have considered fuzzification on many algebraic structures, for example, on semigroups, rings, semirings, near-rings, ordered semigroups, semihypergroups, ordered semihypergroups, and ordered hyperrings .

Inspired by the study on ordered semihyperrings, we study the concept of fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples in this respect. The rest of this paper is organized as follows. In the second section, we review basic concepts regarding the ordered hyperstructures. The third section is dedicated to the fuzzy ordered hyperideals and some properties. In Section 4, we introduce the concept of fuzzy ordered quasi-hyperideals and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples. We give the main theorems which characterize the ordered hyperideals, ordered quasi-hyperideals, and ordered bi-hyperideals in terms of fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals, respectively.

2. Terminology and Basic Properties

In what follows, we summarize some basic notions and facts about semihypergroups, semihyperrings, and ordered semihyperrings.

Let H be a nonempty set and let P(H) be the set of all nonempty subsets of H. A hyperoperation on H is a map :H×HP(H) and the pair (H,) is called a hypergroupoid. For any xH and A,BP(H), we denote (1)AB=xyxA,yB,xA=xA,Ax=Ax.A hypergroupoid (H,) is called a semihypergroup if for all x,y,zH we have (xy)z=x(yz), which means that(2)uxyuz=vyzxvWe say that a semihypergroup (H,) is a hypergroup if, for all xH, we have xH=Hx=H.

Definition 1 (see [<xref ref-type="bibr" rid="B36">8</xref>, <xref ref-type="bibr" rid="B11">21</xref>]).

A semihyperring is an algebraic hyperstructure (R,+,·) which satisfies the following axioms:

(R,+) is a commutative semihypergroup

(R,·) is a semihypergroup

x·(y+z)=x·y+x·z and (x+y)·z=x·z+y·z for all x,y,zR

Let (R,+,·) be a semihyperring. If there exists an element 0R such that x+0=0+x=x and x·0=0·x=0 for all xR, then 0 is called the zero element of R. Throughout this paper we consider a semihyperring (R,+,·) with zero element 0.

A semihyperring R is called commutative if (R,·) is a commutative semihypergroup.

Definition 2 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

A nonempty subset A of a semihyperring (R,+,·) is called a subsemihyperring of R if, for all x,yA, x+yA and x·yA.

A nonempty subset I of a semihyperring (R,+,·) is called a hyperideal of (R,+,·) if, for all x,yI, rR, x+yI and r·xI and x·rI.

Definition 3 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

An ordered semihyperring (R,+,·,) is a semihyperring equipped with a partial order relation such that for all a,b,cR we have the following.

ab implies a+cb+c, meaning that, for any xa+c, there exists yb+c such that xy.

ab and 0c imply a·cb·c, meaning that, for any xa·c, there exists yb·c such that xy. The case c·ac·b is defined similarly.

Note that the concept of ordered semihyperring is a generalization of the concept of ordered semiring.

Semihyperrings are viewed as ordered semihyperrings under the equality order relation . Indeed, let (R,+,·) be a semihyperring. Define the order relation on R by {(a,b)a=b}. Then (R,+,·,) is an ordered semihyperring.

Let A be a nonempty subset of an ordered semihyperring R. Then the set {xRxaforsomeaA} is denoted by the notation (A]. For A={a}, we write (a] instead of ({a}]. An ordered semihyperring (A,+,·,) is an ordered subsemihyperring of (R,+,·,) if A is a subsemihyperring of R and the order on A is the restriction of the order on R. Let (R,+,·) and (S,,) be semihyperrings. A mapping φ:RS is said to be strong homomorphism if φ(x+y)φ(x)φ(y) and φ(x·y)=φ(x)φ(y) for all x,yR. A homomorphism of ordered semihyperrings φ:(R,+,·,1)(S,,,2) is a semihyperring homomorphism such that, for all a,bR, a1b implies φ(a)2φ(b).

Definition 4 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

Let (R,+,·,) be an ordered semihyperring. A nonempty subset I of R is called an ordered hyperideal of R if it satisfies the following conditions:

x+yI for all x,yI

r·xI and x·rI for all xI and rR

If xI and Rrx, then rI

It is clear that {0} and R are ordered hyperideals of R.

Example 5.

Let R=0,a,b,c and let the hyperoperations “” and “” on R be defined as follows: (3) 0 a b c 0 0 a b c a a a 0 , a , b 0 , a , c b b 0 , a , b 0 , b 0 , b , c c c 0 , a , c 0 , b , c 0 , c 0 a b c 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 , a c 0 0 0 , a 0 , b Then, (R,,) is a semihyperring . It is easy to see that I={0,a}, J={0,a,b} are hyperideals of R. K={0,a,c} is not a hyperideal of R.

Definition 6 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

Let (R,+,·,) be an ordered semihyperring. A nonempty subset A of R is called an ordered bi-hyperideal of R if it satisfies the following conditions:

A is a subsemihyperring of R

A·R·AA

If xA and Rrx, then rA

Definition 7 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

Let (R,+,·,) be an ordered semihyperring. A nonempty subset Q of R is called an ordered quasi-hyperideal of R if it satisfies the following conditions:

Q+QQ

(Q·R](R·Q]Q

If xQ and Rrx, then rQ

Example 8.

Consider the semihyperring defined in Example 5. Then R,,, is an ordered semihyperring where the order relation is defined by (4)0,0,a,a,b,b,c,c,0,a,0,b,0,c.The covering relation is given by (5)=0,a,0,b,0,c.Now, it is easy to see that A=0,b is an ordered bi-hyperideal of R but it is not an ordered quasi-hyperideal of R.

The concept of a fuzzy subset of a nonempty set first was introduced by Zadeh in 1965 . Let X be a nonempty set. A fuzzy subset μ of X is a function μ:X[0,1]. Let μ and λ be two fuzzy subsets of X; we say that μ is contained in λ and we write μλ, if μ(x)λ(x) for all xX, and μλ, μλ are defined by (μλ)(x)=minμx,λx and (μλ)(x)=maxμx,λx. The sets μt={xXtμ(x)} and μt>={xXt<μ(x)},t[0,1], are called a level subset and strong level subset of μ, respectively.

3. On Fuzzy Ordered Hyperideals in Ordered Semihyperrings

Notice that the relationships between fuzzy sets and algebraic hyperstructures have been already considered by many researchers [18, 28, 30, 3538]. Recently, ordered ideals in semirings and ordered ideals in Krasner hyperrings have been already considered by Gan and Jiang  and Davvaz and Loeranau-Fotea , respectively. So, it is interesting to study fuzzy ordered hyperideals of ordered semihyperrings.

Definition 9.

Let (R,+,·,) be an ordered semihyperring and let μ be a fuzzy subset of R. μ is called a fuzzy ordered hyperideal of R if the following conditions hold:

minμx,μyinfzx+yμz for all x,yR

maxμx,μyinfzx·yμz for all x,yR

xyμ(x)μ(y) for all x,yR

Example 10.

Let R=0,a,b,c and the hyperoperations “” and “” on R be defined as follows: (6) 0 a b c 0 0 a b c a a a a b b b a 0 , b 0 , b , c c c a 0 , b , c 0 , c 0 a b c 0 0 0 0 0 a 0 a 0 , b 0 b 0 0 0 0 c 0 0 , c 0 0 Then, (R,,) is a semihyperring . Now, the order relation on R is defined by (7)0,0,a,a,b,b,c,c,0,a,0,b,0,c,b,a,c,a.The covering relation is given by (8)=0,b,0,c,b,a,c,aThen, (R,,,) is an ordered semihyperring. Let μ:R[0,1] be defined by (9)μ0=1,μa=0.1,μb=0.2,μc=0.3.Then, μ is a fuzzy ordered hyperideal of R.

Lemma 11.

Any hyperideal of an ordered semihyperring (R,+,·,) can be realized as a level subset of some ordered fuzzy hyperideals of R.

Proof.

Proof is similar to Lemma 4.2 in .

Notice that the characteristic function of a nonempty subset I of an ordered semihyperring R is a fuzzy ordered hyperideal of R if and only if I is an ordered hyperideal of R.

Theorem 12.

A fuzzy subset μ of an ordered semihyperring R is a fuzzy ordered hyperideal of R if and only if the set μt() is an ordered hyperideal of R for all t[0,1].

Proof.

Let μ be a fuzzy ordered hyperideal of R and t[0,1]. Let x,yμt. Then μ(x)t,μ(y)t. Now we have(10)infzx+yμzminμx,μyt.Therefore, for every zx+y, we have μ(z)t; that is, zμt, so x+yμt. Let xμt and rR. Then μ(x)t. So(11)tμxmaxμx,μrinfzr·xμzTherefore, for every zr·x, we have μ(z)t; that is, zμt, so r·xμt. Similarly, x·rμt. Now, let xμt and yR such that yx. Then μ(x)t. Since yx, it follows that μ(y)μ(x)t. This implies that yμt. By Definition 4, μt is an ordered hyperideal of R.

Conversely, let μ be a fuzzy subset of an ordered semihyperring R such that μt() is an ordered hyperideal of R for all 0t1. Let t0=minμx,μy for x,yR. Then obviously x,yμt0. Since every nonempty level set is an ordered hyperideal, x+yμt0. Thus (12)minμx,μy=t0infzx+yμz.Let t1=maxμx,μy for x,yR. Then obviously x,yμt1. Since every nonempty level set is an ordered hyperideal, x·yμt1. Then, we obtain μ(x·y)t1. Thus (13)t1=maxμx,μyμx·yfor all x,yR; that is, maxμx,μyinfzx·yμz.

Finally, let x,yR such that xy. Let t2=μ(y); then yμt2. Since μt2 is an ordered hyperideal of R, xμt2. Thus μ(x)t2=μ(y). This completes the proof.

Corollary 13.

Let μ be a fuzzy set with the upper bound t0 of an ordered semihyperring R. Then the following conditions are equivalent:

μ is a fuzzy ordered hyperideal of R

Each level subset μt, for t[0,t0], is an ordered hyperideal of R

Each strong level subset μt>, for t[0,t0], is an ordered hyperideal of R

Each level subset μt, for tIm(μ), is an ordered hyperideal of R, where Im(μ) denotes the image of μ

Each strong level subset μt>, for tIm(μ)-t0, is an ordered hyperideal of R

Each nonempty level subset of μ is an ordered hyperideal of R

Each nonempty strong level subset of μt> is an ordered hyperideal of R

Let φ be a mapping from an ordered semihyperring R1 to an ordered semihyperring R2. Let μ be a fuzzy subset of R1 and let λ be a fuzzy subset of R2. Then the inverseimageφ-1(λ) of λ is a fuzzy subset of R1 defined by φ-1(λ)(x)=λ(φ(x)) for all xR1. The image φ(μ) of μ is the fuzzy subset of R2 defined by (14)φμy=supxφ-1yμxif  φ-1y,0otherwisefor all yR2.

Lemma 14.

Let R1 and R2 be two ordered semihyperrings and let φ:R1R2 be a strong homomorphism.

If μ is a fuzzy ordered hyperideal of R1, then φ(μ) is a fuzzy ordered hyperideal of R2.

If λ is a fuzzy ordered hyperideal of R2, then φ-1(λ) is a fuzzy ordered hyperideal of R1.

Proof.

It is straightforward.

4. Fuzzy Ordered Bi-Hyperideals and Fuzzy Ordered Quasi-Hyperideals of Ordered Semihyperrings

In this section, we define the concepts of fuzzy ordered bi-hyperideal and fuzzy ordered quasi-hyperideal in ordered semihyperrings and give relationships between them.

Let (R,+,·,) be an ordered semihyperring and aR. We denote (15)Aa=b,cR×Rab·c. For fuzzy subsets μ and λ of a semihyperring R, we define the fuzzy subset μλ of R by letting aR; (16)μλa=supb,cAaminμb,λcif  Aa,0,otherwise.We denote the constant function 1:R[0,1] defined by 1(a)=1 for all aR .

Definition 15.

Let (R,+,·,) be an ordered semihyperring and let μ be a fuzzy subset of R. Then, μ is called a fuzzy ordered bi-hyperideal of R if the following conditions hold:

min{μ(x),μ(y)}infzx+yμz and minμx,μyinfzx·yμz for all x,yR

minμx,μzinfwx·y·zμw for all x,y,zR

xyμ(x)μ(y) for all x,yR

Theorem 16.

A fuzzy subset μ of an ordered semihyperring R is a fuzzy ordered bi-hyperideal of R if and only if the set μt() is an ordered bi-hyperideal of R for all t[0,1].

Proof.

The proof is similar to the proof of Theorem 12.

Definition 17.

Let (R,+,·,) be an ordered semihyperring and let μ be a fuzzy subset of R. Then, μ is called a fuzzy ordered quasi-hyperideal of R if the following conditions hold:

minμx,μyinfzx+yμz for all x,yR

(μ1)(1μ)μ

xyμ(x)μ(y) for all x,yR

The following theorem can be proved in a similar way in the proof of Theorem 4.8 of .

Theorem 18.

A fuzzy subset μ of an ordered semihyperring R is a fuzzy ordered quasi-hyperideal of R if and only if the set μt() is an ordered quasi-hyperideal of R for all t[0,1].

Lemma 19.

Let (R,+,·,) be an ordered semihyperring and let χI be the characteristic function of I. Then, we have the following:

I is an ordered bi-hyperideal of R if and only if χI is a fuzzy ordered bi-hyperideal of R

I is an ordered quasi-hyperideal of R if and only if χI is a fuzzy ordered quasi-hyperideal of R

Proof.

It is straightforward.

Theorem 20.

Let (R,+,·,) be an ordered semihyperring. Then, we have the following:

Every fuzzy ordered hyperideal of R is a fuzzy ordered quasi-hyperideal of R

Every fuzzy ordered quasi-hyperideal of R is a fuzzy ordered bi-hyperideal of R

Proof.

(i) Only we show that the condition (ii) of Definition 17 is satisfied.

Let μ be a fuzzy ordered hyperideal of R and aR. We have (17)μ11μa=minμ1a,1μa.If Aa=, then it is easy to see that (18)minμ1a,1μaμa.If Aa, then there exist x,yR such that ax·y. Then there exists zx·y such that az. Since μ is a fuzzy ordered hyperideal of R, we have(19)μaμzinfzx·yμzminμx,μy.That is, μ(a)minμx,μy. On the other hand, (20)μ11μa=minμ1a,1μa=minsupx,yAaminμx,1y,supx,yAamin1x,μy=minsupx,yAaμx,supx,yAaμy=supx,yAaminμx,μyμa.That is, the condition (ii) of Definition 17 is satisfied. Thus μ is a fuzzy ordered quasi-hyperideal of R.

(ii) Assume that μ is a fuzzy ordered quasi-hyperideal of R. We show that minμx,μyinfzx·yμz and minμx,μzinfwx·y·zμw for all x,y,zR. Let zx·y. We have (21)μ1z=supx,yAzminμx,1y=μx1μz=supx,yAzmin1x,μy=μy.Since μ is a fuzzy ordered quasi-hyperideal, we have (22)μz(μ11μz=minμ1z,1μz=minμx,μy.Hence zx·y; (23)infzx·yμzminμ1z,1μzminμx,μy.Similarly, (24)infwx·y·zμwminμ1w,1μwminμx,μz.Therefore μ is a fuzzy ordered bi-hyperideal of R.

The following example shows that the converse of Theorem 20 is not true in general.

Example 21.

Consider the ordered semihyperring R which is given in Example 8. Now, it is easy to see that A=0,b is an ordered bi-hyperideal of R, but it is not an ordered quasi-hyperideal of R. Let μ be a fuzzy subset of R defined by (25)μ0=0.7,μb=0.7,μa=0.4,μc=0.4.We have (26)μt=R,if  t0,0.4,0,b,if  t0.4,0.7,,if  t0.7,1.Since R and {0,b} are ordered bi-hyperideal of R, then μt is an ordered bi-hyperideal of R for all t[0,1]. Hence μ is a fuzzy ordered bi-hyperideal of R by Theorem 16. But it is not a fuzzy ordered quasi-hyperideal of R.

Definition 22 (see [<xref ref-type="bibr" rid="B11">21</xref>]).

An ordered semihyperring (R,+,·,) is called regular, if, for every aR, there exists xR such that aa·x·a.

Theorem 23.

Let (R,+,·,) be a regular ordered semihyperring and let μ be a fuzzy subset of R. Then, μ is a fuzzy ordered quasi-hyperideal of R if and only if μ is a fuzzy ordered bi-hyperideal of R.

Proof.

” Assume that μ is a fuzzy ordered quasi-hyperideal of R. It is clear that μ is a fuzzy ordered bi-hyperideal of R by Theorem 20 (ii).

” Only we show that the condition (ii) of Definition 17 is satisfied. Let aR. If Aa=, then it is easy to see that (μ1)(1μ)μ. Let Aa.

(1) If (μ1)(a)μ(a), then we have that μ(a)(μ1)(a)minμ1a,1μa; that is, (μ1)(1μ)μ.

(2) If (μ1)(a)>μ(a), then there exists at least one pair (z,w)Aa such that (27)minμz,1w=μz>μa.That is, z,wR, az·w, and μ(z)>μ(a). In this case we will prove that (1μ)(a)μ(a). Let (u,v)Aa; then au·v for some u,vR. Since R is regular and aR there exists xR such that aa·x·a. From aa·x·a, az·w, au·v we get az·w·x·u·v. Since μ is a fuzzy ordered bi-hyperideal R, we obtain that (28)μaμz·w·x·u·vinfαz·w·x·u·vμαminμz,μv.If minμz,μv=μ(z), then μ(a)μ(z). This contradicts with μ(z)>μ(a). Hence, minμz,μv=μ(v) and so μ(a)μ(v)=min1u,μv for all (u,v)Aa. Therefore μ(a)supu,vAamin1u,μv=(1μ)(a). As a result, (29)μ11μa=minμ1a,1μa=minμu,μv=μvμa.That is, (μ1)(1μ)μ.

Now, we have to prove the main characterization theorem for regular semihyperrings. We denote by Qx the ordered quasi-hyperideal of R generated by xxR.

Lemma 24.

Let R be an ordered semihyperring. Then the following conditions are equivalent:

R is regular

A=(ARA] for every ordered bi-hyperideal A of R

Q=(QRQ] for every ordered quasi-hyperideal Q of R

Proof.

( i ) ( i i ) Assume that (i) holds. Let A be any ordered bi-hyperideal of R and let x be any element of A. Since R is regular, there exists aR such that xx·a·x. Then it is easy to see that xx·a·x(ARA]. Hence A(ARA]. On the other hand, since A is an ordered bi-ideal of R, we have (ARA]A and so A=(ARA].

( i i ) ( i i i ) This proof is straightforward.

( i i i ) ( i ) Assume that Q=(QRQ] for every ordered quasi-hyperideal Q of R. Then (Q(x)RQ(x)]=Q(x)(xRx]. Thus x(xRx] implies that R is regular.

Theorem 25.

Let R be an ordered semihyperring. Then, the following conditions are equivalent:

R is regular

μμ1μ for every fuzzy ordered bi-hyperideal μ of R

μμ1μ for every fuzzy ordered quasi-hyperideal μ of R

Proof.

( i ) ( i i ) Assume that (i) holds. Let μ be any fuzzy ordered bi-hyperideal of R and aR. Since R is regular there exists xR such that aa·x·a. Hence (ax,a)Aa. (30)μ1μa=supu,vAaminμ1u,μvminaa·x·aμ1a·x,μx=minsupr,sAa·xminμr,1s,μaminμa,μa=μaTherefore μμ1μ.

( i i ) ( i i i ) This is straightforward from Theorem 20.

( i i i ) ( i ) Assume that (iii) holds and let Q be any ordered quasi-hyperideal of R. Then by Lemma 19, χQ is a fuzzy ordered quasi-hyperideal of R. Now χQχQ1χQ=χ(QRQ]. Therefore Q(QRQ]. On the other hand, since Q is any ordered quasi-hyperideal of R, we have (QRQ]Q and so Q=(QRQ]. By Lemma 24, R is regular.

5. Conclusion

In the structural theory of fuzzy algebraic systems, fuzzy ideals with special properties always play an important role. In this paper, we study fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring. We characterize regular ordered semihyperrings by the properties of these fuzzy hyperideals. As a further work, we will also concentrate on characterizations of different classes of ordered semihyperrings in terms of fuzzy interior hyperideals.

Data Availability

All data generated or analysed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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