Medical, Pharma, Engineering, Science, Technology and Business

^{1}Department of Mathematics, Yazd University, Yazd, Iran

^{2}Faculty 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(+48)-(71)-328-07-51

Fax:

**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

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.

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

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

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

where *σ*∈*S _{n}*.

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:

where *L*=*R*[*x*_{1},…,*x _{n}*] is the vector space of polynomials in

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.

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 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*,*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

*• 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

which implies

Similarly

Hence

This for *μ*(*x*−*y*)=*μ*(0) gives *μ*(*x*)=*μ*(*y*), and *μ*(*x*−*y*)=*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 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 be a collection of fuzzy subsets of *X*. Then, we define the
fuzzy subsets

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}*]∈

A subspace S of an i-ideal of *L* if for all and *y*∈*S* we have

Two *n*-Lie algebras *L*_{1}, *L*_{2} over the same field *F* are isomorphic
if there exists a vector space isomorphism *ϕ*:*L*_{1}→*L*_{2} such that for all

Let *L* be an *n*-Lie algebra. Fixing in [*x*_{1}, *x*_{2},…,*x _{n}*] elements

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

**Definition 3.2**

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

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

(1)

for all *x*_{1},…,*x _{n}*∈

**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*.

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

**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:

• 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*,

• 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

**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 , is a subalgebra (ideal) of *L*.

** Proof.** Let

Conversely, assume that every non-empty is an ideal of *L*.
Therefore, 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, . Therefore, for every * x*_{1},…*x _{n}*∈

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 (with *t*_{1}<*t*_{2}) be any two level
subalgebras of *μ*. Then if and only if there is no *x* in *L* such
that .

**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

is a fuzzy ideal of *L*.

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

Let for some fixed *α*∈[0,1]. Then

In the first case we have , because

In the second, there exists *ε*>0 such that . In this case. Indeed, if, then *x*∈*S*_{λ} for some *λ*≥*α*, which gives *μ*(*x*)≥*λ*≥*α*. Thus.

Conversely, if , then for all *λ*≥*α*, which implies for all *λ*>*α*−*ε*, i.e., if *x*∈*S*_{λ} then *λ*≤*α*−*ε*. Thus *μ*(*x*)≤*α*−*ε*. Therefore. Hence , and consequently . This completes our proof.

**Theorem 3.12**

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

** Proof.** First consider the case when all

It is clear that for arbitrary elements x1,…, xn∈L there exists only one
k such that and only one *k _{i}* such that . Thus.

Suppose for all *i*=1,2,…,*n*. Then, by the assumption, *k _{i}*<

Now, let all Si be ideals and let for some *x*_{1},…*x _{n}*∈

**Corollary 3.13**

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

**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

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

(*c*) 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 *t*_{0}=*μ*(0)≥*μ*(*x*) for all *x*∈*L*.

(*b*) If *x*∈*L*, then *μ*(*x*)=*t _{x}*∈

(*c*) Since , 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*=inf*Im*(*μ*). Then *β*≤*t*∈*Im*(*μ*), i.e., for all * t*∈*Im*(*μ*). But because Ω contains all levels of *μ*. Hence there exists *t*′∈*Im*(*μ*) such that . It follows that so that because every level of *μ* is a subalgebra (resp. ideal) of *L*.

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

In the case *S*≠*L* we consider the fuzzy set *μ _{S}* defined by

Clearly *μ _{S}* is a fuzzy subalgebra (ideal) of

Let

Then contains (by the assumption) all levels of *μ _{S}*. This means that there exists

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

Let *S*={*x*∈*L*|*μ*(*x*)>*α*}. Obviously 0∈*S* and *μ*(*x _{i}*)>

which proves [*x*_{1},…,*x _{n}*]∈

Note that there does not exist z∈L such that *α*≤*μ*(z)<*t*′. This gives Hence . 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.

If , then obviously

If , then for some *p*≥0
and there exists at least one *i*=1,2,…,*n* such that , because implies.

Let *S _{m}* be a maximal ideal of

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 {*t _{i}*} such that

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

is a fuzzy subalgebra (ideal) on *L*.

Indeed, for every *x*_{1},…,*x _{n}*∈

The case when at least one of *x*_{1},*x*_{2},…,*x _{n}* is not in

Now, if all *S _{i}* are ideals, then [

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

Since the chain 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 *x*∈*L*.

**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 *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 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 *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

**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*o*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}*,

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* such 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.

**Proof.** Suppose that *ϕ*:*L*_{1}→*L*_{2} is an *n*-Lie algebra homomorphism, *μ* is a fuzzy ideal of *L*_{1} with the sup property and *v* is the image of *μ* under *ϕ*. Suppose that be such that , respectively. Then,

and

Finally, let and let be such that

Then,

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

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

for all *x*_{1},…,*x _{n}*∈

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 defined by for all *x*∈*L*, is a fuzzy ideal
of .

• 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 *μ _{t}* =

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

Now, we show *μ** is a fuzzy ideal of *L*. Let *x*,*y*∈*L* and *α*∈*F*. Then, we have

and

(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, , because

Finally, *μ**=*v*, since

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 *x*∈*L*. The fuzzy
subset of *L* defined by for *all* *a*∈*L* 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

**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 , is an *n*-Lie algebra
under the following operations:

**Theorem 4.3**

If *μ* is any fuzzy ideal of an *n*-Lie algebra *L*, then the map *ϕ*:*L*→*L*[*μ*]
defined by for all *x*∈*L*, is a homomorphism with kernel , where *t*=*μ*(0).

**Proof.** It is easy to see that *f* is a homomorphism. We show *μ*(*x*)=*μ*(0) implies . 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 . 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*. Consequently . Also, implies that *μ*(*x*)=*μ*(0). Hence, if and only if *μ*(*x*)=*μ*(0). Now, we have

where *t*=*μ*(0) *t* = *μ* (0) .

**Theorem 4.4**

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

**Proof.** If then *μ*(*x*−*y*)=*μ*(0). So

and so Hence, 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*,*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:

**Theorem 4.5 (Fuzzy first isomorphism theorem)**

Let *ϕ*:*L*→*L*′ 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

**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

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.

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