Induced L -bornological vector spaces and L -Mackey convergence 1

1 Introduction

In 1949, Hu [12, 13] first introduced the concept of a bornology and a bornological space to define the concept of boundedness in a topological space as follows:

Definition 1.1. (See [12]). A bornology on a set X is a family B ⊆ 2 X such that

∀x ∈ X, { x } ∈ B ;

The pair ( X , B ) is called a bornological space and the sets belonging to B are viewed as bouneded in this space. Then, most of the research developed the theory of bornological spaces into the context of topological vector spaces (see [10, 19]). Hogle-Nled [10] discussed the theory of bornological vector space and presented the definition of bornological vector space as follows:

Definition 1.2. (See [10]) Let X be a vector space over K , where K represents a field of real or complex numbers. A bornology B on X is said to be a bornology compatible with a vector space structure of X if it satisfies the following conditions:

∀ U , V ∈ B , U + V ∈ B ;

It should be noticed that, the notion of bornological vector spaces is different from that in [19, 21], the former [19] is a space without topology, however, the latter [21] is a space with topology. Recent years, the theory of general bornological spaces plays a key role in research of convergence structures on hyperspaces [2, 14] and optimization theory [4], as well as in study of topologies on function spaces[5–7, 15].

It is worthy noting that Abel and Šostak [1] generalized the theory of bornological spaces to the context of fuzzy sets in 2011. In the following years, Paseka et al. [17] provided the necessary and sufficient condition on a complete lattice for the category L-Born to be topological. Later, Šostak and UI̧jane [20] proposed an alternative approach to fuzzification of the concept of bornology and developed a construction of an L-valued bornology on a set from a family of crisp bornologies on the same set. Thereafter, UI̧jane and Šostak [23] extended the L-valued bornologies to a complete completely distributive lattice. It should be mentioned that, in 2014, Paseka et al. [16] investigated the L-bornological vector spaces and got that for certain complete lattices, L-VBorn is topological over the category Vec of vector spaces. Moreover, they introduced the category L-VBornSys of L-bornological vector systems, and showed that the category L-VBorn is isomorphic to a full reflective subcategory of L-VBornSys in [16]. After that, Zhang and Zhang [24] gave the concept of I-bornological vector spaces and discussed two methods on constructing new I-bornological vector spaces.

As is mentioned in [10], bornological spaces are one class of those spaces that form the general and precise framework in which the fundamental theorems and techniques of Functional Analysis hold. Motivated by [16] and based on [10], we further discuss L-bornological vector spaces and aim to provide some assistance to establish more systematic theory of L-bornological vector spaces. The paper is organized as follows. Firstly, in section 2, we present some necessary notions and fundamental results which are used in the sequel. In section 3, we present a necessary and sufficient condition for an L-bornology to be L-vector bornology and we also prove the L-bornological vector space generated by L-fuzzy bounded sets of an L-topological vector space as an example. Afterwards, the induced L-bornological vector spaces and the quotient L-bornological vector spaces are studied. Finally, in section 4, the notions of L-Mackey convergence and separation in L-bornological vector spaces are discussed, respectively. In particular, we prove that an L-Mackey bornologically convergent sequence is topologically convergent. Meanwhile, the necessary and sufficient conditions for the separation of L-bornological vector spaces, the product and quotient L-bornological vector spaces are investigated, respectively.

2 Preliminaries

In this section, we recall some necessary notions and fundamental results which are used in this paper.

Throughout this paper, X and Y always denote universe of discourse, the symbol θ denotes the neutral element of a vector space and J denotes a directed set. L denotes a Hutton algebra (see [11]), i.e., a completely distributive lattice equipped with an order-reversing involution ^′ : L → L. Note that ⊥ _L and ⊤ _L are its smallest and greatest elements, respectively. L ^X denotes the family of all L-fuzzy sets on X. An L-fuzzy set that takes the constant value λ ∈ L on X is denoted by λ _ . An element λ in L is called an union-irreducible element if λ = α ∨ β implies λ = α or λ = β, λ is called prime if λ ≥ α ∧ β implies λ ≥ α or λ ≥ β, where α, β ∈ L. M (L) and M^* (L ^X ) denote the set of all non-smallest union-irreducible elements in L and L ^X , respectively. The notation Pr (L) denotes the set of all prime elements in L. The elements of M (L) are also called molecules in L (see[22]). For each α ∈ L, β^* (α) denotes a standard minimal family of α. An L-fuzzy set on X is called an L-fuzzy point if it takes the value ⊥ _L for all y ∈ X except a single x ∈ X. If its value at x is λ ∈ L \ {⊥  _L }, we denote this L-fuzzy point by x _λ .

Let f : X → Y be a map. Define f^→ : L ^X  → L ^Y (called L-valued Zadeh function or L-forward powerset operator, [18]) and f^← : L ^Y  → L ^X (called L-backward powerset operator, [18]) by f^→ (A) (y) = ⋁ _f(x)=yA (x) for A ∈ L ^X and y ∈ Y, and f^← (B) = B ∘ f for B ∈ L ^Y respectively.

Let X be a vector space over K , using Zadeh’s extension principle, the addition and scale multiplication operators in L ^X are defined as follows, respectively. For A, B ∈ L ^X and t ∈ K , (A + B) (x) = ⋁ _y+z=x (A (y) ∧ B (z)); ( kA ) ( x ) = A ( x k ) , k ≠ 0; ( 0 A ) ( x ) = { ⋁ y ∈ X A ( y ) , x = θ , ⊥ L , x ≠ θ .

In particular, for L-fuzzy points, we have x _λ  + y _μ  = (x + y) _λ∧μ; kx _λ  = (kx)  _λ , where λ, μ ∈ L.

Definition 2.1. (Fang and Yan [9]) Let X be a vector space over K . An L-fuzzy subset A of X is called balanced if tA ≤ A for each t with |t|≤1.

Definition 2.2. (Fang and Yan [9]) Let X be a vector space over K , λ ∈ M (L). An L-fuzzy subset A of X is called λ-absorbing, if for each x ∈ X there exists t₀ > 0 such that tx _λ ≰A′ for all |t| ≤ t₀. A is called absorbing, if it is λ-absorbing for each λ ∈ M (L).

Definition 2.3. (Liu and Luo [22]) Let (X, τ) be an L-topological space, x _λ  ∈ M^* (L ^X ). A ∈ L ^X is called an R-neighborhood of x _λ , if there exists a Q ∈ τ′ such that x _λ ≰Q and and A ≤ Q. The collection of all R-neighborhood of x _λ is denoted by η (x _λ ).

Definition 2.4. (Liu and Luo [22]) Let (X, τ) be an L-topological space, x _λ  ∈ M^* (L ^X ), A family U of L-fuzzy subsets on X is called an R-neighborhood base of x _λ , if U ⊆ η ( x λ ) and for each W ∈ η (x _λ ) there exists a P ∈ U such that W ≤ P.

Definition 2.5. (Liu and Luo [22]) Let (X, τ) be an L-topological space, S = {S (n) , n ∈ J} a net in (L ^X , τ) and e ∈ M^* (L), then S is said to converge topologically to e (we denote S ⟶ τ e ) if for all P ∈ η (e), S eventually not in P, i.e. there exists n₀ such that S (n) ≰P for n ≥ n₀.

Theorem 2.6. (Fang and Yan [9]) Let (X, τ) be an L-topological vector space, and let U λ (λ ∈ M (L)) be a closed R-neighborhood base of θ _λ . Then U λ ( λ ∈ M ( L ) ) has the following properties: (1) if W ∈ U λ or W = λ _ , then for any x ∈ X and α ∈ M (L) with x _α ≰W, then there exists P ∈ U α such that W ≤ x + P; (2) if P , Q ∈ U λ , then there exists W ∈ U λ such that P ∨ Q ≤ W; (3) if W ∈ U λ , then there exists P ∈ U λ such that P′ + P′ ≤ W′; (4) if W ∈ U λ , then there exists P ∈ U λ such that tP′ ≤ W′ for all t ∈ K with |t|≤1; (5) if W ∈ U λ , then for each x ∈ X there exists t > 0 such that x _λ ≰tW. Conversely, if for each λ ∈ M (L), there is a family U λ of L-fuzzy sets on X satisfying the above conditions (1)-(5), then there exists a unique L-topology τ on X such that (X, τ) is an L-topological vector space, and U λ is a closed R-neighborhood base of θ _λ .

Definition 2.7. (Fang [8]) Let (X, τ) be an L-topological vector space and λ ∈ M (L). An L-fuzzy set A on X is called to be λ-bounded if for each R-neighborhood Q of θ _λ in (X, τ), there exist t> 0 and r ∈ L with r≰λ′ such that A ∧ r _ ≤ tQ ′ ; A is said to be L-fuzzy bounded, if for each λ ∈ M (L), A is λ-bounded. For convenience, the family of all L-fuzzy bounded sets denoted by Bd.

Definition 2.8. (Abel and Šostak [1]) An L-bornological space is a pair ( X , B ) , where X is a set, and B (an L-bornology on X) is a subfamily of L ^X (the elements of which are called bounded L-sets), which satisfy the following axioms: (B1) for every x ∈ X, ⋁ B ∈ B B ( x ) = ⊤ L ; (B2) given B ∈ B and D ∈ L ^X such that D ≤ B, it follows that D ∈ B ; (B3) if S ⊆ B is finite, then ⋁ S ∈ B . Given L-bornological spaces ( X 1 , B 1 ) and ( X 2 , B 2 ) , a map f : X₁ → X₂ is called L-bounded provided that f → ( B 1 ) ∈ B 2 for every B 1 ∈ B 1 .

Definition 2.9. (Paseka et al. [16]) An L-bornological vector space is a tuple ( X , + , * , B ) , where (X, + , *) is a vector space over K , and ( X , B ) is an L-bornological space such that: f : X × X → X, (x, y) ↦ x + y is L-bounded; g : K × X → X , ( k , x ) ↦ kx is L-bounded, where X × X and K × X are equipped with the corresponding product L-bornology B × B and B K × B (here B K is an L-bornology determined by the crisp bornology on K ), respectively.

Theorem 2.10. (Paseka et al. [16]) The product of a family ( X i , B i ) i ∈ J of L-bornological space is given by the source f j : ( ∏ i ∈ J X i , B ) → ( X j , B j ) j ∈ J , where f j : ∏ i ∈ J X i → X j is the j-th projection map and B = { B ∈ L X | B ≤ ⋀ i ∈ J B i } .

3 Induced L-bornological vector spaces

The aim of this section is to introduce and study Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces. At first the necessary and sufficient condition for an L-bornology to be L-vector bornology is presented. The L-bornological vector space generated by L-fuzzy bounded sets of an L-topological vector space is provided as an example. As the focus of this section, the induced L-bornological vector spaces and the quotient L-bornological vector spaces are also discussed.

Theorem 3.1. Let ( X , B ) be an L-bornological space. Then ( X , B ) is an L-bornological vector space ( B is an L-vector bornology) if and only if B satisfies the following conditions: (B4) U , V ∈ B ⇒ U + V ∈ B ; (B5) ∀ t ∈ K , U ∈ B ⇒ tU ∈ B ; (B6) U ∈ B ⇒ ⋁ | t | ≤ 1 tU ∈ B .

Proof. Necessity. (B4) For all U , V ∈ B , it is clear that U × V ∈ B × B . Since ( X , B ) is an L-bornological vector space, thus we obtain f → ( U × V ) = U + V ∈ B , which follows from that f in Definition 2.9 is L-bounded.

(B5) For all t ∈ K and U ∈ B , it is easy to find χ { t } × U ∈ B K × B . Since g : K × X → X is L-bounded, then it deduces that g → ( χ { t } × U ) = tU ∈ B . (B6) Let A₁ = {t | |t|≤1} and for all U ∈ B , it follows that g → ( χ A 1 × U ) = ⋁ | t | ≤ 1 tU ∈ B .

Sufficiency. First, for all A ∈ B × B , there exist B 1 , B 2 ∈ B such that A ≤ B₁ × B₂. Then f → ( A ) ≤ f → ( B 1 × B 2 ) = B 1 + B 2 ∈ B . It is obvious that f → ( A ) ∈ B and f is L-bounded. Next, we will prove that g is L-bounded. For all A ∈ B K × B , there exist B 1 ∈ B K and B 2 ∈ B such that A ≤ B₁ × B₂. Then we get B₁ ≤ {t| |t| ≤ s} = B₃, where s is a constant in K . Hence, g → ( A ) ≤ g → ( B 1 × B 2 ) ≤ g → ( B 3 × B 2 ) = s ⋁ | t | ≤ 1 t B 2 ∈ B , which follows that g is L-bounded. This completes the proof.

Example 3.2. (An L-vector bornology determined by L-fuzzy bounded sets of an L-topological vector space) Let (X, τ) be an L-topological vector space. Then Bd given by Definition 2.7 is an L-vector bornology.

Proof. (B1) It is obvious that ⋁_A∈BdA (x) = ⊤  _L for all x ∈ X. (B2) For all A ∈ Bd, if there exists B ∈ L ^X such that B ≤ A. Then for any λ ∈ M (L), there exist t > 0 and r ∈ L with r≰λ′ such that A ∧ r _ ≤ tQ ′ for every R-neighborhood Q of θ _λ . Thus, we get B ∧ r _ ≤ A ∧ r _ ≤ tQ ′ , which follows that B ∈ Bd. (B3) For all A _i  ∈ Bd and λ ∈ M (L), there exist t _i  > 0 and r _i  ∈ L with r _i ≰λ′ such that A i ∧ r i _ ≤ t i Q ′ for each R-neighborhood Q of θ _λ , where i = 1, 2. Let r₃ = min {r₁, r₂} and t₃ = max {t₁, t₂}. Then we obtain ( A 1 ∨ A 2 ) ∧ r 3 _ = ( A 1 ∧ r 3 _ ) ∨ ( A 2 ∧ r 3 _ ) ≤ ( A 1 ∧ r 1 _ ) ∨ ( A 2 ∧ r 2 _ ) ≤ t 1 Q ′ ∨ t 2 Q ′ ≤ t 3 Q ′ .

This means that A₁ ∨ A₂ ∈ Bd.

(B4) For all A₁, A₂ ∈ Bd, λ ∈ M (L) and any R-neighborhood P of θ _λ , from Theorem 2.6, there exists a R-neighborhood Q of θ _λ such that Q′ + Q′ ≤ P′. Analogous to the proof of (B3), we have ( A 1 + A 2 ) ∧ r 3 _ = ( A 1 ∧ r 3 _ ) + ( A 2 ∧ r 3 _ ) ≤ ( A 1 ∧ r 1 _ ) + ( A 2 ∧ r 2 _ ) ≤ t 1 Q ′ + t 2 Q ′ ≤ t 3 ( Q ′ + Q ′ ) ≤ t 3 P ′ .

So we have A₁ + A₂ ∈ Bd. (B5) For all A ∈ Bd, t ∈ K and λ ∈ M (L), there exists s > 0 and r ∈ L with r≰λ′ such that A ∧ r _ ≤ sQ ′ for all R-neighborhood Q of θ _λ . It is obvious that ( tA ) ∧ r _ = t ( A ∧ r _ ) ≤ tsQ ′ . That means tA ∈ Bd. (B6) For all A ∈ Bd and λ ∈ M (L), there exists s > 0 and r ∈ L with r≰λ′ such that A ∧ r _ ≤ sQ ′ for all R-neighborhood Q of θ _λ . By Theorem 2.6(4), we may put Q′ is balanced. Then ( ⋁ | t | ≤ 1 tA ) ∧ r _ = ⋁ | t | ≤ 1 t ( A ∧ r _ ) ≤ ⋁ | t | ≤ 1 tsQ ′ ≤ sQ ′ , i.e. ⋁_|t|≤1tA ∈ Bd. This completes the proof.

Next, we discuss the induced L-bornology. At the beginning, we construct a family ω ( B ) of fuzzy subsets on X, which satisfies that a fuzzy subset A ∈ ω ( B ) iff A ( α ) ∈ B for each α ∈ Pr (L), where A_(α) = {x|A (x) ≰α}.

Lemma 3.3. For any α ∈ Pr (L) and A ∈ L ^X , the following statements hold.

(1) If A ⁱ  ∈ L ^X , i ∈ J, then ⋃ i ∈ J A ( α ) i = ( ⋁ i ∈ J A i ) ( α ) ; (2) If A, B ∈ L ^X , then A_(α) + B_(α) = (A + B) _(α); (3) For any t ∈ K and A ∈ L ^X , tA_(α) = (tA) _(α).

Theorem 3.4. Let ( X , B ) be a crisp bornological vector space. Then ( X , ω ( B ) ) is an L-bornological vector space.

Proof. (B1) For each B ⊆ X with B ∈ B , it is obvious that χ B ∈ ω ( B ) . From the fact B is a crisp bornology on X, we have ⋃ B ∈ B B = X . Then ⋁ B ∈ B χ B ( x ) = ⊤ L for all x ∈ X. Thus ⋁ A ∈ ω ( B ) A ( x ) ≥ ⋁ B ∈ B χ B ( x ) = ⊤ L , it follows that ⋁ A ∈ ω ( B ) A ( x ) = ⊤ L for all x ∈ X. (B2) For all A ∈ ω ( B ) and B ∈ L ^X with B ≤ A, we have B_(α) ≤ A_(α) and A ( α ) ∈ B for any α ∈ Pr (L). It follows that B ( α ) ∈ B . That means B ∈ ω ( B ) . (B3) For all A , B ∈ ω ( B ) , we have A ( α ) , B ( α ) ∈ B for any α ∈ Pr (L). Then ( A ∨ B ) ( α ) = A ( α ) ∪ B ( α ) ∈ B . That is A ∨ B ∈ ω ( B ) . (B4) For all A , B ∈ ω ( B ) , A ( α ) , B ( α ) ∈ B for any α ∈ Pr (L). Then we have ( A + B ) ( α ) = A ( α ) + B ( α ) ∈ B . Thus, we get A + B ∈ ω ( B ) . (B5) For all A ∈ ω ( B ) , we have A ( α ) ∈ B for any α ∈ Pr (L). Then, ( tA ) ( α ) = tA ( α ) ∈ B , which implies that tA ∈ ω ( B ) . (B6) For all A ∈ ω ( B ) and α ∈ Pr (L), we have ( ⋁ | t | ≤ 1 tA ) ( α ) = ⋃ | t | ≤ 1 t A ( α ) ∈ B . Then, ⋁ | t | ≤ 1 tA ∈ ω ( B ) . This completes the proof.

Theorem 3.5. Let ( X , B ) , ( Y , B Y ) be two crisp bornological vector spaces, ( X , ω ( B ) ) and ( Y , ω ( B Y ) ) be induced L-bornological vector spaces. Then the mapping f : ( X , B ) → ( Y , B Y ) is bounded if and only if f → : ( X , ω ( B ) ) → ( Y , ω ( B Y ) ) is L-bounded.

Proof. Necessity. Suppose that the mapping f : ( X , B ) → ( Y , B Y ) is bounded, for every A ∈ ω ( B ) and any α ∈ Pr (L), (f^→ (A)) _(α) = f (A_(α)). Since A ( α ) ∈ B , we have f ( A ( α ) ) ∈ B Y . This means f → ( A ) ∈ ω ( B Y ) . So f^→ is L-bounded mapping.

Sufficiency. If f → : ( X , ω ( B ) ) → ( Y , ω ( B Y ) ) is L-bounded, for all A ∈ B , χ A ∈ ω ( B ) . Then f → ( χ A ) ∈ ω ( B Y ) . Put α ∈ Pr (L) , α ¬ = ⊤  _L , we have ( f → ( χ A ) ) ( α ) = f ( A ) ∈ B Y . Thus f : ( X , B ) → ( Y , B Y ) is bounded.

By Theorem 3.4 and Theorem 3.5, if we use the notation BVS denotes the category of bornological vector spaces and linear bounded mappings, and the notation L-VBorn denotes the category of L-bornological vector spaces and linear L-bounded mappings. Then the mapping ω : BVS→ L-VBorn is a functor.

Theorem 3.6. Let ( X , B ) be an L-bornological vector space. Then ( X , ι α ( B ) ) is a crisp bornological vector space, where ι α ( B ) = { A ( α ) : A ∈ B } and α ∈ Pr (L).

Proof. (B1) Since ⋁ A ∈ B A ( x ) = ⊤ L for all x ∈ X, then for each α ∈ Pr (L), there exists A ∈ B such that A (x) ≰α. That means x ∈ A_(α). This implies that X ⊆ ⋃ A ( α ) ∈ ι α ( B ) A ( α ) , i.e., ⋃ A ( α ) ∈ ι α ( B ) A ( α ) = X . (B2) Let A ∈ ι α ( B ) and B ⊆ X with B ⊆ A, then there exists A 1 ∈ B such that A = A ( α ) 1 . Set B 1 ( x ) = { A 1 ( x ) , x ∈ B ; ⊥ L , x ∈ X ∖ B . Obviously, B¹ ≤ A¹ and B = B ( α ) 1 . From the fact A 1 ∈ B , we have B 1 ∈ B . Hence, A = A ( α ) 1 ⊇ B ( α ) 1 = B , which means B ∈ ι α ( B ) . (B3) Let A , B ∈ ι α ( B ) , there exist A 1 , B 1 ∈ B such that A = A ( α ) 1 , B = B ( α ) 1 . Since A 1 ∨ B 1 ∈ B , we have A ∪ B = A ( α ) 1 ∪ B ( α ) 1 = ( A 1 ∨ B 1 ) ( α ) , which follows that A ∪ B ∈ ι α ( B ) . The proofs of (B4)-(B6) are similar to the process in Theorem 3.4 and omitted.

By the statements in the section 2.4 [10] and Theorem 3.5, the next Corollary holds evidently.

Corollary 3.7. Let ( X , B ) be an L-bornological vector space, ι ( B ) is a bornology with the base ⋂ α ∈ Pr ( L ) ι α ( B ) , then ( X , ι ( B ) ) is a crisp bornological vector space.

Corollary 3.8. Let ( X , B ) be a crisp bornological vector space and let ( X , B ) be an L-bornological vector space, then (1) B = ι ( ω ( B ) ) ; (2) B ≤ ω ( ι ( B ) ) .

By Corollary 3.7 and Corollary 3.8, it is clear that the mapping ι : L-VBorn→ BVS is a functor, the category BVS is isomorphic to the category ωsd(BVS).

Theorem 3.9. Let ( X i , B i ) i ∈ J be a family of bornological vector spaces indexed by a non-empty set J, then ∏ i ∈ J ω ( B i ) = ω ( ∏ i ∈ J B i ) .

Proof. First, we show ∏ i ∈ J ω ( B i ) ⊇ ω ( ∏ i ∈ J B i ) . For all A ∈ ω ( ∏ i ∈ J B i ) , we have A ( α ) ∈ ∏ i ∈ J B i for all α ∈ Pr (L). Then there exists B i ∈ B i ( i ∈ J ) such that A ( α ) ≤ ∏ i ∈ J B i , which follows that f i ( A ( α ) ) = ( f i → ( A ) ) ( α ) ≤ B i ∈ B i . That implies f i → ( A ) ∈ ω ( B i ) . It is obvious that A ∈ ∏ i ∈ J ω ( B i ) .

Conversely, for all A ∈ ∏ i ∈ J ω ( B i ) , there exists B i ∈ ω ( B i ) such that A ≤ ∏ i ∈ J B i , which implies that f i → ( A ) ≤ B i . Then, we have ( f i → ( A ) ) ( α ) = f i ( A ( α ) ) ≤ B ( α ) i ∈ B i . Hence, A ( α ) ∈ ∏ i ∈ J B i , which means that A ∈ ω ( ∏ i ∈ J B i ) . This completes the proof.

Theorem 3.10. Let ( X , B ) be an L-bornological vector space and let f : X → Y be a linear map of X onto a set Y, then B / f = { f → ( B ) | B ∈ B } is the quotient L-vector bornology on Y.

Proof. First, we need to show that B / f is an L-vector bornology. (B1) For all y ∈ Y, ⋁ A ∈ B / f A ( y ) = ⋁ B ∈ B ⋁ f ( x ) = y B ( x ) = ⊤ L . (B2) If A₁, A₂ ∈ L ^Y , A₁ ≥ A₂ and A 1 ∈ B / f , then there exists B 1 ∈ B such that A₁ = f^→ (B₁). Let B 2 ( x ) = { B 1 ( x ) ∧ A 2 ( f ( x ) ) , A 2 ( f ( x ) ) ≠ ⊥ L ; ⊥ L , A 2 ( f ( x ) ) = ⊥ L .

It is obvious that B 2 ∈ B and A₂ (y) = ⋁ _f(x)=yB₂ (x) = f^→ (B₂) (y). Thus, A 2 ∈ B / f . (B3) For all A 1 , A 2 ∈ B / f , there exist B 1 , B 2 ∈ B such that A₁ = f^→ (B₁) and A₂ = f^→ (B₂). Then (A₁ ∨ A₂) (y) = (f^→ (B₁) ∨ f^→ (B₂)) (y) = ⋁ _f(x)=y (B₁ ∨ B₂) (x) = f^→ (B₁ ∨ B₂) (y), which follows that A 1 ∨ A 2 ∈ B / f . (B4) Similarly, we obtain (A₁ + A₂) (y) = ⋁ _f(x1+x2)=yB₁ (x₁) ∧ B₂ (x₂) = f^→ (B₁ + B₂) (y). Then, A 1 + A 2 ∈ B / f . (B5) For all A ∈ B / f and t ∈ K , there exists B ∈ B such that A = f^→ (B). It follows that tA (y) = tf^→ (B) (y) = ⋁ _f(x)=ytB (x) = f^→ (tB) (y), which means tA ∈ B / f . (B6) For all A ∈ B / f , there exists B ∈ B such that A = f^→ (B). Then, we get ⋁ | t | ≤ 1 tB ∈ B . Hence, ⋁_|t|≤1tA (y) = ⋁ _|t|≤1tf^→ (B) (y) = ⋁ _|t|≤1 ⋁ _f(x)=ytB (x) = f^→ (⋁ _|t|≤1tB) (y). That means ⋁ | t | ≤ 1 tA ∈ B / f . Thus, it is showed that B / f is an L-vector bornology, we remain to prove that B / f is the strongest L-vector bornology. For all B Y with f is L-bounded, then for all B ∈ B , we obtain f → ( B ) ∈ B Y . Thus, f → ( B ) ∈ B / f . It follows that B Y ≤ B / f .

Theorem 3.11. Let B be a crisp vector bornology on vector space X and let f : X → Y be a linear map of X onto a set Y, then ω ( B / f ) = ω ( B ) / f .

Proof. First, we show ω ( B / f ) ⊆ ω ( B ) / f . For all A ∈ ω ( B / f ) , we have A ( α ) ∈ B / f for every α ∈ Pr (L). Then we have f - 1 ( A ( α ) ) = ( f ← ( A ) ) ( α ) ∈ B , i.e., f ← ( A ) ∈ ω ( B ) . Thus, A ∈ ω ( B ) / f , which follows that ω ( B / f ) ⊆ ω ( B ) / f .

Conversely, for all A ∈ ω ( B ) / f , we have f ← ( A ) ∈ ω ( B ) . Then, ( f ← ( A ) ) ( α ) = f - 1 ( A ( α ) ) ∈ B , which means A ( α ) ∈ B / f . Thus, A ∈ ω ( B / f ) , i.e., ω ( B / f ) ⊇ ω ( B ) / f . This completes the proof.

4 L-Mackey convergence and separation in L-bornological vector spaces

In this section, the notions of L-Mackey convergence and separation in L-bornological vector spaces are discussed. Meanwhile, the necessary and sufficient conditions for the separation of L-bornological vector spaces, the product and quotient L-bornological vector spaces are investigated, respectively.

Definition 4.1. Let ( X , B ) be an L-bornological vector space and λ ∈ M (L), S = { x λ ( n ) n } n ∈ ℕ be a λ-sequence in X, where ⋁ n ∈ ℕ ⋀ m ≥ n λ ( n ) ≥ λ . The S is said to converge bornologically to θ _λ if there exists a circled L-fuzzy set B ∈ B and a sequence {s _n } of scalars tending to 0, such that x λ ( n ) n ≰ s n B ′ for all n ∈ ℕ .

Remark 4.2. (1) For historical reasons, bornological convergence is also called Mackey- convergence after G.W. Mackey. Thus we also call the sequence S is L-Mackey convergence in Definition 4.1, and it is denoted x λ ( n ) n ⟶ M θ λ . (2) S = { x λ ( n ) n } is said to convergent to x _λ if (x ⁿ  - x) _λ(n) converges to θ _λ , and it is denoted by x λ ( n ) n ⟶ M x λ .

Theorem 4.3. Let ( X , B ) be an L-bornological vector space and λ ∈ M (L), where L-bornology B is in the sense of Abel and Šostak [1]. If x λ ( n ) n ⟶ M x λ , y λ ( n ) n ⟶ M y λ and t _n  → t in K , then

(1). x λ ( n ) n + y λ ( n ) n ⟶ M x λ + y λ ;

(2). t n x λ ( n ) n ⟶ M tx λ .

Proof. (1) Since x λ ( n ) n ⟶ M x λ and y λ ( n ) n ⟶ M y λ , there exist circled L-fuzzy sets A , B ∈ B and α _n  → 0, β _n  → 0 such that (x ⁿ  - x) _λ(n)≰α _n A′ and (y ⁿ  - y) _λ(n)≰β _n B′ for all n ∈ ℕ . Then we obtain λ (n) ≰α _n A′ (x ⁿ  - x) ∨ β _n B′ (y ⁿ  - y). Thus, λ (n) ≰ max {α _n , β _n } (A + B) ′ (x ⁿ  - x + y ⁿ  - y), i.e. (x ⁿ  - x + y ⁿ  - y) _λ(n)≰ max {α _n , β _n } (A + B) ′. From the fact A , B ∈ B and ( X , B ) is an L-bornological vector space, we have A + B ∈ B . In addition, max {α _n , β _n } →0, it implies that x λ ( n ) n + y λ ( n ) n ⟶ M x λ + y λ .

(2) Since t _n  → t, the set {t _n } is bounded, i.e., there exists k₀ > 0 such that |t _n | ≤ k₀ for all n. Then we get (t _n x ⁿ  - tx) _λ(n) = (t _n x ⁿ  - t _n x + t _n x - tx) _λ(n)≰ max {t _n α _n , t _n  - t} (A + χ_{x}) ′. Hence, from t _n A′ ≥ k₀A′, we obtain (t _n x ⁿ  - t _n x + t _n x - tx) _λ(n)≰ max {k₀α _n , t _n  - t} (A + χ_{x}) ′. Owing to B is an L-bornology in the sense of Abel and Šostak, χ { x } ∈ B , it follows that A + χ { x } ∈ B . It implies that t n x λ ( n ) n ⟶ M tx λ .

Theorem 4.4. Let ( X , B ) be an L-bornological vector space and a linear mapping f : X → Y be L-bounded. If x λ ( n ) n ⟶ M x λ for x _λ  ∈ M^* (L ^X ), then f → ( x λ ( n ) n ) ⟶ M f → ( x λ ) . Proof. Since x λ ( n ) n ⟶ M x λ , then there exist circled L-fuzzy set A ∈ B and α _n  → 0 such that (x ⁿ  - x) _λ(n)≰α _n A′ for all n ∈ ℕ . Thus, we have f → ( x λ ( n ) n ) - f → ( x λ ) = ( f ( x n ) - f ( x ) ) λ ( n ) ≰ f → ( α n A ′ ) = α n f → ( A ′ ) = α n ( f → ( A ) ) ′ . From the boundedness of f, we get f → ( A ) ∈ B , which implies that f → ( x λ ( n ) n ) ⟶ M f → ( x λ ) .

Theorem 4.5. Let (X, τ) be an L-topological vector space and ( X , B ) be an L-bornological vector space determined by L-fuzzy bounded sets in (X, τ). If x λ ( n ) n ⟶ M x λ and λ ∈ M (L), then x λ ( n ) n ⟶ τ x λ .

Proof. From x λ ( n ) n ⟶ M x λ , we know that there exist circled L-fuzzy set A ∈ B and α _n  → 0 such that (x ⁿ  - x) _λ(n)≰α _n A′ for all n ∈ ℕ . Since A is bounded in (X, τ), then for each R-neighborhood Q of θ _λ , there exist t > 0 and r ∈ L with r≰λ′ such that A ∧ r _ ≤ tQ ′ . It is obvious that there exists n 0 ∈ ℕ such that A ∧ r _ ≤ tQ ′ ≤ 1 α n Q ′ and λ(n ≰ r′) for all n ≥ n₀. Thus, we get ( x n - x ) λ ( n ) ≰ α n A ′ ∨ r _ ′ . Hence, (x ⁿ  - x) _λ(n)≰Q for n ≥ n₀, which completes the proof.

Definition 4.6. Let ( X , B ) be an L-bornological vector space, then ( X , B ) is separated if and only if suppM = {θ} for all fuzzy vector subspace M ∈ B , where suppM is the support set of M.

Lemma 4.7. Let ( X , B ) be an L-bornological vector space and M ∈ B . Then M is a fuzzy vector subspace of X if and only if M_(α) is a bounded vector subspace of X for any α ∈ Pr (L).

Proof. Necessity. Let ( X , B ) be an L-bornological vector space, if M is a fuzzy vector subspace of X with M ∈ B , then for any α ∈ Pr (L) and x, y ∈ M_(α), M (x) ≰ α,  M (y) ≰ α. Then for all s , t ∈ K , we have (sM + tM) (sx + ty) ≰α. Since sM + tM ≤ M, it deduces that sx + ty ∈ M_(α). So M_(α) is a bounded vector subspace of X. Sufficiency. Suppose that M_(α) is a bounded vector subspace of X for any α ∈ Pr (L). For all s , t ∈ K , it needs to prove that (sM + tM) _(α) ⊆ M_(α). In fact, if z ∈ (sM + tM) _(α), there exist x, y ∈ M with sx + ty = z such that M (x) ∧ M (y) ≰ α. This means x, y ∈ M_(α). So z = sx + ty ∈ M_(α) which follows from M_(α) is a bounded vector subspace. Then we obtain sM + tM ≤ M. Therefore M is a fuzzy vector subspace of X.

Theorem 4.8. Let ( X , B ) be an L-bornological vector space, then ( X , B ) is separated if and only if ( X , ι α ( B ) ) is separated for any α ∈ Pr (L).

Proof. Necessity. If an L-bornological vector space ( X , B ) is separated, then suppM = {θ} for all fuzzy bounded vector subspace M of X. Thus, M_(α) is a crisp bounded vector subspace of X for any α ∈ Pr (L), which follows from Lemma 4.7. If there exists a vector subspace M₁ of X with M₁ ≠ {θ} and M 1 ∈ ι α ( B ) . Then there exists M ∈ B such that M₁ = M_(α). By decomposition theorem of L-fuzzy sets, M = ⋀ a ∈ L ( a _ ∨ M ( a ) ) , we have α _ ∨ M ( α ) ∈ B . Moreover, it is clear that α _ ∨ M 1 is a fuzzy vector space and ( α _ ∨ M 1 ) ( x ) ¬ = ⊥ L for x ∈ M₁ and x ≠ θ. This deduces a contradiction. Therefore ( X , ι α ( B ) ) is separated for any α ∈ Pr (L). Sufficiency. If ( X , B ) is not a separated L-bornological vector space, then there exists a fuzzy vector subspace M ∈ B such that suppM ≠ {θ}. Let x ∈ suppM and x ≠ θ, then there exists α ∈ Pr (L) such that M (x) ≰α. With the fact that M_(α) is a crisp bounded vector subspace of X and M_(α) ≠ {θ}, contradicting the hypothesis that ( X , ι α ( B ) ) is separated for any α ∈ Pr (L). Hence ( X , B ) is a separated L-bornological vector space.

Theorem 4.9. Let ( X , B ) be an L-bornological vector space and λ ∈ M (L). If x λ ( n ) n ⟶ M x λ and x λ ( n ) n ⟶ M y λ , then ( L X , B ) is separated if and only if x = y.

Proof. Necessity. Since x λ ( n ) n → M x λ and x λ ( n ) n ⟶ M y λ , there exist circled L-fuzzy sets A , B ∈ B , α _n  → 0 and β _n  → 0 such that (x ⁿ  - x) _λ(n)≰α _n A′ and (y ⁿ  - y) _λ(n)≰β _n B′ for all n ∈ ℕ . Then from the proof of Theorem 4.3, we obtain (x - y) _λ(n)≰ max {α _n , β _n } (A + B) ′. From Theorem 3.5, we know that ι α ( B ) is a crisp bornology for any α ∈ Pr (L). Thus, there exists n 0 ∈ ℕ such that x - y ∈ (max {α _n , β _n } (A + B)) _(γ′) for all γ ∈ β^* (λ) and n ≥ n₀. Since ( X , B ) is separated, then ( X , ι γ ′ ( B ) ) is separated, which means {θ} is the only bounded vector subspace of X. If x - y ≠ θ, then the vector subspace spanned by x - y is contained in (A + B) _(γ′), which makes a contradiction.

Sufficiency. If ( X , B ) is not a separated L-bornological vector space, then there exists a fuzzy vector subspace M of X with M ∈ B such that suppM ≠ {θ}. Let x ∈ suppM and x ≠ θ, suppose that M (x) ≠ ⊥  _L , then there exists λ′ ∈ Pr (L) such that M (x) ≰λ′. From Lemma 4.7, we have M_(λ′) is a crisp bounded vector subspace of X. Let x λ = x λ ( n ) n , it is easy to see that x λ ( n ) n ≰ 1 n M ′ , which means that x λ ( n ) n ⟶ M θ λ for λ ∈ M (L). Hence, x = θ, contradicting the hypothesis that x ≠ θ. This completes the proof.

Theorem 4.10. Let ( X i , B i ) be separated L-bornological vector spaces and let the linear mappings f i : ∏ i ∈ J X i → X i be bounded for all i ∈ J. Then ( ∏ i ∈ J X i , ∏ i ∈ J B i ) is separated if and only if for all x ∈ ∏ i ∈ J X i , there exists i ∈ J such that f _i  (x) ≠ θ.

Proof. Necessity. Let ( ∏ i ∈ J X i , ∏ i ∈ J B i ) be a separated L-bornological vector space, then for all λ ∈ M (L) and x ≠ θ, we obtain λ ∧ χ span { x } ∉ B . It implies that there exists i ∈ J such that f i → ( λ ∧ χ span { x } ) ∉ B i . Since ( X i , B i ) is separated for every i ∈ J, then we have f _i  (x) ≠ θ.

Sufficiency. If ( ∏ i ∈ J X i , ∏ i ∈ J B i ) is not separated, there exists a fuzzy vector subspace M with M ∈ ∏ i ∈ J B i such that suppM ¬ = {θ}. Let x ∈ suppM,  x ¬ = θ and denote M (x) = a, then a ∧ χ_span{x} ≤ M. Since x ≠ θ, there exists i ∈ J such that f _i  (x) ≠ θ, then f i → ( a ∧ χ span { x } ) ≠ θ a , which makes a contradiction with ( X i , B i ) is separated for all i ∈ J.

Definition 4.11. Let ( X , B ) be an L-bornological vector space, then an L-fuzzy subset P is L-bornologically closed in ( X , B ) if for all x λ ( n ) n ≤ P and x λ ( n ) n ⟶ M x λ , x _λ  ≤ P holds.

Theorem 4.12. ( X , B ) is a separated L-bornological vector space if and only if θ _λ is L-bornologically closed in ( X , B ) for all λ ∈ M (L).

Proof. Necessity. Suppose that ( X , B ) is a separated L-bornological vector space, let A = θ _λ for all λ ∈ M (L) and let x λ ( n ) n ≤ A . If x λ ( n ) n ⟶ M x α , then α = ⋁ n ∈ ℕ ⋀ m ≥ n λ ( m ) ≤ A ( θ ) = λ and there exists a circled L-fuzzy set B ∈ B and a sequence {s _n } of scalars tending to 0 such that (x ⁿ  - x) _λ(n) = - x_λ(n)≰s _n B′ for all n ∈ ℕ . Thus s_nB(x) ≰ λ (n)′ for all n ∈ ℕ . Put μ ∈ β^* (α) , μ ¬ = ⊤  _L , we have span{x} ⊆ ι_μ′(B). On the other hand, by Theroem and ( X , B ) is a separated L-bornological vector space, ( X , ι μ ′ ( B ) ) is a crisp bornological vector space. This implies x = θ and θ _α  ∈ A. Thus θ _λ is L-bornologically closed in ( X , B ) for all λ ∈ M (L).

Sufficiency. For all x λ ( n ) n ⟶ M x λ and x λ ( n ) n ⟶ M y λ with λ ∈ M (L), we obtain θ λ ( n ) ⟶ M ( x - y ) λ . It is obvious that x = y since θ _λ is L-bornologically closed in ( X , B ) for all λ ∈ M (L). Thus ( X , B ) is separated.

Theorem 4.13. Let ( X , B ) be an L-bornological vector space and M be a vector subspace of X with χ M ∈ B , then quotient space ( X / M , B X / M ) is separated if and only if χ _M is L-bornologically closed in ( X , B ) .

Proof. We only prove the sufficiency. Suppose that χ _M is L-bornologically closed in ( X , B ) . For all λ ∈ M (L) and any sequence x ˆ λ ( n ) n ≤ θ ˆ λ , if x ˆ λ ( n ) n ⟶ M x ˆ α , then α ≤ λ and x λ ( n ) n ≤ χ M . In addition, it is easy to check that x λ ( n ) n ⟶ M x α . Hence x _α  ≤ χ _M , this implies that x ˆ = θ ˆ . That is to say x ˆ α = θ ˆ α ≤ θ ˆ λ . This means that θ ˆ λ is L-bornologically closed in ( X / M , B X / M ) for all λ ∈ M (L), which implies that ( X / M , B X / M ) is separated by Theorem 4.12.