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**J. C. da Motta Ferreira ^{*} and M. G. Bruno Marietto**

Center for Mathematics, Computation and Cognition, Federal University of ABC, 09210-170 Santo Andr´e, SP, Brazil

- *Corresponding Author:
- J. C. da Motta Ferreira

Center for Mathematics,

Computation and Cognition,

Federal University of ABC,

09210-170 Santo Andr´e, SP,

Brazil

E-mail:

**Received date: ** 18 March 2012,** Revised date:** 06 June 2012, **Accepted date:** 11 June 2012

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

In this paper, we apply the concepts of fuzzy sets to Lie algebras in order to introduce and to study the notions of solvable and nilpotent fuzzy radicals. We present conditions to prove the existence and uniqueness of such radicals.

*L*ie algebras were discovered by Sophus *L*ie [4]. There are many applications of *L*ie algebras in several branches
of physics. The notion of fuzzy sets was introduced by Zadeh [8]. Fuzzy set theory has been developed in many
directions by many scholars and has evoked a great interest among mathematicians working in different fields of
mathematics. Many mathematicians have been involved in extending the concepts and results of abstract algebra.
The notions of *fuzzy ideal*s and fuzzy subalgebras of *L*ie algebras over a field were first introduced by Yehia in
[7]. In this paper, we introduce the notion of *solvable* and nilpotent fuzzy radical of a fuzzy algebra of *L*ie algebras
and investigate some of their properties. The results presented in this paper are strongly connected with the results
proved in [1,2,3].

In this section, we present the basic concepts on fuzzy sets which will be used throughout this paper. A new notion is introduced and results are proved for guiding the construction of the main theorems of this work.

**Definition 1. **A mapping of a non-empty set *X* into the closed unit interval [0,1] is called a fuzzy set of *X*. *L*et
μ be any fuzzy set of *X*, then the set {μ(x) | x ∈ *X*} is called the image of μ and is denoted by μ(*X*). The set
{x | x ∈ *X*, μ(x) > 0} is called the support of μ and is denoted by μ*. In particular, μ is called a finite fuzzy set if μ*
is a finite set, and an infinite fuzzy set otherwise. For all real t ∈ [0,1] the subset

is called *a t-level set* of μ.

**Definition 2. ***L*et *X* be a non-empty set and {ν_{i}}_{i∈I} an arbitrary family of fuzzy sets of *X*. One defines the fuzzy
set of *X* ∪_{i∈I} ν_{i}, called union, as

for all x ∈ *X*.

**Remark 3.** *L*et us note that if {ν_{i}}_{i∈I} is a family of fuzzy sets of *X*, then ∪_{i∈I} [ν_{i}]_{t} ⊆ ∪_{i∈I} ν_{i} t, for all t ∈]0,1].

**Definition 4. ***L*et *X* be a non-empty set. One says that a family of fuzzy sets of *X* {ν_{i}}_{i∈I} satisfies the second sup
property if for all x ∈ *X* there is an in dex i0 = i0(x) ∈ I such that (∪_{i∈I} ν_{i})(x) = ν_{i}0 (x).

Thus, a family of fuzzy sets of *X* {ν_{i}}_{i∈I} satisfies the second sup property if, and only if, ∪_{i∈I} ν_{i} (x)∈{ν_{i}(x) |
i ∈ I}, for all x ∈ X.

**Proposition 5.** *L*et *X* be a non-empty set and {ν_{i}}_{i∈I} an arbitrary family of fuzzy sets of *X*. Then ∪_{i∈I} ν_{i} t =
∪_{i∈I} [ν_{i}]_{t} for all t ∈]0,1] if, and only if, the family {ν_{i}}_{i∈I} satisfies the second sup property.

*Proof.* *L*et us take x ∈ *X* and let ∪_{i∈I} ν_{i} (x) = α. If α = 0, then the result is evident. Now if α > 0, then x ∈
∪_{i∈I} ν_{i} α which implies that x ∈ ∪_{i∈I} [ν_{i}]α. It follows that there is an index i0 ∈ I such that x ∈ [ν_{i}0 ]α. Thus,
we have ∪_{i∈I} ν_{i} (x) ≥ ν_{i}0 (x) ≥ α. This implies that ∪_{i∈I} ν_{i} (x) = ν_{i}0 (x). So {ν_{i}}_{i∈I} satisfies the second sup
property.

Now, let us suppose that {ν_{i}}_{i∈I} satisfies the second sup property and let us take t ∈]0,1] and x ∈ ∪_{i∈I} ν_{i} t.
Since there is an index i0 ∈ I such that (∪_{i∈I} ν_{i})(x) = ν_{i}0 (x), then x ∈ [ν_{i}0 ]_{t}. It follows that x ∈ ∪_{i∈I} [ν_{i}]_{t} which
implies that ∪_{i∈I} ν_{i} t
⊆ ∪_{i∈I} [ν_{i}]_{t}. So [∪_{i∈I} ν_{i}]_{t} = ∪_{i∈I} [ν_{i}]_{t}, by Remark 3.

Example 6. *L*et *X* be a non-empty set and Y an arbitrary subset of *X*. *L*et us consider I ⊂ [0,1] an infinite subset
of the rational numbers such that supI =
√
2/2 and let us take {ν_{i}}_{i∈I} the family of the fuzzy sets of *X*, by setting
ν_{i} = iχY, for all i ∈ I. In this case, we have

**Definition 7.** A set S ⊂ [0,1] is said to be an *upper well-ordered set* if for all non-empty subsets C ⊂ S, then
supC ∈ C.

**Definition 8. ***L*et *X* be a non-empty set and S an upper well-ordered set. One defines the set

F(*X*,S) = ν | ν is an arbitrary fuzzy set of *X* such that ν(*X*) ⊆ S.

**Proposition 9. ***L*et X be a non-empty set and S an upper well-ordered set. Then, every family of fuzzy sets {ν_{i}}_{i∈I} of F(X,S) satisfies the second sup property.

*In particular, for every family of fuzzy sets {ν _{i}}_{i∈I} of F(X,S) one has ∪_{i∈I} ν_{i} ∈ F(X,S).*

*L*et *X* be a non-empty set. All fuzzy sets of *X* μ can be wr_{i}tten in terms of its t-level sets [μ]_{t} as [9]

Now, let us consider the converse problem: a family (Aα)α of subsets of *X* is given; is there a fuzzy set μ : *X*→[0,1]
such that [μ]α =, for every α?

The necessary and sufficient conditions are given in the following.

**Theorem 10. ***L*et ()α be a family of subsets of X. The necessary and sufficient conditions for the existence of a
fuzzy set μ : X→[0,1], such that [μ]α = (0 ≤ α ≤ 1), are as follows:

(i) *α ≤ β implies that ⊇,*

(ii) *α1 ≤ α2 ≤ · · · and αn →α imply that *

The proof is given by Negoita and Ralescu [6] and will be omitted here.

In this section, we present the basic concepts of fuzzy *L*ie algebras, and fuzzy *L*ie subalgebras, *fuzzy ideal*s and *solvable* (resp., nilpotent) *fuzzy ideal*s, of a fuzzy *L*ie algebra. Relationships between these concepts with operations
of sum and product of the fuzzy sets are studied.

**Definition 11.** *L*et *L* be a *L*ie algebra over a field F. A fuzzy set μ of *L* is called a fuzzy *L*ie algebra of *L* if

(i) μ(ax+by) ≥ μ(x)∧μ(y),

(ii) μ(xy) ≥ μ(x)∧μ(y),

(iii) μ(0) = 1,

for all a, b ∈ F and x,y ∈ *L*.

A fuzzy set ν of *L* is called a* fuzzy subalgebra* of μ if ν is a fuzzy *L*ie algebra of *L* satisfying ν(x) ≤ μ(x) for
all x ∈ *L*.

Clearly, if μ is a fuzzy algebra of *L*, then μ* is a subalgebra of *L*. Also, μ is a fuzzy *L*ie algebra of *L* if, and only
if, the t-level sets [μ]_{t} are subalgebras of *L*, for all t ∈]0,1]. Moreover, ν is a fuzzy subalgebra of μ, if, and only if,
the t-level sets [ν]_{t} are subalgebras of [μ]_{t}, for all t ∈]0,1].

**Definition 12.** A fuzzy set ν of *L* is called a *fuzzy ideal* of *L* if

(i) ν(ax+by) ≥ ν(x)∧ν(y),

(ii) ν(xy) ≥ ν(x)∨ν(y),

(iii) ν(0) = 1,

for all a, b ∈ F and x,y ∈ *L*.

A fuzzy set ν of *L* is called a *fuzzy ideal* of μ if ν is a fuzzy *L*ie ideal of *L* satisfying ν(x) ≤ μ(x) for all x ∈ *L*.

Clearly, if ν is a *fuzzy ideal* of *L*, then ν* is an ideal of *L*. Also, ν is a *fuzzy ideal* of *L* if, and only if, the t-level
sets [ν]_{t} are ideals of *L*, for all t ∈]0,1]. Moreover, any *fuzzy ideal* of *L* is a fuzzy algebra of *L* and any *fuzzy ideal* of μ is a fuzzy subalgebra of μ.

**Definition 13.** For any fuzzy *L*ie algebra μ of *L*, the fuzzy set of *L*, denoted and defined by

is a fuzzy algebra (resp., *fuzzy ideal*) of μ, called the null fuzzy algebra of μ (resp., *null* *fuzzy ideal* of μ). A fuzzy
set ν of *L* is the null fuzzy algebra of μ if, and only if, [ν]_{t} = {0}, for all t ∈]0,1].

A fuzzy *L*ie algebra μ of *L* is called *abelian* if μ^{2} = o and *non-**abelian* otherwise.

*L*et us observe that if S ⊂ [0,1] is an upper well-ordered set, then o ∈ F(*L*,S) if, and only if, 0,1 ∈ S. Thus,
throughout this paper we will always assume that our upper-ordered set has the real numbers 0 and 1.

**Definition 14. ***L*et *L* be a *L*ie algebra over a field F. One defines the following:

(i) the fuzzy set of *L* (sum), if ν_{1}, . . . , ν^{n} are fuzzy sets of *L*, as

for all x ∈ *L*,

(ii) the fuzzy set ν_{1}ν_{2} of *L* (product), if ν_{1} and ν_{2} are fuzzy sets of *L*, as

for all x ∈ *L*.

**Remark 15.** *L*et us note that if ν_{1},ν_{2} are fuzzy sets of *L*, then [ν_{1}]_{t} +[ν_{2}]_{t} ⊆ [ν_{1}+ν_{2}]_{t} and [ν_{1}]_{t}[ν_{2}]_{t} ⊆ [ν_{1}ν_{2}]_{t}, for
all t ∈]0,1].

*L*emma 16.*Let L be a finite dimensional Lie algebra over a field F and μ a fuzzy Lie algebra of L. If ν _{1} and ν_{2} are
fuzzy subalgebras of μ and t ∈]0,1], then*

*(i) for all x ∈ [ν _{1} +ν_{2}]_{t}, there are elements c ∈ [ν_{1}]_{t} and d ∈ [ν_{2}]_{t} such that x = c+d and (ν_{1} +ν_{2})(x) = ν_{1}(c)∧
ν_{2}(d),*

*(ii) for all x ∈ [ν _{1}ν_{2}]_{t}, there are elements ci ∈ [ν_{1}]_{t} and di ∈ [ν_{2}]_{t} (i = 1, . . . , n) such that and (ν_{1}ν_{2})(x) = *

*In particular, the following holds:*

*(iii) [ν _{1}+ν_{2}]_{t} = [ν_{1}]_{t} +[ν_{2}]_{t},*

*(iv) [ν _{1}ν_{2}]_{t} = [ν_{1}]_{t}[ν_{2}]_{t}, for all t ∈]0,1].*

*Proof.* (i) and (iii) Since *L* is finite dimensional, then both ν_{1} and ν_{2} take finite values in [0,1], by [5]. This implies
that there are two finite sets of real numbers {r0 = 0 < r1 < · · · < rm−1 < 1 = rm} and {s_{0} = 0 < s_{1} < · · · <
sn−1 < 1 = sn} such that [ν_{1}]r = [ν_{1}]r_{i}, for all r ∈]r_{i}−1,r_{i}] (i = 1,...,m), and [ν_{2}]s = [ν_{2}]s_{j}, for all s ∈]s_{j}−1,s_{j} ]
(j = 1, . . . , n). Thus, for t ∈]0,1] and for x ∈ [ν_{1} +ν_{2}]_{t}, let (ν_{1} +ν_{2})(x) = α. Then, there are integers 1 ≤ i ≤ m
and 1 ≤ j ≤ n such that r_{i}−1 < α ≤ r_{i} and s_{j}−1 < α ≤ s_{j} which implies (r_{i}−1 ∨s_{j}−1) < α ≤ (r_{i} ∧s_{j}). It follows
that there are elements c,d ∈ *L* such that x = c +d and ν_{1}(c) ∧ ν_{2}(d) > r_{i}−1 ∨ s_{j}−1. Hence, ν_{1}(c) > r_{i}−1 and
ν_{2}(d) > s_{j}−1, which implies that ν_{1}(c) ≥ r_{i} and ν_{2}(d) ≥ s_{j}, that is, ν_{1}(c)∧ν_{2}(d) ≥ r_{i} ∧s_{j} ≥ α. This implies that
(ν_{1} +ν_{2})(x) = ν_{1}(c)∧ν_{2}(d). Moreover, since c ∈ [ν_{1}]_{t} and d ∈ [ν_{2}]_{t}, then [ν_{1} +ν_{2}]_{t} ⊂ [ν_{1}]_{t} +[ν_{2}]_{t}. From Remark
15 we obtain [ν_{1}+ν_{2}]_{t} = [ν_{1}]_{t} +[ν_{2}]_{t}.

Similarly, we prove the cases (ii) and (iv).

**Definition 17. ***L*et *L* be a finite dimensional *L*ie algebra over a field F and μ a fuzzy *L*ie algebra of *L*. For any fuzzy
subalgebra ν of μ we define inductively the de_{ri}ved se_{ri}es of ν as the descending chain of fuzzy subalgebras of μ
ν^{(1)} ≥ ν^{(2)} ≥ ν^{(3)} ≥ · · · , by setting ν^{(1)} = ν and ν^{(n+1)} = (ν^{(n)})^{2} for every n ≥ 1, and the lower central se_{ri}es of
ν as the descending chain of fuzzy algebras of μ ν_{1} ≥ ν_{2} ≥ ν3 ≥ · · · , by defying ν_{1} = ν and ν^{n} = νν^{n−1} for every
n ≥ 2.

The fuzzy subalgebra ν is said *solvable* (resp., nilpotent) if there exists an integer k =k(ν)≥1 such that ν^{(k)} =o
(resp., ν^{k} =o). The smallest strict positive integer k such that ν^{(k)} =o (resp., ν^{k} =o) is called the index of solvability
(resp., index of nilpotency) of ν.

Clearly, Definition 17 generalizes the concept of solvability (resp., nilpotency) as defined in the class of the *L*ie
algebras [4].

**Corollary 18. ***Let L be a finite dimensional Lie algebra over a field F and μ a fuzzy Lie algebra of L. If ν is a fuzzy
subalgebra of μ, then*

*for all integer n ≥ 1 and t ∈]0,1].*

Thus, ν is *solvable* (resp., nilpotent) if, and only if, there exists an integer k = k(ν) ≥ 1 such that [ν]_{t} (k) = {0}
(resp., [ν]_{t} k = {0}), for all t ∈]0,1].

**Proposition 19. ***Let L be a finite dimensional Lie algebra over a field F and μ a fuzzy Lie algebra of L. If ν _{1},ν_{2} are fuzzy ideals of μ, then the sum ν_{1}+ν_{2} and the product ν_{1}ν_{2} are fuzzy ideals of μ. Moreover,*

(i) [ν_{1}]_{t} +[ν_{2}]_{t} (2l+1) ⊆ [ν_{1}](l)
t +[ν_{2}](l)
t , for all t ∈]0,1] and integer l ≥ 1,

(ii) each non-associative product of l (= lν_{1} +lν_{2} ) terms [ν_{i1} ]_{t} · · · [ν_{il} ]_{t}, ij = 1 or 2 (1 ≤ j ≤ l), where lν_{1} terms
are formed by the t-level sets [ν_{1}]_{t} and lν_{2} terms are formed by the t-level sets [ν_{2}]_{t}, is a subset of both the sets

*Proof.* The first part of the proposition can be easily shown. Next, by the Jacobi identity and using the Principle of
Mathematical Induction, we can also demonstrate that

(a) (i = 1,2), for all integer l ≥ 1,

(b) (i, j = 1, 2; i ≠ j), for all integer l ≥ 1.

Now

Then using again the Principle of Mathematical Induction, (a), and (b) we have, for an integer l ≥ 1,

So for all integer l ≥ 1.

Next, by the Jacobi identity and the Principle of Mathematical Induction again, we can demonstrate that

(c) (i = 1,2), for any integers l_{1}, l_{2} ≥ 1,

Now (ii) is evident, for l = 2. Hence, let us consider an integer l (= lν_{1} +lν_{2} ) ≥ 2 and a non-associative product of
l terms [ν_{i1} ]_{t} · · · [ν_{il} ]_{t}, where ij = 1 or 2 (1 ≤ j ≤ l), lν_{1} terms are formed by the t-level sets [ν_{1}]_{t} and lν_{2} terms are
formed by the t-level sets [ν_{2}]_{t}. It follows that we can wr_{i}te the previous product as a product of two non-associative
products

pi,qj = 1 or 2 (1 ≤ i ≤ r;1 ≤ j ≤ s), where [ν_{p1} ]_{t} · · · [ν_{pr} ]_{t} and [ν_{q1} ]_{t} · · · [νqs ]_{t} are products of r (= r_{ν1} +r_{ν2} ) and
s (= sν_{1} +sν_{2} ) terms, respectively, with r_{ν1} and sν_{1} terms formed by the t-level sets [ν_{1}]_{t} and, r_{ν2} and sν_{2} terms
formed by the t-level sets [ν_{2}]_{t}. It follows that l = r+s, lν_{1} = r_{ν1} +sν_{1}, and lν_{2} = r_{ν2} +sν_{2} . From the Principle of Mathematical Induction, we have that [ν_{p1} ]_{t} · · · [ν_{pr} ]_{t} is a subset of both t-level sets and
[ν_{q1} ]_{t} · · · [νqs ]_{t} is a subset of both t-level sets This implies that [ν_{i1} ]_{t} · · · [ν_{il} ]_{t} is a subset of both the sets and From the condition (c), we conclude that [ν_{i1} ]_{t} · · · [ν_{il} ]_{t} is a subset of both
the sets

The proposition is proved.

**Proposition 20.** *Let μ be a fuzzy Lie algebra of L. If ν _{1} and ν_{2} are solvable (resp., nilpotent) fuzzy ideals of μ, then
ν_{1}+ν_{2} and the ν_{1}ν_{2} are also solvable (resp., nilpotent) fuzzy ideals of μ.*

*Proof.* *L*et kν_{1} = kν_{1} (t) ≥ 1 and kν_{2} = kν_{2} (t) ≥ 1 be the indices of solvability of ν_{1} and ν_{2}, respectively. *L*et us take
k = k(t) = max{kν_{1},kν_{2} }. It follows that (ν_{1} +ν_{2})(2k+1)
t = {0}, for all t ∈]0,1], by Proposition 19(i), *L*emma
16(iii) and Corollary 18. Thus ν_{1}+ν_{2} is a *solvable* *fuzzy ideal* of μ.

Now, let kν_{1} = kν_{1} (t) ≥ 1 and kν_{2} = kν_{2} (t) ≥ 1 be the indices of nilpotency of ν_{1} and ν_{2}, respectively. *L*et us
take k = k(t) = kν_{1} +kν_{2} . Then, for all t ∈]0,1], we have (ν_{1} +ν_{2})k
t = {0}, Proposition 19(ii), *L*emma 16(iii)
and Corollary 18. So ν_{1}+ν_{2} is a nilpotent *fuzzy ideal* of μ.

**Corollary 21.** *Let L be a finite dimensional Lie algebra over a field F and μ a fuzzy Lie algebra of L. If S is an
upper well-ordered set and ν _{1} and ν_{2} are solvable (resp., nilpotent) fuzzy ideals of μ in F(L,S), then ν_{1} +ν_{2} and
ν_{1}ν_{2} are solvable (resp., nilpotent) fuzzy ideals of μ in F(L,S).*

In this section, we present the main results of this paper.We prove that every fuzzy *L*ie algebra has a unique maximal *solvable* (resp., nilpotent) *fuzzy ideal*, called the *solvable* (resp., nilpotent) fuzzy radical.

*L*et us begin by introducing the following definition.

**Definition 22. ***Let μ be a fuzzy Lie algebra of L and S an upper well-ordered set. One says that a fuzzy subalgebra
ν of μ is a maximal element of μ in F(L,S) if ν ∈ F(L,S) and for every fuzzy algebra ν* of μ in F(L,S) such that
ν ⊂ ν*, then ν = ν*. In this case, one says that the fuzzy algebra μ has a maximum in F(L,S).*

**Theorem 23. ***L*et *L* be a finite dimensional *L*ie algebra over a field F and S an upper well-ordered set. Then each
fuzzy algebra μ of *L* in F(*L*,S) has a maximal *solvable* (resp., nilpotent) *fuzzy ideal* in F(*L*,S).

*Proof.* *L*et μ be a fuzzy algebra of *L* in F(*L*,S) and let us consider the set

Ξ = ν | ν is a *solvable* *fuzzy ideal* of μ in F(*L*,S).

Obviously, the set Ξ is non-empty and partially ordered by ≤. *L*et us take a subset {ν_{i}}_{i∈I} of Ξ totally ordered by
≤. *L*et us show that ν = ∪_{i∈I} ν_{i} is an upper bound of {ν_{i}}_{i∈I} in Ξ. In fact, for every i ∈ I, we have ν_{i}(x) ≤ μ(x)
for all x ∈ *L*. Hence

for all x ∈ *L*. Now let us consider arbitrary elements i, j ∈ I. As the set {ν_{i}}_{i∈I} is totally ordered by ≤, then either
ν_{i} ≤ νj or νj ≤ ν_{i} implies either

νj(x)∧νj(y) ≥ ν_{i}(x)∧νj(y) or ν_{i}(x)∧ν_{i}(y) ≥ ν_{i}(x)∧νj(y),

for all x,y ∈ *L*. Thus, for all x,y ∈ *L* we have

Next, for all a ∈ F and x ∈ *L* we have

Also, for all x,y ∈ *L* we have

Finally, we have

So ν is a *fuzzy ideal* of μ.

Now, let us consider t ∈]0,1]. Since *L* is finite dimensional, then *L* has a unique maximal *solvable* ideal. It
follows that each t-level set [ν_{i}]_{t}, for i ∈ I, is also a *solvable* ideal of *L* which implies that ∪_{i∈I} [ν_{i}]_{t} is a *solvable* ideal of *L*, because the family {ν_{i}}_{i∈I} of Ξ is totally ordered by ≤. By Propositions 5 and 9, we have that ∪_{i∈I} ν_{i} t
is a *solvable* ideal of *L*. Therefore, ν is a *solvable* *fuzzy ideal* of μ in F(*L*,S) and so an upper bound of {ν_{i}}_{i∈I} in
Ξ. From Zorn’s lemma, Ξ possesses at least one maximal element.

Similarly, we prove the nilpotent case.

**Theorem 24. ***L*et *L* be a finite dimensional *L*ie algebra over a field F and S an upper well-ordered set. Then, every *solvable* (resp., nilpotent) *fuzzy ideal* ν of μ in F(*L*,S) is contained in a unique maximal *solvable* (resp., nilpotent) *fuzzy ideal* of μ in F(*L*,S), called *solvable* (resp., nilpotent) fuzzy radical of μ in F(*L*,S) and denoted by R(μ,S)
(resp., N(μ,S)).

*Proof.* *L*et be a maximal *solvable* (resp., nilpotent) *fuzzy ideal* of μ in F(*L*,S). If ν is a *solvable* (resp., nilpotent) *fuzzy ideal* of μ in F(*L*,S), then S+ν is a *solvable* (resp., nilpotent) *fuzzy ideal* of μ in F(*L*,S), by Corollary 21,
and (x) = (x)∧ν(0) ≤ {c)∧ν(d) | x = c+d} = (Sν)(x), for all x ∈ *L*. Thus ≤+ν. Since S is
maximal, then +ν ≤ . So +ν = . Hence ν(x) = (0)∧ν(x) ≤ {(c)∧ν(d) | x = c+d} = (+ν)(x),
for all x ∈ *L*. So ν ≤ . *L*et R(μ,) = (resp., N(μ,) = ) be.

In this section, we introduce the notions of simple and semisimple *fuzzy ideal*s and establish relations with the *solvable* fuzzy radical.

**Definition 25.** *L*et *L* be a *L*ie algebra over a field F and μ a fuzzy *L*ie ideal of *L*. One says that μ is a simple fuzzy
ideal if

(i) μ is a *fuzzy ideal* *non-**abelian*;

(ii) for all *fuzzy ideal*s ν of μ, one has either [ν]_{t} = [μ]_{t} or [ν]_{t} = {0}, for all t ∈]0,1].

**Definition 26.** *L*et *L* be a finite dimensional *L*ie algebra over a field F, S an upper well-ordered set and μ a fuzzy
ideal of *L* in F(*L*,S). One says that μ is a semisimple *fuzzy ideal* in F(*L*,S) if

(i) μ is a *fuzzy ideal* *non-**abelian*;

(ii) its *solvable* fuzzy radical in F(*L*,S) is o, that is, R(μ,S) = o.

**Theorem 27. ***L*et *L* be a finite dimensional *L*ie algebra over a field F and μ a fuzzy *L*ie ideal of *L* *non-**abelian*. If μ
is a simple *fuzzy ideal*, then μ* is a non *solvable* ideal of *L*.

Moreover, μ is a simple *fuzzy ideal* if, and only if, μ* is a minimal ideal of *L*.

*Proof.* First, let us observe that μ* is not a zero ideal, since μ^{2} is non-null. Since *L* is finite dimensional, there is
a finite set of real numbers {r0 = 0 < r1 < · · · < rm−1 < 1 = rm} such that [μ]_{t} = [μ]r_{i} for all t ∈]r_{i}−1,r_{i}] (i =
1,...,m). This implies that [μ]_{t} = [μ]r1 = μ*
, for all t ∈]0,r1]. If μ* is a *solvable* ideal of *L*, then and there is an integer k ≥ 1 such that (μ*)^{(k)} = {0}. Thus, by Theorem 10, we can construct the fuzzy set ν of *L* defined by t-level sets; [ν]_{0} = *L*, [ν]_{t} = (μ*)^{(2)} for all t ∈]0,r1], and [ν]_{t} = {0} for all t ∈]r1,1]. It follows that ν is
a *solvable* *fuzzy ideal* of *L* such that [ν]_{t} = [μ]_{t} for some t ∈]0,r1]. This an absurd.

Now let us consider I an ideal of *L* with I ⊆ μ*. By Theorem 10, let us consider the fuzzy set ν of *L* defined by
t-level sets; [ν]_{0} = *L*, [ν]_{t} = I for all t ∈]0,r1], and [ν]_{t} = {0} for all t ∈]r1,1]. Clearly, ν is a *fuzzy ideal* of μ and
so we have either [ν]_{t} = [μ]_{t} or [ν]_{t} = {0}, for all t ∈]0,1]. It follows that I = {0} or I = μ*. So μ* is a minimal
ideal of *L*. Reciprocally, let us consider a *fuzzy ideal* ν of μ. Then the t-level set [ν]_{t} is an ideal of the algebra *L* and [ν]_{t} ⊆ [μ]_{t} ⊆ μ*, for all t ∈]0,1]. Since μ* is a minimal ideal of *L*, then either [ν]_{t} = μ* or [ν]_{t} = {0}, for all
t ∈]0,1]. This implies that [ν]_{t} = [μ]_{t} or [ν]_{t} = {0}, for all t ∈]0,1]. So μ is a simple *fuzzy ideal*.

**Corollary 28. ***Let L be a finite dimensional Lie algebra over a field F and μ a fuzzy Lie ideal of L. If μ is a fuzzy
simple ideal, then [μ] _{t} = μ* or [μ]_{t} = {0}, for all t ∈]0,1].*

**Theorem 29. ***L*et *L* be a finite dimensional *L*ie algebra over a field F and S an upper well-ordered set. Then any
simple *fuzzy ideal* μ of *L* in F(*L*,S) is semisimple in F(*L*,S).

*Proof.* *L*et us consider ν a *solvable* *fuzzy ideal* of μ in F(*L*,S). Then [ν]_{t} ⊆ μ* is a *solvable* ideal of *L*, for all
t ∈]0,1]. Since μ is simple, then [ν]_{t} = {0} for all t ∈]0,1], by Theorem 27, which implies ν = o. Consequently,
R(μ,S) = o.

**Theorem 30. ***L*et *L* be a finite dimensional *L*ie algebra over a field F, S an upper well-ordered set, and μ a fuzzy
ideal of *L* in F(*L*,S) *non-**abelian*. If μ is semisimple in F(*L*,S), then μ* is a non-*solvable* ideal of *L*.

Moreover, μ is semisimple in F(*L*,S) if, and only if, μ* does not contain non-trivial *solvable* ideals of *L*.

*Proof.* First, from a similar construction, as shown in the demonstration of Theorem 27, if μ* is a *solvable* ideal of *L*, then we can construct the fuzzy set ν of *L* defined by t-level sets; [ν]_{0} = *L*, [ν]_{t} = (μ*)^{(2)} for all t ∈]0,r1], and
[ν]_{t} = {0} for all t ∈]r1,1], by Theorem 10. Its follows that ν is a *solvable* *fuzzy ideal* of μ in F(*L*,S). This implies
that ν = o and so μ is *abelian*. This is absurd.

Now, let us consider J as a *solvable* ideal of *L* in F(*L*,S) with I ⊆ μ*. By Theorem 10 again, let us consider
the fuzzy set ν of μ defined by [ν]_{0} = *L*, [ν]_{t} = J for all t ∈]0,r1], and [ν]_{t} = {0} for all t ∈]r1,1]. Clearly, ν is a *solvable* *fuzzy ideal* of μ in F(*L*,S) which implies that ν = 0. Thus, we have J = 0. So μ* does not contain nontri_{}vial *solvable* ideals of *L* in F(*L*,S). Reciprocally, let us consider a *solvable* *fuzzy ideal* ν of μ in F(*L*,S). Then,
for all t ∈]0,1] the t-level set [ν]_{t} is a *solvable* ideal of *L* such that [ν]_{t} ⊆ [μ]_{t} ⊆ μ*. This implies that [ν]_{t} = {0}, for
all t ∈ [0,1]. This implies that ν = o. So R(μ,S) = o.

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