Fuzzy Hom-Lie Ideals of Hom-Lie Algebras

. In this paper, we introduce the cocept of fuzzy Hom-Lie ideals of Hom-Lie algebras and we investigate some of their properties. We study the relationship between fuzzy Hom-Lie subalgebras (resp. ideals) and Hom-Lie subalgebras (resp. ideals). Finally, The properties of fuzzy Hom-Lie subalgebras and fuzzy Hom-Lie ideals under morphisms of Hom-Lie algebras are studied.


Introduction
The motivations to study Hom-Lie structures are related to physics and to deformations of Lie algebras, especially Lie algebras of vector fields. The Hom-Lie algebras were originally studied by Hartwig, Larsson, and Silvestrov in [7] as part of a study of deformations of the Witt and the Virasoro algebras. It is one of generalizations of the notion of Lie algebras. In recent years, they have become an interesting subject of mathematics and physics. For more details on Hom-Lie algebras we refer the readers to [5,9,10,11,14].
The notion of fuzzy sets was firstly introduced by Zadeh [17]. The fuzzy set theory states that there are propositions with an infinite number of truth values, assuming two extreme values, 1 (totally true), 0 (totally false) and a continuum in between, that justify the term "fuzzy". Applications of this theory can be found, for example, in artificial intelligence, computer science, control engineering, decision theory, logic and management science.
The study of fuzzy Lie subalgebras of Lie algebras was initiated by Yehia [15] in 1996. Later fuzzy sets (and more generally intuitionistic fuzzy sets and complex fuuzy sets) have been applied in various directions in Lie algebras by mant authors (see e.g. [2,3,4,6,12,13], and references therein) . In this paper we describe fuzzy Hom-Lie algebras.

Preliminaries
In this section we give some relevant definitions, notations, and results that will be used frequently throughout the paper. Let X be a nonempty set. A fuzzy set A on X is an object having the form A = {(x, µ(x)) | x ∈ X}, where µ denotes the degree of membership function that assigns each element x ∈ X a real number µ(x) ∈ [0, 1].
Let A = {x, µ(x)) | x ∈ X} be a fuzzy set. For the sake of simplicity, we shall use the notation A = µ. The complement of A is defined by Let F be a ground field. A Hom-Lie algebra over F is a triple (L, [ , ], α) where L is a vector space over F , α : L → L is a linear map, and [ , ] : L × L → L is a bilinear map (called a bracket), satisfying the following properties: for all x, y ∈ L (skew-symmetry property).
It is clear that every Lie algebra is a Hom-Lie algebra by setting α = id L (The identity map). For a Hom-Lie algebra L over a field (ii) ϕ • α 1 = α 2 • ϕ.
Throughout this paper, L is a Hom-Lie algebra over F .

Fuzzy Hom-Lie Subalgebras and Fuzzy Hom-Lie Ideals
For the sake of simplicity, we shall use the symbols a∧b = min {a, b} and a∨b = max {a, b}.
Definition 3.1 A fuzzy set µ on L is a fuzzy Hom-Lie subalgebra if the following conditions are satisfied for all x, y ∈ L, and c ∈ F : If the condition (iii) is replaced by µ([x, y]) ≥ µ(x)∨µ(y), then µ is called a fuzzy Hom-Lie ideal of L. Note that the second condition implies µ(x) ≤ µ(0) and It is clear that if µ is a a fuzzy Hom-Lie ideal of L, then it is a fuzzy Hom-Lie subalgebra of L.
This implies that the Hom-Jacobi identity is satisfied. We define µ as follows: Then µ is a fuzzy Hom-Lie ideal of L.

Relations Between Fuzzy Hom-Lie Ideals and Hom-Lie Ideals
Let V be a vector space and µ be a fuzzy set on it. For t ∈ [0, 1] the set U(µ, t) = {x ∈ V | µ(x) ≥ t} is called an upper level of µ. The following theorem will show a relation between fuzzy Hom-Lie subalgebras of L and Hom-Lie subalgebras of L. (ii) the non empty set U(µ, t) is a Hom-Lie subalgebra of L for every t ∈ Im(µ).
Proof. Let t ∈ Im(µ), and let x, y ∈ U(µ, t), and c ∈ F . As µ is a fuzzy Hom-Lie subalgebra of L, we have µ( and so x + y, αx, and [x, y] are elements in U(µ, t). Conversely, let U(µ, t) be Hom-Lie subalgebras of L for every t ∈ Im(µ). Let x, y ∈ L and c ∈ F . We may assume µ(y) ≥ µ(x) = t 1 , so x, y ∈ U(µ, t 1 ). As U(µ, t 1 ) is a subspace of L, we have c.
Using almost the same argument one can show the following result.

On Morphism of Hom-Lie algebras
Let f : X → Y be a function. If B = µ B is a fuzzy set of Y , then the preimage of B is defined to be a fuzzy set Also if A = µ A is a fuzzy set on X, then the image of A is defined to be a fuzzy set (See for example [14]). The following theorem was obtained by Kim and Lee in [8] in the setting of Lie algebras. We extend it to Hom-Lie algebra case. If B = µ B is a fuzzy Hom-Lie subalgebra (resp. ideal) of L 2 , then the fuzzy set f −1 (B) is also a fuzzy Hom-Lie subalgebra (resp. ideal) of L 1 .
Proof. Let x 1 , x 2 ∈ L 1 . Then Let x ∈ L 1 and c ∈ F . Then The case of fuzzy Hom-Lie ideal is similar to show. If f : L 1 → L 2 is a Lie algebra homomorphism and A = µ A is a fuzzy subalgebra of L 1 , then the image of A, f (A) is a fuzzy subalgebra of f (L 1 ) ( [8]). In the following theorem we establish an analogue result for the case of Hom-Lie algebras. Proof. Let y 1 , y 2 ∈ L 2 . As f is onto, there are x 1 , x 2 ∈ L 1 such that f (x 1 ) = y 1 and f (x 2 ) = y 2 . We have Also, For y ∈ L 2 and c ∈ F , we find and and so also, Chung-Gook Kim and Dong-Soo Lee ( [8]) proved if ϕ : L → L ′ is a surjective Lie algebra homomorphism and A = µ A is a fuzzy ideal of L, then ϕ(A) is a fuzzy ideal of L ′ . We will extend the result to fuzzy Hom-Lie algebra case. Proof. The proof is similar to the proof of the theorem above. We only need to show that µ f (A) ([y 1 , y 2 ] 2 ) ≥ µ f (A) (y 1 ) ∨ µ f (A) (y 2 ) for all y 1 , y 2 ∈ L 2 . Let y 1 , y 2 ∈ L 2 , and assume, by contradiction, that µ f (A) ([y 1 , y 2 ] 2 ) < µ f (A) (y 1 ) ∨ µ f (A) (y 2 ). Then µ f (A) ([y 1 , y 2 ] 2 ) < µ f (A) (y 1 ) or µ f (A) ([y 1 , y 2 ] 2 ) < µ f (A) (y 2 ). We may assume, without loss of generality, that µ f (A) ([y 1 , y 2 ] 2 ) < µ f (A) (y 1 ). Choose a number t ∈ [0, 1] such that µ f (A) ([y 1 , y 2 ] 2 ) < t < µ f (A) (y 2 ). There is a ∈ f −1 (y 1 ) with µ A (a) > t. As f is onto, there exists b ∈ f −1 (y 2 ).