Journal of Generalized Lie Theory and Applications

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.


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 (x 1 ,…,x n ) of elements in L another element [x 1 ,…,x n ] which is n-linear, skew-symmetric: (  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: where L=R[x 1 ,…,x n ] 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].
The study of fuzzy Lie algebras was initiated in refs.[8,9], and continued in various directions by many authors (for example [10][11][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 x∈X.The set

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,y∈V and α∈F, the following conditions are satisfied: • µ(x+y)≥min{µ(x), µ(y)} for all x,y∈V, • µ(αx)≥µ(x) for all x∈V, α∈F.
Note that the second condition implies, µ(−x)≥ µ(x) for all x∈V,

Lemma 2.2
If µ is a fuzzy subspace of a vector space V, then µ(x)≤ µ(0) for all x∈V, and for all x,y∈V.

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 t µ 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].
µ ∈ be a collection of fuzzy subsets of X.Then, we define the

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 [x 1 ,…,x n ]∈S for all x 1 ,…,x n ∈S.
Let L be an n-Lie algebra.Fixing in [x 1 , x 2 ,…,x n ] elements x 2 ,…, x {n-1} we obtain a new binary operation 〈x,y〉=[x,x 2 ,…,x n−1 ,y] with the property 〈x k ,y〉=〈y,x k 〉=0 for all k=2,…,n−1 and all y∈L.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 x 2 ,…,x n-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.
Let L be an n-Lie algebra.A fuzzy ideal of L is a fuzzy subspace µ such that ([ , , ]) ( ) for all , , and 1 .
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 for all x 1 ,…,x n ∈L.
Proposition 3.4: If µ is a fuzzy ideal of an n-Lie algebra L, then ∈ is an ideal of L contained in every non-empty level subset of µ.

Theorem 3.6
Let ϕ:L→L′ be an n-Lie algebra homomorphism of an n-Lie algebra L onto an n-Lie algebra L′.Then the following conditions hold: 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 {µ n } be a chain of fuzzy subalgebras (ideals) of an n-Lie algebra L. Then n n µ  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.
Proof.Let µ be a fuzzy ideal of L. Since µ is a fuzzy subspace of L, by Theorem 2.3, each non-empty t µ is a subspace of L. Therefore, it is enough to prove that 1 [ , , , , , , ] . For every . Since µ is a fuzzy ideal, we have , , , , , , ]) and so 1 1 1 [ , , , , , , ] Conversely, assume that every non-empty t µ is an ideal of L.
Therefore, t µ is a subspace of L and so by Theorem 2.3, µ is a fuzzy subspace of L. Now, for every y∈L, we put t 0 =µ(y).Then,  , , , , , , ]) 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 1 t µ and 2 t µ (with t 1 <t 2 ) be any two level subalgebras of µ.Then

Theorem 3.11
Let {S| λ∈λ}, where ∅≠λ⊆[0,1], be a collection of ideals of an n-Lie algebra L such that Then µ defined by Proof.By Theorem 3.9, it is sufficient to show that every non-empty In the first case we have
Proof.First consider the case when all S i are subalgebras.If  because in the opposite case x 1 ,…,x n and [x 1 ,…,x n ] will be in some S k .So, in this case t for all i=1,2,…,n.Then, by the assumption, k i <k and . Since µ also is a fuzzy subspace of a vector space L, it is a fuzzy subalgebra of L. Now, let all S i be ideals and let [x 1 ,…x n ]∈S k \S k−1 for some x 1 ,…x n ∈L.Then these x 1 ,…x n are in L\S k−1 .If not, then there exists . This completes the proof that µ is a fuzzy ideal.

Corollary 3.13
For any chain S 0 ⊂S 1 ⊂S 2 … of subalgebras (ideals) of an n-Lie algebra L and any chain of reals 1≥t 0 >t 1 >…≥0 there exists a fuzzy subalgebra .

Theorem 3.14
Let Im(µ)={t 1 |i∈I} be the image of a fuzzy subalgebra (ideal) µ of an n-Lie algebra L. Then (a) There exists a unique t 0 ∈Im(µ) such that t 0 ≥ t i for all t i ∈Im(µ), (b) L is the set-theoretic union of all t i µ , 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.
µ Ω ∈ contains (by the assumption) all levels of µ S .This means that there exists x 0 ∈S such that µ µ ∈ i.e., µ(x 0 )=µ S (x) for some x∈S.Hence µ attains its infimum on all subalgebras (ideals) of L.
To prove the converse let α µ be a level subalgebra of µ.If α=t for some t∈Im(µ), then α µ ∈ Ω .If α≠t for all t∈Im(µ), then there does not exist x∈L such that µ(x)=α.
Note that there does not exist z∈L such that α≤µ(z)<t′.This gives 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 of ideals of L which does not terminate at finite step.We prove that µ defined by


where k=0,1,2,… and S 0 =L, is a fuzzy ideal with an infinite number of values.
for some p≥0 and there exists at least one i=1,2,…,n Let S m be a maximal ideal of L such that at least one of x 1 ,…,x n belongs to S m \S m+1 .Then m≤p.Indeed, for m>p we have 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 {t i } such that t i =µ(x i ) for some x i ∈L.But in this case t i µ form a strictly ascending chain of subalgebras (ideals) of L, which is a contradiction.
The case when at least one of x 1 ,x 2 ,…,x n is not in M is obvious.Hence µ is a fuzzy subalgebra.Now, if all S i are ideals, then [x 1 ,…,x n ]∈S m for m=min{k 1 ,…,k n }.Thus p≤m.Hence which means that in this case µ is a fuzzy ideal.
Since the chain S 1 ⊂S 2 ⊂S 3 ⊂… 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 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.If µ(x)=1 for all x∈L, 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 a∈L, then a fuzzy subset v defined by ( ) every x∈L.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 µ h defined on L by µ h (x)=h(µ(x)) is a fuzzy subalgebra (ideal).Moreover, µ h 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 for all y∈f(L) is called the image of µ under f.Similarly, if v is a fuzzy subset in f(L), then the fuzzy set µ=vf 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:L 1 →L 2 be an n-Lie algebra homomorphism, and v be a fuzzy ideal of L 2 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 , for all x 1 ,…,x n ,y∈L and α∈F.
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 x 0 ∈S that µ(x 0 )=sup x∈S {µ(x)}.

Theorem 3.23
An n-Lie algebra homomorphism image of a fuzzy ideal having the sup property is a fuzzy ideal.

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 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 an n-Lie algebra L. For any x,y∈L, we define a binary relation ∼ on L by x∼y if and only if µ(x−y)=µ(0).Then ∼ is a congruence relation on L. We denote [x] µ the equivalence class containing x, and L/µ={[x]µ|x∈L} the set of all equivalence classes of L.Then, L/µ is an n-Lie algebra under the following operations: 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.
generalized Jacobi identity (called also the Filippov identity): structures are also called n-Lie algebras or Filippov algebras.For n=2 we obtain a classical Lie algebras.
for i,j∈I, 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) S≠L.For S=L let β=infIm(µ).Then β≤t∈Im(µ), i.e., Ω contains all levels of µ.Hence there exists t′∈Im(µ) such that = of µ is a subalgebra (resp.ideal) of L.Now it sufficient to show that β=t′.If β<t′, then there exists t″∈Im(µ) such that β≤t″<t′.This implies = a contradiction.Therefore β=t′∈Im(µ).In the case S≠L we consider the fuzzy set µ S defined by f ,Clearlyµ S is a fuzzy subalgebra (ideal) of L if S is a subalgebra (ideal).Let Volume 11 • Issue 2 • 1000268 J Generalized Lie Theory Appl, an open access journal ISSN: 1736-4337 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.

Proposition 3 .
20: A maximal normal fuzzy subalgebra of an n-Lie algebra L takes only two values: 0 and 1.
x∈L, is a fuzzy ideal of / t L µ .• 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 x∈I, then there exists a fuzzy ideal µ of L such that = t I µ , where t=µ(0); and v=µ * .

χTheorem 4 . 2
2).We define a fuzzy subset µ of L by µ(x)=v(x+I) for all x∈L.A routine computation shows that µ is a fuzzy ideal of L. Now, 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 a a x or all a L µ µ − ∈ is called the fuzzy coset determined by x and µ.Let I be an ideal of L. If χ I is the characteristic function of I, then it is easy to see that is the characteristic function of x+I.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 If µ is any fuzzy ideal of an n-Lie algebra L, then the map ϕ:L→L[µ] defined by * ( ) = x x ϕ µ for all x∈L, is a homomorphism with kernel t µ , where t=µ(0).Proof.It is easy to see that f is a homomorphism.We show µ(x)=µ(0) implies * * 0 = x µ µ .For this, let a∈L.Then, µ(a)≤µ(0)=µ(x).If µ(a)<µ(x), then µ(a−x)=µ(a), by Lemma 2.2.On the other hand, if µ(a)=µ(x), then a,x∈{y∈L|µ(y)=µ(0)}.Hence, µ(a−x)=µ(0)=µ(x)=µ(a).Therefore, in either case, we have shown that µ(a−x)=µ(a) for all a∈L.that µ(x)=µ(0).Hence, and only if µ(x)=µ(0).Now, we have

Theorem 4 . 6 (Theorem 4 . 7 (Conclusion
Fuzzy second isomorphism theorem)Let µ and λ be two fuzzy ideals of an n-Lie algebra L with µ(0)=λ(0).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).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.
every x 1 ,…x n ∈L we have 1 Volume 11 • Issue 2 • 1000268 J Generalized Lie Theory Appl, an open access journal ISSN: 1736-4337 for fuzzy subalgebra (ideal) on L. , and a minimal number p such that [x 1 ,…,x n ]∈S p .If all S i are subalgebras, then for k=max{k 1 ,k 2 ,…,k n } all x 1 ,…,x n and [x 1 ,…,x n ] are in S k .Thus k≥p.Consequently,