L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

Abstract In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.


Introduction
Since Rosenfeld [1] introduced fuzzy sets in the realm of the group theory, many researchers are engaged in extending the concepts and results of abstract algebra to the boarder framework of the fuzzy set. Liu [2] de ned the concepts of fuzzy rings and fuzzy ideals in a ring. Katsaras and Liu [3] introduced the concept of a fuzzy subspace of a vector space. In [4,5] Nanda used fuzzy sets to develop the theory of fuzzy elds. Negoita and Ralescu [6] introduced the notion of fuzzy modules, etc. However, not all results can be extended to the fuzzy set [7][8][9][10][11] and fuzzi cation develops slowly in algebra theory. On the other hand, algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, information sciences, coding theory and so on. This provides su cient motivations for us to review various concepts and results from the realm of abstract algebra to a broader framework of a fuzzy set.
In this paper, the concept of a fuzzy subspace is extended to a Novikov algebra. In section 2, we de ne L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras, and discuss some fundamental properties. In section 3, we show that addition, product and intersection of L-fuzzy ideals are L-fuzzy ideals [resp. L-fuzzy subalgebras], but the union of L-fuzzy ideals may not be an L-fuzzy ideal. In section 4, we show that the quotient algebra A/µ of the L-fuzzy ideal µ is isomorphic to the algebra A/Aµ of the non-fuzzy ideal Aµ. In section 5, we show that if f : A → A is an L-fuzzy Novikov algebra homomorphism, then the preimage of an L-fuzzy ideal is an L-fuzzy ideal [resp. L-fuzzy subalgebra]. When f is surjective, a homomorphic image is an L-fuzzy ideal. Moreover, the addition, product and intersection of L-fuzzy ideals in A are preserved by f .

Preliminaries
Let X be any set and L be a non-trivial complete distributive lattice (in particular L could be [ , ]). Then an L-fuzzy set µ in X is characterised by a map µ : X → L. L X will be denoted as all the L-fuzzy subsets in X, it can be given whatever operations L has, and these operations in L X will obey any law valid in L which extends point by point [12].
A pre-Lie algebra A is a vector space with a binary operation (x, y) → x · y satisfying for all x, y, z ∈ A. The algebra is called Novikov algebra, if (x · y) · z = (x · z) · y is satis ed. Throughout this paper A will be denote as a Novikov algebra over a eld F, unless explicitly stated otherwise.
De nition 2.1. [3] Let V be a vecter space over a eld F. An L-fuzzy subspace is an L-fuzzy subset µ : V → L,

Lemma 2.2. [3] Let V be a vecter space over a eld F, an L-fuzzy subset µ : V → L is an L-fuzzy subspace if and only if
( µ( ) = for all k, l ∈ F, x, y ∈ V.

De nition 2.3. Let
A be a Novikov algebra over a eld F with a bilinear product (x, y) → x · y. An L-fuzzy subspace µ : A → L is called an L-fuzzy subalgebra of A, if the inequation is satis ed for all x, y ∈ A.
An L-fuzzy subspace µ : is satis ed for all x, y ∈ A.

Remark 2.4. L-Fuzzy subalgebras and L-Fuzzy ideals of a Novikov algebra
Example 2.5. Let (A, *) be a commutative associative algebra, and D be its derivation. Then the new product makes (A, ·) become a Novikov algebra for a = by Gelfand and Do man [13], for a ∈ F by Filippov [14] and for a xed element a ∈ A by Xu [15]. If µ is an L-fuzzy subspace of (A, *), then µ is an L-fuzzy subalgebra under the conditions of Gelfand [13] and Filippov [14], but µ may not be an L-fuzzy subalgebra under the condition of a ∈ A by Xu [15].
Thus µ is an L-fuzzy subalgebra of A.
If a ∈ A, then Thus µ is not necessarily an L-fuzzy subalgebra of A.
The addition and the multiplication of A are extended by means of Zadeh's extension principle [16], to two operations on L A denoted by ⊕ and ⊗ as follows: The scalar multiplication kx for k ∈ F and x ∈ A is extended to an action of the eld F on L A denoted by as follows: Proof.

L-fuzzy ideals and subalgebras
(1) µ ⊕ ρ is an L-fuzzy subspace of A by Proposition 3.3 of [3]. Let x, y ∈ A. Then Thus µ ⊕ ρ is an L-fuzzy subalgebra of A.
(2) i∈I µ i is an L-fuzzy subspace of A by Proposition 3.4 of [3]. Let x, y ∈ A. Then Thus i∈I µ i is an L-fuzzy subalgebra of A.

Theorem 3.2. (1) Let µ, ρ be L-fuzzy ideals of A. Then µ ⊕ ρ is also an L-fuzzy ideal of A.
(2) Let {µ i : i ∈ I} be a set of L-fuzzy ideals of A. Then the intersection i∈I µ i of A is also an L-fuzzy ideal of A. Proof.
(1) µ ⊕ ρ is an L-fuzzy subspace of A by Proposition 3.3 of [3]. Let x, y ∈ A. Then Similarly, we can prove that i∈I µ i is an L-fuzzy subspace of A by Proposition 3.4 of [3]. Let x, y ∈ A. Then

Theorem 3.4. Let µ be an L-fuzzy subspace of A. Then µ is an L-fuzzy ideal of A if and only if χ
Thus µ is an L-fuzzy ideal of A.

Theorem 3.5. Let µ, ρ be L-fuzzy ideals of A. Then µ ⊗ ρ is also an L-fuzzy ideal of A.
Proof. By Proposition 3.3 (2), we have

By Proposition 3.3 (1), it is obvious that
By Theorem 3.4, µ ⊗ ρ is an L-fuzzy ideal of A.

Coset of an L-fuzzy ideals
De nition 4.1. [17] Let µ be an L-fuzzy ideal of A. For each x ∈ A, the L-fuzzy subset x + µ : A → L de ned by (x + µ)(y) = µ(y − x) is called a coset of the L-fuzzy ideal µ. Proof. If x+µ = y+µ, then evaluating both side of this equation at x we get µ( for all z ∈ A. Thus x + µ ≥ y + µ. On the other hand, we have for all z ∈ A.
Thus y + µ ≥ x + µ. As is clear from the above descriptions, we get the equation x + µ = y + µ.
It is easy to see that Aµ is an ideal of A.
The addition, the scalar multiplication and the multiplication operation of the cosets in De nition 4.6 are well de ned by Proposition 4.5.

Theorem 4.7. The Novikov quotient algebra A/µ is isomorphic to the algebra A/Aµ.
Proof. Consider the surjective algebra homomorphism π : A → A/µ de nes by π(x) = x + µ. By Theorem 4.2, Ker(π) = Aµ. By the fundamental theorem of homomorphisms, there exists an isomorphism from A/Aµ to A/µ. The isomorphic correspondence is given by x + µ = x + Aµ for x ∈ A.