Generalizing Two Structure Theorems of Lie Algebras to the Fuzzy Lie Algebras

Lie algebras were proposed by Sophus Lie [1] and there are many applications of them in several branches of physics [2]. The notion of fuzzy sets was introduced by Zadeh [3] and many mathematicians have been involved in extending the concepts and results of abstract Lie algebra to fuzzy theory. This paper is the continuation of the results obtained in [4], where we presented conditions to generalize the concepts of solvable and nilpotent radicals of Lie algebras (called of solvable and nilpotent fuzzy radicals, respectively) to a class of fuzzy Lie algebras. In this article we use the solvable fuzzy radical to generalize the structure theorem of semisimple Lie algebras and the Levi’s decomposition theorem to a class of the fuzzy Lie algebras. The results presented in this paper are still strongly connected with results proved in [5-10].


Introduction
Lie algebras were proposed by Sophus Lie [1] and there are many applications of them in several branches of physics [2]. The notion of fuzzy sets was introduced by Zadeh [3] and many mathematicians have been involved in extending the concepts and results of abstract Lie algebra to fuzzy theory. This paper is the continuation of the results obtained in [4], where we presented conditions to generalize the concepts of solvable and nilpotent radicals of Lie algebras (called of solvable and nilpotent fuzzy radicals, respectively) to a class of fuzzy Lie algebras. In this article we use the solvable fuzzy radical to generalize the structure theorem of semisimple Lie algebras and the Levi's decomposition theorem to a class of the fuzzy Lie algebras. The results presented in this paper are still strongly connected with results proved in [5][6][7][8][9][10].

Fuzzy Sets, Fuzzy Lie Algebras and Fuzzy Lie Ideals
In this section we present the basic concepts of fuzzy sets, fuzzy Lie algebras, fuzzy ideals among others which will be used throughout this paper. More details referring to these concepts and its properties can be found in [4].
A mapping of a non-empty set X into the closed unit interval [0,1] is called a fuzzy set of X and the set {µ (x)| x ∈ X} is called the image of denoted by µ (X). For all real t∈ [0,1] the subset [µ] t = {x ∈ X| µ (x) ≥ t} is called a t-level set of µ and the set { x | x ∈ X, µ (x) > 0} is called the support of µ denoted by µ * .
A set S ⊂ [0,1] is said to be an upper well ordered set if for all nonempty subsets C ⊂ S, then supC∈C. One defines the set F(X, S) = {v | v is an fuzzy set of X such that v(X) ⊆ S}.
Let L be a Lie algebra over a field F. A fuzzy set µ of L is called a fuzzy Lie algebra of L if satisfies the following conditions:  1]. Moreover, if µ is a fuzzy algebra of L then µ * is a subalgebra of L.
A fuzzy set v of L is called a fuzzy ideal of L if satisfies the following conditions: and (iii) )v(0) = 1, for all a, b∈F and x, y∈L. A fuzzy set v of L is called a fuzzy ideal of µ if v is a fuzzy Lie ideal of L satisfying v(x) ≤ µ (x) for all x∈L. One has that v is a fuzzy ideal of L if, and only if, the t-level sets [v] t are ideals of L, for all t∈]0,1]. Moreover, any fuzzy ideal of L is a fuzzy algebra of L, any fuzzy ideal of µ is a fuzzy subalgebra of µ and if v is a fuzzy ideal of L then v * is an ideal of L.
We define the null fuzzy algebra of µ as the fuzzy set of L 1, = 0, and as a consequence of this definition we assume that our upper ordered set S has the real numbers 0 and 1.
A fuzzy Lie algebra µ of L is called abelian if µ 2 = 0 and non-abelian otherwise.
If v 1 ,…,v n are fuzzy sets of L, one defines: To understand the main result of [4] which will be used at the end of this section and in the remainder of this article, we present the following definition.
For any fuzzy subalgebra v of a fuzzy algebra µ one defines inductively the derived series of v as the descending chain of fuzzy subalgebras of µ v (1) ≥v (2) algebra over a field F and S an upper well ordered set. Then every solvable (resp., nilpotent) fuzzy ideal v 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)).
Let µ be a fuzzy Lie ideal of L. One says that µ is a simple fuzzy ideal if: (i) µ is a non-abelian fuzzy ideal and (ii) for all fuzzy ideals v of µ, one has either To conclude this section, we extend the notion of a semisimple fuzzy ideal [4, Definition 26] for a semisimple fuzzy algebra.
Let L be a finite dimensional Lie algebra over a field F, S an upper well ordered set and µ a fuzzy algebra of L in F(L,S). One says that µ is a semisimple fuzzy algebra in F(L,S) if: (i) µ is a non-abelian fuzzy algebra and (ii) its solvable fuzzy radical in

Semisimple Fuzzy Ideals
In this section we generalize the theorem of decomposition of a semisimple Lie algebra as a direct sum of simple Lie ideals for the case of a semisimple fuzzy ideal, similarly to the crisp case. For this, we begin with the following definition.
Definition 3.1. Let L be a Lie algebra over a field F, S an upper well ordered set and µ a fuzzy ideal of L. One says that a fuzzy set π of L is a fuzzy ideal of µ relative to µ * if the following conditions are satisfied: In this case, π * is also an ideal of µ * .
If π is a fuzzy ideal of µ relative to µ * , then one says that a fuzzy set σ of L is a fuzzy ideal of π relative to µ * if the following conditions are satisfied: In this case, σ * is also an ideal of π * .
One says that a fuzzy ideal π of µ relative to µ * is a simple fuzzy ideal of µ relative to µ * if the following conditions are satisfied: (v) π is a non-abelian fuzzy ideal in µ * (that is, π 2 ≠ o in µ * ); (vi) for all fuzzy ideal σ of π relative to µ * , one has either Hereafter, we exemplify each of the fuzzy ideals defined above.
It is easy to check that S is an upper well ordered set and µ a fuzzy ideal of L in F(L,S) such that µ * = M. Next, from the representation of a fuzzy set, according to [4, Section 1], if P is an ideal of M, then the fuzzy set π of L defined by its t-level sets as:[π] 0 = L, [π] t = P, for all , is a fuzzy ideal of µ relative to µ * in F(L,S) such that π * = P. Yet, if S is an ideal of P, then the fuzzy set σ of L defined by its t-level sets as: , is a fuzzy ideal of π relative to µ * in F(L,S) such that σ * = S. Now, let us suppose that P is simple. Then P (2) (=PP) ≠ 0 which implies that there are nonzero elements a 1 ,…,a n ,b 1 ,…,b n in P such that ∑ is a nonzero element belongs to M(=µ * ). It follows that , from the representation of a fuzzy set which shows that π is a non-abelian fuzzy ideal in µ * . Moreover, for all fuzzy ideal σ of π relative toµ * , we have [σ] t an ideal of π * (= P) for . Thus, π is is a simple fuzzy ideal of µ relative to µ * . Theorem 3.3. Let L be a finite dimensional Lie algebra over a field F, µ a fuzzy Lie ideal of L and π a non-abelian (in µ * ) fuzzy ideal of µ relative to µ * . If π is a simple fuzzy ideal of µ relative to µ * , then π * is not a solvable ideal of µ * .
Moreover, π is a simple fuzzy ideal of µ relative to µ * if, and only if, π * is a simple ideal of µ * .
Proof. First, let us observe that * (2) ( ) π is not a zero ideal of µ * since π 2 is non-null in µ * . Since L is finite dimensional, there is a finite set of real numbers 0 , by [11]. This implies that  (ii) for each fuzzy ideal π of µ relative to µ * there are fuzzy ideals and each ideal of µ * is a sum of certain simple ideals of µ * , by [1,Theorem and Corollary,5.2,pp. 23]. This implies that (i =1,…,n), by [13,14]. This shows that v 1 ,….,v n are fuzzy ideals in F(L,S). Yet, for all fuzzy ideal σ i of v i relative to * Next, let us show that Firstly, let us observe that it is sufficient to consider only the case when x ∈ µ * . Hence, let us fix an index i(1 ≤ I ≤ n) and let us consider an element x (≠ 0) ∈ µ * . If Now, let π be a simple fuzzy ideal of µ relative to µ * . Then π * is a sum of certain simple ideals of µ * , by [

Levi Fuzzy Decomposition
In this section we generalize the theorem of Levi decomposition of a Lie algebra for the case of a class of fuzzy ideal, similarly to the crisp case. (ii) for all t ∈]0,1].
In the following example we show that the conditions of the Definition 4.1 are not artificial.

Open Questions
The history of the class of fuzzy Lie algebras proposed in [4] and in this paper is far from over. In fact, there are many unanswered