Nilpotent fuzzy lie ideals

1 Introduction

Sophus Lie introduced Lie algebras (1842-1899). Problems about the group replaced with the problems about the Lie algebra which are more tractable. Some applications of Lie algebras are spectroscopy of molecules and atoms. Nilpotent Lie algebras have a vital role over Lie algebras, and differential geometry. The study of Engel Lie algebras allowed us to solve existing problems on nilpotent Lie algebras. Every 2-Engel Lie algebra is nilpotent of class at most 3. In [4], it is shown that 3-Engel Lie algebras with char k ≠ 2, 5 are nilpotent of class at most 4. The notion of fuzzy sets was first introduced by L. A. Zadeh [24]. Later on, the notion of fuzzy sets has been extensively studied by various authors on several sciences [1, 14–18]. Many researchers have applied the fuzzy sets theory in different fields (see for e.g., [8–13, 22]). Moreover, Kim and Lee in [14] investigated some properties of fuzzy Lie algebra and fuzzy Lie ideal. In [23], nilpotent fuzzy Lie algebras were defined. In that paper, Yahia assigned to a fuzzy Lie ideal a descending central series to define the nilpotency of a fuzzy Lie ideal. Now, in this paper, by using a new definition for ascending series of a fuzzy Lie idea, we define the concept of nilpotent fuzzy Lie ideal, which we name it a good nilpotent fuzzy Lie ideal. Also, we get some critical results of fuzzy Lie algebra. The appropriateness of our definition can be seen from the fact that, for a fuzzy Lie ideal ρ of L, {a ∈ L ∣ ρ(a ∘ b₁ ∘ . . . ∘ b _n ) = ρ(0) ,   for  any  b₁, . . . , b _n  ∈ L} is equal to the n-th term of ascending series. Therefore, by this definition the fuzzification of nilpotent Lie algebras is possible. Moreover, with this definition, for fuzzy Lie ideals, we get some significant results on fuzzy Lie ideals that have not been considered before. Also, we define an Engel fuzzy Lie ideal of L. Then we prove that each Engel fuzzy Lie ideal of finite Lie algebra L is good nilpotent. In particular, using the notion of good nilpotent fuzzy Lie ideals, we show that with some additional properties if ρ is a good nilpotent fuzzy Lie ideal of L and A is a finite maximal ideal of L, then ρ is finite. There is another way we can justify our definition of a good nilpotent fuzzy Lie ideal. In the classical Lie algebra, each finite-dimensional Engel Lie algebra over a field is nilpotent. This initial study can verify, as we have done in Theorem 4.11, that each Engel fuzzy Lie ideal of a finite Lie algebra is good nilpotent.

2 Preliminary

In this section, we recall some basic definitions which are proposed by the pioneers of this subject.

Definition 2.1. [5] Let L be a vector space over a field F. Consider the operation ∘ : L × L ⟶ L defined by (x, y) ⟶ x ∘ y. Then L is called a Lie algebra if the following axioms are satisfied.:

x ∘ y is bilinear,

Notation: From now one in this paper, let L be a Lie algebra.

A subspace H of L is called a Lie subalgebra, if for all x, y ∈ H we have (x ∘ y) ∈ H. The Lie ideal of L, denoted by I, is the subspace of L with the property (x ∘ y) ∈ I, for any x ∈ L, y ∈ I. (Since x ∘ y = -(y ∘ x), the condition could just as well be written: (y ∘ x) ∈ I).

For a Lie ideal I, quotient algebra L/I is a vector space L/I, where Lie multiplication is defined by (x + I) ∘(y + I) =(x ∘ y) + I, for any x, y ∈ L. Derived algebra of L, denoted by L ∘ L, consists of all linear combinations of commutators x ∘ y. Clearly L ∘ L is a Lie ideal of L. L is called Abelian if for any x, y ∈ L, x ∘ y = 0. Evidently, L is Abelian if and only if L ∘ L = 0. Descending central series of L is a sequence of Lie ideals of L defined by L⁰ = L, L¹ = L ∘ L,...,Lⁱ⁺¹ = L ⁱ  ∘ L, for any i ∈ ℕ . Also the ascending central series is the sequence of Lie ideals {0} = Z₀(L)⊂ Z₁(L) ⊂ . . . ⊂, where Z₀(L) = {0}, Z₁(L) = Z(L) = {z ∈ L ∣ z ∘ x = 0,   for  any  x ∈ L}) and recursively, the Lie ideal Z _i (L) is defined by Z _i (L)/Z_i-1(L) = Z(L/Z_i-1(L)). (See [5])

Definition 2.2. [5] L is called nilpotent if L ^m  = 0 (or Z _m (L) = L), for some nonnegative integer m.

Theorem 2.3. [5] Let n ∈ ℕ . Then

x ∈ Z _n (L) if and only if for any y₁, . . . , y _n  ∈ L, ((x ∘ y₁) ∘ . . . ∘ y _n ) =0,

Definition 2.4. L is called n-Engel if ( x ∘ y ∘ . . . ∘ y ︸ n ) = 0 , where (x ∘ ₀y) = x,   (x ∘ ₁y) = x ∘ y and inductively for n ∈ ℕ , x ∘  _n y =((x ∘ _n-1y) ∘ y.

Definition 2.5. [24] A fuzzy subset α of X is a function α : X → [0, 1].

Definition 2.6. [18] For a function g : X → Y and a fuzzy subset ρ of X the fuzzy subset g(ρ) of Y is defined by the following ( g ( ρ ) ) ( y ) = { ⋁ x ∈ g - 1 ( y ) ρ ( x ) , g - 1 ( y ) ≠ ϕ 0 , otherwise

Definition 2.7. [18] A fuzzy subset γ : L ⟶ [0, 1] is said to be a fuzzy Lie subalgebra of L if for all x, y ∈ L and α ∈ F it satisfies:

γ(x + y) ≥ γ(x) ∧ γ(y).

Definition 2.8. [23] A fuzzy subset γ : L ⟶ [0, 1], satisfying the conditions (i) and (ii) of Definition 2.7 and γ(x ∘ y) ≥ γ(x), for any x, y ∈ L, is called a fuzzy lie ideal of L.

Theorem 2.9. [14] Let γ be a fuzzy Lie subalgebra of L. Then, for any x, y ∈ L;

γ(0) ≥ γ(x).

Notation: From now one in this paper, let I be a Lie ideal of L and γ be a fuzzy Lie ideal of L, unless otherwise state.

3 Nilpotent fuzzy Lie ideals: revision

Nilpotency is a vital concepts in the study of Lie algebras. In [23] Yehia used the concept of fuzzy sets to Lie algebras and defined the notion of nilpotent fuzzy Lie ideal. Now in this section, a new definition for nilpotent fuzzy Lie ideal is proposed, which is a well-defined extension of that. In Lie algebras, for a nilpotent Lie algebra L and a non-zero ideal of it, say I, we have I ∩ Z(L) ≠0. In Theorem 3.15, we proved that for a good nilpotent fuzzy Lie ideal L and a non-zero ideal of it, say I, I ∩ C(γ) ≠0. Using this helps us to get the primary result of this section (Corollary 3).

Take C₀(γ) = {0}. Let C₁(γ) = {x ∈ L ∣ γ(x ∘ y) = γ(0), for any y ∈ L}. Then C₁(γ) is a Lie ideal of L. For this, let x ∈ C₁(γ) and y ∈ L. Then for any c ∈ L, γ((x ∘ y) ∘ c) ≥ γ(x ∘ y) = γ(0) and so (x ∘ y) ∈ C₁(γ). Hence, C₁(γ) is an ideal of L. Define a subalgebra C₂(γ) of L such that C 2 ( γ ) C 1 ( γ ) = Z ( L C 1 ( γ ) ) . Clearly C₁(γ) is a Lie ideal of C₂(γ). Now, we show that (C₂(γ) ∘ L) ⊆ C₂(γ). For this, let x ∈ C₂(γ) and g ∈ L. Thus x + C 1 ( γ ) ∈ C 2 ( γ ) C 1 ( γ ) = Z ( L C 1 ( γ ) ) . Then (x + C₁(γ)) ∘(g + C₁(γ)) = C₁(γ) for any g ∈ L. Therefore (x ∘ g) ∈ C₁(γ) ⊆ C₂(γ). Hence (C₂(γ) ∘ L) ⊆ C₂(γ). Thus C₂(γ) is a Lie ideal of L. Similarly for k ≥ 2 we define a Lie ideal C _k (μ) such that C k ( γ ) C k - 1 ( γ ) = Z ( L C k - 1 ( γ ) ) . It is clear that C₀(γ) ⊆ C₁(γ) ⊆ C₂(γ) ⊆ . . ..

Definition 3.1. A fuzzy Lie ideal γ of L is called a good nilpotent fuzzy Lie ideal of L if for a non-negative integer m, we have C _m (γ) = L. The smallest integer such that C _m (γ) = L is called the class of γ. We abbrrviate by GNFLI, the "good nilpotent fuzzy Lie ideal" of L.

Example 3.2. Consider W be a vector space over a field F with basis {e₁, . . . , e₈} and define Lie bracket operation ∘ : W × W ⟶ W as follows:

e₁ ∘ e₂ = e₅  e₁ ∘ e₃ = e₆  e₁ ∘ e₄ = e₇  e₁ ∘ e₅ = e₈  e₂ ∘ e₃ = e₈

e₂ ∘ e₄ = e₆  e₂ ∘ e₆ = - e₇  e₃ ∘ e₄ = - e₅  e₃ ∘ e₅ = - e₇  e₄ ∘ e₆ = - e₈,

and for all other i ≤ j, e i ∘ e j = 0 e i ∘ e j = - ( e j ∘ e i ) Then W is a Lie algebra over a field F. Consider a fuzzy set γ defined as follows: γ ( a ) = { 1 if a = 0 , e 1 , e 5 , e 6 , e 7 , e 8 , 0 if otherwise . Then γ is a fuzzy Lie ideal of W. We prove that C(γ) = W. Since for any a ∈ W, γ(e₈ ∘ a) =1 = γ(0), we conclude that e₈ ∈ C(γ). Similarly we can see that e₁, . . . , e₇ ∈ C(γ). Consequently, γ is a GNFLI.

Example 3.3. Let W be as in Example 3.2. We define the fuzzy set λ of W as follows λ ( a ) = { 1 if a = 0 , 0.5 if a = e 2 , e 5 , e 6 , e 7 , e 8 , 0 if otherwise . Then λ is a fuzzy Lie ideal of W. We prove that λ is a GNFLI of class 3. Since λ ( e 1 ∘ b ∘ c ∘ r ) = λ ( t ∘ c ∘ r ) , ∀ t ∈ { e 5 , e 6 , e 7 , e 8 , 0 } and so λ ( e 1 ∘ b ∘ c ∘ r ) = λ ( t ∘ c ∘ r ) = λ ( g ∘ r ) , ∀ g ∈ { e 8 , 0 } Therefore, λ(e₁ ∘ b ∘ c ∘ r) =1 = λ(0). Also, λ ( e 2 ∘ b ∘ c ∘ r ) = λ ( t ∘ c ∘ r ) , ∀ t ∈ { e 5 , e 6 , e 7 , e 8 } and so λ ( e 2 ∘ b ∘ c ∘ r ) = λ ( t ∘ c ∘ r ) = λ ( g ∘ r ) , ∀ g ∈ { e 8 , 0 , e 7 } Hence λ(e₂ ∘ b ∘ c ∘ r) =1 = λ(0). Similarly, λ ( a ∘ b ∘ c ∘ t ) = 1 = λ ( 0 ) , ∀ a , b , c , t ∈ V Therefore, λ is a GNFLI of class 3.

Now, we state that for n ∈ ℕ , each Lie ideal C _n (γ), defined by C n + 1 ( γ ) C n ( γ ) = Z ( L C n ( γ ) ) , is equal to { a ∈ L ∣ γ ( a ∘ b 1 ∘ . . . ∘ b n ) = γ ( 0 ) , for any b 1 , b 2 , . . . , b n ∈ L }

Lemma 3.4. For any i ∈ ℕ , we have C i ( γ ) = { a ∈ L ∣ γ ( a ∘ b 1 ∘ . . . ∘ b i ) = γ ( 0 ) } .(1) for any b₁, b₂, . . . , b _i  ∈ L

Proof. We prove it by induction on i. If i = 1, then C 1 ( γ ) = { a ∈ L ∣ γ ( a ∘ b ) = γ ( 0 ) , for any b ∈ L } Consider the result is true for i ≤ n, then a ∈ C_n+1(γ) ⇔ a + C n ( γ ) ∈ C n + 1 ( γ ) C n ( γ ) = Z ( L C n ( γ ) ) , ⇔(a + C _n (γ)) ∘(b₁ + C _n (γ)) = C _n (γ) , for any b₁ ∈ L, ⇔ ( a ∘ b 1 ) + C n ( γ ) = C n ( γ ) , for any b 1 ∈ L , ⇔ ( a ∘ b 1 ) ∈ C n ( γ ) , for any b 1 ∈ L , ⇔γ((a ∘ b₁) ∘ b₂ ∘ . . . ∘ b_n+1) = γ(0) , for any b₁, . . . , b_n+1 ∈ L.□

Lemma 3.5. For any k ∈ ℕ , we have C k ( γ ) C ( γ ) = Z k - 1 ( L C ( γ ) ) where C(γ) = C₁(γ).

Proof. Note that for i ∈ ℕ , a ∈ C _i (L) if and only if (a ∘ b₁ ∘ . . . ∘ b _i ) =0, for any b₁, b₂, . . . , b _i  ∈ L. Hence for any b₁, b₂, . . . , b_k-1 ∈ L,

a + C ( γ ) ∈ Z k - 1 ( L C ( γ ) )

⇔(a + C(γ)) ∘(b₁ + C(γ)) ∘ . . . ∘(b_k-1 + C(γ)) = C(γ) , byTheorem 2.3, ⇔ ( a ∘ b 1 ∘ . . . ∘ b k - 1 ) + C ( γ ) = C ( γ ) , ⇔ ( a ∘ b 1 ∘ . . . ∘ b k - 1 ) ∈ C ( γ ) , ⇔ γ ( a ∘ b 1 ∘ . . . ∘ b k - 1 ∘ b k ) = γ ( 0 ) , by Lemma 3.4 , ⇔ a ∈ C k ( γ ) , by Lemma 3.4 , ⇔ a + C ( γ ) ∈ C k ( γ ) C ( γ ) . Therefore, C k ( γ ) C ( γ ) = Z k - 1 ( L C ( γ ) ) .□

A linear map θ: L → L′, where L and L′ are two Lie algebras, is called a Lie homomorphism if for any x, y ∈ L, θ(x ∘ y) = θ(x) ∘ θ(y).

Lemma 3.6. Let K = L C ( γ ) , γ ¯ be a fuzzy Lie ideal of K and I = C ( γ ¯ ) . If for some n ∈ ℕ , C _n (K) = K, then there exist m ≤ n such that C m ( K I ) = K I . Moreover nilpotency of K implies the nilpotency of K I .

Proof. Since K is nilpotent, we conclude that C _n (K) = K, for some n ∈ ℕ . We claim that there exist m ≤ n such that C m ( K I ) = K I . Also, K I is a homomorphic image of K. Then by Theorem 2.3, we have K I is nilpotent of class at most m.□

Now, we prove that by a GNFLI of L we obtain a GNFLI of L C ( γ ) .

Theorem 3.7. Let γ ¯ be a fuzzy Lie ideal of L C ( γ ) . If γ is a GNFLI of class m, then γ ¯ is a GNFLI of class n, where n ≤ m.

Proof. Since γ is a GNFLI of class m, we have C _m (γ) = L. We claim that C n ( γ ¯ ) = L C ( γ ) where n ≤ m. By Lemma 3.5, C m ( γ ) = L ⇔ L C ( γ ) = C m ( γ ) C ( γ ) = Z m - 1 ( L C ( γ ) ) . Now put n instead of m and γ ¯ instead of γ, then C n ( γ ¯ ) = L C ( γ ) ⇔ Z n - 1 ( L C ( γ ) C ( γ ¯ ) ) = L C ( γ ) C ( γ ¯ ) . If Z m - 1 ( L C ( γ ) ) = L C ( γ ) , then by Lemma 3.6 we have

Z n - 1 ( L C ( γ ) C ( γ ¯ ) ) = L C ( γ ) C ( γ ¯ ) i.e. if C _m (γ) = L, then C n ( γ ¯ ) = L C ( γ ) This complete the result.□

Theorem 3.8. [14] K and L be two Lie algebras and g : L ⟶ K be a Lie homomorphism. Consider γ and ν be fuzzy Lie ideals of L and K, respectively. Then

the fuzzy set g(γ) is a fuzzy Lie ideal of K.

Theorem 3.9. Let g : L ⟶ K be an epimorphism of Lie algebras. If γ is a GNFLI of L, then g(γ) is a GNFLI of K.

Proof. By Theorem 3.8, g(γ) is a fuzzy Lie ideal of K. We claim that for any i ∈ ℕ , g(C _i (γ)) ⊆ C _i (g(γ)). Let a ∈ g(C _i (γ)). Then there exists u ∈ C _i (γ) such that a = g(u). Since g is an epimorphism, we conclude that for all b₁, . . . , b _n  ∈ K, b _i  = g(v _i ) for some v _i  ∈ L where 1 ≤ i ≤ n. Then (a ∘ b₁ ∘ . . . ∘ b _n ) =(g(u) ∘ g(v₁) ∘ . . . ∘ g(v _n )) and so

(g(γ))(a ∘ b₁ ∘ . . . ∘ b _n ) = ⋁ g ( c ) = ( a ∘ b 1 ∘ . . . ∘ b n ) γ ( c ) = ⋁ g ( c ) = g ( u ∘ v 1 ∘ . . . ∘ v n ) γ ( c ) .(2) Now, by Lemma 3.4 and u ∈ C _i (γ), we get that γ(u ∘ v₁ ∘ . . . ∘ v _n ) = γ(0 _L ). Therefore, ( g ( γ ) ) ( a ∘ b 1 ∘ . . . ∘ b n ) = γ ( 0 L ) = ( g ( γ ) ) ( 0 K ) .(3) Hence by Lemma 3.4, a ∈ C _i (g(γ)). Consequently, g(C _i (γ)) ⊆ C _i (g(γ)). Now if γ is GNFLI, then there exists nonnegative integer n such that C _n (γ) = L and so g(C _n (γ)) = g(L). Therefore, C _n (g(γ)) = K and so g(γ) is GNFLI.□

Theorem 3.10. Consider g : L ⟶ K be an epimorphism of Lie algebras. Then λ is a GNFLI of K if and only if g^-1(λ) is a GNFLI of L.

Proof. We claim that for any i ∈ ℕ , C _i (g^-1(λ)) = g^-1(C _i (λ)). By Lemma 3.4,

x ∈ C _i (g^-1(λ))

⇔(g^-1(λ))(x ∘ x₁ ∘ . . . ∘ x _i ) =(g^-1(λ))(0) , forany x₁, x₂, . . . , x _i  ∈ L,

⇔λ(g(x) ∘ g(x₁)) ∘ . . . ∘ g(x _i )) = λ(0) , forany x₁, x₂, . . . , x _i  ∈ L,

⇔g(x) ∈ C _i (λ) , ⇔ x ∈ g^-1(C _i (λ)) .

Then λ is GNFLI if and only if there exists nonnegative integer n such that C _n (λ) = K if and only if g^-1(C _n (λ)) = g^-1(K) if and only if C _n (g^-1(λ)) = L if and only if, g^-1(λ) is GNFLI.□

Proposition 3.11. Consider two fuzzy Lie ideals γ and λ of L such that γ ⊆ λ and γ(0) = λ(0). Then C(γ) ⊆ C(λ).

Proof.Take x ∈ C(γ), then for any y ∈ L, we have γ(x ∘ y) = γ(0),. Also, λ ( 0 ) = γ ( 0 ) = γ ( x ∘ y ) ≤ λ ( x ∘ y ) ≤ λ ( 0 ) .(4) Hence λ(0) = λ(x ∘ y), then x ∈ C(λ). Consequently, C(γ) ⊆ C(λ).□

Lemma 3.12. Let i > 1. Then for any y ∈ L, (x ∘ y) ∈ C_i-1(γ) if and only if x ∈ C _i (γ).

Proof. (⇒) Take x ∈ L and (x ∘ y) ∈ C_i-1(γ) for any y ∈ L. Then by Lemma 3.4, γ((x ∘ y) ∘ y₁ ∘ . . . ∘ y_i-1)) = γ(0) for any y, y₁, . . . , y_i-1 ∈ L . Hence x ∈ C _i (γ). (⟸) In the similar way we have the converse.□

Now we present a connection between nilpotency of a Lie algebra and its fuzzy Lie ideals.

Theorem 3.13. L is nilpotent if and only if any fuzzy Lie ideal γ of L is GNFLI.

Proof. (⇒) Consider γ be a fuzzy Lie ideal of L. Since L is nilpotent we conclude that Z _n (L) = L for some n ∈ ℕ . We show that for any nonnegative integer i, Z _i (L) ⊆ C _i (γ). For i = 0 or 1, the result is immediate. Let for i > 1, Z _i (L) ⊆ C _i (γ) and a ∈ Z_i+1(L). Then for any b ∈ L we have (a ∘ b) ∈ Z _i (L) ⊆ C _i (γ). Then by Lemma 3.12, a ∈ C_i+1(γ). Hence for any i ≥ 0 we have Z_i+1(L) ⊆ C_i+1(γ), and so C _n (γ) = L. Therefore γ is GNFLI. (⟸) Consider any fuzzy Lie ideal of L be GNFLI. Define fuzzy Lie ideal γ on L as follows:

γ ( a ) = { 1 if a ∈ Z 0 ( L ) , 1 i + 1 if a ∈ Z i ( L ) - Z i - 1 ( L ) , 0 otherwise . Then γ is a fuzzy Lie ideal of L. We show that for any nonnegative integer i C _i (γ) ⊆ Z _i (L). For i = 0, the proof is clear. If i = 1 and a ∈ C₁(γ), then for any b ∈ L we have γ(a ∘ b) = γ(0) =1. Then by definition of γ, (a ∘ b) ∈ Z₀(L) = {0} and so a ∈ Z₁(L). Now let C_i-1(γ) ⊆ Z_i-1(L), for i ≥ 2. Then by Lemma 3.12, a ∈ C _i (γ) implies that for any b ∈ L; (a ∘ b) ∈ C_i-1(γ) ⊆ Z_i-1(L). Hence, for any b, b₁, . . . , b_i-1 ∈ L, (a ∘ b ∘ b₁ ∘ . . . ∘ b_i-1) =0 which implies that a ∈ Z _i (L). Then C _i (γ) ⊆ Z _i (L), for any nonnegative integer i. It is clear that Z _i (L) ⊆ C _i (γ) for any nonnegative integer i. Then C _i (γ) = Z _i (L). Now by the assumption there exists n ∈ ℕ such that L = C _n (γ). Hence L = Z _n (L) and so L is nilpotent.□

Theorem 3.14. Let γ₁ and γ₂ be two GNFLI of L. Then γ₁ × γ₂ is a GNFLI too.

Proof. It is easy to show that γ₁ × γ₂, denoted by γ, is a fuzzy Lie ideal of L × L. Now by hypotheses there exist n 1 , n 2 ∈ ℕ such that C _{n 1} (γ₁) = L and C _{n 2} (γ₂) = L. Take n = max {n₂, n₁} and (x, y) ∈ L × L. Then for any x₁ . . . , x _n , y₁ . . . , y _n  ∈ L we have γ 1 ( x ∘ x 1 ∘ . . . ∘ x n ) = γ 1 ( 0 ) , γ 2 ( y ∘ y 1 ∘ . . . ∘ y n ) = γ 2 ( 0 ) . Then (γ₁ × γ₂)((x, y) ∘ . . . ∘(x _n , y _n )) = γ 1 ( x ∘ x 1 ∘ . . . ∘ x n ) ∧ γ 2 ( y ∘ y 1 ∘ . . . ∘ y n ) = ( γ 1 × γ 2 ) ( 0 , 0 ) . Therefore, C _n (γ₁ × γ₂) = L × L and so γ is a GNFLI of L × L.□

Lemma 3.15. Let γ be a GNFLI of L of class n ≥ 2 and H, be a nontrivial Lie ideal of L. Then H ∩ C(γ) ≠ {0}.

Proof. By hypotheses we have C _n (γ) = L. Thus { 0 } = C 0 ( γ ) ⊆ C 1 ( γ ) ⊆ . . . ⊆ C n ( γ ) = L Since H ∩ C _n (γ) = H ∩ L = H ≠ {0}, then there is j ∈ ℕ such that H ∩ C _j (γ) ≠ {0}. Let i be the smallest index such that H ∩ C _i (γ) ≠ {0} (so H ∩ C_i-1(γ) = {0}). We claim that (H ∩ C _i (γ) ∘ L) ⊆ H. Take w ∈(H ∩ C _i (γ) ∘ L). Then w = a ∘ g for some a ∈ H ∩ C _i (γ) and g ∈ L such that. Since H is an ideal we conclude that w ∈ H. Thus (H ∩ C _i (γ) ∘ L) ⊆ H. Also by Lemma 3.12 and a ∈ H ∩ C _i (γ) we get that (a ∘ g) ∈ C_i-1(γ). Thus ((H ∩ C _i (γ)) ∘ L) ⊆ C_i-1(γ). Hence (H ∩ C _i (γ) ∘ L) ⊆ H ∩ C_i-1(γ) = {0}. Therefore H ∩ C _i (γ) ⊆ Z(L) ⊆ C(γ). Then H ∩ C _i (γ) ⊆ H ∩ C(γ). Now if H ∩ C(γ) = {0}, then H ∩ C _i (γ) = {0} which is a contradiction. Therefore, H ∩ C(γ) ≠ {0}.□

In what followes we show that if γ is a GNFLI, then every minimal ideal of L is contained in C(γ).

Theorem 3.16. Let γ be a GNFI of class n ≥ 2 and H be a minimal Lie ideal of L. Then H ⊆ C(γ).

Proof. Note that C(γ) is Lie ideals of L. Then H ∩ C(γ) is an ideal of L. Also, by Lemma 3.15, {0} ≠ H ∩ C(γ). Since H is a minimal ideal of L, we get that H ∩ C(γ) ⊆ H. Then H ⊆ C(γ).□

Theorem 3.17. Let I be a Lie ideal of L, γ be a fuzzy Lie ideal of L and τ : L/I ⟶ [0, 1] defined by τ(x + I) = dis ⋁ _u∈I  γ(x + u) . Then τ is a fuzzy Lie ideal of L/I.

Proof. It is clear that, τ is well define. Take x + I, y + I ∈ L/I. Then τ(α(x + I)) = τ(αx + I) = dis ⋁ _z∈Iγ(αx + z) ≥ dis ⋁ _z∈Iγ(x + z) = τ(x + I) . Also τ((x + I) ∘(y + I)) = τ(x ∘ y) + I = dis ⋁ _z∈Iγ(x ∘ y) + z) ≥ dis ⋁ _z∈Iγ(x + z) = τ(x + I) . In the similar way τ((x + I) +(y + I)) = τ(x + I) ∧ τ(y + I).□

Lemma 3.18. [20] For any x, y ∈ L if γ(x) ≠ γ(y), then γ(x + y) = γ(x) ∧ γ(y).

Lemma 3.19. Let A be a maximal Abelian ideal of L. Consider a GNFLI γ of L such that γ(a) = γ(0) for any a ∈ A and γ(a) ≠ γ(0) for any a ∈ L - A. Then A = C L ( A ) = { a ∈ L ∣ ( a ∘ d ) = 0 for any d ∈ A . }(5)

Proof. First we show that C _L (A) is an ideal of L. Take a ∈ C _L (A) and x ∈ L. Then, for any d ∈ A we obtain ((a ∘ x) ∘ d) =(x ∘ d ∘ a) +(d ∘ a ∘ x) = {0}. Thus C _L (A) is an ideal of L. Suppose A ⊊ C _L (A). Then { 0 } ≠ C L ( A ) A is an ideal of L A . Take γ ¯ the fuzzy Lie ideal of L A . Thus by Lemma 3.15, C L ( A ) A ∩ C ( γ ¯ ) ≠ { 0 } . Then there exists a non-trivial element x + A ∈ C L ( A ) A ∩ Z ( γ ¯ ) and so x ∈ C _L (A). Therefore for any a ∈ L we have γ ¯ ( ( x + A ) ∘ ( a + A ) ) = γ ¯ ( 0 + A ) i.e. by Theorem 3.17, ⋁_d∈Aγ((x ∘ a) + d) = γ(0). Now if γ(x ∘ a) = γ(d) for some d ∈ A, then by definition of γ, we obtain (x ∘ a) ∈ A. And for the case, γ(x ∘ a) ≠ γ(d) for any d ∈ A, by Lemma 3.18, we get that γ ( 0 ) = ⋁ d ∈ A γ ( ( x ∘ a ) + d ) = ⋁ d ∈ A ( γ ( x ∘ a ) ∧ γ ( d ) ) = γ ( x ∘ a ) And so x ∘ a ∈ A. Consider B consists of all linear combinations of A and x. By x ∈ C _L (A) we get that B is Abelian. Also since for all a ∈ L we have (g ∘ a) ∈ A ⊆ B we conclude that B is an ideal of L. Therefore, A ⊊ B is an ideal of L. Then, B is an Abelian ideal of L, a contradiction. Therefore, A = C _L (A).□

Note that, the set of all Lie homomorphisms of L, denoted by Aut(L), is a Lie algebra with the following Lie product: ( f ∘ h ) ( x ) = f ( h ( x ) ) - h ( f ( x ) ) , ∀ f , h ∈ Aut ( L ) . In what follows, we present conditions that make a GNFLI, be finite.

Corollary 3.20. Let A be a finite maximal Abelian Lie ideal of L. Consider a GNFLI γ of L such that γ(a) = γ(0), for any a ∈ A, and γ(a) ≠ γ(0), for any a ∈ L - A. Then γ is finite.

Proof. Since A is an ideal of L, then for any x ∈ L and a ∈ A we have x ∘ a ∈ A. Define a map φ as follows: φ : L ⟶ Aut ( A ) x ⟶ φ x : A ⟶ A a ⟶ x ∘ a . φ is a Lie homomorphism. It is enough to show that φ(x₁ ∘ x₂) = φ(x₁) oφ(x₂). Let x₁, x₂ ∈ L. Then for d ∈ A ( φ ( x 1 ∘ x 2 ) ) ( d ) = ( φ ( x 1 ∘ x 2 ) ) ( d ) = ( x 1 ∘ x 2 ∘ d ) = - ( d ∘ ( x 1 ∘ x 2 ) ) = ( x 1 ∘ ( x 2 ∘ d ) ) + [ ( x 2 ∘ ( d ∘ x 1 ) ) = ( x 1 ∘ ( x 2 ∘ d ) ) - ( x 2 ∘ ( d ∘ x 1 ) ) = ( x 1 ∘ ( x 2 ∘ d ) ) - ( x 2 ∘ ( x 1 ∘ a ) ) = ( φ x 1 ) ( x 2 ∘ d ) - ( φ x 2 ) ( x 1 ∘ a ) = ( φ x 1 o φ x 2 ) d - ( φ x 2 o φ x 1 ) d = ( ( φ x 1 ) ∘ ( φ x 2 ) ) d = ( φ ( x 1 ) o φ ( x 2 ) ) d . Now we prove that Ker(φ) = C _L (A). Take x ∈ Ker(φ). Then φ(x) =0 where 0 is the zero homomorphism. Then for any a ∈ A we have (φ(x))(a) =0(a), and so x ∘ a = 0. Thus, x ∈ C _L (A). Therefore, Ker(φ) = C _L (A). Now by Theorem 3.19, we obtain that L Ker ( φ ) = L A ⊆ Aut ( A ) . Since A and so Aut(A) is finite, we conclude that L is finite, and so γ is finite.□

In the classical Lie algebra, each finite-dimensional Engel Lie algebra over a field is nilpotent. This initial study can verify, as we have done in Theorem 4.11.

Lemma 4.1. Let ∽  be a binary relation defined by the following x ∽ y ⇔ γ ( x - y ) = γ ( 0 ) . Then ∽ is a congruence relation.

Proof. It is clear that ∽  is reflexive and symmetric. We prove that it is transitive. Take x ∽  y and y ∽  c, where x, y, c ∈ L. Then γ(x - y) = γ(y - c) = γ(0). Since γ is a fuzzy Lie algebra of L, we conclude that γ ( x - c ) ≥ γ ( x - y ) ∧ γ ( y - c ) = γ ( 0 ) Hence γ(x - c) = γ(0), then x  ∽   c. Therefore ∼ is an equivalence relation. Consider x  ∽   y and c ∈ L. Then, γ((x + c) -(y + c)) = γ(x - y) = γ(0) and so (x + c)   ∽   (y + c). Now by definition, we get that γ((c - x) -(c - y)) = γ(-(c + y) +(c + x)) = γ(- y + x) = γ(x - y) = γ(0) and so (c - x)   ∽  (c - y).

Also γ(αx - αy) = γ(α(x - y)) ≥ γ(x - y) = γ(0), then αx ∽ αy.

Also, γ((x ∘ c) -(y ∘ t) = γ((x - y) ∘(c - t)) ≥ γ(x - y) ∧ γ(c - t) = γ(0) and so (x ∘ c) ∽(y ∘ t). Thus ∽ is a congruence relation.□

Note: Let L/γ be the set of all equivalence class containing x, denoted by x + γ, then L/γ by the following operations is a Lie algebra (x + γ) +(y + γ) =(x + y) + γ and(x + γ) ∘(y + γ) =(x ∘ y) + γ forany x + γ, y + γ ∈ L/γ where 0 + γ is unit of L/γ and (- x) + γ = -(x + γ).

Definition 4.2. A fuzzy Lie ideal γ of L is called an Engel fuzzy Lie ideal if for each x, y ∈ L there exists a nonnegative integer m such that γ(x ∘  _m y) = γ(0). We abbreviate by EFLI, the "Engel fuzzy Lie ideal”.

Example 4.3. Every fuzzy Lie ideal of an Abelian Lie algebra is 1-EFLI.

Example 4.4. Consider W be as Example 3.2. We define the following fuzzy set λ on W λ ( a ) = { 1 if a = 0 , 0.5 if a = e 2 , e 5 , e 6 , e 7 , e 8 , 0 if otherwise . Then λ is a 2-EFLI of W.

Notation: Let b ∈ L and n ∈ ℕ . Define C_0,b(γ) = {0} and C_n,b(γ) to be the subalgebra generated by {a ∈ L ∣ γ(a ∘  _n b) = γ(0)}.

Theorem 4.5. If for any y ∈ L, C_n,y(γ) = L, then γ is n-EFLI.

Proof. Suppose x be an arbitrary element of L. By hypotheses we get that γ(x ∘  _i y) = γ(0) for any y ∈ L. Therefore γ is n-EFLI.□

Theorem 4.6. Every fuzzy Lie ideal γ is a GNFLI(EFLI) of L if and only if L/γ is a nilpotent (Engel) Lie algebra.

Proof. (⇒) Let γ be a GNFLI of L. Since for any a + γ, b + γ ∈ L/γ, we have (a + γ) ∘(b + γ) =(a ∘ b) + γ, then for any n ∈ ℕ and a₁, . . . , a _n  ∈ L, we get that (a ∘ a₁ ∘ . . . ∘ a _n ) + γ =(a + γ) ∘(a₁ + γ) ∘ . . . ∘(a _n  + γ). If γ is a GNFLI, then there exist n ∈ ℕ such that C _n (γ) = L. Then, by Lemma 3.4, { a ∈ L ∣ γ ( a ∘ a 1 ∘ . . . ∘ a n ) = γ ( 0 ) } = L .(6) for all a₁, a₂, a₃, . . . , a _n  ∈ L

Moreover,

γ ( a ) = γ ( 0 ) if and only if a ∼ 0(7) if and only if  a + γ = 0 + γ .

Thus by (6) and (7), we have L / γ = { a + γ ∣ a ∈ L } = { a + γ ∣ γ ( a ∘ a 1 ∘ . . . ∘ a n ) = γ ( 0 ) , ∀ a 1 , a 2 , a 3 , . . . , a n ∈ L } = { a + γ ∣ ( a + γ ) ∘ ( a 1 + γ ) ∘ . . . ∘ ( a n + γ ) = 0 + γ , ∀ a 1 , a 2 , a 3 , . . . , a n ∈ L } = Z n ( L / γ ) . Consequently, L/γ is a nilpotent Lie algebra of class n. (⟸) Consider L/γ be a nilpotent Lie algebra of class n, then

L/γ = Z _n (L/γ) = {a + γ ∣(a + γ) ∘(a₁ + γ) ∘ . . . ∘(a _n  + γ) =0 + γ, ∀  a₁, a₂, a₃, . . . , a _n  ∈ L .}

Take a ∈ L therefore for any a₁, a₂, a₃, . . . , a _n  ∈ L, (a + γ) ∘(a₁ + γ) ∘ . . . ∘(a _n  + γ) =0 + γ. Then by (7) we have γ(a ∘ a₁ ∘ . . . ∘ a _n ) = γ(0). Thus by Lemma 3.4, a ∈ C _n (γ). Thus L = C _n (γ) and so γ is GNFLI. By the same manipulation, we can prove that γ is an EFLI if and only if L/γ is an Engel Lie algebra.□

Theorem 4.7. Consider fuzzy Lie ideals γ and λ of L such that γ ⊆ λ and γ(0) = λ(0). If γ is a GNFLI of class m, then λ is a GNFLI n, where n ≤ m.

Proof. First, we show that for any i ∈ ℕ , C _i (γ) ⊆ C _i (λ). By Theorem 3.11, for i = 1, the result is immediate. Let for i ≥ 2, C _i (γ) ⊆ C _i (λ) and a ∈ C_i+1(γ). Then by Lemma 3.12, for any y ∈ L, we have a ∘ b ∈ C _i (γ) ⊆ C _i (λ). Thus by Lemma 3.12, a ∈ C_i+1(λ). Hence C_i+1(γ) ⊆ C_i+1(λ). Now since γ is a GNFLI of class m we conclude that L = C _m (γ) ⊆ C _m (λ) ⊆ L. Thus, L = C _m (λ), and so λ is a GNFLI of class at most m.□

Theorem 4.8. Every GNFLI γ of L is an EFLI.

Proof. Let γ be a GNFLI of L. Then by Theorem 4.6, L γ is a nilpotent Lie algebra. It is clear that, each nilpotent Lie algebra is Engel. Therefore L γ is Engel. Then by Theorem 4.6, γ is EFLI.□

Note: Let L be an Engel Lie algebra that is not nilpotent. Then by Theorem 4.6, we can see that the converse of Theorem 4.8, is not true in general.

Theorem 4.9. [4, 5] (i) Every finite-dimensional Engel Lie algebra over a field is nilpotent.

(ii) Each 4-Engel Lie algebras over a field with characteristic not equal to 2,3,5 is nilpotent of class at most 7.

The fuzzy analogies of Theorem 4.9, are the following two theorems.

Theorem 4.10. Let L be a Lie algebra of characteristic k ≠ 2, 3, 5, and γ be a 4-EFLI of L. Then γ is a GNFLI.

Proof. Consider γ be a 4-EFLI of L. Therefore by Theorem 4.6, L/γ is a 4-Engel Lie algebra of characteristic k ≠ 2, 3, 5, Then by Theorem 4.9(ii), L/γ is nilpotent. By Theorem 4.6, we have γ is a GNFLI.□

Theorem 4.11. Every EFLI of a finite Lie algebra is GNFLI.

Proof. Consider γ be an EFLI of L. Then by Theorem 4.6, L/γ is a finite Engel Lie ideal, and so by Theorem 4.9(ii), L/γ is nilpotent. Now using Theorem 4.6, we have γ is a GNFLI.□

In this paper, we shorten and simplify some previous results in this topic by the notion of good nilpotent fuzzy Lie ideals. Then, fuzzification of some significant consequences on nilpotent Lie ideals is obtained. Specially, we proved that a Lie algebra L is nilpotent if and only if any fuzzy Lie ideal of L is a good nilpotent fuzzy Lie ideal. Also, we showed that every Engel fuzzy Lie ideal of a finite Lie algebra is a good nilpotent fuzzy Lie ideal. We think it could be useful to solve some problems with fuzzy Lie algebras.