Applications of fuzzy inclusion orders between L -subsets in fuzzy topological structures

Abstract

Based on fuzzy inclusion order between L-subsets, stratified L-ordered uniform convergence spaces and stratified L-ordered limit spaces are introduced. It is shown that the resulting categories are all Cartesian closed topological categories. Also, the relationship among stratified L-ordered limit spaces, stratified L-ordered Cauchy spaces and stratified L-ordered uniform convergence spaces are investigated.

Keywords

Bireflective subcategory Cartesian closed category

1. Introduction

Since Zadeh introduced the concept of fuzzy sets, many mathematical structures have been combined with fuzzy set theory, including fuzzy topology [1 , 30], fuzzy convexity [22–25 , 33] and so on. In the framework of fuzzy topology, many related mathematical structures have also been studied, such as fuzzy convergence structures [3 , 26], fuzzy Cauchy (filter) structures [13 , 20] and fuzzy (semi-, quasi-, pre-)uniform convergence structures [4–6 , 12].

Considering fuzzy convergence structures that are compatible with fuzzy inclusion order between L-subsets, Fang [2] and Li [15] modified the definition of Jäger’s stratified L-generalized convergence structures to obtain so called stratified L-ordered convergence structures. Following this idea, Wu and Fang [32] introduced L-ordered fuzzifying convergence structures and showed that the resulting category, as a bireflective full subcategory of the category of L-fuzzifying convergence spaces [34], is also a Cartesian closed topological category. Afterwards, Fang defined stratified L-ordered semiuniform convergence spaces [4], stratified L-ordered quasiuniform limit spaces [5] and stratified L-preuniform convergence spaces [6], and studied their categorical properties. In [20], Pang introduced the concept of stratified L-ordered filter spaces and studied its relationship with stratified L-filter spaces [18] from a categorical aspect.

In fuzzy set theory, fuzzy inclusion order between L-subsets follows in line with the intuition more than classical inclusion order between L-subsets. As stated above, fuzzy inclusion order has been endowed with some topological structures. Motivated by this, we will apply fuzzy inclusion order between L-subsets to define stratified L-ordered uniform convergence spaces and stratified L-ordered limit spaces. Then we will investigate their Cartesian-closedness and establish their categorical relationship with stratified L-ordered Cauchy spaces [20].

2. Preliminaries

We consider in this paper complete lattice L where finite meets distributive over arbitrary joins, i.e., a ∧ ⋁ _i∈Ib_i = ⋁ _i∈I (a ∧ b_i) holds for all a, b_i (i ∈ I) . These lattices are called complete Heyting algebras (or frames). The bottom (resp. top) element of L is denoted by ⊥ (resp. ⊤). We can then define a residual implication by $a \to b = ⋁ {c \in L ∣ a \land c ⩽ b} .$ We will often use, without explicitly mentioning, the following properties of the residual implication.

Lemma 2.1. ([8]) Let L be a complete Heyting algebra. Then the following statements hold:

(H1) ⊤ → a = a.

(H2) a ⩽ b if and only if a→ b = ⊤.

(H3) (a ∧ b) → (a ∧ c) ⩾ b → c.

(H4) (a → b) → (a → c) ⩾ b → c.

(H5) (a → b) ∧ (c → d) ⩽ (a ∧ c) → (b ∧ d).

(H6) ⋀_j∈Ja_j → ⋀ _j∈Jb_j ⩾ ⋀ _j∈J (a_j → b_j).

(H7) ⋁_j∈Ja_j → ⋁ _j∈Jb_j ⩾ ⋀ _j∈J (a_j → b_j).

For a nonempty set X, L^X denotes the set of all L-subsets on X. The smallest element and the largest element in L^X are denoted by $\underline{⊥}$ and $\underline{⊤}$ , respectively. For each a ∈ L, $\underline{a}$ denotes the constant map X → L, x ↦ a.

Definition 2.2. ([2]) The map $S (-, -) : L^{X} \times L^{X} \to L$ defined by $\forall C, D \in L^{X}, S (C, D) = \underset{x \in X}{⋀} (C (x) \to D (x))$ is called the fuzzy inclusion order of L-subsets.

Definition 2.3. ([7]) A map $F : L^{X} \to L$ is called a stratified L-filter on X if it satisfies

(F1) $F (\underline{⊥}) = ⊥, F (\underline{⊤}) = ⊤$ ;

(F2) $A ⩽ B \Rightarrow F (A) ⩽ F (B)$ ;

(F3) $F (A \land B) ⩾ F (A) \land F (B)$ ;

(Fs) $a \land F (A) ⩽ F (\underline{a} \land A)$ . The family of all stratified L-filters on X will be denoted by $ℱ_{L}^{s} (X) .$ For every x ∈ X, $[x] \in F_{L}^{s} (X)$ is defined by [x] (A) = A (x) for all A ∈ L^X.

On the set $ℱ_{L}^{s} (X)$ of all stratified L-filters on X, an order ⩽ defined by $ℱ ⩽ G iff ℱ (A) ⩽ G (A)$ for all A ∈ L^X, was introduced in [7]. For a nonempty family ${ℱ_{λ}}$ _λ∈Λ of stratified L-filters, the infimum Λ_λ ∈Λ $ℱ$ λ is given by ( $Λ$ _{$λ$
∈Λ} $ℱ$ _λ) (A) = $\land$ _λ ∈Λ $ℱ$ _λ (A) for all A ∈ L^X. In order to guarantee the least upper bound for a family { $ℱ$ λ }_λ∈Λ, Höhle and Šostak presented the following lemma.

Lemma 2.4. ([7]) For a family { $ℱ$ _λ}_λ∈Λ of stratified L-filters on X, there exists a stratified L-filter $ℱ$ such that $ℱ$ _$λ$ ⩽ $ℱ$ (∀ λ ∈ Λ), if and only if

$ℱ_{λ 1} (A_{1}) \land . . . \land ℱ_{λ n} (A_{n})$ $= ⊥$ Whenevre $A_{1} \land \dots \land A_{n} = \underline{⊥},$

for $n \in ℕ$ , A₁, ⋯ , A_n ∈ L^X, {λ₁, ⋯ , λ_n} ⊆ Λ. In the case of existence, the supremum $\lor$ λ ∈Λ_{$ℱ$

λ} of a nonempty family {_{$ℱ$

λ}} _λ∈Λ of stratified L-filters is given by $\begin{matrix} (\underset{λ \in Λ}{\lor} ℱ_{λ}) (A) & = & \lor_{n \in ℕ} \lor {ℱ_{λ_{1}} (A_{1}) \land \dots \land ℱ_{λ_{n}} (A_{n}) ∣ \\ A_{1} \land \dots \land A_{n} ⩽ A} \end{matrix}$ $φ :$ for all A ∈ L^X.

Let φ : X→Y be a map and $ℱ$ be a stratified L-filter on X. Define φ^→ : L^X→L^Y and φ^← : L^Y→L^X by φ^$\to$ (A) (y) = _{$\lor$
φ (x) = y}A (x) for A ∈ L^X and y ∈ Y, and φ^← (B) = B ∘ φ for B ∈ L^Y, respectively. Then the map φ^⇒ ( $ℱ$ ) : L^Y $\to$ L defined by φ^⇒ ( $ℱ$ ) (A) = $ℱ$ (φ^← (A)) for A ∈ L^Y, is a stratified L-filter on Y, which is called the image of $ℱ$ under φ.

In [12], Jäger defined the composition of two stratified L-filters. Let $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X) .$ , A ∈ L^X×X and (x, y) ∈ X × X. We define ( $ℱ$ ∘ $G$ ) (A) = ⋁ { $ℱ$ (B) ∧ $G$ (C) ∣ B ∘ C ⩽ A}, where (B ∘ C) (x, y) = ⋁ {B (x, z) ∧ C (z, y) ∣ z ∈ X}. Then $ℱ$ ∘ $G$ ∈ $ℱ_{L}^{s} (X) .$ if and only if $B \circ C = \underline{⊥}$ implies $ℱ$ (B)∧ $G$ (C) = ⊥. Let $ℋ$ ∈ $ℱ_{L}^{s} (X) .$ . We define ^{$ℋ$
- 1} (A) = $ℋ$ (A^-1) for each A ∈ L^X×X, where A^-1 (x, y) = A (y, x) for each (x, y) ∈ X × X.

In [9], Jäger proposed the product of two stratified L-filters $F \in ℱ_{L}^{s} (X) .$ , $G \in ℱ_{L}^{s} (Y) (Y)$ , which is denoted by $F \times G$ . Concretely, for each A ∈ L^X×Y, $(F \times G) (A) = sqj {ℱ (B) \land G (C) ∣ B \times C ⩽ A}$ . For a map φ : X $\to$ Y, (φ × φ) ^⇒ ( $ℱ$ × $G$ ) = φ^⇒ ( $ℱ$ ) × φ^⇒ ( $G$ ).

Let $S_{F} (-, -)$ denote the fuzzy inclusion order on $ℱ_{L}^{s} (X)$ , i.e., for any $F, G \in F_{L} (X),$ $S_{F} (F, G) = \underset{A \in L^{X}}{⋀} (F (A) \to G (A)) .$ Then we have the following lemma, which can be found in [20].

Lemma 2.5. ([20]) Let φ : X $\to$ Y be a map. Then

(1) $S_{F} (φ^{\Rightarrow} (F), φ^{\Rightarrow} (G)) ⩾ S_{F} (F, G) .$

(2) $S_{F} (F \times H, G \times ℋ) ⩾ S_{F} (F, G) .$

(3) $S_{F} (F \land H, G \land ℋ) ⩾ S_{Fitsc} (F, G) .$

(4) $S_{F} (F \times F, G \times G) ⩾ S_{F} (F, G) .$

Definition 2.6. [10] A map $lim : F_{L}^{s} (X) \to L^{X}$ is called a stratified L-limit structure on X if it satisfies:

(LGC1) ∀x ∈ X, lim [x] (x) = ⊤ ;

(LGC2) $\forall F, G \in F_{L}^{s} (X), F ⩽ G$ implies $lim F ⩽ lim G;$

(LGLC) ∀ $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X) .$ , lim( $ℱ$ ∧ $G$ ) = lim $ℱ$ ∧ lim $G$

The pair (X, lim) is called a stratified L-limit space.

A map φ : (X, lim _X) $\to$ (Y, lim _Y) between stratified L-limit spaces is called continuous provided that $\lim_{X} F (x) \leq \lim_{Y φ} \Rightarrow (F) (φ (x))$ for each $F \in F_{L}^{s} (X)$ and x ∈ X. The category of stratified L-limit spaces and their continuous maps is denoted by SL-LC.

Definition 2.7. ([12]) A map Λ : $ℱ_{L}^{s} (X) . \to$ L is called a stratified L-uniform convergence structure on X if it satisfies:

(LUC1) Λ (dl× dl) = ⊤;

(LUC2) $ℱ$ ⩽ $G$ implies Λ ( $ℱ$ ) ⩽ Λ ( $G$ );

(LUC3) Λ ( $ℱ$ ) ⩽ Λ (^{$ℱ$
- 1});

(LUC4) Λ ( $ℱ$ ) ∧ Λ ( $G$ ) ⩽ Λ ( $ℱ$ ∧ $G$ );

(LUC5) Λ ( $ℱ$ ) ∧ Λ ( $G$ ) ⩽ Λ ( $ℱ$ ∘ $G$ ) whenever $ℱ$ ∘ $G$ exists.

The pair (X, Λ) is called a stratified L-uniform convergence space.

A map φ : (X, Λ_X) $\to$ (Y, Λ_Y) between stratified L-uniform convergence spaces is called uniformly continuous provided that Λ_X ( $ℱ$ ) ⩽ Λ_Y ((φ × φ) ^⇒ ( $ℱ$ )) for each $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ . The category of stratified L-uniform convergence spaces and uniformly continuous maps is denoted by SL-UC.

Definition 2.8. ([13]) A map γ : $ℱ_{L}^{s} (X) .$ $\to$ L is called a stratified L-Cauchy structure on X if it satisfies:

(LC1) γ (dl) =⊤;

(LC2) $ℱ$ ⩽ $G$ implies γ ( $ℱ$ ) ⩽ γ ( $G$ );

(LC3) If $ℱ$ ∨ $G$ exists, then γ ( $ℱ$ ∧ $G$ ) = γ ( $ℱ$ ) ∧ γ ( $G$ ).

The pair (X, γ) is called a stratified L-Cauchy space.

A map φ : (X, _γX) $\to$ (Y, _γY) between stratified L-Cauchy spaces is called Cauchy continuous provided that _γX ( $ℱ$ ) ⩽ _γY (φ^⇒ ( $ℱ$ )) for each $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ . The category of stratified L-Cauchy spaces and Cauchy continuous maps is denoted by SL-Chy.

Definition 2.9. ([20]) A stratified L-Cauchy structure γ on X is called a stratified L-ordered Cauchy structure if it satisfies:

(LOC) $S_{F} (ℱ, G) ⩽ γ (ℱ) \to γ (G)$ .

The pair (X, γ) is called a stratified L-ordered Cauchy space. Let SL-OChy denote the full subcategory of SL-Chy consisting of stratified L-ordered Cauchy spaces.

Proposition 2.10. ([20]) The category SL-OChy is a bireflective subcategory of SL-Chy.

3. Stratified L-ordered uniform convergence spaces

In this section, we will apply fuzzy inclusion order to stratified L-uniform convergence spaces and propose the notion of stratified L-ordered uniform convergence spaces. Then we will study its relationship with stratified L-uniform convergence spaces and its Cartesian-closedness.

Definition 3.1. A stratified L-uniform convergence structure Λ on X is called a stratified L-ordered uniform convergence structure if it satisfies:

(LOUC) $S_{F} (ℱ, G) ⩽ Λ (ℱ) \to Λ (G)$ .

The pair (X, Λ) is called a stratified L-ordered uniform convergence space. Let SL-OUC denote the full subcategory of SL-UC consisting of stratified L-ordered uniform convergence spaces.

We put E_{Λ
_X} = {Λ ∣ (X, Λ) is a stratified L-ordered uniform convergence space and Λ_X ( $ℱ$ ) ⩽ Λ ( $ℱ$ ) forall $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ } .

Example 3.2. Let Λ : $ℱ_{L}^{s} (X) .$ $\to$ L be a principle stratified L-uniform convergence structure in the sense of Jäger [12]. According to the proof that Λ satisfies (LOUC) in [4], we know that Λ is a stratified L-ordered uniform convergence structure on X.

We do not go into details here, but only remark that (LOUC) implies (LUC2). This means that each stratified L-ordered uniform convergence structure must be a stratified L-uniform convergence structure. But the converse is not true. The next example shows there exists a stratified L-uniform convergence structure which is not a stratified L-ordered uniform convergence structure.

Example 3.3. Let X = {x, y} and L be the chain L = {0, α, 1} such that 0 < α < 1. Suppose that Λ_d : $ℱ_{L}^{s} (X) .$ $\to$ L is the discrete stratified L-uniform convergence structure in the sense of Jäger [12]. That is, for each $ℱ$ ∈ $ℱ_{L}^{s} (X)$ , $\begin{matrix} Λ_{d} (ℱ) \\ = {\begin{matrix} 1, & if ℱ ⩾ \underset{x \in E}{⋀} [x] \times [x] for some finite E \subseteq X; \\ 0, & else . \end{matrix} \end{matrix}$ Next, we show that Λ_d is not a stratified L-ordered uniform convergence structure. For this, we introduce the stratified L-filter ^{$ℱ$
*} : L^X $\to$ L as follows: $\begin{matrix} \forall A \in L^{X}, F^{*} (A) = {\begin{matrix} 1, & if A = 1_{X}, \\ α, & if A (x) = 1, A (y) \neq 1, \\ α, & if A (x) = α, \\ 0, & if A (x) = 0 . \end{matrix} \end{matrix}$ This stratified L-filter can be found in [10].

In this case, define an L-subset A ∈ L^X×X by A (x, x) = A (y, y) =1, A (y, x) = A (x, y) = α. Using the definition of $F^{*}$ , we observe that $(F^{*} \times F^{*}) (A) = ⋁ {F^{*} (B) \land F^{*} (C) ∣ B \times C ⩽ A} ⩽ α .$ because B (x) ∧ C (y) ⩽ α or B (y) ∧ C (x) ⩽ α is true whenever B, C ∈ L^X fulfill B × C ⩽ A. However it holds that [x] × [x] (A) = [y] × [y] (A) =1, which implies that $F^{*} \times F^{*} [x] \times [x]$ , $F^{*} \times F^{*} [y] \times [y]$ and $F^{*} \times F^{*} ([x] \times [x]) \land ([y] \times [y])$ . Therefore, $Λ_{d} ([x] \times [x]) \to Λ_{d} (F^{*} \times F^{*}) = 1 \to 0 = 0 .$

On the other hand, for the point L-filter [x] of x, it holds that $\begin{matrix} S_{F} ([x], F^{*}) & = & \underset{C \in L^{X}}{⋀} ([x] (C) \to F^{*} (C)) \\ = & (1 \to 1) \land (1 \to α) \land (α \to α) \\ \land (0 \to 0) = α, \end{matrix}$ which can be checked by considering the nine different L-sets C ∈ L^X. Further we observe that $\begin{matrix} S_{F} ([x] \times [x], F^{*} \times F^{*}) \\ ⩾ S_{F} ([x], F^{*}) \land S_{F} ([x], F^{*}) = α \land α = α . \end{matrix}$ It follows that $\begin{matrix} Λ_{d} ([x] \times [x]) \to Λ_{d} (F^{*} \times F^{*}) \\ ≱ S_{F} ([x] \times [x], F^{*} \times F^{*}), \end{matrix}$ which means that the discrete structure Λ_d fails to satisfy the axiom (LOUC). Thus, Λ_d is not a stratified L-ordered uniform convergence structure.

Next we discuss the relationship between stratified L-uniform convergence structures and stratified L-ordered uniform convergence structures. Firstly, we give the following lemmas.

Lemma 3.4. Let (X, Λ_X) be a stratified L-uniform convergence space and define ${\bar{Λ}}_{X} : ℱ_{L}^{s} (X) . \to L$ by ${\bar{Λ}}_{X} (ℱ) = \underset{Λ \in E_{Λ_{X}}}{\land} Λ (ℱ) .$ Then ${\bar{Λ}}_{X}$ is a stratified L-ordered uniform convergence structure on X.

Proof. (LUC1)–(LUC5) are straightforward.

(LOUC) Take any $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X)$ . Then $\begin{matrix} {\bar{Λ}}_{X} (ℱ) \to {\bar{Λ}}_{X} (G) & = & \underset{Λ \in E_{Λ_{X}}}{\land} Λ (ℱ) \to \underset{Λ \in E_{Λ_{X}}}{\land} Λ (G) \\ \overset{(H 6)}{⩾} & \underset{Λ \in E_{Λ_{X}}}{\land} Λ (ℱ) \to Λ (G) \\ ⩾ & S_{F} (ℱ, G) . \end{matrix}$ Thus, ${\bar{Λ}}_{X}$ is a stratified L-ordered uniform convergence structure on X.□

Lemma 3.5. Let (Y, Λ_Y) be a stratified L-ordered uniform convergence space, φ : X $\to$ Y be a map and define $Λ_{X}^{*} : ℱ_{L}^{s} (X) . \to L$ as follows: $Λ_{X}^{*} (ℱ) = Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ)) .$ Then $Λ_{X}^{*}$ is a stratified L-ordered uniform convergence structure on X.

Proof. (LUC1)–(LUC5) are straightforward.

(LOUC) Take any $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X)$ . Then $\begin{matrix} Λ_{X}^{*} (ℱ) \to Λ_{X}^{*} (ℱ) \\ = Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ)) \to Λ_{Y} ((φ \times φ)^{\Rightarrow} (G)) \\ ⩾ S_{F} ((φ \times φ)^{\Rightarrow} (ℱ)), (φ \times φ)^{\Rightarrow} (G))) \\ ⩾ S_{F} (ℱ, G) (by Lemma 2) . \end{matrix}$ This proves that $Λ_{X}^{*}$ is a stratified L-ordered uniform convergence structure on X.

Proposition 3.6. The category SL-OUC is a bireflective subcategory of SL-UC.

L(e) (X, Λ_X) be a stratified L-uniform convergence space and define ${\bar{Λ}}_{X} : ℱ_{L}^{s} (X) . \to L$ as follows: $\forall ℱ \in ℱ_{L}^{s} (X) ., {\bar{Λ}}_{X} (ℱ) = \underset{Λ \in E_{Λ_{X}}}{\land} Λ (ℱ) .$ By Lemma 3, we know $(X, {\bar{Λ}}_{X})$ is a stratified L-ordered uniform convergence space. Further, we claim that ${id}_{X} : (X, Λ_{X}) \to (X, {\bar{Λ}}_{X})$ is the SL-OUC-bireflector.

For this it suffices to prove:

${id}_{X} : (X, Λ_{X}) \to (X, {\bar{Λ}}_{X})$ is uniformly continuous.

For each stratified L-ordered uniform convergence space (Y, Λ_Y) and each map φ : X → Y, the uniformly continuity of φ : (X, Λ_X) → (Y, Λ_Y) implies the uniformly continuity of $φ : (X, {\bar{Λ}}_{X}) \to (Y, Λ_{Y})$ .

For (1), by Lemma 3, we know $Λ_{X} (ℱ) ⩽ {\bar{Λ}}_{X} (ℱ)$ for all $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ . Hence, the uniformly continuity of ${id}_{X} : (X, Λ_{X}) \to (X, {\bar{Λ}}_{X})$ is proved.

For (2), we need only show that ${\bar{Λ}}_{X} (ℱ) ⩽ Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ))$ for all $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ . By Lemma 3, we know $(X, Λ_{X}^{*})$ is a stratified L-ordered uniform convergence space. Since φ : (X, Λ_X) → (Y, Λ_Y) is uniformly continuous, it follows that $Λ_{X} (ℱ) ⩽ Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ) = Λ_{X}^{*} (ℱ)$ . Thus, ${id}_{X} : (X, Λ_{X}) \to (X, Λ_{X}^{*})$ is uniformly continuous. By the definition of E_{Λ
_X}, we obtain $Λ_{X}^{*} \in E_{Λ_{X}}$ . Therefore, ${\bar{Λ}}_{X} (ℱ) ⩽ Λ_{X}^{*} (ℱ) = Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ))$ , as desired. □

In order to show the Cartesian-closedness of SL-OUC, we first give the following lemma.

Lemma 3.7. The category SL-OUC is closed under formations of power objects in SL-UC.

Proof. Let (X, Λ_X) and (Y, Λ_Y) be stratified L-ordered uniform convergence spaces and (UC, Λ_UC) be the corresponding power space in SL-UC (see [12]), i.e., for each $Φ \in F_{L}^{s} (UC \times UC),$ $\begin{matrix} Λ_{UC} (Φ) & = & \underset{ℱ \in ℱ_{L}^{s} (X) .}{⋀} (Λ_{X} (ℱ) \to Λ_{Y} ((ev \times ev)^{\Rightarrow} \\ (η^{\Rightarrow} (Φ \times ℱ)))) . \end{matrix}$ Here, UC = UC (X, Y) = {φ : (X, Λ_X) $\to$ (Y, Λ_Y) ∣ φ isuniformlycontinuous}, ev : UC (X, Y) × X $\to$ Y : (φ, x) ↦ φ (x) is the evaluation map and $\begin{matrix} η : {\begin{matrix} (UC \times UC) \times (X \times X) & \to & (UC \times X) \times (UC \times X) \\ ((φ, ψ), (x, y)) & \mapsto & ((φ, x), (ψ, y)) . \end{matrix} \end{matrix}$

In order to show Λ_UC is a stratified L-ordered uniform convergence structure on UC (X, Y), we need only verify that Λ_UC satisfies (LOUC). Take any $Φ_{1}, Φ_{2} \in ℱ_{L}^{s} (UC \times UC)$ . Then $\begin{matrix} Λ_{UC} (Φ_{1}) \to Λ_{UC} (Φ_{2}) \\ = & \underset{ℱ \in ℱ_{L}^{s} (X) .}{⋀} (Λ_{X} (ℱ) \to Λ_{Y} ((ev \times ev)^{\Rightarrow} (η^{\Rightarrow} (Φ_{1} \times ℱ)))) \\ \to \underset{G \in ℱ_{L}^{s} (X) .}{⋀} (Λ_{X} (G) \to Λ_{Y} ((ev \times ev)^{\Rightarrow} (η^{\Rightarrow} (Φ_{2} \times G)))) \\ \overset{(H 6)}{⩾} & \underset{ℱ \in ℱ_{L}^{s} (X) .}{⋀} Λ_{Y} ((ev \times ev)^{\Rightarrow} (η^{\Rightarrow} (Φ_{1} \times ℱ))) \\ \to Λ_{Y} ((ev \times ev)^{\Rightarrow} (η^{\Rightarrow} (Φ_{2} \times ℱ))) \\ ⩾ & \underset{ℱ \in ℱ_{L}^{s} (X) .}{⋀} S_{F} ((ev \times ev)^{\Rightarrow} (η^{\Rightarrow} (Φ_{1} \times ℱ))), (ev \times ev)^{\Rightarrow} \\ (η^{\Rightarrow} (Φ_{2} \times ℱ)))) \\ ⩾ & \underset{ℱ \in ℱ_{L}^{s} (X) .}{⋀} S_{F} (Φ_{1} \times ℱ, Φ_{2} \times ℱ) \\ ⩾ & S_{F} (Φ_{1}, Φ_{2}) . (by Lemma 2) \end{matrix}$ This proves that Λ_UC is a stratified L-ordered uniform convergence structure on UC (X, Y).

Proposition 3.8. ([12]) The category SL-UC is a Cartesian closed topological category.

By Propositions 3,3 and Lemma 3, we obtain

Proposition 3.9. The category SL-OUC is a Cartesian closed topological category.

Remark 3.10. In [31], Wang and Fang had claimed that the category SL-OUC is Cartesian closed, but didn’t give the concrete proof. Here, we presented the details to show the Cartesian-closedness of SL-OUC by using the relationship between SL-OUC and SL-UC.

4. Stratified L-ordered limit spaces

In this section, we will define stratified L-ordered limit spaces by using fuzzy inclusion order betweenL-subsets. Then we will study its Cartesian-closedness and establish its categorical relationship with stratified L-limit spaces and stratified L-ordered Cauchy spaces.

Definition 4.1. A stratified L-limit structure lim on X is called a stratified L-ordered limit structure if it satisfies:

(LGOC) $S_{F} (F, G) ⩽ S (lim F, lim G) .$

The pair (X, lim) is called a stratified L-ordered limit space. Let SL-OLC denote the full subcategory of SL-LC consisting of all stratified L-ordered limit spaces.

Example 4.2. ([9]) Let X be a nonempty set. If a map lim : $ℱ_{L}^{s} (X) . \to$ L^X is a stratified L-principle generalized convergence structure on X, in the sense that lim satisfies (LGC1)–(LGC2) and for each $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ , x ∈ X, $(Lp) lim ℱ (x) = S_{F} (U^{x}, F),$ where $U^{X} : L^{x} \to L$ is defined by $U^{X} (A) = \underset{G \in ℱ_{L}^{s} (X) .}{⋀} (lim G (x) \to G (A)),$ then (X, lim) is a stratified L-ordered limit space.

Since (LGOC) implies (LGC2), each stratified L-ordered limit structure must be a stratified L-limit strutcture. But the converse is not true. In order to show this, we present the following example.

Example 4.3. ([2]) Let X = {x, y} and L = {0, α, 1} such that 0 < α < 1. We define the stratified L-limit structure lim on X by $\begin{matrix} \forall ℱ \in ℱ_{L}^{s} (X) ., \forall z \in X, lim ℱ (z) = {\begin{matrix} 1, & if ℱ ⩾ [z], \\ 0, & otherwise . \end{matrix} \end{matrix}$

In [2], Example 4.9 shows this lim is a stratified L-generalized convergence structure on X and fails to fulfil (LOGC). In fact, this lim is a stratified L-limit structure on X. Therefore, we point out that this stratified L-limit structure lim is not a stratified L-ordered limit structure.

Adopting the similar proofs of Lemmas 3, 3 and Proposition 3, we can obtain

Proposition 4.4. The category SL-OLC is a bireflective subcategory of SL-LC.

In order to show the Cartesian-closedness of SL-OLC, we give the following lemma.

Lemma 4.5. The category SL-OLC is closed under formations of power objects in SL-LC.

Proof. Let (X, lim _X) and (Y, lim _Y) be stratified L-ordered limit spaces and ([X $\to$ Y] , lim _{[X
$\to$
Y]}) be the corresponding power space in SL-LC, i.e., $\begin{matrix} {lim}_{[X \to Y]} ℱ (φ) & = & \underset{(H, x) \in ℱ_{L}^{s} (X) . \times X}{⋀} ({lim}_{X} H (x) \\ \to {lim}_{Y} {ev}^{\Rightarrow} (F \times H) (φ (x))) . \end{matrix}$ Here, [X $\to$ Y] = {φ : (X, lim _X) $\to$ (Y, lim _Y) ∣ φ iscontinuous} and ev : [X→Y] × X→Y : (φ, x) ↦ φ (x) is the evaluation map.

In order to show lim _[X→Y] is a stratified L-ordered limit structure on [X→Y], we need only verify that lim _[X→Y] satisfies (LGOC). Take any $ℱ_{1}, ℱ_{2} \in ℱ_{L}^{s} ([X \to Y])$ . Then $\begin{matrix} S ({lim}_{[X \to Y]} ℱ_{1}, {lim}_{[X \to Y]} ℱ_{2}) \\ = & \underset{φ \in [X \to Y]}{\land} {lim}_{[X \to Y]} ℱ_{1} (φ) \to {lim}_{[X \to Y]} ℱ_{2} (φ) \\ = & \underset{φ \in [X \to Y]}{\land} (\underset{(H, x) \in ℱ_{L}^{s} (X) . \times X}{⋀} ({lim}_{X} H (x) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{1} \times H) (φ (x))) \\ \to \underset{(G, y) \in ℱ_{L}^{s} (X) . \times X}{⋀} ({lim}_{X} G (y) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{2} \times G) (φ (y)))) \\ \overset{(H 6)}{⩾} & \underset{φ \in [X \to Y]}{\land} \underset{(H, x) \in ℱ_{L}^{s} (X) . \times X}{⋀} (({lim}_{X} H (x) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{1} \times H) (φ (x))) \\ \to ({lim}_{X} H (x) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{2} \times H) (φ (x)))) \end{matrix}$ $\begin{matrix} (H 4) ⩾ & \underset{φ \in [X \to Y]}{\land} \underset{(H, x) \in ℱ_{L}^{s} (X) . \times X}{⋀} ({lim}_{Y} {ev}^{\Rightarrow} (F_{1} \times H) (φ (x)) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{2} \times H) (φ (x))) \\ ⩾ & \underset{ℋ \in ℱ_{L}^{s} (X) .}{\land} \underset{y \in Y}{\land} ({lim}_{Y} {ev}^{\Rightarrow} (F_{1} \times H) (y) \to {lim}_{Y} {ev}^{\Rightarrow} (F_{2} \times H) (y)) \\ = & \underset{ℋ \in ℱ_{L}^{s} (X) .}{\land} S ({lim}_{Y} {ev}^{\Rightarrow} (F_{1} \times H), {lim}_{Y} {ev}^{\Rightarrow} (F_{2} \times H)) \\ ⩾ & \underset{ℋ \in ℱ_{L}^{s} (X) .}{\land} S_{F} ({ev}^{\Rightarrow} (F_{1} \times H), {ev}^{\Rightarrow} (F_{2} \times H)) \\ ⩾ & \underset{H \in ℱ_{L}^{s} (X) .}{⋀} S_{F} (F_{1} \times H, F_{2} \times H) (by Lemma 5) \\ ⩾ & S_{F} (F_{1}, F_{2}) . (by Lemma 2) \end{matrix}$ Hence, lim _[X→Y] is a stratified L-ordered limit structure on [X→Y].

Proposition 4.6. [[12]]The category SL-LC is a Cartesian closed topological category.

By Propositions 4, 4 and Lemma 4, we obtain

Proposition 4.7. The category SL-OLC is a Cartesian closed topological category.

Next we discuss the relationship between stratified L-ordered limit spaces and stratified L-ordered Cauchy spaces.

Lemma 4.8. Let (X, γ) be a stratified L-ordered Cauchy space and define by $\forall ℱ \in ℱ_{L}^{s} (X), \forall x \in X, \lim_{γ} ℱ (x) = γ ([x] \land ℱ) .$ Then lim_γ is a stratified L-ordered limit structure on X.

Proof. (LGC1) Obviously.

(LGCL) Take any $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X) .$ , x ∈ X. Since [x] ∧ $ℱ$ ⩽ [x] and [x] ∧ $G$ ⩽ [x], by Lemma 2, we know that (dl ∧ $ℱ$ ) ∨ (dl ∧ $G$ ) exists. Then $\begin{matrix} {lim}_{γ} (ℱ \land G) (x) & = & γ ((dl \land ℱ) \land (dl \land G)) \\ ⩾ & γ (dl \land ℱ) \land γ (dl \land G) \\ = & {lim}_{γ} ℱ (x) \land {lim}_{γ} G (x) . \end{matrix}$

(LGOC) Take any $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X) .$ . Then $\begin{matrix} S ({lim}_{γ} ℱ, {lim}_{γ} G) & = & \land_{x \in X} {lim}_{γ} ℱ (x) \to {lim}_{γ} G (x) \\ = & \land_{x \in X} γ (dl \land ℱ) \to γ (dl \land G) \end{matrix}$ $\begin{matrix} ⩾ & \land_{x \in X} S_{F} (dl \land ℱ, dl \land G) \\ ⩾ & S_{F} (ℱ, G) . (by Lemma 2) \end{matrix}$ This shows $lim_{γ}$ is a stratified L-ordered limit structure on X.□

Lemma 4.9. If φ : (X, _γX) → (Y, _γY) is Cauchy continuous with respect to stratified L-ordered Cauchy structures _γX and _γY, then $φ : (X, lim_{γ_{X}}) \to (Y, lim_{γ_{Y}})$ is continuous with respect to stratified L-ordered limit structures $lim_{γ_{X}}$ and $lim_{γ_{Y}}$ .

Proof. Since φ : (X, _γX) → (Y, _γY) is Cauchy continuous, we have $\forall ℱ \in ℱ_{L}^{s} (X) ., γ_{X} (ℱ) ⩽ γ_{Y} (φ^{\Rightarrow} (ℱ)) .$ Then for any x ∈ X, it follows that $\begin{matrix} {lim}_{γ_{X}} ℱ (x) & = & γ_{X} (dl \land ℱ) ⩽ γ_{Y} ([φ (x)] \land φ^{\Rightarrow} (ℱ)) \\ = & {lim}_{γ_{Y}} φ^{\Rightarrow} (ℱ) (φ (x)) . \end{matrix}$ Therefore, $φ : (X, lim_{γ_{X}}) \to (Y, lim_{γ_{Y}})$ is continuous.□

From Lemmas 4 and 4, we obtain a concrete functor $ℍ$ as follows:

$\begin{matrix} ℍ : {\begin{matrix} SL - OChy & \to & SL - OLC, \\ (X, γ) & \mapsto & (X, lim_{γ}), \\ φ & \mapsto & φ . \end{matrix} \end{matrix}$

5. Relations among SL-OUC, SL-OLC and SL-OChy

In this section, we will focus on the relationship among SL-OUC, SL-OLC and SL-OChy. Since some proofs parallel to those in [13], we only present some necessary proofs. In this section, we will restrict to the case that L is a complete Boolean algebra. Then every stratified L-filter is tight, i.e., $F (\underline{a}) = a$ for all a ∈ L (see e.g. [11]). The reason why we demand the tightness of all stratified L-filters is the following result.

Lemma 5.1. ([13]) Let $ℱ$ , $G$ , $ℋ$ , $K$ ∈ $ℱ_{L}^{s} (X) .$ . If $ℱ$ ∨ $ℋ$ ∈ $ℱ_{L}^{s} (X) .$ is tight, then ( $ℱ$ × $G$ ) ∘ ( $ℋ$ × $K$ ) = $ℱ$ × $K$ .

Lemma 5.2. Let (X, Λ) be a stratified L-ordered uniform convergence space and define _γΛ : $ℱ_{L}^{s} (X) .$ →L as follows: $\forall ℱ \in ℱ_{L}^{s} (X) ., γ_{Λ} (ℱ) = Λ (ℱ \times ℱ) .$ Then γ_⋀ is a stratified L-ordered Cauchy structure on X.

Proof. The verifications of (LC1) and (LC3) can be found in [12] (Lemma 5.2), where Lemma 5.1 is necessary.

(LOC) Take any $ℱ$ , $γ$ ∈ $ℱ_{L}^{s} (X) .$ . Then $\begin{matrix} γ_{Λ} (ℱ) \to γ_{Λ} (G) & = & Λ (ℱ \times ℱ) \to Λ (G \times G) \\ ⩾ & S_{F} (ℱ \times ℱ, G \times G) \\ ⩾ & S_{F} (ℱ, G) . (by Lemma 2) \end{matrix}$ This means _γΛ is a stratified L-ordered Cauchy structure on X.□

Lemma 5.3. If φ : (X, Λ_X) → (Y, Λ_Y) is uniformly continuous with respect to stratified L-ordered uniform convergence structures Λ_X and Λ_Y, then φ : (X, _{γΛ_X}) → (Y, _{γΛ_Y}) is Cauchy continuous with respect to stratified L-ordered Cauchy structures _{γΛ_X} and _{γΛ_Y}.

Proof. Take any $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ . Then $\begin{matrix} γ_{Λ_{X}} (ℱ) & = & Λ_{X} (ℱ \times ℱ) \\ ⩽ & Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ \times ℱ)) \\ = & Λ_{Y} (φ^{\Rightarrow} (ℱ) \times φ^{\Rightarrow} (ℱ)) \\ = & γ_{Λ_{Y}} (φ^{\Rightarrow} (ℱ)) . \end{matrix}$ Hence, φ : (X, _{γΛ_X}) → (Y, _{γΛ_Y}) is Cauchy continuous.□

From Lemmas 5.2 and 5.3, we obtain a concrete functor $𝕂$ as follows:

$\begin{matrix} 𝕂 : {\begin{matrix} SL - OUC & \to & SL - OChy, \\ (X, Λ) & \mapsto & (X, γ_{Λ}), \\ φ & \mapsto & φ . \end{matrix} \end{matrix}$

Lemma 5.4. Let (X, Λ) be a stratified L-ordered uniform convergence space and define lim _Λ : $ℱ_{L}^{s} (X) .$ →L^X as follows: $\forall ℱ \in ℱ_{L}^{s} (x), x \in X, \lim_{Λ} ℱ (x) = Λ (ℱ \times [x])$ Then lim _Λ is a stratified L-ordered limit structure on X.

Proof. The verifications of (LGC1) and (LGC3) can be found in [12] (Lemma 6.1).

(LGOC) Take any $ℱ$ , $G$ ∈ $ℱ_{L}^{s} (X) .$ . Then $\begin{matrix} S ({lim}_{Λ} ℱ, {lim}_{Λ} G) & = & ⋀_{x \in X} Λ (ℱ \times [x]) \to Λ (G \times [x]) \\ ⩾ & ⋀_{x \in X} S_{F} (ℱ \times [x], G \times [x]) \\ ⩾ & S_{F} (ℱ, G) . (by Lemma 2) \end{matrix}$ Hence, lim _Λ is a stratified L-ordered limit structure on X.□

Lemma 5.5. If φ : (X, Λ_X) → (Y, Λ_Y) is uniformly continuous with respect to stratified L-ordered uniform convergence structures Λ_X and Λ_Y, then $φ : (X, lim_{Λ_{X}}) \to (Y, lim_{Λ_{Y}})$ is continuous with respect to stratified L-ordered limit structures $lim_{Λ_{X}}$ and $lim_{Λ_{Y}}$ .

Proof. Take any $ℱ$ ∈ $ℱ_{L}^{s} (X) .$ and x ∈ X. Then $\begin{matrix} {lim}_{Λ_{X}} ℱ (x) & = & Λ_{X} (ℱ \times [x]) \\ ⩽ & Λ_{Y} ((φ \times φ)^{\Rightarrow} (ℱ \times [x])) \\ = & Λ_{Y} (φ^{\Rightarrow} (ℱ) \times [φ (x)]) \\ = & {lim}_{Λ_{Y}} φ^{\Rightarrow} (ℱ) (φ (x)) . \end{matrix}$ Hence, $φ : (X, lim_{Λ_{X}}) \to (Y, lim_{Λ_{Y}})$ is continuous.□

From Lemmas 5.4 and 5.5, we obtain a concrete functor $𝔾$ as follows: $\begin{matrix} 𝔾 : {\begin{matrix} SL - OUC & \to & SL - OLC, \\ (X, Λ) & \mapsto & (X, lim_{Λ}), \\ φ & \mapsto & φ . \end{matrix} \end{matrix}$

Proposition 5.6. Let (X, Λ) be a stratified L-ordered uniform convergence space. Then $lim_{Λ} = lim_{γ_{Λ}}$ .

Proof. The proof parallel to Lemma 5.5 in [13] and we omit it.□

Combining the results in [13], we obtain the following diagram.

6. Conclusions

In comparison with the classical inclusion order between L-subsets, fuzzy inclusion order between L-subsets possesses more logical flavor. Inspired by this, we make full use of fuzzy inclusion order between L-subsets to stratified L-uniform convergence structures and stratified L-limit structures, and obtained so called stratified L-ordered uniform convergence structures and stratified L-ordered limit structures. Besides, considering stratified L-ordered Cauchy structures [20], which was proposed by applying fuzzy inclusion order between L-subsets to stratified L-Cauchy structures, we established the relationship among stratified L-ordered uniform convergence structures, stratified L-ordered limit structures and stratified L-ordered Cauchy structures. Moreover, combining the results in [13], we provided a common framework of the relationship among difference kinds of fuzzy spatial structures.

With the fuzzy inclusion order between L-subsets, several new types of fuzzy spatial structures were proposed. In the future, we will explore their spatial properties, such as the completion of stratified L-ordered uniform convergence spaces, categorical properties of stratified L-limit spaces and so on. Also, we can consider how to apply fuzzy inclusion order between L-subsets to new fuzzy spatial structures and obtain new types of “fuzzy-ordered” spatial structures.

Footnotes

Acknowledgments

The authors would like to thank the referees for their helpful comments. This work is supported by the Project of Shandong Province Higher Educational Science and Technology Program (NO. J18KA245).

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