alexa Fuzzy n-Lie Algebras | Open Access Journals
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Fuzzy n-Lie Algebras

Davvaz B1 and Dudek WA2*

1Department of Mathematics, Yazd University, Yazd, Iran

2Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland

*Corresponding Author:
Dudek WA
Faculty of Pure and Applied Mathematics
Wroclaw University of Science and Technology
Wyspian’skiego 27, 50-370 Wroclaw, Poland
Tel: 71 320-31-62
Fax:
(+48)-(71)-328-07-51
E-mail: [email protected]

Received Date: April 28, 2017; Accepted Date: June 05, 2017; Published Date: June 12, 2017

Citation: Davvaz B, Dudek WA (2017) Fuzzy n-Lie Algebras. J Generalized Lie Theory Appl 11: 268. doi: 10.4172/1736-4337.1000268

Copyright: © 2017 Davvaz B, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

Properties of fuzzy subalgebras and ideals of n-ary Lie algebras are described. Methods of construction fuzzy ideals are presented. Connections with various fuzzy quotient n-Lie algebras are proved.

Keywords

Fuzzy set; n-ary Lie algebra; Subalgebra; Ideal; Fuzzy ideal

Introduction

In 1985 Filippov [1] proposed a generalization of the concept of a Lie algebra by replacing the binary operation by n-ary one. He defined an n-ary Lie algebra structure on a vector space L as an operation which associates with each n–tuple (x1,…,xn) of elements in L another element [x1,…,xn] which is n-linear, skew-symmetric:

Equation

and satisfies the generalized Jacobi identity (called also the Filippov identity):

Equation

where σSn.

Now, such structures are also called n-Lie algebras or Filippov algebras. For n=2 we obtain a classical Lie algebras.

Note that such an n-ary operation, realized on the smooth function algebra of a manifold and additionally assumed to be an n-derivation, is an n-Poisson structure. This general concept, however, was not introduced neither by Filippov, nor by other mathematicians that time. It was done much later in 1994 by Takhtajan [2] in order to formalize mathematically the n-ary generalization of Hamiltonian mechanics proposed by Nambu [3]. Apparently Nambu was motivated by some problems of quark dynamics and the n-bracket operation he considered was:

Equation

where L=R[x1,…,xn] is the vector space of polynomials in n-variables.

Nambu does not mentions that the n-bracket operation satisfies the generalized Jacobi identity but Filippov reports this operation in his paper [1] among other examples of n-Lie algebras. The formal proof is given in [4].

Ternary Lie algebras were studied [5,6]. For other generalizations and applications see ref. [7].

The study of fuzzy Lie algebras was initiated in refs. [8,9], and continued in various directions by many authors (for example [10- 12]). The study of fuzzy n-ary algebras was initiated by Dudek [13]. Davvaz and Dudek described fuzzy n-ary groups as a generalization of Rosenleld’s fuzzy groups [14].

In this paper we describe fuzzy n-ary Lie algebras.

Preliminaries

Let X be a non-empty 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, if μ(x)≤ λ(x) for all xX. The set Equation is called a level subset of μ.

Definition 2.1

Let V be a vector space over a field F. A fuzzy subset μ of V is called a fuzzy subspace of V if for all x,yV and αF, the following conditions are satisfied:

μ(x+y)≥min{μ(x), μ(y)} for all x,yV,

μ(αx)≥μ(x) for all xV, αF.

Note that the second condition implies, μ(−x)≥ μ(x) for all xV,

Lemma 2.2

If μ is a fuzzy subspace of a vector space V, then μ(x)≤ μ(0) for all xV, and

• μ(x)=μ(−x),

• μ(x−y)=μ(0)⇒ μ(x)=μ(y),

• μ(x)< μ(y)⇒ μ(x−y)=μ(x)=μ(y−x)

for all x,y∈V.

Proof. Directly from the definition we obtain μ(x)≤ μ(0) and μ(x)=μ(−x). Moreover, for all x,y∈V we have

Equation

which implies

Equation

Similarly

Equation

Hence

Equation

This for μ(xy)=μ(0) gives μ(x)=μ(y), and μ(xy)=μ(x) for μ(x)< μ(y).

Theorem 2.3

For a fuzzy subset μ of a vector space V, the following statements are equivalent.

μ is a fuzzy subspace of V.

• Each non-empty Equation is a subspace of V.

This theorem firstly proved in ref. [15] is a consequence of the Transfer Principle for fuzzy sets described in ref. [16].

Let Equation be a collection of fuzzy subsets of X. Then, we define the fuzzy subsets Equation

Equation

Fuzzy Subalgebras and Ideals

Recall that a non-empty subset S of an n-Lie algebra L is its subalgebra if it is a subspace of a vector space L and [x1,…,xn]∈S for all x1,…,xnS.

A subspace S of an i-ideal of L if for all Equation and yS we have Equation

Two n-Lie algebras L1, L2 over the same field F are isomorphic if there exists a vector space isomorphism ϕ:L1L2 such that for all Equation

Let L be an n-Lie algebra. Fixing in [x1, x2,…,xn] elements x2,…, x{n-1} we obtain a new binary operation Equation with the property Equation for all k=2,…,n−1 and all yL. It is easily to see that L with respect to this new operation is an classical Lie algebra. It is called a binary retract. Fixing various x2,…,xn-1 we obtain various (generally non-isomorphic) retracts. Obviously, any subalgebra (ideal) of an n-Lie algebra is a subalgebra (ideal) of each binary retract of L. The converse is not true. Hence results obtained for n-Lie algebras are essential generalizations of results proved for Lie algebras.

Basing on the idea of fuzzyfications of algebras with one n-ary operation proposed in ref. [13] we present a fuzzyfication of n-Lie algebras.

Definition 3.1

Let L be an n-Lie algebra. A fuzzy subalgebra of L is a fuzzy subspace μ such that

Equation

Definition 3.2

Let L be an n-Lie algebra. A fuzzy ideal of L is a fuzzy subspace μ such that

Equation

The following facts are obvious. Their proofs are very similar to the proofs of analogous results for fuzzy n-ary systems [13] and fuzzy Lie algebras [9].

Proposition 3.3:

A fuzzy subspace μ of an n-Lie algebra L is its fuzzy ideal if and only if

Equation (1)

for all x1,…,xnL.

Proposition 3.4: If μ is a fuzzy ideal of an n-Lie algebra L, then

Equation

is an ideal of L contained in every non-empty level subset of μ.

Proposition 3.5: Let μ and λ be two fuzzy ideals of an n-Lie algebra L such that μ(0)=λ(0). Then Equation.

Theorem 3.6

Let ϕ:LL′ be an n-Lie algebra homomorphism of an n-Lie algebra L onto an n-Lie algebra L′. Then the following conditions hold:

• if μ is a fuzzy ideal of L, then ϕ(μ) is a fuzzy ideal of L′,

• if v is a fuzzy ideal of L′ then ϕ−1(v) is a fuzzy ideal of L,

Equation for every t∈[0,1] and every fuzzy ideal v of L′.

Proposition 3.7: Let L be an n-Lie algebra. Then the intersection of any family of fuzzy subalgebras (ideals) of L is again a fuzzy subalgebra (ideal) of L.

It is easy to see that the union of fuzzy subalgebras (ideals) of an n-Lie algebra L is not a fuzzy subalgebra (ideal) of L, in general. But we have the following proposition on the union of fuzzy subalgebras (ideals) of L.

Proposition 3.8: Let {μn} be a chain of fuzzy subalgebras (ideals) of an n-Lie algebra L. Then Equation is a fuzzy subalgebra (ideal) of L.

Theorem 3.9

For a fuzzy subset μ of an n-Lie algebra L, the following statements are equivalent.

μ is a fuzzy subalgebra (ideal) of L.

• Each non-empty Equation, is a subalgebra (ideal) of L.

Proof. Let μ be a fuzzy ideal of L. Since μ is a fuzzy subspace of L, by Theorem 2.3, each non-empty Equation is a subspace of L. Therefore, it is enough to prove that Equation. For every Equation and x1,…,xnL we show that Equation. Since μ is a fuzzy ideal, we have

Equation

Conversely, assume that every non-empty Equation is an ideal of L. Therefore, Equation is a subspace of L and so by Theorem 2.3, μ is a fuzzy subspace of L. Now, for every yL, we put t0=μ(y). Then, Equation. Therefore, for every x1,…xnL we have Equation which implies that Equation. So, μ is a fuzzy ideal.

For subalgebras the proof is analogous.

Proposition 3.10: Let L be an n-Lie algebra and μ be a fuzzy subalgebra of L. Let Equation (with t1<t2) be any two level subalgebras of μ. Then Equation if and only if there is no x in L such that Equation.

Theorem 3.11

Let {S| λ∈λ}, where ∅≠λ⊆[0,1], be a collection of ideals of an n-Lie algebra L such that

Equation

Then μ defined by

Equation

is a fuzzy ideal of L.

Proof. By Theorem 3.9, it is sufficient to show that every non-empty level Equation is an ideal of L.

Let Equation for some fixed α∈[0,1]. Then

Equation

In the first case we have Equation, because

Equation

In the second, there exists ε>0 such that Equation. In this caseEquation. Indeed, ifEquation, then xSλ for some λα, which gives μ(x)≥λα. ThusEquation.

Conversely, if Equation, then Equation for all λα, which implies Equation for all λ>αε, i.e., if xSλ then λαε. Thus μ(x)≤αε. ThereforeEquation. Hence Equation, and consequently Equation. This completes our proof.

Theorem 3.12

Let μ be a fuzzy subset defined on an n-Lie algebra L and let Im(μ)={t0,t1,t2,…}, where 1≥t0>t1>t2…≥0. If S0S1S2… are subalgebras (ideals) of L such that Equation for k=0,1,2,…, where S−1=∅, then μ is a fuzzy subalgebra (ideal) of L.

Proof. First consider the case when all Si are subalgebras. If Equation then also at least one of x1,…,xn is in Equation because in the opposite case x1,…,xn and [x1,…,xn] will be in some Sk. So, in this case

Equation

It is clear that for arbitrary elements x1,…, xn∈L there exists only one k such that Equation and only one ki such that Equation. ThusEquation.

Suppose Equation for all i=1,2,…,n. Then, by the assumption, ki<k and Equation, where s=max{k1,…,kn}. Hence x1,…xnSk−1 and, in the consequence, [x1,…xn]∈Sk−1 because Sk−1 is a subalgebra. This is a contradiction. Therefore there is at least one Equation. In this case Equation. Since μ also is a fuzzy subspace of a vector space L, it is a fuzzy subalgebra of L.

Now, let all Si be ideals and let Equation for some x1,…xnL. Then these x1,…xn are in L\Sk−1. If not, then there exists xiSk−1. But in this case [x1,…xn]∈Sk−1 because Sk−1 is an ideal. This is a contradiction. So, all Equation. Hence Equation. Now, if Equation, then also all x1,…xn are in Equation. Thus Equation. This completes the proof that μ is a fuzzy ideal.

Corollary 3.13

For any chain Equation of subalgebras (ideals) of an n-Lie algebra L and any chain of reals Equation there exists a fuzzy subalgebra (ideal) μ of L such that Equation.

Theorem 3.14

Let Im(μ)={t1|iI} be the image of a fuzzy subalgebra (ideal) μ of an n-Lie algebra L. Then

(a) There exists a unique t0Im(μ) such that t0ti for all tiIm(μ),

(b) L is the set-theoretic union of all Equation,

(c) Equation is linearly ordered by inclusion,

(d) Ω contains all level subalgebras (ideals) of μ if and only if μ attains its infimum on all subalgebras (ideals) of L.

Proof. (a) Follows from the fact that t0=μ(0)≥μ(x) for all xL.

(b) If xL, then μ(x)=txIm(μ). Thus Equation, where tiIm(μ), which proves (b).

(c) Since Equation, then Ω linearly ordered by inclusion.

(d) Suppose that Ω contains all levels of μ. Let S be a subalgebra (ideal) of L. If μ is constant on S, then we are done. Assume that μ is not constant on S. We have two cases: (1) S=L and (2) SL. For S=L let β=infIm(μ). Then βtIm(μ), i.e., Equation for all tIm(μ). But Equation because Ω contains all levels of μ. Hence there exists t′∈Im(μ) such that Equation. It follows that Equation so that Equation because every level of μ is a subalgebra (resp. ideal) of L.

Now it sufficient to show that Equation then there exists Equation such that Equation. This implies Equation, which is a contradiction. Therefore Equation.

In the case SL we consider the fuzzy set μS defined by

Equation

Clearly μS is a fuzzy subalgebra (ideal) of L if S is a subalgebra (ideal).

Let

Equation

Then Equation contains (by the assumption) all levels of μS. This means that there exists x0S such that Equation, i.e., Equation for some xS. Hence μ attains its infimum on all subalgebras (ideals) of L.

To prove the converse let Equation be a level subalgebra of μ. If α=t for some tIm(μ), then Equation If α≠t for all tIm(μ), then there does not exist xL such that μ(x)=α.

Let S={xL|μ(x)>α}. Obviously 0∈S and μ(xi)>α for all xiS. From the fact that μ is a fuzzy subalgebra we obtain

Equation

which proves [x1,…,xn]∈S. Hence S is a subalgebra. By hypothesis, there exists yS such that μ(y)=inf{μ(x)|xS}. But μ(y)∈Im(μ) implies μ(y)=t′ for some t′∈Im(μ). Hence inf{μ(x)|xS}=t′>α.

Note that there does not exist z∈L such that αμ(z)<t′. This gives Equation Hence Equation. Thus Ω contains all level subalgebras of μ.

Theorem 3.15

If every fuzzy subalgebra (ideal) μ defined on an n-Lie algebra L has a finite number of values, then every descending chain of subalgebras (ideals) of L terminates at finite step.

Proof. Suppose there exists a strictly descending chain

Equation

of ideals of L which does not terminate at finite step. We prove that μ defined by

Equation

where k=0,1,2,… and S0=L, is a fuzzy ideal with an infinite number of values.

If Equation, then obviously

Equation

If Equation, thenEquation for some p≥0 and there exists at least one i=1,2,…,n such that Equation, because Equation impliesEquation.

Let Sm be a maximal ideal of L such that at least one of x1,…,xn belongs to Sm\Sm+1. Then mp. Indeed, for m>p we have Equation and, consequently Equation, which is impossible. Thus mp and

Equation

This proves that μ is a fuzzy ideal and has an infinite number of different values. This is a contradiction. Hence every descending chain of ideals terminates at finite step.

For subalgebras the proof is analogous.

Theorem 3.16

Every ascending chain of subalgebras (ideals) of an n-Lie algebra L terminates at finite step if and only if the set of values of any fuzzy subalgebra (ideal) of L is a well-ordered subset of [0,1].

Proof. If the set of values of a fuzzy subalgebra (ideal) μ is not wellordered, then there exists a strictly decreasing sequence {ti} such that ti=μ(xi) for some xiL. But in this case Equation form a strictly ascending chain of subalgebras (ideals) of L, which is a contradiction.

In order to prove the converse suppose that there exists a strictly ascending chain Equation of subalgebras (ideals) of L. Then Equation is a subalgebra (ideal) of L and μ defined by

Equation

is a fuzzy subalgebra (ideal) on L.

Indeed, for every x1,…,xnM there exist a minimal number ki such that Equation, and a minimal number p such that Equation. If all Si are subalgebras, then for Equation are in Sk. Thus kp. Consequently,

Equation

The case when at least one of x1,x2,…,xn is not in M is obvious. Hence μ is a fuzzy subalgebra.

Now, if all Si are ideals, then [x1,…,xn]∈Sm for m=min{k1,…,kn}. Thus pm. Hence

Equation

which means that in this case μ is a fuzzy ideal.

Since the chain Equation is not terminating, μ has a strictly descending sequence of values. This contradicts that the set of values of any fuzzy subalgebra (ideal) is well-ordered. The proof is complete.

Definition 3.17

A fuzzy subset μ of an n-Lie algebra L is said to be normal if μ(0)=1.

The following lemma is obvious.

Lemma 3.18

If μ is a fuzzy subalgebra (ideal) of an n-Lie algebra L, then μ+ defined by

μ+(x)=μ(x)+1−μ(0)

is a normal fuzzy subalgebra (ideal) of L.

Corollary 3.19

Any fuzzy subalgebra (ideal) of an n-Lie algebra L is contained in some normal fuzzy subalgebra (ideal) of it.

Proof. Indeed, μ (x) ≤ μ (x) +1−μ (0) = μ + (x) for every xL.

Proposition 3.20: A maximal normal fuzzy subalgebra of an n-Lie algebra L takes only two values: 0 and 1.

Proof. If μ(x)=1 for all xL, then obviously μ is a maximal normal fuzzy subalgebra of L. If μ is a maximal normal fuzzy subalgebra of L and 0<μ(a)<1 for some aL, then a fuzzy subset v defined by Equation is a fuzzy subalgebra of L. Moreover, v+ is a non-constant normal fuzzy subalgebra of L such that μ(x)≤v+(x) for every xL. Thus, μ is not maximal. Obtained contradiction shows that μ(a)=0 for all μ(a)<1.

Proposition 3.21: Let μ be a fuzzy subalgebra (ideal) of an n-Lie algebra L. If h:[0,μ(0)]→[0,1] is an increasing function, then a fuzzy subset μh defined on L by μh(x)=h(μ(x)) is a fuzzy subalgebra (ideal). Moreover, μh is normal if and only if h(μ (0))=1.

Proof. Straightforward.

If μ is a fuzzy subset of an n-Lie algebra L, and f is a function defined on L, then the fuzzy subset v of f(L) defined by Equation, for all yf(L) is called the image of μ under f. Similarly, if v is a fuzzy subset in f(L), then the fuzzy set μ=vof in L is called preimage of v under f.

Theorem 3.22

An n-Lie algebra homomorphic preimage of a fuzzy ideal is a fuzzy ideal.

Proof. Let ϕ:L1L2 be an n-Lie algebra homomorphism, and v be a fuzzy ideal of L2 and μ be the preimage of v under ϕ. Then, as it is not difficult to see, μ is a fuzzy subspace of L and

Equation

for all x1,…,xn,yL and αF.

A fuzzy set μ of a set X is said to possess sup property if for every non-empty subset S of X, there exists x0S such that μ(x0)=supxS{μ(x)}.

Theorem 3.23

An n-Lie algebra homomorphism image of a fuzzy ideal having the sup property is a fuzzy ideal.

Proof. Suppose that ϕ:L1L2 is an n-Lie algebra homomorphism, μ is a fuzzy ideal of L1 with the sup property and v is the image of μ under ϕ. Suppose that Equation be such that Equation, respectively. Then,

Equation

and

Equation

Finally, let Equation and let Equation be such that

Equation

Then,

Equation

This proves that v is a fuzzy ideal of ϕ(L).

Fuzzy Quotient n-Lie Algebras

If I is an ideal of an n-Lie algebra L, then we can define a new n-Lie algebra on the quotient space L/I with the n-linear map

Equation

for all x1,…,xnL.

If I is an ideal of an n-Lie algebra L, then the quotient space L/I is also an n-Lie algebra and is the quotient n-Lie algebra.

Theorem 4.1

Let L be an n-Lie algebra.

• Let μ be a fuzzy ideal of L and let t=μ(0). Then the fuzzy subset μ* of Equation defined by Equation for all xL, is a fuzzy ideal of Equation.

• If I is an ideal of L and v is a fuzzy ideal of L/I such that v(x+I)=v(I) only when xI, then there exists a fuzzy ideal μ of L such that μt = I , where t=μ(0); and v=μ*.

Proof. (1). Since μ is a fuzzy ideal of L, Equation is an ideal of L. Now, μ* is well-defined, because if Equation for x,yL, then Equation and so μ(xy)=μ(0). Hence, μ(x)=μ(y) which implies that Equation.

Now, we show μ* is a fuzzy ideal of L. Let x,yL and αF. Then, we have

Equation

and

Equation

(2). We define a fuzzy subset μ of L by μ(x)=v(x+I) for all xL. A routine computation shows that μ is a fuzzy ideal of L. Now, Equation, because

Equation

Finally, μ*=v, since Equation

Let μ be any fuzzy ideal of an n-Lie algebra L and let x∈L. The fuzzy subset * x μ of L defined by *( ) = ( ) f x μ a μ a − x or all a∈L is called the

Let μ be any fuzzy ideal of an n-Lie algebra L and let xL. The fuzzy subset Equation of L defined by Equation for all aL is called the fuzzy coset determined by x and μ.

Let I be an ideal of L. If χI is the characteristic function of I, then it is easy to see that Equation is the characteristic function of x+I.

Theorem 4.2

Let μ be any fuzzy ideal of an n-Lie algebra L. Then the set of all fuzzy cosets of μ in L, i.e., the set Equation, is an n-Lie algebra under the following operations:

Equation

Theorem 4.3

If μ is any fuzzy ideal of an n-Lie algebra L, then the map ϕ:LL[μ] defined byEquation for all xL, is a homomorphism with kernel Equation, where t=μ(0).

Proof. It is easy to see that f is a homomorphism. We show μ(x)=μ(0) implies Equation . For this, let aL. Then, μ(a)≤μ(0)=μ(x). If μ(a)<μ(x), then μ(ax)=μ(a), by Lemma 2.2. On the other hand, if μ(a)=μ(x), then Equation. Hence, μ(ax)=μ(0)=μ(x)=μ(a). Therefore, in either case, we have shown that μ(ax)=μ(a) for all aL. Consequently Equation. Also, Equation implies that μ(x)=μ(0). Hence, Equation if and only if μ(x)=μ(0). Now, we have

Equation

where t=μ(0) t = μ (0) .

Theorem 4.4

Given a homomorphism of n-Lie algebras :LL′ and fuzzy ideal μ of L and μ′ of L′ such that Equation. Then, there is a homomorphism of n-Lie algebras Equation, where Equation, such that the following diagram is commutative.

Equation

Proof. If Equation then μ(xy)=μ(0). So

Equation

and so Equation Hence,Equation holds. Thus, ϕ* is well-defined. It is easily seen that ϕ* is a homomorphism.

Let μ be a fuzzy ideal of an n-Lie algebra L. For any x,yL, we define a binary relation ∼ on L by xy if and only if μ(xy)=μ(0). Then ∼ is a congruence relation on L. We denote [x]μ the equivalence class containing x, and L/μ={[x]μ|xL} the set of all equivalence classes of L. Then, L/μ is an n-Lie algebra under the following operations:

Equation

Theorem 4.5 (Fuzzy first isomorphism theorem)

Let ϕ:LL′ be an epimorphism of n-Lie algebras and λ be a fuzzy ideal of L′. Then L/ϕ−1(λ)≅L′|λ.

Let I be an ideal and μ a fuzzy ideal of an n-Lie algebra L. If μ is restricted to I, then μ is a fuzzy ideal of I and I|μ is an ideal of L/μ.

Theorem 4.6 (Fuzzy second isomorphism theorem)

Let μ and λ be two fuzzy ideals of an n-Lie algebra L with μ(0)=λ(0). Then Equation

Theorem 4.7 (Fuzzy third isomorphism theorem)

Let μ and λ be two fuzzy ideals of an n-Lie algebra L with λ⊆μ and μ(0)=λ(0). Then Equation

Conclusion

Methods of construction fuzzy ideals are presented. Connections with various fuzzy quotient n-Lie algebras are proved. Properties of fuzzy subalgebras and ideals of n-ary Lie algebras are described.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Article Usage

  • Total views: 299
  • [From(publication date):
    August-2017 - Nov 19, 2017]
  • Breakdown by view type
  • HTML page views : 261
  • PDF downloads : 38
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2017-18
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri & Aquaculture Journals

Dr. Krish

[email protected]

1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2017 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
adwords