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

Xin Zhou; Liangyun Chen; Yuan Chang

doi:10.1515/math-2019-0126

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

Xin Zhou , Liangyun Chen und Yuan Chang

Veröffentlicht/Copyright: 26. Dezember 2019

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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/A_μ of the non-fuzzy ideal A_μ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.

Keywords: fuzzy set; fuzzy algebra; fuzzy subalgebra; fuzzy ideal

MSC 2010: 03E72; 08A72

1 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] defined 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 fields. 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 fuzzification 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 sufficient 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 define 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/A_μ of the non-fuzzy ideal A_μ. In section 5, we show that if f : A₁ → A₂ 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 A₁ are preserved by f.

2 Preliminaries

Let X be any set and L be a non-trivial complete distributive lattice (in particular L could be [0, 1]). 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

(x⋅y)⋅z−x⋅(y⋅z)=(y⋅x)⋅z−y⋅(x⋅z)

for all x, y, z ∈ A. The algebra is called Novikov algebra, if

(x⋅y)⋅z=(x⋅z)⋅y

is satisfied.

Throughout this paper A will be denote as a Novikov algebra over a field F, unless explicitly stated otherwise.

Definition 2.1

[3] Let V be a vecter space over a field F. An L-fuzzy subspace is an L-fuzzy subset μ : V → L, satisfying

μ(x + y) ≥ μ(x) ∧ μ(y),
μ(kx) ≥ μ(x),
μ(0) = 1

for all k ∈ F, x, y ∈ V.

Lemma 2.2

[3] Let V be a vecter space over a field F, an L-fuzzy subset μ : V → L is an L-fuzzy subspace if and only if

μ(kx + ly) ≥ μ(x) ∧ μ(y),
μ(0) = 1

for all k, l ∈ F, x, y ∈ V.

Definition 2.3

Let A be a Novikov algebra over a field 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

μ(x⋅y)≥μ(x)∧μ(y)

is satisfied for all x, y ∈ A.

An L-fuzzy subspace μ : A → L is called an L-fuzzy ideal of A, if the inequation

μ(x⋅y)≥μ(x)∨μ(y)

is satisfied for all x, y ∈ A.

Remark 2.4

L-Fuzzy subalgebras and L-Fuzzy ideals of a Novikov algebra A are L-fuzzy subspaces of A, then for all k ∈ F, x ∈ V we have μ(kx) = μ x.

Example 2.5

Let (A, ∗) be a commutative associative algebra, and D be its derivation. Then the new product

x⋅y=x∗Dy+a∗x∗y for all x,y∈A,

makes (A, ⋅) become a Novikov algebra for a = 0 by Gelfand and Doffman [13], for a ∈ F by Filippov [14] and for a fixed 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].

Let x, y ∈ A. If a ∈ F, then

μ(x⋅y)=μ(x∗Dy+a∗x∗y)≥μ(x)∧sup{μ(t):t∈D−1(y)}∧μ(x)∧μ(y)=μ(x)∧μ(y).

Thus μ is an L-fuzzy subalgebra of A.

If a ∈ A, then

μ(x⋅y)≥μ(x)∧sup{μ(t):t∈D−1(y)}∧μ(a)∧μ(x)∧μ(y)=μ(a)∧μ(x)∧μ(y).

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:

(μ ⊕ ρ)(x) = sup{μ(a) ∧ ρ(b) : a + b = x},
(μ ⊗ ρ)(x) = sup{μ(a) ∧ ρ(b) : a ⋅ b = x}

for all μ, ρ ∈ L^A, x, a, b ∈ A.

The scalar multiplication kx for k ∈ F and x ∈ A is extended to an action of the field F on L^A denoted by ⊙ as follows:

(k⊙μ)(x)=μ(k−1x)if k≠0,1if k=0,x=0,0if k=0,x≠0.

3 L-fuzzy ideals and subalgebras

Theorem 3.1

Let μ be an L-fuzzy ideal and ρ be an L-fuzzy subalgebra of A. Then μ ⊕ ρ is also an L-fuzzy subalgebra of A.
Let {μ_i : i ∈ I} be a set of L-fuzzy subalgebras of A. Then the intersection ⋂i∈I μ_i of A is also an L-fuzzy subalgebra of A.

Proof

μ ⊕ ρ is an L-fuzzy subspace of A by Proposition 3.3 of [3]. Let x, y ∈ A. Then

(μ⊕ρ)(x⋅y)=sup{μ(x1⋅y)∧ρ(x2⋅y):x1+x2=x}≥sup{(μ(x1)∨μ(y))∧(ρ(x2)∧ρ(y)):x1+x2=x}=sup{(μ(x1)∧ρ(x2)∧ρ(y))∨(μ(y)∧ρ(x2)∧ρ(y)):x1+x2=x}≥sup{(μ(x1)∧ρ(x2)∧ρ(y)):x1+x2=x}=(μ⊕ρ)(x)∧(ρ)(y)≥(μ⊕ρ)(x)∧(μ⊕ρ)(y) for x1,x1∈A.

Thus μ ⊕ ρ is an L-fuzzy subalgebra of A.
⋂i∈I μ_i is an L-fuzzy subspace of A by Proposition 3.4 of [3]. Let x, y ∈ A. Then

(⋂i∈Iμi)(x⋅y)=infi∈I{μi(x⋅y)}≥infi∈I{μi(x)∧μi(y)}=(⋂i∈Iμi)(x)∧(⋂i∈Iμi)(y).

Thus ⋂i∈I μ_i is an L-fuzzy subalgebra of A. □

Theorem 3.2

Let μ, ρ be L-fuzzy ideals of A. Then μ ⊕ ρ is also an L-fuzzy ideal of A.
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

μ ⊕ ρ is an L-fuzzy subspace of A by Proposition 3.3 of [3].

Let x, y ∈ A. Then

(μ⊕ρ)(x⋅y)=sup{μ(x1⋅y)∧ρ(x2⋅y):x1+x2=x}≥sup{(μ(x1)∨μ(y))∧(ρ(x2)∨ρ(y)):x1+x2=x}≥sup{[(μ(x1)∨μ(y))∧(ρ(x2)]∨[(μ(x1)∨μ(y))∧ρ(y)]:x1+x2=x}≥sup{(μ(x1)∧ρ(x2))∨(μ(y)∧ρ(y)):x1+x2=x}=(μ⊕ρ)(x)∨(μ∧ρ)(y)≥(μ⊕ρ)(x) for x1,x2∈A.

Similarly, we can prove that (μ ⊕ ρ)(x ⋅ y) ≥ (μ ⊕ ρ)(y). Thus (μ ⊕ ρ)(x ⋅ y) ≥ (μ ⊕ ρ)(x) ∨ (μ ⊕ ρ)(y).
⋂i∈I μ_i is an L-fuzzy subspace of A by Proposition 3.4 of [3]. Let x, y ∈ A. Then

(⋂i∈Iμi)(x⋅y)=infi∈I{μi(x⋅y)}≥infi∈I{μi(x)∨μi(y)}≥∨{infi∈I(μi(x)),infi∈I(μi(y))}=(⋂i∈Iμi)(x)∨(⋂i∈Iμi)(y).

□

Proposition 3.3

Let μ, ρ, σ be L-fuzzy ideals of A. Then

(μ ⊗ ρ) ⊗ σ = (μ ⊗ σ) ⊗ ρ;
μ ⊗ (ρ ⊗ σ) ⊆ ((μ ⊗ ρ) ⊗ σ) ⊕ (ρ ⊗ (μ ⊗ σ)) ⊕ ((ρ ⊗ μ) ⊗ σ).

Proof

Let x ∈ A. Then

((μ⊗ρ)⊗σ)(x)=sup{(μ⊗ρ)(m)∧σ(n):m⋅n=x}=sup{sup{μ(a)∧ρ(b):a⋅b=m}∧σ(n):m⋅n=x}=sup{((μ(a)∧ρ(b))∧σ(n):(a⋅b)⋅n=x}=sup{((μ(a)∧σ(n))∧ρ(b):(a⋅n)⋅b=x}=sup{sup{μ(a)∧σ(n):a⋅n=c}∧ρ(b):c⋅b=x}=sup{(μ⊗σ)(c)∧ρ(b):c⋅b=x}=((μ⊗σ)⊗ρ)(x) for a,b,c,m,n∈A.
Let x ∈ A. Then

(μ⊗(ρ⊗σ))(x)=sup{μ(m)∧(ρ⊗σ)(n):m⋅n=x}=sup{μ(m)∧sup{ρ(b)∧σ(c):b⋅c=n}:m⋅n=x}=sup{sup{μ(m)∧(ρ(b)∧σ(c)):m⋅(b⋅c)=x}}=sup{μ(m)∧(ρ(b)∧σ(c)):(m⋅b)⋅c+b⋅(m⋅c)−(b⋅m)⋅c=x}=sup{((μ(m)∧ρ(b))∧σ(c))∧(ρ(b)∧(μ(m)∧σ(c)))∧((ρ(b)∧μ(m))∧σ(c)):(m⋅b)⋅c+b⋅(m⋅c)−(b⋅m)⋅c=x}≤sup{(sup(μ(m)∧ρ(b))∧σ(c))∧(ρ(b)∧sup(μ(m)∧σ(c)))∧(sup(ρ(b)∧μ(m))∧σ(c)):(m⋅b)⋅c+b⋅(m⋅c)−(b⋅m)⋅c=x}=sup{((μ⊗ρ)(r)∧σ(c))∧(ρ(b)∧(μ⊗σ)(s))∧((ρ⊗μ)(t)∧σ(c)):r⋅c+b⋅s+t⋅c=x,r=m⋅b,s=m⋅c,t=−b⋅m}≤sup{sup((μ⊗ρ)(r)∧σ(c))∧sup(ρ(b)∧(μ⊗σ)(s))∧sup((ρ⊗μ)(t)∧σ(c)):r⋅c+b⋅s+t⋅c=x,r=m⋅b,s=m⋅c,t=−b⋅m}=sup{((μ⊗ρ)⊗σ)(u)∧(ρ⊗(μ⊗σ))(v)∧((ρ⊗μ)⊗σ)(w):u+v+w=x,u=r⋅c,v=b⋅s,w=t⋅c}=(((μ⊗ρ)⊗σ)⊕(ρ⊗(μ⊗σ))⊕((ρ⊗μ)⊗σ))(x) for m,n,b,c,r,s,t,u,v,w∈A.

□

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_A ⊗ μ ⊆ μ and μ ⊗ χ_A ⊆ μ, where χ_A(x) = 1 for all x ∈ A.

Proof

(⇐) Suppose that χ_A ⊗ μ ⊆ μ. Let x, y ∈ A. Then

μ(x⋅y)≥(χA⊗μ)(x⋅y)=sup{χA(a)∧μ(b):a⋅b=x⋅y)}≥χA(x)∧μ(y)≥μ(y) for a,b∈A.

Suppose that μ ⊗ χ_A ⊆ μ. Let x, y ∈ A. Then

μ(x⋅y)≥(μ⊗χA)(x⋅y)=sup{μ(c)∧χA(d):c⋅d=x⋅y)}≥μ(x)∧χA(y)≥μ(x) for c,d∈A.

Thus μ is an L-fuzzy ideal of A. (⇒) Suppose μ is an L-fuzzy ideal of A. Let x ∈ A. Then

(χA⊗μ)(x)=sup{χA(a)∧μ(b):a⋅b=x}=sup{μ(b):a⋅b=x}≤μ(x) for a,b∈A.

Similarly, we can prove (μ ⊗ χ_A)(x) ≤ μ(x). □

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

χA⊗(μ⊗ρ)⊆((χA⊗μ)⊗ρ)⊕(μ⊗(χA⊗ρ))⊕((μ⊗χA)⊗ρ)⊆(μ⊗ρ)⊕(μ⊗ρ)⊕(μ⊗ρ)⊆(μ⊗ρ).

By Proposition 3.3 (1), it is obvious that

(μ⊗ρ)⊗χA=(μ⊗χA)⊗ρ⊆μ⊗ρ.

By Theorem 3.4, μ ⊗ ρ is an L-fuzzy ideal of A. □

Remark 3.6

Let {μ_i : i ∈ I} be a set of L-fuzzy ideals [resp. L-fuzzy subalgebras] in A. Then the union ⋃i∈I μ_i may not be an L-fuzzy ideal [resp. L-fuzzy subalgebra]. It can be proved by the same method as the proof of Proposition 5.7 in [1].

4 Coset of an L-fuzzy ideals

Definition 4.1

[17] Let μ be an L-fuzzy ideal of A. For each x ∈ A, the L-fuzzy subset x + μ : A → L defined by (x + μ)(y) = μ(y − x) is called a coset of the L-fuzzy ideal μ.

Theorem 4.2

If μ is an L-fuzzy ideal of A, then x + μ = y + μ if and only if μ(x − y) = μ(0) = 1. In that case μ(x) = μ(y).

Proof

If x + μ = y + μ, then evaluating both side of this equation at x we get μ(x − y) = μ(x − x) = μ(0) for x, y ∈ A.

Conversely, if μ(x − y) = μ(0) = 1, then

(x+μ)(z)=μ(z−x)=μ(z−y+y−x)≥μ(z−y)∧μ(y−x)=μ(z−y)=(y+μ)(z)

for all z ∈ A. Thus x + μ ≥ y + μ. On the other hand, we have

(y+μ)(z)=μ(z−y)=μ(z−x+x−y)≥μ(z−x)∧μ(x−y)=μ(z−x)=(x+μ)(z)

for all z ∈ A.

Thus y + μ ≥ x + μ. As is clear from the above descriptions, we get the equation x + μ = y + μ. □

Remark 4.3

Let A_μ = {x ∈ A | μ(x) = 1}. It is easy to see that A_μ is an ideal of A.

Remark 4.4

If μ is an L-fuzzy ideal in A, then (x + μ)(z) = μ(y − x) for all z ∈ y + A_μ. In particular, (x + μ)(z) = μ(x) for all z ∈ A_μ.

Proposition 4.5

Let μ be an L-fuzzy ideal and x₁, x₂, y₁, y₂ be any elements in A. If x₁ + μ = y₁ + μ and x₂ + μ = y₂ + μ, then

(x₁ + x₂) + μ = (y₁ + y₂) + μ,
(x₁ ⋅ x₂) + μ = (y₁ ⋅ y₂) + μ,
kx₁ + μ = ky₁ + μ for all k ∈ F.

Proof

The proof of (1), (2) by Proposition 3.4 in [17]. It is sufficient to prove (3).

Since μ (x₁ − y₁) = μ (k(x₁ − y₁)) = μ (kx₁ − ky₁) = 0, we get that kx₁ + μ = ky₁ + μ by Theorem 4.2. □

Definition 4.6

The algebra A/μ of the L-fuzzy ideal μ is called the quotient algebra of Novikov algebra A. We define an addition, a scalar multiplication and a multiplication operations of the cosets as follows:

(x + μ) ⊕ (y + μ) = (x + y) + μ,
k ⊙ (x + μ) = kx + μ,
(x + μ) ⊗ (y + μ) = (x ⋅ y) + μ for all k ∈ F, x, y ∈ A.

The addition, the scalar multiplication and the multiplication operation of the cosets in Definition 4.6 are well defined by Proposition 4.5.

Theorem 4.7

The Novikov quotient algebra A/μ is isomorphic to the algebra A/A_μ.

Proof

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

5 L-fuzzy ideals on homomorphism

Definition 5.1

[5] Let X₁ and X₂ be sets. A map f : X₁ → X₂ has a natural extension f̃ : L^X₁ → L^X₂ defined by

f~(μ)(y)=sup{μ(x):x∈f−1(y)}f−1(y)≠∅,0f−1(y)=∅

for all μ ∈ L^X₁, y ∈ X₂. f̃(μ) is called the homomorphic image of the L-fuzzy set μ.

Let ρ ∈ L^X₂, we define the L-fuzzy set μ ∈ L^X₁ by μ(x) = ρ(f(x)) for all x ∈ X₁, where μ is called the preimage of ρ and denoted by f⁻¹(ρ).

Definition 5.2

[1] Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be an algebra homomorphism. An L-fuzzy subset μ in A₁ is called f-invariant if for any x, y ∈ A₁, f(x) = f(y) implies μ(x) = μ(y).

Theorem 5.3

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be an algebra homomorphism. If μ is an L-fuzzy subalgebra of A₂, then f̃⁻¹(μ) is also an L-fuzzy subalgebras of A₁.

Proof

By Definition 5.1, we have

f̃⁻¹(μ)(x + y) = μ(f(x + y)) = μ(f(x) + f(y)) ≥ μ(f(x)) ∧ μ(f(y)) ≥ f̃⁻¹(μ)(x) ∧ f̃⁻¹(μ)(y).
f̃⁻¹(μ)(kx) = μ(f(kx)) = μ(kf(x)) ≥ μ(f(x))f̃⁻¹(μ)(x).
f̃⁻¹(μ)(x ⋅ y) = μ(f(x ⋅ y)) = μ(f(x) ⋅ f(y)) ≥ μ(f(x)) ∧ μ(f(y)) ≥ f̃⁻¹(μ)(x) ∧ f̃⁻¹(μ)(y) for all k ∈ F, x, y ∈ A₁. □

Theorem 5.4

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be an algebra homomorphism. If μ is an L-fuzzy ideal of A₂, then f̃⁻¹(μ) is also an L-fuzzy ideal of A₁.

Proof

Similar with the proof of Theorem 5.3. □

Theorem 5.5

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be an algebra homomorphism. If μ is an L-fuzzy subalgebra of A₁, then f̃(μ) is also an L-fuzzy subalgebra of A₂.

Proof

Since f(0) = 0 and μ(0) = 1, it is clear that f̃(μ)(0) = 1. By Proposition 3.2 of [3], f̃(μ) is an L-fuzzy subspace of A₂.

Let x, y ∈ A₂. It is enough to show that f̃(μ)(x ⋅ y) ≥ f̃(μ)(x) ∧ f̃(μ)(y).

If x ⋅ y ∈ f(A₁), assume that f̃(μ)(x ⋅ y) < f̃(μ)(x) ∧ f̃(μ)(y). Then f̃(μ)(x ⋅ y) < f̃(μ)(x) and f̃(μ)(x ⋅ y) < f̃(μ)(y). We can choose a number t ∈ [0, 1] such that f̃(μ)(x ⋅ y) < t < f̃(μ)(x) and f̃(μ)(x ⋅ y) < t < f̃(μ)(y). There exist a ∈ f⁻¹(x) ⊆ A₁, b ∈ f⁻¹(y) ⊆ A₁ such that μ(a) > t, μ(b) > t.

Since f(a ⋅ b) = f(a) ⋅ f(b) = x ⋅ y, we have f⁻¹(x ⋅ y) ≠ ∅, and

f~(μ)(x⋅y)=sup{μ(z):z∈f−1(x⋅y)}≥μ(a⋅b)≥μ(a)∧μ(b)>t>f~(μ)(x⋅y).

This is a contradiction. Similarly, we can prove the other case.

If x ⋅ y ∉ f(A₁), we have x ∉ f(A₁) or y ∉ f(A₁). By Definition 5.1, f̃(μ)(x) = 0 or f̃(μ)(y) = 0, it is obvious that f̃(μ)(x ⋅ y) ≥ f̃(μ)(x) ∧ f̃(μ)(y).

Hence, f̃(μ) is an L-fuzzy subalgebra of A₂. □

Theorem 5.6

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be a surjective algebra homomorphism. If μ is an L-fuzzy ideal of A₁, then f̃(μ) is also an L-fuzzy ideal of A₂.

Proof

Since f(0) = 0 and μ(0) = 1, it is clear that f̃(μ)(0) = 1. By Proposition 3.2 of [3], f̃(μ) is an L-fuzzy subspace of A₂.

Let x, y ∈ A₂. It is enough to show that f̃(μ)(x ⋅ y) ≥ f̃(μ)(x) ∨ f̃(μ)(y).

Assume that f̃(μ)(x ⋅ y) < f̃(μ)(x) ∨ f̃(μ)(y). Then f̃(μ)(x ⋅ y) < f̃(μ)(x) or f̃(μ)(x ⋅ y) < f̃(μ)(y). Without loss of generality, we can choose a number t ∈ [0, 1] such that f̃(μ)(x ⋅ y) < t < f̃(μ)(x). There exists an a ∈ f⁻¹(x) ⊆ A₁ such that μ(a) > t.

Since f is surjective, there exists b ∈ A₁ such that μ(b) = y.

Since f(a ⋅ b) = f(a) ⋅ f(b) = x ⋅ y, we have f⁻¹(x ⋅ y) ≠ ∅, and

f~(μ)(x⋅y)=sup{μ(z):z∈f−1(x⋅y)}≥μ(a⋅b)≥μ(a)>t>f~(μ)(x⋅y).

This is a contradiction.

Similarly, we can prove the other case. Hence, f̃(μ) is an L-fuzzy ideal in A₂. □

Theorem 5.7

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be an algebra homomorphism. Then

if μ, ρ are L-fuzzy subalgebras of A₁, then f̃(μ ⊕ ρ) = f̃(μ)⊕f̃(ρ),
if {μ_i : i ∈ I} is a set of L-fuzzy subalgebras of A₁, then f̃( ⋂i∈I μ_i) = ⋂i∈I f̃(μ_i),
if μ, ρ are L-fuzzy subalgebras of A₁, then f̃(μ ⊗ ρ) = f̃(μ) ⊗ f̃(ρ).

Proof

(1) and (2) can be proved by the same method as the proof of Theorem 5.1 in [10]. It is sufficient to prove (3).

Let x ∈ A₂. We prove that f̃(μ ⊗ ρ)(x) = (f̃(μ) ⊗ f̃(ρ))(x). If x = y ⋅ z ∉ f(A₁), we have y ∉ f(A₁) or z ∉ f(A₁). By the proof of Theorem 5.5, we get f̃(μ ⊗ ρ)(x) = 0 and (f̃(μ) ⊗ f̃(ρ))(x) = f̃(μ)(x) ⊗ f̃(ρ)(x) = sup{f̃(μ)(y) ∧ f̃(ρ)(z) : x = y ⋅ z} = 0.

Let x = y ⋅ z ∉ f(A₁). Assume that f̃(μ ⊗ ρ)(x) < (f̃(μ) ⊗ f̃(ρ))(x). We can choose an element t ∈ L such that f̃(μ ⊗ ρ)(x) < t < f̃(μ)(x) ⊗ f̃(ρ)(x).

Since f̃(μ)(x) ⊗ f̃(ρ)(x) = sup{f̃(μ)(y) ∧ f̃(ρ)(z) : x = y ⋅ z}, there exist y, z ∈ A₂ such that x = y ⋅ z with f̃(μ)(y) > t and f̃(ρ)(z) > t. Since x ∈ f(A₁), there exists an x₁ ∈ A₁ such that f(x₁) = x and x₁ = y₁ ⋅ z₁ for y₁ ∈ f⁻¹(y), z₁ ∈ f⁻¹(z) with μ(y₁) > t and ρ(z₁) > t.

Since f(y₁ ⋅ z₁) = f(y₁) ⋅ f(z₁) = y ⋅ z = x, we have

f~(μ⊗ρ)(x)=sup{(μ⊗ρ)(x1):f(x1)=x}=sup{μ(a)∧ρ(b):f(x1)=f(a⋅b)=x}≥μ(y1)∧ρ(z1)>t.

This is a contradiction.

Similarly, for the case f̃(μ ⊗ ρ)(x) > (f̃(μ) ⊗ f̃(ρ))(x), we get a contradiction. Hence, f̃(μ ⊗ ρ) = f̃(μ) ⊗ f̃(ρ). □

Theorem 5.8

Let A₁ and A₂ be Novikov algebras, and f : A₁ → A₂ be a surjective algebra homomorphism. Then

if μ, ρ are L-fuzzy ideals of A₁, then f̃(μ ⊕ ρ) = f̃(μ) ⊕ f̃(ρ),
if {μ_i : i ∈ I} is a set of L-fuzzy ideals of A₁, then f̃( ⋂i∈I μ_i) = ⋂i∈I f̃(μ_i),
if μ, ρ are L-fuzzy ideals of A₁, then f̃(μ ⊗ ρ) = f̃(μ) ⊗ f̃(ρ).

Proof

(1) and (2) can be proved by the same method as the proof of Theorem 5.1 in [10]. It is sufficient to prove (3).

Let x ∈ A₂. We prove that f̃(μ ⊗ ρ)(x) = (f̃(μ) ⊗ f̃(ρ))(x). Assume that f̃(μ ⊗ ρ)(x) < (f̃(μ) ⊗ f̃(ρ))(x). We can choose an element t ∈ L such that f̃(μ ⊗ ρ)(x) < t < f̃(μ)(x) ⊗ f̃(ρ)(x).

Sincef̃(μ)(x) ⊗ f̃(ρ)(x) = sup{f̃(μ)(y) ∧ f̃(ρ)(z) : x = y ⋅ z}, there exist y, z ∈ A₂ such that x = y ⋅ z with f̃(μ)(y) > t and f̃(ρ)(z) > t. Since f is surjective, there exists an x₁ ∈ A₁ such that f(x₁) = x and x₁ = y₁ ⋅ z₁ for y₁ ∈ f⁻¹(y), z₁ ∈ f⁻¹(z) with μ(y₁) > t and ρ(z₁) > t.

Since f(y₁ ⋅ z₁) = f(y₁) ⋅ f(z₁) = y ⋅ z = x, we have

f~(μ⊗ρ)(x)=sup{(μ⊗ρ)(x1):f(x1)=x}=sup{sup{μ(a)∧ρ(b):f(x1)=f(a⋅b)=x}}=sup{μ(a)∧ρ(b):f(a⋅b)=x}≥μ(y1)∧ρ(z1)>t.

This is a contradiction.

Similarly, for the case f̃(μ ⊗ ρ)(x) > (f̃(μ) ⊗ f̃(ρ))(x), we get a contradiction. Hence, f̃(μ ⊗ ρ) = f̃(μ) ⊗ f̃(ρ). □

Acknowledgement

Supported by NNSF of China (No. 11771069), NSF of Jilin province (No. 20170101048JC), the project of jilin province department of education (No. JJKH20180005K) and University Science Programming of Xin Jiang (Grant No. XJEDU2016S080).

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Received: 2019-07-06

Accepted: 2019-11-04

Published Online: 2019-12-26

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https://doi.org/10.1515/math-2019-0126

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fuzzy set; fuzzy algebra; fuzzy subalgebra; fuzzy ideal

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