/ / Article

Research Article | Open Access

Volume 2019 |Article ID 3693926 | 7 pages | https://doi.org/10.1155/2019/3693926

# On Fuzzy Ordered Hyperideals in Ordered Semihyperrings

Accepted12 Dec 2018
Published03 Feb 2019

#### Abstract

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 [1]. Since then many researchers have worked on algebraic hyperstructures and developed it [2, 3]. A short review of this theory appears in [48].

The notion of semiring was introduced by Vandiver [9] 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 [10] for the information we need concerning semiring theory. In [1113], quasi-ideals of semirings are studied and some properties and related results are given. In [8], 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 [14], studies the notion of semihyperring in a general form. Ameri and Hedayati define k-hyperideals in semihyperrings in [15]. In 2011, Heidari and Davvaz [16] studied a semihypergroup 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 [17]. Also, the properties of fuzzy hyperideals in an ordered semihypergroup are investigated in [18, 19]. Yaqoop and Gulistan [20] study the concept of ordered LA-semihypergroup. In [21], Davvaz and Omidi introduce the basic notions and properties of ordered semihyperrings and prove some results in this respect. In 2018, Omidi and Davvaz [22] studied on special kinds of hyperideals in ordered semihyperrings. Some properties of hyperideals in ordered Krasner hyperrings can be found in [23].

After the introduction of fuzzy sets by Zadeh [24], 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 [25] 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 [28]. 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 [2934].

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 be a nonempty set and let be the set of all nonempty subsets of . A hyperoperation on is a map and the pair is called a hypergroupoid. For any and , we denote A hypergroupoid is called a semihypergroup if for all we have , which means thatWe say that a semihypergroup is a hypergroup if, for all , we have

Definition 1 (see [8, 21]). semihyperring is an algebraic hyperstructure which satisfies the following axioms: (i) is a commutative semihypergroup(ii) is a semihypergroup(iii) and for all

Let be a semihyperring. If there exists an element such that and for all , then is called the zero element of . Throughout this paper we consider a semihyperring with zero element

A semihyperring is called commutative if is a commutative semihypergroup.

Definition 2 (see [21]). (i)A nonempty subset of a semihyperring is called a subsemihyperring of if, for all , and (ii)A nonempty subset of a semihyperring is called a hyperideal of if, for all , , and and .

Definition 3 (see [21]). An ordered semihyperring is a semihyperring equipped with a partial order relation such that for all we have the following. (i) implies , meaning that, for any , there exists such that .(ii) and imply , meaning that, for any , there exists such that . The case 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 [21]. Indeed, let be a semihyperring. Define the order relation on by Then is an ordered semihyperring.

Let be a nonempty subset of an ordered semihyperring . Then the set is denoted by the notation . For , we write instead of . An ordered semihyperring is an ordered subsemihyperring of if is a subsemihyperring of and the order on is the restriction of the order on . Let and be semihyperrings. A mapping is said to be strong homomorphism if and for all A homomorphism of ordered semihyperrings is a semihyperring homomorphism such that, for all , implies .

Definition 4 (see [21]). Let be an ordered semihyperring. A nonempty subset of is called an ordered hyperideal of if it satisfies the following conditions: (i) for all (ii) and for all and (iii)If and , then It is clear that and are ordered hyperideals of

Example 5. Let and let the hyperoperations “” and “” on be defined as follows: Then, is a semihyperring [35]. It is easy to see that , are hyperideals of . is not a hyperideal of .

Definition 6 (see [21]). Let be an ordered semihyperring. A nonempty subset of is called an ordered bi-hyperideal of if it satisfies the following conditions: (i) is a subsemihyperring of (ii)(iii)If and , then

Definition 7 (see [21]). Let be an ordered semihyperring. A nonempty subset of is called an ordered quasi-hyperideal of if it satisfies the following conditions:(i)(ii)(iii)If and , then

Example 8. Consider the semihyperring defined in Example 5. Then is an ordered semihyperring where the order relation is defined by The covering relation is given by Now, it is easy to see that is an ordered bi-hyperideal of but it is not an ordered quasi-hyperideal of .

The concept of a fuzzy subset of a nonempty set first was introduced by Zadeh in 1965 [36]. Let be a nonempty set. A fuzzy subset of is a function Let and be two fuzzy subsets of ; we say that is contained in and we write , if for all , and , are defined by and . The sets and , are called a level subset and strong level subset of , 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 [39] and Davvaz and Loeranau-Fotea [30], respectively. So, it is interesting to study fuzzy ordered hyperideals of ordered semihyperrings.

Definition 9. Let be an ordered semihyperring and let be a fuzzy subset of . is called a fuzzy ordered hyperideal of if the following conditions hold: (i) for all (ii) for all (iii) for all

Example 10. Let and the hyperoperations “” and “” on be defined as follows: Then, is a semihyperring [21]. Now, the order relation on is defined by The covering relation is given by Then, is an ordered semihyperring. Let be defined by Then, is a fuzzy ordered hyperideal of .

Lemma 11. Any hyperideal of an ordered semihyperring can be realized as a level subset of some ordered fuzzy hyperideals of

Proof. Proof is similar to Lemma 4.2 in [30].

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

Theorem 12. A fuzzy subset of an ordered semihyperring is a fuzzy ordered hyperideal of if and only if the set is an ordered hyperideal of for all .

Proof. Let be a fuzzy ordered hyperideal of and . Let . Then . Now we haveTherefore, for every , we have ; that is, , so . Let and . Then . SoTherefore, for every , we have ; that is, , so . Similarly, . Now, let and such that Then . Since , it follows that . This implies that . By Definition 4, is an ordered hyperideal of .
Conversely, let be a fuzzy subset of an ordered semihyperring such that is an ordered hyperideal of for all . Let for . Then obviously . Since every nonempty level set is an ordered hyperideal, . Thus Let for . Then obviously . Since every nonempty level set is an ordered hyperideal, . Then, we obtain . Thus for all ; that is, .
Finally, let such that . Let ; then . Since is an ordered hyperideal of , . Thus . This completes the proof.

Corollary 13. Let be a fuzzy set with the upper bound of an ordered semihyperring . Then the following conditions are equivalent: (i) is a fuzzy ordered hyperideal of (ii)Each level subset , for , is an ordered hyperideal of (iii)Each strong level subset , for , is an ordered hyperideal of (iv)Each level subset , for , is an ordered hyperideal of , where denotes the image of (v)Each strong level subset , for , is an ordered hyperideal of (vi)Each nonempty level subset of is an ordered hyperideal of (vii)Each nonempty strong level subset of is an ordered hyperideal of Let be a mapping from an ordered semihyperring to an ordered semihyperring . Let be a fuzzy subset of and let be a fuzzy subset of . Then the of is a fuzzy subset of defined by for all The image of is the fuzzy subset of defined by for all

Lemma 14. Let and be two ordered semihyperrings and let be a strong homomorphism. (i)If is a fuzzy ordered hyperideal of , then is a fuzzy ordered hyperideal of .(ii)If is a fuzzy ordered hyperideal of , then is a fuzzy ordered hyperideal of .

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 be an ordered semihyperring and . We denote [30]. For fuzzy subsets and of a semihyperring , we define the fuzzy subset of by letting ; We denote the constant function defined by for all [30].

Definition 15. Let be an ordered semihyperring and let be a fuzzy subset of . Then, is called a fuzzy ordered bi-hyperideal of if the following conditions hold: (i) and for all (ii) for all (iii) for all

Theorem 16. A fuzzy subset of an ordered semihyperring is a fuzzy ordered bi-hyperideal of if and only if the set is an ordered bi-hyperideal of for all .

Proof. The proof is similar to the proof of Theorem 12.

Definition 17. Let be an ordered semihyperring and let be a fuzzy subset of . Then, is called a fuzzy ordered quasi-hyperideal of if the following conditions hold: (i) for all (ii)(iii) for all The following theorem can be proved in a similar way in the proof of Theorem 4.8 of [30].

Theorem 18. A fuzzy subset of an ordered semihyperring is a fuzzy ordered quasi-hyperideal of if and only if the set is an ordered quasi-hyperideal of for all .

Lemma 19. Let be an ordered semihyperring and let be the characteristic function of . Then, we have the following: (i) is an ordered bi-hyperideal of if and only if is a fuzzy ordered bi-hyperideal of (ii) is an ordered quasi-hyperideal of if and only if is a fuzzy ordered quasi-hyperideal of

Proof. It is straightforward.

Theorem 20. Let be an ordered semihyperring. Then, we have the following: (i)Every fuzzy ordered hyperideal of is a fuzzy ordered quasi-hyperideal of (ii)Every fuzzy ordered quasi-hyperideal of is a fuzzy ordered bi-hyperideal of

Proof. (i) Only we show that the condition (ii) of Definition 17 is satisfied.
Let be a fuzzy ordered hyperideal of and . We have If , then it is easy to see that If , then there exist such that . Then there exists such that . Since is a fuzzy ordered hyperideal of , we haveThat is, . On the other hand, That is, the condition (ii) of Definition 17 is satisfied. Thus is a fuzzy ordered quasi-hyperideal of .
(ii) Assume that is a fuzzy ordered quasi-hyperideal of . We show that and for all Let . We have Since is a fuzzy ordered quasi-hyperideal, we have Hence ; Similarly, Therefore is a fuzzy ordered bi-hyperideal of .

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

Example 21. Consider the ordered semihyperring which is given in Example 8. Now, it is easy to see that is an ordered bi-hyperideal of , but it is not an ordered quasi-hyperideal of . Let be a fuzzy subset of defined by We have Since and are ordered bi-hyperideal of , then is an ordered bi-hyperideal of for all . Hence is a fuzzy ordered bi-hyperideal of by Theorem 16. But it is not a fuzzy ordered quasi-hyperideal of .

Definition 22 (see [21]). An ordered semihyperring is called regular, if, for every , there exists such that .

Theorem 23. Let be a regular ordered semihyperring and let be a fuzzy subset of . Then, is a fuzzy ordered quasi-hyperideal of if and only if is a fuzzy ordered bi-hyperideal of .

Proof. ” Assume that is a fuzzy ordered quasi-hyperideal of . It is clear that is a fuzzy ordered bi-hyperideal of by Theorem 20 (ii).
” Only we show that the condition (ii) of Definition 17 is satisfied. Let . If , then it is easy to see that . Let
(1) If , then we have that ; that is, .
(2) If , then there exists at least one pair such that That is, , , and . In this case we will prove that . Let ; then for some . Since is regular and there exists such that . From , , we get . Since is a fuzzy ordered bi-hyperideal , we obtain that If , then . This contradicts with . Hence, and so for all . Therefore . As a result, That is,

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

Lemma 24. Let be an ordered semihyperring. Then the following conditions are equivalent: (i) is regular(ii) for every ordered bi-hyperideal of (iii) for every ordered quasi-hyperideal of

Proof. Assume that (i) holds. Let be any ordered bi-hyperideal of and let be any element of . Since is regular, there exists such that Then it is easy to see that . Hence . On the other hand, since is an ordered bi-ideal of , we have and so
This proof is straightforward.
Assume that for every ordered quasi-hyperideal of . Then . Thus implies that is regular.

Theorem 25. Let be an ordered semihyperring. Then, the following conditions are equivalent: (i) is regular(ii) for every fuzzy ordered bi-hyperideal of (iii) for every fuzzy ordered quasi-hyperideal of

Proof. Assume that (i) holds. Let be any fuzzy ordered bi-hyperideal of and . Since is regular there exists such that Hence . Therefore .
This is straightforward from Theorem 20.
Assume that (iii) holds and let be any ordered quasi-hyperideal of . Then by Lemma 19, is a fuzzy ordered quasi-hyperideal of . Now . Therefore . On the other hand, since is any ordered quasi-hyperideal of , we have and so . By Lemma 24, 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.

#### References

1. F. Marty, “Sur une generalization de la notion de groupe,” in Proceedings of the 8th Congress Math. Scandinaves, pp. 45–49, Stockholm, Sweden, 1934. View at: Google Scholar
2. O. Kazanci, S. Yamak, and B. Davvaz, “On n-ary hypergroups and fuzzy n-ary homomorphism,” Iranian Journal of Fuzzy Systems, vol. 8, no. 1, pp. 65–76, 2011. View at: Google Scholar | MathSciNet
3. I. Rehman, N. Yaqoob, and S. Nawaz, “Hyperideals and hypersystems in LA-hyperrings,” Songklanakarin Journal of Science and Technology, vol. 39, no. 5, pp. 651–657, 2017. View at: Google Scholar
4. P. Corsini, Prolegomena of Hypergroup Theory, Aviain editore, Tricesimo, Italy, 2nd edition, 1993. View at: MathSciNet
5. P. Corsini and V. Leoreanu-Fotea, Applications of Hyperstructure Theory, Advances in Mathematics, Kluwer Academic, Dordrecht, The Netherlands, 2003. View at: Publisher Site | MathSciNet
6. B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, Fla, USA, 2007.
7. M. Krasner, “A class of hyperrings and hyperfields,” International Journal of Mathematics and Mathematical Sciences, vol. 6, pp. 307–312, 1983. View at: Google Scholar
8. T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Florida, USA, 1994. View at: MathSciNet
9. H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition does not hold,” Bulletin (New Series) of the American Mathematical Society, vol. 40, no. 12, pp. 914–920, 1934. View at: Publisher Site | Google Scholar | MathSciNet
10. J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 1999. View at: Publisher Site | MathSciNet
11. C. Donges, “On quasi-ideals of semirings,” International Journal of Mathematical Sciences, vol. 17, no. 1, pp. 47–58, 1994. View at: Publisher Site | Google Scholar
12. K. Iséki, “Quasi-ideals in semirings without zero,” Proceedings of the Japan Academy, Series A: Mathematical Sciences, vol. 34, pp. 79–81, 1958. View at: Publisher Site | Google Scholar | MathSciNet
13. M. Shabir, A. Ali, and S. Batool, “A note on quasi-ideals in semirings,” Southeast Asian Bulletin of Mathematics, vol. 29, pp. 923–928, 2004. View at: Google Scholar | MathSciNet
14. B. Davvaz, “Rings derived from semihyper-rings,” Algebras, Groups and Geometries, vol. 20, no. 2, pp. 245–252, 2003. View at: Google Scholar | MathSciNet
15. R. Ameri and H. Hedayati, “On k-hyperideals of semihyperrings,” Journal of Discrete Mathematical Sciences & Cryptography, vol. 10, no. 1, pp. 41–54, 2007. View at: Publisher Site | Google Scholar | MathSciNet
16. D. Heidari and B. Davvaz, “On ordered hyperstructures,” Politehnica University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics, vol. 73, no. 2, pp. 85–96, 2011. View at: Google Scholar | MathSciNet
17. T. Changphas and B. Davvaz, “Properties of hyperideals in ordered semihypergroups,” Italian Journal of Pure and Applied Mathematics, no. 33, pp. 425–432, 2014. View at: Google Scholar | MathSciNet
18. B. Pibaljommee and B. Davvaz, “On fuzzy bi-hyperideals in ordered semihypergroups,” Journal Intelligent and Fuzzy Systems, vol. 28, no. 5, pp. 2141–2148, 2015. View at: Google Scholar
19. B. Pibaljommee, K. Wannatong, and B. Davvaz, “An investigation on fuzzy hyperideals of ordered semihypergroups,” Quasigroups and Related Systems, vol. 23, no. 2, pp. 297–308, 2015. View at: Google Scholar | MathSciNet
20. N. Yaqoob and M. Gulistan, “Partially ordered left almost semihypergroups,” Journal of the Egyptian Mathematical Society, vol. 23, no. 2, pp. 231–235, 2015. View at: Publisher Site | Google Scholar | MathSciNet
21. B. Davvaz and S. Omidi, “Basic notions and properties of ordered semihyperrings,” Categories and General Algebraic Structures with Applications, vol. 4, no. 1, pp. 43–62, 3 (Persian pp.), 2016. View at: Google Scholar | MathSciNet
22. S. Omidi and B. Davvaz, “Contribution to study special kinds of hyperideals in ordered semihyperrings,” Journal of Taibah University for Science, vol. 11, pp. 1083–1094, 2018. View at: Google Scholar
23. S. Omidi, B. Davvaz, and P. Corsini, “Operations on hyperideals in ordered Krasner hyperrings,” Analele Mathematical Journal of the Ovidius University of Constantza, vol. 24, no. 3, pp. 275–293, 2016. View at: Publisher Site | Google Scholar | MathSciNet
24. L. A. Zadeh, “Fuzzy sets,” Inform Control, vol. 8, pp. 338–353, 1965. View at: Google Scholar | MathSciNet
25. A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, pp. 512–517, 1971. View at: Publisher Site | Google Scholar | MathSciNet
26. P. S. Das, “Fuzzy groups and level subgroups,” Journal of Mathematical Analysis and Applications, vol. 85, no. 1, pp. 264–269, 1981. View at: Publisher Site | Google Scholar | MathSciNet
27. W. J. Liu, “Fuzzy invariant subgroups and fuzzy ideals,” Fuzzy Sets and Systems, vol. 8, no. 2, pp. 133–139, 1982. View at: Publisher Site | Google Scholar | MathSciNet
28. B. Davvaz and I. Cristea, Fuzzy Algebraic Hyperstructures, An Introduction, Springer, 2015. View at: Publisher Site | MathSciNet
29. A. Asokkumar, “Class of semihyperrings from partitions of a set,” Ratio Mathematica, vol. 25, pp. 3–14, 2013. View at: Google Scholar
30. B. Davvaz and V. Leoreanu-Fotea, “Fuzzy ordered Krasner hyperrings,” Journal of Intelligent & Fuzzy Systems. Applications in Engineering and Technology, vol. 29, no. 3, pp. 1057–1064, 2015. View at: Publisher Site | Google Scholar | MathSciNet
31. O. Kazanci and S. Yamak, “Generalized fuzzy bi-ideals of semigroups,” Soft Computing, vol. 12, no. 11, pp. 1119–1124, 2008. View at: Publisher Site | Google Scholar
32. N. Kehayopulu, S. Lajos, and M. Tsingelis, “On intra-regular ordered semigroups,” Pure Mathematics and Applications, vol. 4, no. 3, pp. 317–327, 1993. View at: Google Scholar | MathSciNet
33. N. Kehayopulu and M. Tsingelis, “Fuzzy right, left, quasi-ideals, bi-ideals in ordered semigroups,” Lobachevskii Journal of Mathematics, vol. 30, no. 1, pp. 17–22, 2009. View at: Publisher Site | Google Scholar | MathSciNet
34. D. Mandal, “Fuzzy bi-ideals and fuzzy quasi-ideals in ordered semirings,” Gulf Journal of Mathematics, vol. 2, pp. 60–67, 2014. View at: Google Scholar
35. X. Huang, Y. Yin, and J. Zhan, “Characterizations of semihyperrings by their (γ,γqγ)-fuzzy hyperideals,” Journal of Applied Mathematics, vol. 2013, Article ID 348203, 13 pages, 2013. View at: Google Scholar | MathSciNet
36. M. M. Zahedi, M. Bolurian, and A. Hasankhani, “On polygroups and fuzzy subpolygroups,” Journal of Fuzzy Mathematics, vol. 3, no. 1, pp. 1–15, 1995. View at: Google Scholar | MathSciNet
37. M. Azar, M. Gulistan, N. Yaqoob, and S. Kadry, “On Fuzzy Ordered LA-Semihypergroups,” International Journal of Nonlinear Analysis and Applications (IJNAA ), vol. 16, no. 2, pp. 276–289, 2018. View at: Google Scholar
38. M. Azhar, N. Yaqoob, M. Gulistan, and M. M. Khalaf, “On ϵ,ϵqk-fuzzy hyperideals in ordered LA-semihypergroups,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 9494072, 13 pages, 2018. View at: Publisher Site | Google Scholar | MathSciNet
39. A. P. Gan and Y. L. Jiang, “On ordered ideals in ordered semirings,” Journal of Mathematical Research with Applications (JMRA) (Formerly Journal of Mathematical Research and Exposition (JMRE)), vol. 31, no. 6, pp. 989–996, 2011. View at: Google Scholar | MathSciNet

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