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ISRN Algebra
Volume 2011 (2011), Article ID 953124, 8 pages
http://dx.doi.org/10.5402/2011/953124
Research Article

On Hyperideals in Left Almost Semihypergroups

Department of Mathematics & Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra 6001, Albania

Received 7 June 2011; Accepted 11 July 2011

Academic Editors: A. V. Kelarev and A. Kiliçman

Copyright © 2011 Kostaq Hila and Jani Dine. 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.

Abstract

This paper deals with a class of algebraic hyperstructures called left almost semihypergroups (LA-semihypergroups), which are a generalization of LA-semigroups and semihypergroups. We introduce the notion of LA-semihypergroup, the related notions of hyperideal, bi-hyperideal, and some properties of them are investigated. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks, and so forth. We define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.

1. Introduction and Preliminaries

The applications of mathematics in other disciplines, for example in informatics, play a key role, and they represent, in the last decades, one of the purposes of the study of the experts of hyperstructures theory all over the world. Hyperstructure theory was introduced in 1934 by a French mathematician Marty [1], at the 8th Congress of Scandinavian Mathematicians, where he defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics and computer science by many mathematicians. 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. Some principal notions about hyperstructures and semihypergroups theory can be found in [17].

The Theory of ideals, in its modern form, is a contemporary development of mathematical knowledge to which mathematicians of today may justly point with pride. Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such as in informatics, physics, and others. As an example of applications of the concept of an ideal in informatics, let us mention that ideals of algebraic structures have been used recently to design efficient classification systems, see [812].

The study of LA-semigroup as a generalization of commutative semigroup was initiated in 1972 by Kazim and Naseeruddin [13]. They have introduced the concept of an LA-semigroup and have investigated some basic but important characteristics of this structure. They have generalized some useful results of semigroup theory. Since then, many papers on LA-semigroups appeared showing the importance of the concept and its applications [1323]. In this paper, we generalize this notion introducing the notion of LA-semihypergroup which is a generalization of LA-semigroup and semihypergroup, proposing so a new kind of hyperstructure for further studying. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks etc. Although the hyperstructure is nonassociative and noncommutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic hyperstructures. A several properties of hyperideals of LA-semihypergroup are investigated. In this note, we define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.

Recall first the basic terms and definitions from the hyperstructure theory.

Definition 1.1. A map 𝐻×𝐻𝒫(𝐻) is called hyperoperation or join operation on the set 𝐻, where 𝐻 is a nonempty set and 𝒫(𝐻)=𝒫(𝐻){} denotes the set of all nonempty subsets of 𝐻.

Definition 1.2. A hyperstructure is called the pair (𝐻,), where is a hyperoperation on the set 𝐻.

Definition 1.3. A hyperstructure (𝐻,) is called a semihypergroup if for all 𝑥,𝑦,𝑧𝐻, (𝑥𝑦)𝑧=𝑥(𝑦𝑧), which means that 𝑢𝑥𝑦𝑢𝑧=𝑣𝑦𝑧𝑥𝑣.(1.1)
If 𝑥𝐻 and A,𝐵 are nonempty subsets of 𝐻, then 𝐴𝐵=𝑎𝐴,𝑏𝐵𝑎𝑏,𝐴𝑥=𝐴{𝑥},𝑥𝐵={𝑥}𝐵.(1.2)

Definition 1.4. A nonempty subset 𝐵 of a semihypergroup 𝐻 is called a sub-semihypergroup of 𝐻 if 𝐵𝐵𝐵, and 𝐻 is called in this case super-semihypergroup of 𝐵.

Definition 1.5. Let (𝐻,) be a semihypergroup. Then 𝐻 is called a hypergroup if it satisfies the reproduction axiom, for all 𝑎𝐻, 𝑎𝐻=𝐻𝑎=𝐻.

Definition 1.6. A hypergrupoid (𝐻,) is called an LA-semihypergroup if, for all 𝑥,𝑦,𝑧𝐻, (𝑥𝑦)𝑧=(𝑧𝑦)𝑥.(1.3)
Every LA-semihypergroup (𝐻,) satisfies the medial law, that is, for all 𝑥,𝑦,𝑧,𝑤𝐻, (𝑥𝑦)(𝑧𝑤)=(𝑥𝑧)(𝑦𝑤).(1.4)
In every LA-semihypergroup with left identity, the following law holds: (𝑥𝑦)(𝑧𝑤)=(𝑤𝑦)(𝑧𝑥),(1.5) for all 𝑥,𝑦,𝑧,𝑤𝐻.

An element 𝑒 in an LA-semihypergroup 𝐻 is called identity if 𝑥𝑒=𝑒𝑥={𝑥},forall𝑥𝐻. An element 0 in a semihypergroup 𝐻 is called zero element if 𝑥0=0𝑥={0},forall𝑥𝐻. A subset 𝐼 of an LA-semihypergroup 𝐻 is called a right (left) hyperideal if 𝐼𝐻𝐼(𝐻𝐼𝐼) and is called a hyperideal if it is two-sided hyperideal, and if 𝐼 is a left hyperideal of 𝐻, then 𝐼𝐼=𝐼2 becomes a hyperideal of 𝐻. By a bi-hyperideal of an LA-semihypergroup 𝐻, we mean a sub-LA-semihypergroup 𝐵 of 𝐻 such that (𝐵𝐻)𝐵𝐵. It is easy to note that each right hyperideal is a bi-hyperideal. If 𝐻 has a left identity, then it is not hard to show that 𝐵2 is a bi-hyperideal of 𝐻 and 𝐵2𝐻𝐵2=𝐵2𝐻. If 𝐸(𝐵𝐻) denotes the set of all idempotents subsets of 𝐻 with left identity 𝑒, then 𝐸(𝐵𝐻) forms a hypersemilattice structure, also if 𝐶=𝐶2, then (𝐶𝐻)𝐶𝐸(𝐵𝐻). The intersection of any set of bi-hyperideals of an LA-semihypergroup 𝐻 is either empty or a bi-hyperideal of 𝐻. Also the intersection of prime bi-hyperideals of an LA-semihypergroup 𝐻 is a semiprime bi-hyperideal of 𝐻.

2. Main Results

Proposition 2.1. Let 𝐻 be an LA-semihypergroup with left identity, 𝑇 a left hyperideal, and 𝐵 a bi-hyperideal of 𝐻. Then 𝐵𝑇 and 𝑇2𝐵 are bi-hyperideals of 𝐻.
Proof. Using the medial law (1.4), we get ((𝐵𝑇)𝐻)(𝐵𝑇)=((𝐵𝑇)𝐵)(𝐻𝑇)((𝐵𝐻)𝐵)𝑇𝐵𝑇,(2.1) also (𝐵𝑇)(𝐵𝑇)=(𝐵𝐵)(𝑇𝑇)𝐵𝑇.(2.2) Hence, 𝐵𝑇 is a bi-hyperideal of 𝐻. we obtain 𝑇2𝑇𝐵𝐻2=𝑇𝐵2𝑇𝐻(𝐵𝐻)2𝑇𝐵2𝑇(𝐵𝐻)2=𝑇𝐵2𝑇2((𝐵𝐻)𝐵)𝑇2𝐵,(2.3) also 𝑇2𝑇𝐵2=𝑇𝐵2𝑇2(𝐵𝐵)𝑇2𝐵.(2.4) Hence, 𝑇2𝐵 is a bi-hyperideal of 𝐻.

Proposition 2.2. Let 𝐻 be an LA-semihypergroup with left identity and 𝐵1,𝐵2 two bi-hyperideals of 𝐻. Then 𝐵1𝐵2 is a bi-hyperideal of 𝐻.

Proof. Using (1.4), we get 𝐵1𝐵2𝐵𝐻1𝐵2=𝐵1𝐵2𝐵(𝐻𝐻)1𝐵2=𝐵1𝐵𝐻2𝐵𝐻1𝐵2=𝐵1𝐻𝐵1𝐵2𝐻𝐵2𝐵1𝐵2.(2.5)

By the above, if 𝐵1 and 𝐵2 are nonempty, then 𝐵1𝐵2 and 𝐵2𝐵1 are connected bi-hyperideals. Proposition 2.1 leads us to an easy generalization, that is, if 𝐵1,𝐵2,𝐵3,,𝐵𝑛 are bi-hyperideals of an LA-semihypergroup 𝐻 with left identity, then 𝐵1𝐵2𝐵3𝐵𝑛,𝐵21𝐵22𝐵23𝐵2𝑛(2.6) are bi-hyperideals of 𝐻, consequently the set 𝒞(𝐻𝐵) of bi-hyperideals forms an LA-semihypergroup.

If 𝐻 is an LA-semihypergroup with left identity 𝑒, then 𝑎𝐿=𝐻𝑎,𝑎𝑅=𝑎𝐻 and 𝑎𝐻=(𝐻𝑎)𝐻 are bi-hyperideals of 𝐻. It can be easily shown that 𝑎𝑏𝐿=𝑎𝐿𝑏𝐿, 𝑎𝑏𝑅=𝑎𝑅𝑏𝑅, and 𝑎𝑏𝑅=𝑏𝐿𝑎𝐿. Hence, this implies that 𝑎𝑅𝑏𝑅=𝑏𝐿𝑎𝐿 and 𝑎𝐿𝑏𝐿=𝑏𝑅𝑎𝑅. Also, 𝑎𝐿𝑏𝑅=𝑏𝐿𝑎𝑅, 𝑎𝑎𝐿=𝑎2𝐿, 𝑎𝑎𝑅=𝑎2𝑅, 𝑎𝑎𝐿=𝑎𝑎𝑅, and 𝑎𝐿=𝑎𝑅 (if 𝑎 is an idempotent), consequently 𝑎𝑎𝐿=𝑎𝑎𝑅. It is easy to show that 𝑎𝑅𝑎2=𝑎2𝑎𝐿.

Lemma 2.3. Let 𝐻 be an LA-semihypergroup with left identity, and let 𝐵 be an idempotent bi-hyperideal of 𝐻. Then 𝐵 is a hyperideal of 𝐻.
Proof. By the definition of LA-semihypergroup (1.3), we have 𝐵𝐻=(𝐵𝐵)𝐻=(𝐻𝐵)𝐵=𝐻𝐵2𝐵𝐵=2𝐻𝐵=(𝐵𝐻)𝐵,(2.7) and every right hyperideal in 𝐻 with left identity is left.

Lemma 2.4. Let 𝐻 be an LA-semihypergroup with left identity 𝑒, and let 𝐵 be a proper bi-hyperideal of 𝐻. Then 𝑒𝐵.

Proof. Let us suppose that 𝑒𝐵. Since 𝑏(𝑒)𝑏𝐵, using (1.3), we have (𝑒𝑒)=(𝑒)𝑒(𝑆𝐵)𝐵𝐵. It is impossible. So, 𝑒𝐵.

It can be easily noted that {𝑥𝐻(𝑥𝑎)𝑥=𝑒}𝐵.

Proposition 2.5. Let 𝐻 be an LA-semihypergroup with left identity, and let 𝐴,𝐵 be bi-hyperideals of 𝐻. Then the following statements are equivalent: (1)every bi-hyperideal is idempotent, (2)𝐴𝐵=𝐴𝐵,(3)the hyperideals of 𝐻 form a hypersemilattice (𝐿𝐻,), where 𝐴𝐵=𝐴𝐵.

Proof. (1)(2). Using Lemma 2.3, it is easy to note that 𝐴𝐵𝐴𝐵. Since 𝐴𝐵𝐴,𝐵 implies (𝐴𝐵)2𝐴𝐵, hence 𝐴𝐵𝐴𝐵.
(2)(3). 𝐴𝐵=𝐴𝐵=𝐴𝐵=𝐵𝐴=𝐵𝐴 and 𝐴𝐴=𝐴𝐴=𝐴𝐴=𝐴. Similarly, associativity follows. Hence, (𝐿𝐻,) is a hypersemilattice.
(3)(1). 𝐴=𝐴𝐴=𝐴𝐴.

A bi-hyperideal 𝐵 of an LA-semihypergroup 𝐻 is called a prime bi-hyperideal if 𝐵1𝐵2𝐵 implies either 𝐵1𝐵 or 𝐵2𝐵 for every bi-hyperideal 𝐵1 and 𝐵2 of 𝐻. The set of bi-hyperideals of 𝐻 is totally ordered under the set inclusion if for all bi-hyperideals 𝐼,𝐽 either 𝐼𝐽 or 𝐽𝐼.

Theorem 2.6. Let 𝐻 be an LA-semihypergroup with left identity. Every bi-hyperideal of 𝐻 is prime if and only if it is idempotent and the set of the bi-hyperideals of 𝐻 is totally ordered under the set inclusion.

Proof. Let us assume that every bi-hyperideal of 𝐻 is prime. Since 𝐵2 is a hyperideal and so is prime which implies that 𝐵𝐵𝐵, hence 𝐵 is idempotent. Since 𝐵1𝐵2 is a bi-hyperideal of 𝐻 (where 𝐵1 and 𝐵2 are bi-hyperideals of 𝐻) and so is prime. Now by Lemma 2.3, either 𝐵1𝐵1B2 or 𝐵2𝐵1𝐵2 which further implies that either 𝐵1𝐵2 or 𝐵2𝐵1. Hence, the set of bi-hyperideals of 𝐻 is totally ordered under set inclusion.
Conversely, let us assume that every bi-hyperideals of 𝐻 is idempotent and the set of bi-hyperideals of 𝐻 is totally ordered under set inclusion. Let 𝐵1,𝐵2 and 𝐵 be the bi-hyperideals of 𝐻 with 𝐵1𝐵2𝐵 and without loss of generality assume that 𝐵1𝐵2. Since 𝐵1 is an idempotent, so 𝐵1=𝐵1𝐵1𝐵1𝐵2𝐵 implies that 𝐵1𝐵, and, hence, every bi-hyperideal of 𝐻 is prime.

A bi-hyperideal 𝐵 of an LA-semihypergroup 𝐻 is called strongly irreducible bi-hyperideal if 𝐵1𝐵2𝐵 implies either 𝐵1𝐵 or 𝐵2𝐵 for every bi-hyperideal 𝐵1 and 𝐵2 of 𝐻.

Theorem 2.7. Let 𝐻 be an LA-semihypergroup with zero. Let 𝐷 be the set of all bi-hyperideals of 𝐻, and Ω the set of all strongly irreducible proper bi-hyperideals of 𝐻, then Γ(Ω)={𝑂𝐵𝐵𝐷} forms a topology on the set Ω, where 𝑂𝐵={𝐽Ω;𝐵𝐽} and 𝜙 : Bi-hyperideal(𝐻)Γ(Ω) preserves finite intersection and arbitrary union between the set of bi-hyperideals of 𝐻 and open subsets of Ω.

Proof. Since {0} is a bi-hyperideal of 𝐻 and 0 belongs to every bi-hyperideal of 𝐻, then 𝑂𝐵={𝐽Ω,{0}𝐽}={}, also 𝑂𝐻={𝐽Ω,𝐻𝐽}=Ω which is the first axiom for the topology. Let {𝑂𝐵𝛼𝛼𝐼}Γ(Ω), then 𝑂𝐵𝛼={𝐽Ω,𝐵𝛼𝐽,forsome𝛼𝐼}={𝐽Ω,𝐵𝛼𝐽}=𝑂𝐵𝛼, where 𝐵𝛼 is a bi-hyperideal of 𝐻 generated by 𝐵𝛼. Let 𝑂𝐵1 and 𝑂𝐵2Γ(Ω), if 𝐽𝑂𝐵1𝑂𝐵2, then 𝐽Ω and 𝐵1,𝐵2𝐽. Let us suppose 𝐵1𝐵2𝐽, this implies that either 𝐵1𝐽 or 𝐵2𝐽. It is impossible. Hence, 𝐵1𝐵2𝐽 which further implies that 𝐽𝑂𝐵1𝐵2. Thus 𝑂𝐵1𝑂𝐵2𝑂𝐵1𝐵2. Now if 𝐽𝑂𝐵1𝐵2, then 𝐽Ω and 𝐵1𝐵2𝐽. Thus 𝐽𝑂𝐵1 and 𝐽𝑂𝐵2, therefore 𝐽𝑂𝐵1𝑂𝐵2, which implies that 𝑂𝐵1𝐵2𝑂𝐵1𝑂𝐵2. Hence Γ(Ω) is the topology on Ω. Define 𝜙 : Bi-hyperideal(𝐻)Γ(Ω) by 𝜙(𝐵)=𝑂𝐵, then it is easy to note that 𝜙 preserves finite intersection and arbitrary union.

A hyperideal 𝑃 of an LA-semihypergroup 𝐻 is called prime if 𝐴𝐵𝑃 implies that either 𝐴𝑃 or 𝐵𝑃 for all hyperideals 𝐴 and 𝐵 in 𝐻.

Let 𝑃𝐻 denotes the set of proper prime hyperideals of an LA-semihypergroup 𝐻 absorbing 0. For a hyperideal 𝐼 of 𝐻, we define the sets Θ𝐼={𝐽𝑃𝐻𝐼𝐽} and Γ(𝑃𝐻)={Θ𝐼,𝐼isahyperidealof𝐻}.

Theorem 2.8. Let 𝐻 be an LA-semihypergroup with zero. The set Γ(𝑃𝐻) constitutes a topology on the set P𝐻.

Proof. Let Θ𝐼1,Θ𝐼2Γ(𝑃𝐻), if 𝐽Θ𝐼1Θ𝐼2, then 𝐽𝑃𝐻 and 𝐼1𝐽 and 𝐼2𝐽. Let 𝐼1𝐼2𝐽 which implies that either 𝐼1𝐽 or 𝐼2𝐽, which is impossible. Hence, 𝐽Θ𝐼1𝐼2. Similarly Θ𝐼1𝐼2Θ𝐼1Θ𝐼2. The remaining proof follows from Theorem 2.7.

The assignment 𝐼Θ𝐼 preserves finite intersection and arbitrary union between the hyperideal(𝐻) and their corresponding open subsets of Θ𝐼.

Let 𝑃 be a left hyperideal of an LA-semihypergroup 𝐻. 𝑃 is called quasiprime if for left hyperideals 𝐴,𝐵 of 𝐻 such that 𝐴𝐵𝑃, we have 𝐴𝑃 or 𝐵𝑃.

Theorem 2.9. Let 𝐻 be an LA-semihypergroup with left identity 𝑒. Then a left hyperideal 𝑃 of 𝐻 is quasiprime if and only if (𝐻𝑎)𝑏𝑃 implies that either 𝑎𝑃 or 𝑏𝑃.

Proof. Let 𝑃 be a left hyperideal of 𝐻. Let us assume that (𝐻𝑎)𝑏𝑃, then 𝐻((𝐻𝑎)𝑏)𝐻𝑃𝑃,(2.8) that is, 𝐻((𝐻𝑎)𝑏)=(𝐻𝑎)(𝐻𝑏).(2.9) Hence, either 𝑎𝑃 or 𝑏𝑃.
Conversely, let us assume that 𝐴𝐵𝑃, where 𝐴 and 𝐵 are left hyperideal of 𝐻 such that 𝐴𝑃. Then there exists 𝑥𝐴 such that 𝑥𝑃. Now, by the hypothesis, we have (𝐻𝑥)𝑦(𝐻𝐴)𝐵𝐴𝐵𝑃 for all 𝑦𝐵. Since 𝑥𝑃, so by hypothesis, 𝑦𝑃 for all 𝑦𝐵, we obtain 𝐵𝑃. This shows that 𝑃 is quasiprime.

An LA-semihypergroup 𝐻 is called an antirectangular if 𝑎(𝑏𝑎)𝑏, for all 𝑎,𝑏𝐻. It is easy to see that 𝐻=𝐻𝐻. In the following results for an antirectangular LA-semihypergroup 𝐻, 𝑒𝐻.

Proposition 2.10. Let 𝐻 be an LA-semihypergroup. If 𝐴,𝐵 are hyperideals of 𝐻, then 𝐴𝐵 is a hyperideal.

Proof. Using (1.4), we have (𝐴𝐵)𝐻=(𝐴𝐵)(𝐻𝐻)=(𝐴𝐻)(𝐵𝐻)𝐴𝐵,(2.10) also 𝐻(𝐴𝐵)=(𝐻𝐻)(𝐴𝐵)=(𝐻𝐴)(𝐻𝐵)𝐴𝐵,(2.11) which shows that 𝐴𝐵 is a hyperideal.

Consequently, if 𝐼1,𝐼2,,𝐼𝑛 are hyperideals of 𝐻, then 𝐼1𝐼2𝐼3𝐼𝑛,𝐼21𝐼22𝐼23𝐼2𝑛(2.12) are hyperideals of 𝐻 and the set (𝐼) of hyperideals of 𝐻 form an antirectangular LA-semihypergroup.

Lemma 2.11. Let 𝐻 be an antirectangular LA-semihypergroup. Any subset of 𝐻 is left hyperideal if and only if it is right.

Proof. Let 𝐼 be a right hyperideal of 𝐻, then using (1.3), we get 𝑖((𝑘)𝑘)𝑖(𝑖𝑘)(𝑘)𝐼.
Conversely, let us suppose that 𝐼 is a left hyperideal of 𝐻, then using (1.3), we have 𝑖((𝑡𝑖)𝑡)(𝑡)(𝑡𝑖)𝐼.

It is fact that 𝐻𝐼=𝐼𝐻. From the above lemma, we remark that every quasiprime hyperideal becomes prime in an antirectangular LA-semihypergroup.

Lemma 2.12. Let 𝐻 be an anti-rectangular LA-semihypergroup. If 𝐼 is a hyperideal of 𝐻, then 𝑆(𝑎)={𝑥𝐻𝑎(𝑥𝑎)𝑥,for𝑎𝐼}𝐼.

Proof. Let 𝑦𝑆(𝑎), then 𝑦(𝑦𝑎)𝑦(𝐻𝐼)𝐻𝐼. Hence 𝐻(𝑎)𝐼. Also, 𝑆(𝑎)={𝑥𝐻𝑥(𝑥𝑎)𝑥,for𝑎𝐼}𝐼.

An hyperideal 𝐼 of an LA-semihypergroup 𝐻 is called an idempotent if 𝐼𝐼=𝐼. An LA-semihypergroup 𝐻 is said to be fully idempotent if every hyperideal of 𝐻 is idempotent.

Proposition 2.13. Let 𝐻 be an antirectangular LA-semihypergroup, and, 𝐴,𝐵 be hyperideals of 𝐻. Then the following statements are equivalent: (1)𝐻 is fully idempotent, (2)𝐴𝐵=𝐴𝐵,(3)the hyperideals of 𝐻 form a hypersemilattice (𝐿𝐻,) where 𝐴𝐵=𝐴𝐵.

The proof follows from Proposition 2.5.

The set of hyperideals of 𝐻 is totally ordered under set inclusion if for all hyperideals 𝐼,𝐽 either 𝐼𝐽 or 𝐽𝐼 and denoted by hyperideal(𝐻).

Theorem 2.14. Let 𝐻 be an antirectangular LA-semihypergroup. Then every hyperideal of 𝐻 is prime if and only if it is idempotent and hyperideal (𝐻) is totally ordered under set inclusion.

Proof. The proof follows from Theorem 2.6.

In conclusion, let us mention that it would be interesting to investigate whether it is possible to apply hyperideals of hyperstructures to the construction of classification systems similar to those introduced in [812].

Acknowledgment

The authors are highly grateful to referees for their valuable comments and suggestions.

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