Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Abstract

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

1. Introduction

In [1], Goguen replaced the structure of membership values $[0,1]$ in Zadeh fuzzy sets [2] with an arbitrary lattice and introduced the concept of an L-fuzzy set that allows us to study various lattice-valued structures under the auspices of the general approach to fuzziness. Afterwards, a complete residuated lattice, introduced by Ward and Dilworth [3], became an important source of membership values of fuzzy sets. In [4], Höhle& Šostak used quantales [5,6] and $MV$-algebras [7] as other algebraic structures for L-fuzzy topologies. The latter extend an ordinary topology, Chang fuzzy topology [8], Lowen fuzzy topology [9], and the Šostak fuzzy topological structure [10].

In [11,12], rough sets were introduced by Pawlak, and they soon became a popular tool used to analyze various intelligent systems based on imprecise, insufficient, incomplete, and vague information. As a result, the corresponding direction was remarkably developed due to its predominant use. After that, the equivalence relation in the theory of rough sets was replaced by an arbitrary relation (cf., [13,14,15]) to handle a wider scope of uncertainty. Yao and Lin [16] showed that, under certain conditions, the rough upper and lower approximations of a set are nothing more than a closure and interior of it. Consequently, they proposed several models of rough sets.

Šostak in [17] proposed the concept of a fuzzy category, in which, unlike the usual category, objects and morphisms are given up to a degree, and the latter is an element of the corresponding lattice.

A graded approach to topology and many related structures (L-topology, L-fuzzifying topology, and L-fuzzy topology) is an essential characteristic of the spaces where sets are identified with their characteristic or lattice-valued membership functions. Even the mappings among these spaces are endowed with degrees. As for examples, graded continuous mappings between L-fuzzifying topological spaces were proposed and studied by Pang in [18], or a graded approach to study mappings between M-fuzzifying convex spaces was proposed by Xiu and Pang in [19].

Our main motivation was to propose and study an approach that was different from the axiomatic way of introducing a fuzzy pretopology and associated spaces of (lattice) fuzzy-valued functions. In particular, we focused on better understanding the concepts of closing and opening. Recall the paper by Fang and Yue [20], in which the relationship between L-fuzzy closure systems and L-fuzzy topological spaces was discussed from a categorical viewpoint.

For a long time, closing and opening were closely related to the corresponding Kuratowski operators. In a fuzzy scientific literature, this correspondence has lead to a thorough study of approximation spaces with the so-called preorder fuzzy relations. With the development of a weaker concept of fuzzy pretopology, the focus was changed to topological structures where the closing and opening operators are not idempotent. As far as we know, the first publication in which non-idempotent closure and opening were identified with the Čech operators appeared in [21].

Recall that in [22], it was shown by Čech that the category of co-topological spaces can be embedded into the category of (now known as) Čech closure spaces as a reflective subcategory. After this groundbreaking step, there were a lot of discussions about these types of spaces, see, for example, [23,24,25,26,27,28,29,30]. In fuzzy-valued topological structures, Qiao [31] showed that a Galois connection exists between the category of stratified L-Čech closure spaces and the category of reflexive L-fuzzy relation sets.

Our contribution discusses these weaker structures from the categorical point of view and establishes the relationships between them. The narrow goal of this contribution was to establish the relationship between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces, and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We aimed to show that there is a Galois connection between the corresponding categories.

The structure of this paper is as follows. In Section 2, we recall background algebraic concepts and some useful properties of a complete lattice L, where finite meets distribute over arbitrary joins. In Section 3, we analyze the relationship between L-fuzzy Čech closure spaces and L-fuzzy co-topological spaces from the categorical viewpoint. In Section 4, the relationship between L-fuzzy approximation spaces and L-fuzzy co-topological spaces is studied. In the last section, we formulate conclusions.

2. Preliminaries

In this section, we recall some useful concepts and their properties that will be needed in the sequel.

We consider a complete lattice L where finite meets distribute over arbitrary joins, i.e., $x\wedge \left({\bigvee }_{i\in \mathsf{\Gamma }}{y}_i\right)={\bigvee }_{i\in \mathsf{\Gamma }}(x\wedge y_i)$ for each $x,y_i\in L$$(i\in \mathsf{\Gamma })$. This lattice is called a frame. The bottom (resp. top) element of L is denoted by ⊥(resp. ⊤). We can then define a residual implication by

$$\begin{array}{c}x\to y=\bigvee \{z\in L\mid x\wedge z\le y\}.\end{array}$$

The following properties of the residual implication are used.

Lemma 1

([4,32,33,34]). For each $x,y,z,w,x_i,y_i\in L$, the following properties hold. (1) If $y\le z$, then $x\to y\le x\to z$ and $z\to x\le y\to x$, (2) $(x\to y)\to y\ge x$, and $x\to y=\top ⟺x\le y$, (3) $x\to \top =\top$ and $\top \to x=x$, (4) $x\to \left({\bigwedge }_{i\in \mathsf{\Gamma }}{y}_i\right)={\bigwedge }_{i\in \mathsf{\Gamma }}(x\to y_i)$ and $\left({\bigvee }_{i\in \mathsf{\Gamma }}{x}_i\right)\to y={\bigwedge }_{i\in \mathsf{\Gamma }}(x_i\to y)$, (5) ${\bigvee }_{i\in \mathsf{\Gamma }}{x}_i\to {\bigvee }_{i\in \mathsf{\Gamma }}{y}_i\ge {\bigwedge }_{i\in \mathsf{\Gamma }}(x_i\to y_i)$ and ${\bigwedge }_{i\in \mathsf{\Gamma }}{x}_i\to {\bigwedge }_{i\in \mathsf{\Gamma }}{y}_i\ge {\bigwedge }_{i\in \mathsf{\Gamma }}(x_i\to y_i)$, (6) $(x\to y)\wedge x\le y$ and $(x\to y)\wedge (y\to z)\le (x\to z)$, (7) $x\to y\le (y\to z)\to (x\to z)$ and $x\to y\le (z\to x)\to (z\to y)$, (8) $(x\wedge y)\to z=x\to (y\to z)=y\to (x\to z)$, (9) $y\to z\le x\wedge y\to x\wedge z$ and $(x\to z)\wedge (y\to w)\le x\wedge y\to z\wedge w$, (10) $(x\to y)\vee (z\to w)\le (x\wedge z)\to (y\vee w)$, (11) $(x\to y)\wedge (z\to w)\le (x\vee z)\to (y\vee w)$.

$L^X$ refers to the set of all L-valued subsets (fuzzy sets) of X, so that the corresponding lattice $L^X$ is completely distributive because it inherits the structure of the lattice L. By the pointwise definition of $\bigvee ,\bigwedge$ and a complement ^*, where $a^{*}=a\to \perp$, we can introduce on $L^X$ the operation $(\lambda \to \mu )$ as $(\lambda \to \mu )\left(x\right)=\lambda \left(x\right)\to \mu \left(x\right)$. Moreover, the L-subsets ${\top }_X$ and ${\perp }_X$ defined by ${\top }_X\left(x\right)=\top$ and ${\perp }_X\left(x\right)=\perp ,\forall x\in X$, are the universal upper and lower bounds in $L^X$, respectively. We do not distinguish between an element $\alpha \in L$ and the constant L-valued set $\alpha \in L^X$, such that $\alpha \left(x\right)=\alpha$. The characteristic function of any subset A of X is denoted by ${\top }_A$. For any subset A of X and $\alpha \in L$, we denote the fuzzy set $\alpha \wedge {\top }_A$ by $\alpha \wedge A$.

Definition 1

([32,33]). Let X be a set. An L-fuzzy relation on a set X is a map $R:X\times X\to L$. Then, R is said to be

(E1) reflexive if $R(x,x)=\top$ for all $x\in X$,

(E2) transitive if $R(x,y)\wedge R(y,z)\le R(x,z)$ for all $x,y,z\in X.$

A L-fuzzy relation R is called an L-fuzzy preorder if it is reflexive and transitive.

Lemma 2

([20,32]). For a given set X, define a binary mapping $S:L^X\times L^X\to L$ by

$$\begin{array}{c}S(\lambda ,\mu )={\bigwedge }_{x\in X}(\lambda \left(x\right)\to \mu \left(x\right)).\end{array}$$

Then, for each $\lambda ,\mu ,\rho ,\nu \in L^X$, the following properties hold:

(1) $\lambda \le \mu ⟺S(\lambda ,\mu )\ge \top$,

(2) if $\lambda \le \mu$, then $S(\rho ,\lambda )\le S(\rho ,\mu )$ and $S(\lambda ,\rho )\ge S(\mu ,\rho )$,

(3) $S(\lambda ,\mu )\wedge S(\nu ,\rho )\le S(\lambda \wedge \nu ,\mu \wedge \rho )$ and $S(\lambda ,\mu )\vee S(\nu ,\rho )\le S(\lambda \vee \nu ,\mu \vee \rho )$,

(4) $S(\mu ,\rho )\le S(\lambda ,\mu )\to S(\lambda ,\rho )$,

(5) $S(\lambda ,\rho )\le S(\lambda ,\mu )\wedge S(\mu ,\rho )$,

(6) if $\phi :X\to Y$ is a map, then $S(\lambda ,\mu )\le S({\phi }^{\leftarrow }\left(\lambda \right),{\phi }^{\leftarrow }\left(\mu \right))$.

Theorem 1

([35,36]). Suppose that $F:\mathcal{D}\to \mathcal{C},G:\mathcal{C}\to \mathcal{D}$ are concrete functors. Then, the following conditions are equivalent:

(1) $\{i{d}_Y:F\circ G\left(Y\right)\to Y\mid Y\in \mathcal{C}\}$ is a natural transformation from the functor $F\circ G$ to the identity functor on $\mathcal{C},$ and $\{i{d}_X:X\to G\circ F\left(X\right)\mid X\in \mathcal{D}\}$ is a natural transformation from the identity functor $\mathcal{D}$ to the functor $G\circ F$,

(2) For each $Y\in \mathcal{C},$$i{d}_Y:F\circ G\left(Y\right)\to Y$ is a $\mathcal{C}$-morphism, and for each $X\in \mathcal{D}$, $i{d}_X:X\to G\circ F\left(X\right)$ is a $\mathcal{D}$-morphism.

In this case, $(F,G)$ is called a Galois connection between $\mathcal{C}$ and $\mathcal{D}$. If $(F,G)$ is a Galois connection, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F.

Definition 2

([13,37,38]). Let R be a binary L-fuzzy relation on X, the ordered pair $(X,R)$ is called an L-fuzzy approximation space based on the L-fuzzy relation R. A mapping $\overline{R}:L^X\to L^X$ defined as $\overline{R}\left(\lambda \right)\left(x\right)={\bigvee }_{y\in X}R(x,y)\wedge \lambda \left(y\right)$, for any $\lambda \in L^X$ and $x\in X$ is called an L-fuzzy upper approximation operator on X.

For readers’ convenience, we list some of the main properties of the upper approximation operator in the following sentence, see e.g., [13,21].

Proposition 1.

(A) 

Let $(X,R)$ be an L-fuzzy approximation space and $\overline{R}:L^X\to L^X$, L-fuzzy upper approximation operator on X. Then, for all $\lambda ,{\lambda }_j\in L^X$, the following properties hold:

(1) $\overline{R}\left({\top }_X\right)={\top }_X$ and $\overline{R}\left({\perp }_X\right)={\perp }_X$ if R is reflexive,

(2) $\lambda \le \overline{R}\left(\lambda \right)$ if R is reflexive and $\overline{R}\left(\alpha \right)\le \alpha$,

(3) $S(\lambda ,\mu )\le S(\overline{R}\left(\lambda \right),\overline{R}\left(\mu \right))$,

(4) $\overline{R}(\lambda \vee \mu )\le \overline{R}\left(\lambda \right)\vee \overline{R}\left(\mu \right)$,

(5) $\overline{R}\left({\bigvee }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)={\bigvee }_{i\in \mathsf{\Gamma }}\overline{R}\left({\lambda }_i\right)$,

(6) $\overline{R}(\alpha \wedge \lambda )=\alpha \wedge \overline{R}\left(\lambda \right),$ for all $\lambda \in L^X$ and $\alpha \in L$.

(B) 

If $\mathit{2}=\{0,1\}$, then $\overline{R}:{\mathit{2}}^X\to L^X$ defined as $\overline{R}\left(A\right)\left(x\right)={\bigvee }_{y\in A}R(x,y)$ is an L-fuzzifying upper approximation operator on X and $(X,R)$ is called an L-fuzzifying approximation space [39]. If R is reflexive, then for all $A,B,A_i\in 2^X$ we have

(1) $\overline{R}\left(X\right)={\top }_X$ and $\overline{R}\left(\varphi \right)={\perp }_X$,

(2) ${\top }_A\le \overline{R}\left(A\right)$,

(3) If $A\subseteq B,$ then $\overline{R}\left(A\right)\le \overline{R}\left(B\right)$,

(4) $\overline{R}(A\cup B)\le \overline{R}\left(A\right)\vee \overline{R}\left(B\right)$,

(5) $\overline{R}\left({\bigcup }_{i\in \mathsf{\Gamma }}{A}_i\right)={\bigvee }_{i\in \mathsf{\Gamma }}\overline{R}\left(A_i\right)$.

Definition 3

([31,34,40]). A mapping $\mathcal{F}:L^X\to L$ is called an L-fuzzy co-topology on X if it satisfies the following conditions:

(LF1) $\mathcal{F}\left({\top }_X\right)=\mathcal{F}\left({\perp }_X\right)=\top$,

(LF2) $\mathcal{F}(\lambda \vee \mu )\ge \mathcal{F}\left(\lambda \right)\wedge \mathcal{F}\left(\mu \right)$ for all $\lambda ,\mu \in L^X$,

(LF3) $\mathcal{F}\left({\bigwedge }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)\ge {\bigwedge }_{i\in \mathsf{\Gamma }}\mathcal{F}\left({\lambda }_i\right)\forall {\left\{{\lambda }_i\right\}}_{i\in \mathsf{\Gamma }}\subseteq L^X$.

The pair $(X,\mathcal{F})$ is called an L-fuzzy co-topological space. An L-fuzzy co-topological space is said to be

(1) strong if $\mathcal{F}(\alpha \to \lambda )\ge \mathcal{F}\left(\lambda \right)$,

(2) Alexandrov if $\mathcal{F}\left({\bigvee }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)\ge {\bigwedge }_{i\in \mathsf{\Gamma }}\mathcal{F}\left({\lambda }_i\right)\forall {\left\{{\lambda }_i\right\}}_{i\in \mathsf{\Gamma }}\subseteq L^X$.

An L-co-topology on X is a subset $\eta \subseteq L^X$ such that

(F1) ${\top }_X\in \eta$ and ${\perp }_X\in \eta$,

(F2) $\lambda \vee \mu \in \eta$ for all $\lambda ,\mu \in \eta$,

(F3) ${\bigwedge }_{i\in \mathsf{\Gamma }}{\lambda }_i\in \eta$ for all ${\left\{{\lambda }_i\right\}}_{i\in \mathsf{\Gamma }}\subseteq \eta$.

An L-co-topology is strong if $\alpha \to \lambda \in \eta$ for all $\alpha \in L$ and $\lambda \in \eta$.

If $L=\mathit{2}=\{0,1\}$, then $F:2^X\to L$ is called an L-fuzzifying co-topology on X [41] if it satisfies the following conditions:

(1) $F\left(X\right)=F\left(\varphi \right)=\top$,

(2) $F(A\cup B)\ge F\left(A\right)\wedge F\left(B\right)$,

(3) $F\left({\bigcap }_{i\in \mathsf{\Gamma }}{A}_i\right)\ge {\bigwedge }_{i\in \mathsf{\Gamma }}F\left(A_i\right)\forall {\left\{A_i\right\}}_{i\in \mathsf{\Gamma }}\subseteq 2^X$.

The pair $(X,F)$ is called an L-fuzzifying co-topological space. An L-fuzzifying co-topological space is said to be Alexandrov if $F\left({\bigcup }_{i\in \mathsf{\Gamma }}{A}_i\right)\ge {\bigwedge }_{i\in \mathsf{\Gamma }}F\left(A_i\right)\forall {\left\{A_i\right\}}_{i\in \mathsf{\Gamma }}\subseteq 2^X$.

Definition 4

([15,18,42]). Let $(X,{\mathcal{F}}_X)$ and $(Y,{\mathcal{F}}_Y)$ be two L-fuzzy co-topological spaces and $\phi :X\to Y$ be a mapping. Then, $D_{\mathcal{F}}\left(\phi \right)$ defined by

$$\begin{array}{c}{D}_{\mathcal{F}}\left(\phi \right)={\bigwedge }_{\lambda \in L^Y}\left({\mathcal{F}}_Y\left(\lambda \right)\to {\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\right)\end{array}$$

is called the degree of $LF$-continuous map φ.

If $D_{\mathcal{F}}\left(\phi \right)=\top$, then ${\mathcal{F}}_Y\left(\lambda \right)\le {\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)$ for all $\lambda \in L^Y$, which is exactly the definition of $LF$-continuous map between L-fuzzy co-topological spaces.

A map $\phi :(X,{\eta }_X)\to (Y,{\eta }_Y)$ between L-co-topological spaces is L-continuous if ${\phi }^{\leftarrow }\left(\lambda \right)\in {\eta }_X$ for all $\lambda \in {\eta }_Y$.

A map $\phi :(X,F_X)\to (Y,F_Y)$ between L-fuzzifying co-topological spaces is $LFy$-continuous if $F_Y\left(A\right)\subseteq F_X\left({\phi }^{\leftarrow }\left(A\right)\right)$ for all $A\in 2^Y$.

The category of L-fuzzy co-topological spaces with $LF$-continuous maps as morphisms is denoted by L-FCTop.

The category of Alexandrov L-fuzzy co-topological spaces with $LF$-continuous maps as morphisms is denoted by AL-FCTop.

Write SL-FCTop for the full subcategory of L-FCTop composed of objects of all strong L-fuzzy co-topological spaces.

The category of L-co-topological spaces with L-continuous maps as morphisms is denoted by L-CTop.

The category of L-fuzzifying co-topological spaces with $LFy$-continuous maps as morphisms is denoted by L-FYCTop.

Remark 1. 

(1) If $id:X\to X$ is the identity mapping, then $D_{\mathcal{F}}\left(id\right)=\top$,

(2) if $\phi :X\to Y$ and $\psi :Y\to Z$ be two mappings, then $D_{\mathcal{F}}(\psi \circ \phi )\ge D_{\mathcal{F}}\left(\phi \right)\bigwedge D_{\mathcal{F}}\left(\psi \right)$.

Definition 5

([40,43,44]). The map $\mathcal{C}:L^X\to L^X$ is called an L-fuzzy Čech closure operator on X if for every $\lambda ,\mu ,{\lambda }_i,\alpha \in L^X$, it satisfies:

(LC1) $\mathcal{C}\left({\perp }_X\right)={\perp }_X$,

(LC2) $\mathcal{C}\left(\lambda \right)\ge \lambda$,

(LC3) if $\lambda \le \mu$, then $\mathcal{C}\left(\lambda \right)\le \mathcal{C}\left(\mu \right)$,

(LC4) $\mathcal{C}(\lambda \vee \mu )\le \mathcal{C}\left(\lambda \right)\vee \mathcal{C}\left(\mu \right)$.

The pair $(X,\mathcal{C})$ is called an L-fuzzy Čech closure space. A L-fuzzy Čech closure space is said to be:

(1) strong if $\mathcal{C}(\alpha \wedge \lambda )\ge \alpha \wedge \mathcal{C}\left(\lambda \right)$ for every $\alpha \in L$,

(2) Alexandrov if $\mathcal{C}\left({\bigvee }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)={\bigvee }_{i\in \mathsf{\Gamma }}\mathcal{C}\left({\lambda }_i\right)$.

If $\mathit{2}=\{0,1\},$ then the mapping $cl:{\mathit{2}}^X\to L^X$ is called an L-fuzzifying Čech closure operator on X if for every $A,B\in {\mathrm{2}}^X$, it satisfies:

(1) $cl\left(\varphi \right)={\perp }_X$,

(2) $cl\left(A\right)\ge {\top }_A$,

(3) if $A\subseteq B$, then $cl\left(A\right)\le cl\left(B\right)$,

(4) $cl(A\cup B)\le cl\left(A\right)\vee cl\left(B\right)$.

The pair $(X,cl)$ is called an L-fuzzifying Čech closure space. An L-fuzzifying Čech closure space is Alexandrov if $cl\left({\bigcup }_{i\in \mathsf{\Gamma }}{A}_i\right)={\bigvee }_{i\in \mathsf{\Gamma }}cl\left(A_i\right).$

3. L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topologies

In this section, we discuss the categorical aspect of the relationship between L-fuzzy Čech closure spaces and L-fuzzy co-topological spaces.

Theorem 2.

Let $(X,cl)$ be an L-fuzzifying Čech closure space. Then, a mapping ${\mathcal{C}}_{cl}:L^X\to L^X$ defined by ${\mathcal{C}}_{cl}\left(\lambda \right)\left(x\right)={\bigvee }_{\alpha \in L}\alpha \wedge cl\left({\lambda }_{\alpha }\right)\left(x\right)$ is a strong L-fuzzy Čech closure operator on X, where ${\lambda }_{\alpha }=\{x\in X\mid \lambda \left(x\right)\ge \alpha \}$.

Proof. 

It is easy to prove that $(X,{\mathcal{C}}_{cl})$ is an L-fuzzy Čech closure space. Thus, we only check that it is strong. For any $\lambda \in L^X$ and $\alpha \in L$, it follows that

$$\begin{array}{cc}\alpha \wedge {\mathcal{C}}_{cl}\left(\lambda \right)\left(x\right)& =\alpha \wedge {\bigvee }_{r\in L}r\wedge cl\left({\lambda }_r\right)\left(x\right)\hfill \\ & ={\bigvee }_{r\in L}\alpha \wedge (r\wedge cl\left({\lambda }_r\right))\left(x\right)\hfill \\ & \le {\bigvee }_{r\in L}r\wedge cl\left({(\alpha \wedge \lambda )}_r\right)\left(x\right)={\mathcal{C}}_{cl}(\alpha \wedge \lambda )\left(x\right).\hfill \end{array}$$

□

Theorem 3.

Let $(X,\mathcal{F})$ be an L-fuzzy co-topological space. For any $x\in X$ we define a mapping ${\mathcal{C}}_{\mathcal{F}}:L^X\to L^X$ as

$$\begin{array}{c}{\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\left(x\right)={\bigwedge }_{\mu \in L^X}(\mathcal{F}\left(\mu \right)\wedge S(\lambda ,\mu )\to \mu \left(x\right)).\end{array}$$

Then, $(X,{\mathcal{C}}_{\mathcal{F}})$ is a strong L-fuzzy Čech closure space.

Proof. 

(LC1) By Lemma 1, we have

$$\begin{array}{c}{\mathcal{C}}_{\mathcal{F}}\left({\perp }_X\right)\left(x\right)={\bigwedge }_{\mu \in L^X}\left(\mathcal{F}\left(\mu \right)\wedge S({\perp }_X,\mu )\to \mu \left(x\right)\right)\le \mathcal{F}\left({\perp }_X\right)\wedge S({\perp }_X,{\perp }_X)\to {\perp }_X\left(x\right)=\perp .\end{array}$$

(LC2) By Lemma 1(11), we have

$$\begin{array}{cc}S(\lambda ,{\mathcal{C}}_{\mathcal{F}}\left(\lambda \right))\hfill & ={\bigwedge }_{x\in X}\left(\lambda \left(x\right)\to {\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\left(x\right)\right)\hfill \\ & ={\bigwedge }_{x\in X}\left(\lambda \left(x\right)\to {\bigwedge }_{\mu \in L^X}(\mathcal{F}\left(\mu \right)\left(\mu \right)\wedge S(\lambda ,\mu )\to \mu \left(x\right))\right)\hfill \\ & ={\bigwedge }_{x\in X}{\bigwedge }_{\mu \in L^X}\left(\mathcal{F}\left(\mu \right)\wedge S(\lambda ,\mu )\wedge \left(\lambda \right(x)\to \mu (x\left)\right)\right)\hfill \\ & ={\bigwedge }_{x\in X}{\bigwedge }_{\mu \in L^X}\left(\mathcal{F}\left(\mu \right)\wedge S(\lambda ,\mu )\to \left(\lambda \right(x)\to \mu (x\left)\right)\right)\hfill \\ & ={\bigwedge }_{\mu \in L^X}\left((\mathcal{F}\left(\mu \right)\wedge S(\lambda ,\mu ))\to {\bigwedge }_{x\in X}(\lambda \left(x\right)\to \mu \left(x\right))\right)\hfill \\ & ={\bigwedge }_{\mu \in L^X}\left(\left(\mathcal{F}\right(\mu )\wedge S(\lambda ,\mu \left)\right)\to S(\lambda ,\mu )\right)\ge \top .\hfill \end{array}$$

Hence, ${\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\ge \lambda .$

(LC3) If $\lambda \le \mu$, then

$$\begin{array}{cc}S({\mathcal{C}}_{\mathcal{F}}\left(\lambda \right),{\mathcal{C}}_{\mathcal{F}}\left(\mu \right))& ={\bigwedge }_{x\in X}\left({\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\left(x\right)\to {\mathcal{C}}_{\mathcal{F}}\left(\mu \right)\left(x\right)\right)\hfill \\ & ={\bigwedge }_{x\in X}\left({\bigwedge }_{\rho \in L^X}(\mathcal{F}\left(\rho \right)\wedge S(\lambda ,\rho )\to \rho \left(x\right))\to {\bigwedge }_{\nu \in L^X}(\mathcal{F}\left(\nu \right)\wedge S(\mu ,\nu )\to \nu \left(x\right))\right)\hfill \\ & \ge {\bigwedge }_{\rho \in L^X}\left({\bigwedge }_{\rho \in L^X}(\mathcal{F}\left(\rho \right)\wedge S(\lambda ,\rho )\to \rho \left(x\right))\to (\mathcal{F}\left(\rho \right)\wedge S(\mu ,\rho )\to \rho \left(x\right))\right)\hfill \\ & \ge {\bigwedge }_{\rho \in L^X}\left(\mathcal{F}\left(\rho \right)\wedge S(\lambda ,\rho )\to \left(\mathcal{F}\right(\rho )\wedge S(\mu ,\rho \left)\right)\right)\hfill \\ & \ge {\bigwedge }_{\rho \in L^X}\left(S(\lambda ,\rho )\to S(\mu ,\rho )\right)\ge S(\lambda ,\mu ).\hfill \end{array}$$

Therefore, ${\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\le {\mathcal{C}}_{\mathcal{F}}\left(\mu \right)$.

(LC4) By Lemma 1(6) and Lemma 2(4), we have

$$\begin{array}{cc}{\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\vee {\mathcal{C}}_{\mathcal{F}}\left(\mu \right)& ={\bigwedge }_{\rho \in L^X}(\mathcal{F}\left(\rho \right)\wedge S(\lambda ,\rho )\to \rho )\vee {\bigwedge }_{\nu \in L^X}(\mathcal{F}\left(\nu \right)\wedge S(\mu ,\nu )\to \nu )\hfill \\ & ={\bigwedge }_{\rho \in L^X}{\bigwedge }_{\nu \in L^X}\left(\left(\mathcal{F}\right(\rho )\wedge S(\lambda ,\rho )\to \rho )\vee \left(\mathcal{F}\right(\nu )\wedge S(\mu ,\nu )\to \nu )\right)\hfill \\ & ={\bigwedge }_{\rho ,\nu \in L^X}\left(\mathcal{F}\left(\rho \right)\wedge \mathcal{F}\left(\nu \right)\wedge S(\lambda ,\rho )\wedge S(\mu ,\nu )\to (\rho \vee \nu )\right)\hfill \\ & \ge {\bigwedge }_{\rho ,\nu \in L^X}\left(\mathcal{F}(\rho \vee \nu )\wedge S(\lambda \vee \nu ,(\rho \vee \nu \left)\right)\to (\rho \vee \nu )\right)\ge {\mathcal{C}}_{\mathcal{F}}(\lambda \vee \mu ).\hfill \end{array}$$

Since $\alpha \le S(\lambda ,\alpha \wedge \lambda )\le S({\mathcal{C}}_{\mathcal{F}}\left(\lambda \right),{\mathcal{C}}_{\mathcal{F}}(\alpha \wedge \lambda )).$ Hence, $\alpha \wedge {\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\le {\mathcal{C}}_{\mathcal{F}}(\alpha \wedge \lambda )$ and ${\mathcal{C}}_{\mathcal{F}}$ is strong. □

Remark 2.

Let $(X,\eta )$ be an L-co-topological space. Then, the mapping ${\mathcal{C}}_{\eta }:L^X\to L^X$ defined by

$$\begin{array}{c}{\mathcal{C}}_{\eta }\left(\lambda \right)=\bigwedge \{\mu \in L^X\mid \lambda \le \mu ,\mu \in \eta \}\end{array}$$

is an L-fuzzy Čech closure operator on X and $(X,{\mathcal{C}}_{\eta })$ is an L-fuzzy Čech closure space. If η is strong, then ${\mathcal{C}}_{\eta }$ is strong.

Corollary 1.

Let $(X,F)$ be an L-fuzzifying co-topological space, then $c{l}_F\left(A\right)={\bigwedge }_{x\notin B\supseteq A}F\left(B\right)$ is an L-fuzzifying Čech closure operator on X.

Theorem 4.

Let $(X,\mathcal{C})$ be an L-fuzzy Čech closure space. Define a map ${\mathcal{F}}_{\mathcal{C}}:L^X\to L$ by

$$\begin{array}{c}{\mathcal{F}}_{\mathcal{C}}\left(\lambda \right)=S(\mathcal{C}\left(\lambda \right),\lambda ).\end{array}$$

Then, ${\mathcal{F}}_{\mathcal{C}}$ is an L-fuzzy co-topology on X.

If $\mathcal{C}$ is strong, then ${\mathcal{F}}_{\mathcal{C}}$ is a strong L-fuzzy co-topology on X.

Proof. 

(LF1) ${\mathcal{F}}_{\mathcal{C}}\left({\top }_X\right)={\bigwedge }_{x\in X}(\mathcal{C}\left({\top }_X\right)\left(x\right)\to {\top }_X\left(x\right))=\top$. Similarly, ${\mathcal{F}}_{\mathcal{C}}\left({\perp }_X\right)=\top$.

(LF2) By Lemma 1, we have

$$\begin{array}{cc}{\mathcal{F}}_{\mathcal{C}}(\lambda \vee \mu )& ={\bigwedge }_{x\in X}\left(\mathcal{C}(\lambda \vee \mu )\left(x\right)\to (\lambda \vee \mu )\left(x\right)\right)\hfill \\ & \ge {\bigwedge }_{x\in X}\left(\mathcal{C}\left(\lambda \right)\left(x\right)\vee \mathcal{C}\left(\mu \right)\left(x\right)\to \left(\lambda \right(x)\vee \mu (x\left)\right)\right)\hfill \\ & \ge {\bigwedge }_{x\in X}(\mathcal{C}\left(\lambda \right)\to \lambda \left(x\right))\wedge {\bigwedge }_{x\in X}(\mathcal{C}\left(\mu \right)\to \mu \left(x\right))={\mathcal{F}}_{\mathcal{C}}\left(\lambda \right)\wedge {\mathcal{F}}_{\mathcal{C}}\left(\mu \right).\hfill \end{array}$$

(LF3) By Lemma 1, we have

$$\begin{array}{cc}{\mathcal{F}}_{\mathcal{C}}\left({\bigwedge }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)& ={\bigwedge }_{x\in X}\left(\mathcal{C}\left({\bigwedge }_i{\lambda }_i\right)\left(x\right)\to \left({\bigwedge }_{i\in \mathsf{\Gamma }}{\lambda }_i\right)\left(x\right)\right)\hfill \\ & \ge {\bigwedge }_{x\in X}\left({\bigwedge }_{i\in \mathsf{\Gamma }}\mathcal{C}\left({\lambda }_i\right)\left(x\right)\to {\bigwedge }_{i\in \mathsf{\Gamma }}{\lambda }_i\left(x\right)\right)\hfill \\ & \ge {\bigwedge }_{i\in \mathsf{\Gamma }}{\bigwedge }_{x\in X}\left(\mathcal{C}\left({\lambda }_i\right)\to {\lambda }_i\left(x\right)\right)={\bigwedge }_{i\in \mathsf{\Gamma }}{\mathcal{F}}_{\mathcal{C}}\left({\lambda }_i\right).\hfill \end{array}$$

If $\mathcal{C}$ is strong and by Lemma 1, we have

$$\begin{array}{cc}{\mathcal{F}}_{\mathcal{C}}({\alpha }_X\to \lambda )& ={\bigwedge }_{x\in X}\left(\mathcal{C}({\alpha }_X\to \lambda )\left(x\right)\to ({\alpha }_X\to \lambda )\left(x\right)\right)\hfill \\ & ={\bigwedge }_{x\in X}\left({\alpha }_X\wedge \mathcal{C}({\alpha }_X\to \lambda )\left(x\right)\to \lambda \left(x\right)\right)\hfill \\ & ={\bigwedge }_{x\in X}\left(\mathcal{C}({\alpha }_X\wedge ({\alpha }_X\to \lambda ))\left(x\right)\to \lambda \left(x\right)\right)\hfill \\ & \ge {\bigwedge }_{x\in X}\left(\mathcal{C}\left(\lambda \right)\left(x\right)\to \lambda \left(x\right)\right)={\mathcal{F}}_{\mathcal{C}}\left(\lambda \right).\hfill \end{array}$$

□

Corollary 2.

Let $(X,\mathcal{C})$ be an L-fuzzy Čech closure space. Define a subset ${\eta }_{\mathcal{C}}\subseteq L^X$ by ${\eta }_{\mathcal{C}}=\{\lambda \mid \mathcal{C}\left(\lambda \right)=\lambda \}$, then ${\eta }_{\mathcal{C}}$ is an L-co-topology on X. If $\mathcal{C}$ is strong, then ${\eta }_{\mathcal{C}}$ is a strong L-co-topology on X.

Corollary 3.

Let $(X,cl)$ be an L-fuzzifying Čech closure space. Define a map $F_{cl}:2^X\to L$ by $F_{cl}\left(A\right)={\bigwedge }_{x\notin A}cl{\left(A\right)}^{*}$. Then, $F_{cl}$ is an L-fuzzifying co-topology on X.

Definition 6.

Let $(X,{\mathcal{C}}_X)$ and $(Y,{\mathcal{C}}_X)$ be two L-fuzzy Čech closure spaces and $\phi :X\to Y$ be a mapping. Then, $D_{\mathcal{C}}\left(\phi \right)$ defined by

$$\begin{array}{c}{D}_{\mathcal{C}}\left(\phi \right)={\bigwedge }_{\lambda \in L^Y}S\left({\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\mathcal{C}}_Y\left(\lambda \right)\right)\right)\end{array}$$

is called the degree of $LF$-closure map for φ.

If $D_{\mathcal{C}}\left(\phi \right)=\top$, then ${\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\le {\phi }^{\leftarrow }\left({\mathcal{C}}_Y\left(\lambda \right)\right)$ for all $\lambda \in L^Y$, which is exactly the definition of $LF$-closure map between L-fuzzy Čech closure spaces.

Let $\phi :(X,c{l}_X)\to (Y,c{l}_Y)$ be a mapping between L-fuzzifying Čech closure spaces. Then, φ is an $LFy$-closure map if $c{l}_X\left({\phi }^{\leftarrow }\left(A\right)\right)\subseteq {\phi }^{\leftarrow }\left(c{l}_Y\left(A\right)\right)$ for all $A\in 2^Y$.

The category of L-fuzzy Čech closure spaces with $LF$-closure maps as morphisms is denoted by L-FCCs.

The category of Alexandrov L-fuzzy Čech closure spaces with $LF$-closure maps as morphisms is denoted by AL-FCCs.

Write SL-FCCs for the full subcategory of  L-FCCs composed of objects of all strong L-fuzzy Čech closure spaces.

The category of L-fuzzifying Čech closure spaces with $LFy$-closure maps as morphisms is denoted by L-FYCCs, and the category of Alexandrov L-fuzzifying Čech closure spaces with $LFy$-closure maps as morphisms is denoted by AL-FYCCs.

The following theorem shows that the correspondence $(X,\mathcal{C})⊢(X,{\mathcal{F}}_{\mathcal{C}})$ induces a concrete functor ${\rm Y}:\mathbf{L}-\mathbf{FCCs}\to \mathbf{L}-\mathbf{FCTop}$ with ${\rm Y}(X,\mathcal{C})=(X,{\mathcal{F}}_{\mathcal{C}}),{\rm Y}\left(\phi \right)=\phi$.

Theorem 5.

Let $(X,{\mathcal{C}}_X)$ and $(Y,{\mathcal{C}}_Y)$ be two L-fuzzy Čech closure spaces. Then, $D_{\mathcal{C}}\left(\phi \right)\le D_{{\mathcal{F}}_{\mathcal{C}}}\left(\phi \right)$.

Proof. 

By Lemmas 1 and 2, we have

$$\begin{array}{cc}{D}_{{\mathcal{F}}_{\mathcal{C}}}\left(\phi \right)& ={\bigwedge }_{\lambda \in L^Y}\left({\mathcal{F}}_{{\mathcal{C}}_Y}\left(\lambda \right)\to {\mathcal{F}}_{{\mathcal{C}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}[{\bigwedge }_{y\in Y}\left({\mathcal{C}}_Y\left(\lambda \right)\left(y\right)\to h\left(y\right))\to {\bigwedge }_{x\in X}\left({\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}\left[{\bigwedge }_{x\in X}\left(\lambda \left(\phi \left(x\right)\right)\to {\mathcal{C}}_Y\left(\lambda \right)\left(\phi \left(x\right)\right)\right)\to {\bigwedge }_{x\in X}\left({\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\to {\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left[\left({\phi }^{\leftarrow }\left({\mathcal{C}}_Y\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\to \left({\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left({\phi }^{\leftarrow }\left({\mathcal{C}}_Y\left(\lambda \right)\right)\left(x\right)\to {\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}S\left({\mathcal{C}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\mathcal{C}}_Y\left(\lambda \right)\right)\right)=D_{\mathcal{C}}\left(\phi \right).\hfill \end{array}$$

The next theorem shows that the correspondence $(X,{\mathcal{F}}_X)⊢(X,{\mathcal{C}}_{{\mathcal{F}}_X})$ induces a concrete functor $\mathsf{\Delta }:\mathbf{L}-\mathbf{FCTop}\to \mathbf{L}-\mathbf{FCCs}$ with $\mathsf{\Delta }(X,{\mathcal{F}}_X)=(X,{\mathcal{C}}_{{\mathcal{F}}_X}),\mathsf{\Delta }\left(\phi \right)=\phi$. □

Theorem 6.

Let $(X,{\mathcal{F}}_X)$ and $(Y,{\mathcal{F}}_Y)$ be two L-fuzzy co-topological spaces. Then, $D_{\mathcal{F}}\left(\phi \right)\le D_{{\mathcal{C}}_{\mathcal{F}}}\left(\phi \right).$

Proof. 

By Lemmas 1 and 2, we have

$$\begin{array}{l}{D}_{{\mathcal{C}}_{\mathcal{F}}}\left(\phi \right)={\bigwedge }_{\lambda \in L^Y}S\left({\mathcal{C}}_{{\mathcal{F}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\mathcal{C}}_{{\mathcal{F}}_Y}\left(\lambda \right)\right)\right)\\ ={\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left({\mathcal{C}}_{{\mathcal{F}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left({\mathcal{C}}_{{\mathcal{F}}_Y}\left(\lambda \right)\right)\left(x\right)\right)\\ ={\bigwedge }_{h\in L^Y}{\bigwedge }_{x\in X}\left({\mathcal{C}}_{{\mathcal{F}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\mathcal{C}}_{{\mathcal{F}}_Y}\left(\lambda \right)\left(\phi \left(x\right)\right)\right)\\ ={\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left[{\bigwedge }_{\mu \in L^X}(({\mathcal{F}}_X\left(\mu \right)\wedge S({\phi }^{\leftarrow }\left(\lambda \right),\mu ))\to \mu \left(x\right))\to {\bigwedge }_{\rho \in L^Y}(({\mathcal{F}}_Y\left(\rho \right)\wedge S(\lambda ,\rho ))\to \rho \left(\phi \left(x\right)\right))\right]\\ \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{\rho \in L^Y}{\bigwedge }_{x\in X}\left[(({\mathcal{F}}_X\left(\mu \right)\wedge S({\phi }^{\leftarrow }\left(\lambda \right),\mu ))\to \mu \left(x\right))\to (({\mathcal{F}}_Y\left(\rho \right)\wedge S(\lambda ,\rho ))\to {\phi }^{\leftarrow }\left(\rho \right)\left(x\right))\right]\\ \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{\rho \in L^Y}{\bigwedge }_{x\in X}[(({\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\rho \right)\right)\wedge S({\phi }^{\leftarrow }\left(\lambda \right),{\phi }^{\leftarrow }\left(\rho \right)))\to {\phi }^{\leftarrow }\left(\rho \right)\left(x\right))\\ \to (({\mathcal{F}}_Y\left(\rho \right)\wedge S({\phi }^{\leftarrow }\left(\lambda \right),{\phi }^{\leftarrow }\left(\rho \right)))\to {\phi }^{\leftarrow }\left(\rho \right)\left(x\right))]\ge {\bigwedge }_{\rho \in L^Y}\left({\mathcal{F}}_Y\left(\rho \right)\to {\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\rho \right)\right)\right)=D_{\mathcal{F}}\left(\phi \right).\end{array}$$

The theorem below shows a close relationship between L-FCTop (SL-FCTop) and L-FCCs (SL-FCCs) categorically. □

Theorem 7. 

$(\mathsf{\Delta },{\rm Y})$ forms a Galois connection between the categoryL-FCTopand the categoryL-FCCs. Moreover, Υ is a left inverse of $\mathsf{\Delta },$ i.e., ${\rm Y}\circ \mathsf{\Delta }(X,{\mathcal{C}}_X)=(X,{\mathcal{C}}_X)$ for any $(X,{\mathcal{C}}_X)\in \mathbf{L}-\mathbf{FCCs}.$

Proof. 

Firstly, we show that $D_{\mathcal{F}}\left(i{d}_X\right)=\top$, where, $i{d}_X:(X,{\rm Y}\circ \mathsf{\Delta }\left({\mathcal{F}}_X\right))\to (X,{\mathcal{F}}_X).$

For any $(X,{\mathcal{F}}_X)\in \mathbf{L}-\mathbf{FCTop}$, we have

$$\begin{array}{cc}{D}_{\mathcal{F}}\left(i{d}_X\right)& ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\mathcal{F}}_{{\mathcal{C}}_{{\mathcal{F}}_X}}\left(\lambda \right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to S({\mathcal{C}}_{{\mathcal{F}}_X}\left(\lambda \right),\lambda )\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to S(\left({\bigwedge }_{\mu \in L^X}((({\mathcal{F}}_X\left(\mu \right)\wedge S(\lambda ,\mu ))\to \mu ),\lambda )\right)\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to S(({\mathcal{F}}_X\left(\lambda \right)\wedge S(\lambda ,\lambda ))\to \lambda ),\lambda \left)\right)\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to ({\mathcal{F}}_X\left(\lambda \right)\wedge S(\lambda ,\lambda ))\right)={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\mathcal{F}}_X\left(\lambda \right)\right)=\top .\hfill \end{array}$$

It follows that $i{d}_X:(X,{\rm Y}\circ \mathsf{\Delta }\left({\mathcal{F}}_X\right))=(X,{\mathcal{F}}_{{\mathcal{C}}_{{\mathcal{F}}_X}})\to (X,{\mathcal{F}}_X)$ is $LF$-continuous.

Second, we show that $D_{\mathcal{C}}\left(i{d}_X\right)=\top ,$ where $i{d}_X:(X,\mathsf{\Delta }\circ {\rm Y}\left({\mathcal{C}}_X\right))=(X,{\mathcal{C}}_{{\mathcal{F}}_{{\mathcal{C}}_X}})\to (X,{\mathcal{F}}_X)$.

For any $(X,{\mathcal{C}}_X)\in \mathbf{L}-\mathbf{FCCs}$, we have

$$\begin{array}{cc}{D}_{\mathcal{C}}\left(i{d}_X\right)& ={\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\mathcal{C}}_{{\mathcal{F}}_{{\mathcal{C}}_X}}\left(\lambda \right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\bigwedge }_{\mu \in L^X}(({\mathcal{F}}_{{\mathcal{C}}_X}\left(\mu \right)\wedge S(\lambda ,\mu ))\to \mu )\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\bigwedge }_{\mu \in L^X}((S({\mathcal{C}}_X\left(\mu \right),\mu )\wedge S(\lambda ,\mu ))\to \mu )\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\bigwedge }_{\mu \in L^X}((S({\mathcal{C}}_X\left(\mu \right),\mu )\wedge S({\mathcal{C}}_X\left(\lambda \right),{\mathcal{C}}_X\left(\mu \right)))\to \mu )\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\bigwedge }_{\mu \in L^X}(S({\mathcal{C}}_X\left(\lambda \right),\mu )\to \mu )\right)\ge {\bigwedge }_{\lambda \in L^X}S\left({\mathcal{C}}_X\left(\lambda \right),{\mathcal{C}}_X\left(\lambda \right)\right)=\top .\hfill \end{array}$$

It follows that $\mathsf{\Delta }\left({\rm Y}(X,{\mathcal{C}}_X)\right)=(X,{\mathcal{C}}_X)$ and then $i{d}_X:(X,{\rm Y}\circ \mathsf{\Delta }\left({\mathcal{C}}_X\right))=(X,{\mathcal{C}}_{{\mathcal{F}}_{{\mathcal{C}}_X}})\to (X,{\mathcal{C}}_X)$ is an $LF$-closure map. □

Example 1.

Let $L=[0,1]$ with the implication → defined by $x\to y=(1-x+y)\wedge 1$. Let $X=\{x,y,z\}$ be a set and $\rho \in L^{*}$ defined as follows: $\rho \left(x\right)=0.1,\rho \left(y\right)=0.8,\rho \left(z\right)=0.7$.

(1) Define an L-fuzzy co-topology $\mathcal{F}:L^X\to L$ as follows:

$$\mathcal{F}\left(\lambda \right)=\left\{\begin{array}{cc}1,\hfill & if\lambda =1_X,\lambda =0_X,\hfill \\ 0.5\hfill & if\lambda =\rho ,\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

From Theorem 3, we obtain an L-fuzzy Čech closure operator ${\mathcal{C}}_{\mathcal{F}}:L^X\to L^X$ as

$$\begin{array}{cc}{\mathcal{C}}_{\mathcal{F}}\left(\lambda \right)\left(x\right)& ={\bigwedge }_{\mu \in L^X}(\mathcal{F}\left(\mu \right)\wedge S(\lambda ,\mu )\to \mu \left(x\right))\hfill \\ & =(S(\lambda ,0_X)\to 0_X)\wedge (S(\lambda ,1_X)\to 1_X)\wedge (0.5\wedge S(\lambda ,\mu )\to \rho ).\hfill \end{array}$$

For ${\lambda }_1=(0.6,0.4,0.2)$, we have ${\mathcal{C}}_{\mathcal{F}}\left({\lambda }_1\right)=(0.5,0.6,0.6)$.

From Theorems 3 and 4, we have ${\mathcal{F}}_{{\mathcal{C}}_{\mathcal{F}}}\left({\lambda }_1\right)=0.4\ge \mathcal{F}\left({\lambda }_1\right)=0.$

(2) Define an L-fuzzy Čech closure operator $\mathcal{C}:L^X\to L^X$ as follows:

$$\mathcal{C}\left(\lambda \right)=\left\{\begin{array}{cc}{0}_X,\hfill & if\lambda =0_X,\hfill \\ \rho \hfill & if{0}_X\ne \lambda \le \rho ,\hfill \\ 1_X,\hfill & otherwise.\hfill \end{array}\right.$$

From Theorem 4, we obtain an L-fuzzy co-topology ${\mathcal{F}}_{\mathcal{C}}:L^X\to L$ as

$${\mathcal{F}}_{\mathcal{C}}\left(\lambda \right)=\left\{\begin{array}{cc}1,\hfill & if\lambda =1_X,\hfill \\ S(\rho ,\lambda )\hfill & if\lambda \le \rho ,\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Moreover, from Theorems 3 and 4, we have

$$\begin{array}{c}{\mathcal{C}}_{{\mathcal{F}}_{\mathcal{C}}}\left(\lambda \right)={\bigwedge }_{\mu \in L^X}({\mathcal{F}}_{\mathcal{C}}\left(\mu \right)\wedge S(\lambda ,\mu )\to \mu )=(S(\lambda ,1_X)\to 1_X)\wedge (S(\lambda ,\rho )\to \rho ).\end{array}$$

Hence, ${\mathcal{C}}_{{\mathcal{F}}_{\mathcal{C}}}\left({\lambda }_1\right)=(0.4,0.8,0.7)\le \mathcal{C}\left({\lambda }_1\right)=(1,1,1)$.

Corollary 4. 

$(\mathsf{\Delta },{\rm Y})$ forms a Galois connection between the categoryL-FYCTopand the categoryL-FYCCs. Moreover, Υ is a left inverse of $\mathsf{\Delta }.$

4. L-Fuzzy Approximations Spaces and L-Fuzzy Co-Topological Spaces

We devote this section to the categorical aspect of the relationship between L-fuzzy co-topological spaces and L-fuzzy approximation spaces.

Let $(X,\mathcal{F})$ be an Alexandrov L-fuzzy co-topological space. Then, it is easily seen that the mapping $R_{\mathcal{F}}:X\times X⟶L$ defined by $R_{\mathcal{F}}(x,y)={\bigwedge }_{\lambda \in L^X}(\mathcal{F}\left(\lambda \right)\to (\lambda \left(y\right)\to \lambda \left(x\right)))$ is a reflexive L-fuzzy relation.

Conversely, Let $(X,R)$ be an L-fuzzy approximation space, where R is reflexive, then the mapping ${\mathcal{F}}_R:L^X\to L$ defined by ${\mathcal{F}}_R\left(\lambda \right)={\bigwedge }_{x,y\in X}R(x,y)\to (\lambda \left(y\right)\to \lambda \left(x\right))$ is a strong Alexandrov L-fuzzy co-topological space on X.

Definition 7.

Let $(X,R_X)$ and $(Y,R_Y)$ be two L-fuzzy approximation spaces. Then,

(1) $D_R\left(\phi \right)$ defined by $D_R\left(\phi \right)={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to R_Y(\phi \left(x\right),\phi \left(y\right))\right)$ is called the degree of $LF$-order preserving map for φ.

If $D_R\left(\phi \right)=\top$, then $R_X(x,y)\le R_Y(\phi \left(x\right),\phi \left(y\right))$ for all $x,y\in X$, which is exactly the definition of $LF$-order preserving map between L-fuzzy approximation spaces.

(2) $D_{\overline{R}}\left(\phi \right)$ defined by $D_{\overline{R}}\left(\phi \right)={\bigwedge }_{\lambda \in L^Y}S\left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\overline{R}}_Y\left(\lambda \right)\right)\right)$ is called the degree of $LF$-approximation map for φ.

If $D_{\overline{R}}\left(\phi \right)=\top$, then ${\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\le {\phi }^{\leftarrow }\left({\overline{R}}_Y\left(\lambda \right)\right)$ for all $\lambda \in L^Y$, which is exactly the definition of $LF$-approximation map between L-fuzzy approximation spaces.

Let $\phi :(X,R_X)\to (Y,R_Y)$ be a mapping between L-fuzzifying approximation spaces. Then, $\phi$ is an $LFy$-approximation map if ${\overline{R}}_X\left({\phi }^{\leftarrow }\left(A\right)\right)\subseteq {\phi }^{\leftarrow }\left({\overline{R}}_Y\left(A\right)\right)$ for all $A\in 2^Y$.

The category of L-fuzzy approximation spaces based on reflexive L-fuzzy relations with $LF$-approximation maps as morphisms defined above is denoted by L-FAPP.

The category of L-fuzzifying approximation spaces based on reflexive L-fuzzy relations with $LFy$-approximation maps as morphisms defined above is denoted by L-FYAPP.

Theorem 8.

Let $(X,{\mathcal{F}}_X)$ and $(Y,{\mathcal{F}}_Y)$ be two Alexandrov L-fuzzy co-topological spaces, then

(1) $D_{\mathcal{F}}\left(\phi \right)\le D_{R_{\mathcal{F}}}\left(\phi \right)$,

(2) $D_{R_{\mathcal{F}}}\left(\phi \right)\le D_{{\overline{R}}_{\mathcal{F}}}\left(\phi \right)$.

Proof. 

(1) By Lemmas 1 and 2, we have

$$\begin{array}{l}{D}_{R_{\mathcal{F}}}\left(\phi \right)={\bigwedge }_{x,y\in X}\left(R_{{\mathcal{F}}_X}(x,y)\to R_Y(\phi \left(x\right),\phi \left(y\right))\right)\\ ={\bigwedge }_{x,y\in X}\left({\bigwedge }_{\lambda \in L^X}({\mathcal{F}}_X\left(\lambda \right)\to (\lambda \left(y\right)\to \lambda \left(x\right)))\to \left({\bigwedge }_{\mu \in L^Y}\right({\mathcal{F}}_Y\left(\mu \right)\to (\mu \left(\phi \left(y\right)\right)\to \mu \left(\phi \left(x\right)\right))\right)\\ \ge {\bigwedge }_{x,y\in X}{\bigwedge }_{\mu \in L^Y}(({\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\mu \right)\right)\to ({\phi }^{\leftarrow }\left(\mu \right)\left(y\right)\to {\phi }^{\leftarrow }\left(\mu \right)\left(x\right))))\\ \to \left(({\mathcal{F}}_Y\left(\mu \right)\to \left({\phi }^{\leftarrow }\left(\mu \right)\left(y\right)\right)\to {\phi }^{\leftarrow }\left(\mu \right)\left(x\right))\right))\ge {\bigwedge }_{\mu \in L^Y}\left({\mathcal{F}}_Y\left(\mu \right)\to {\mathcal{F}}_X\left({\phi }^{\leftarrow }\left(\mu \right)\right)\right)=D_{\mathcal{F}}\left(\phi \right).\end{array}$$

(2) By Lemmas 1 and 2, we have

$$\begin{array}{cc}{D}_{{\overline{R}}_{\mathcal{F}}}\left(\phi \right)& ={\bigwedge }_{\lambda \in L^Y}S\left({\overline{R}}_{{\mathcal{F}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\overline{R}}_{{\mathcal{F}}_Y}\left(\lambda \right)\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left({\overline{R}}_{{\mathcal{F}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\overline{R}}_{{\mathcal{F}}_Y}\left(\lambda \right)\left(\phi \left(x\right)\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left[{\bigvee }_{z\in X}(R_{{\mathcal{F}}_X}(x,z)\wedge {\phi }^{\leftarrow }\left(\lambda \right)\left(z\right))\to {\bigvee }_{y\in Y}(R_{{\mathcal{F}}_Y}(\phi \left(x\right),y)\wedge \lambda \left(y\right))\right]\hfill \\ & \ge {\bigwedge }_{h\in L^Y}{\bigwedge }_{x,z\in X}\left[(R_{{\mathcal{F}}_X}(x,z)\wedge \lambda \left(\phi \left(z\right)\right))\to (R_{{\mathcal{F}}_Y}(\phi \left(x\right),\phi \left(z\right))\wedge \lambda \left(\phi \left(z\right)\right))\right]\hfill \\ & \ge {\bigwedge }_{x,z\in X}\left(R_{{\mathcal{F}}_X}(x,z)\to R_{{\mathcal{F}}_Y}(\phi \left(x\right),\phi \left(z\right))\right)=D_{R_{\mathcal{F}}}\left(\phi \right).\hfill \end{array}$$

The above theorem shows that the correspondence $(X,{\mathcal{F}}_X)⊢(X,R_{{\mathcal{F}}_X})$ induces a concrete functor $\mathsf{\Gamma }:\mathbf{AL}-\mathbf{FCTop}\to \mathbf{L}-\mathbf{FAPP}$ with $\mathsf{\Gamma }(X,{\mathcal{F}}_X)=(X,R_{{\mathcal{F}}_X}),\mathsf{\Gamma }\left(\phi \right)=\phi .$

Let $(X,R)$ be a reflexive L-fuzzy approximation space, then it is easily seen that the mapping ${\mathcal{F}}_{\overline{R}}:L^X\to L$ defined by $\forall \lambda \in L^X,{\mathcal{F}}_{\overline{R}}\left(\lambda \right)=S(\overline{R}\left(\lambda \right),\lambda )$ is a strong Alexandrov L-fuzzy co-topology on X. □

Theorem 9.

Let $(X,R_X)$ and $(Y,R_Y)$ be two L-fuzzy approximation spaces, then $D_{\overline{R}}\left(\phi \right)\le D_{{\mathcal{F}}_{\overline{R}}}\left(\phi \right).$

Proof. 

$$\begin{array}{cc}{D}_{{\mathcal{F}}_{\overline{R}}}\left(\phi \right)& ={\bigwedge }_{\lambda \in L^Y}\left({\mathcal{F}}_{{\overline{R}}_Y}\left(\lambda \right)\to {\mathcal{F}}_{{\overline{R}}_X}\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}[{\bigwedge }_{y\in Y}\left({\overline{R}}_Y\left(\lambda \right)\left(y\right))\to \lambda \left(y\right)\to {\bigwedge }_{x\in X}\left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}\left[{\bigwedge }_{x\in X}\left({\overline{R}}_Y\left(\lambda \right)\left(\phi \left(x\right)\right)\to \lambda \left(\phi \left(x\right)\right)\right){\bigwedge }_{x\in X}\left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left[{\phi }^{\leftarrow }\left({\overline{R}}_Y\left(\lambda \right)\right)\left(x\right)\to \left({\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\to \left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left(\lambda \right)\left(x\right)\right)\right]\hfill \\ & \ge {\bigwedge }_{\lambda \in L^Y}{\bigwedge }_{x\in X}\left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right)\left(x\right)\to {\phi }^{\leftarrow }\left({\overline{R}}_Y\left(\lambda \right)\right)\left(x\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^Y}S\left({\overline{R}}_X\left({\phi }^{\leftarrow }\left(\lambda \right)\right),{\phi }^{\leftarrow }\left({\overline{R}}_Y\left(\lambda \right)\right)\right)=D_{\overline{R}}\left(\phi \right).\hfill \end{array}$$

The above theorem shows that the correspondence $(X,{\overline{R}}_X)⊢(X,{\mathcal{F}}_{{\overline{R}}_X})$ induces a concrete functor $\overline{\mathsf{\Delta }}:\mathbf{L}-\mathbf{FAPP}\to \mathbf{AL}-\mathbf{FCTop}$ with $\overline{\mathsf{\Delta }}(X,R_X)=(X,{\mathcal{F}}_{{\overline{R}}_X}),\overline{\mathsf{\Delta }}\left(\phi \right)=\phi .$ □

Corollary 5.

$(\overline{\mathsf{\Delta }},\mathsf{\Gamma })$ forms a Galois connection between the categoryL-FAPPand the categoryAL-FCTop. Moreover, Γ is a left inverse of $\overline{\mathsf{\Delta }}$, i.e., $\mathsf{\Gamma }\circ \overline{\mathsf{\Delta }}(X,R_X)=(X,R_X)$ for any $(X,R_X)\in \mathbf{L}-\mathbf{FAPP}$.

Proof. 

Firstly, we show that $D_R\left(i{d}_X\right)=\top ,$ where $i{d}_X:(X,\mathsf{\Gamma }\circ \overline{\mathsf{\Delta }}\left(R_X\right))\to (X,R_{{\mathcal{F}}_{{\overline{R}}_X}}).$

For any $(X,R_X)\in \mathbf{L}-\mathbf{FAPP}$, we have

$$\begin{array}{cc}{D}_R\left(i{d}_X\right)& ={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to R_{{\mathcal{F}}_{{\overline{R}}_X}}(x,y)\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}({\mathcal{F}}_{{\overline{R}}_X}\left(\lambda \right)\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}({\mathcal{F}}_{{\overline{R}}_X}\left(\lambda \right)\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}(S({\overline{R}}_X\left(\lambda \right),\lambda )\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & \ge {\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}(({\overline{R}}_X\left(\lambda \right)\left(x\right)\to \lambda \left(x\right))\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & \ge {\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}((R_X(x,y)\wedge \lambda \left(y\right))\to \lambda \left(x\right))\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left(R_X(x,y)\to {\bigwedge }_{\lambda \in L^X}(R_X(x,y)\to (\lambda \left(y\right)\to \lambda \left(x\right))\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & \ge {\bigwedge }_{x,y\in X}\left(R_X(x,y)\to R_X(x,y)\right)=\top .\hfill \end{array}$$

It follows that $\mathsf{\Gamma }\left(\overline{\mathsf{\Delta }}(X,R_X)\right)=(X,R_X)$ and then $i{d}_X:(X,R_X)\to (X,\mathsf{\Gamma }\circ \overline{\mathsf{\Delta }}\left(R_X\right))$ is an $LF$-order preserving map.

Second, we show that $D_{\mathcal{F}}\left(i{d}_X\right)=\top ,$ where $i{d}_X:(X,\overline{\mathsf{\Delta }}\circ \mathsf{\Gamma }\left({\mathcal{F}}_X\right))=(X,{\mathcal{F}}_{{\overline{R}}_{{\mathcal{F}}_X}})\to (X,{\mathcal{F}}_X).$

For any $(X,{\mathcal{F}}_X)\in \mathbf{AL}-\mathbf{FCTop},$ we have

$$\begin{array}{cc}{D}_{\mathcal{F}}\left(i{d}_X\right)& ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\mathcal{F}}_{{\overline{R}}_{{\mathcal{F}}_X}}\left(\lambda \right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to S({\overline{R}}_{{\mathcal{F}}_X}\left(\lambda \right),\lambda )\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\bigwedge }_{x\in X}\left(({\bigvee }_{y\in X}(R_{{\mathcal{F}}_X}(x,y)\wedge \lambda \left(y\right))\to \lambda \left(x\right))\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\bigwedge }_{x,y\in X}(R_{{\mathcal{F}}_X}(x,y)\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\bigwedge }_{x,y\in X}\left(({\mathcal{F}}_X\left(\lambda \right)\to (\lambda \left(y\right)\to \lambda \left(x\right))\to (\lambda \left(y\right)\to \lambda \left(x\right)))\right)\right)\hfill \\ & \ge {\bigwedge }_{\lambda \in L^X}\left({\mathcal{F}}_X\left(\lambda \right)\to {\mathcal{F}}_X\left(\lambda \right)\right)=\top .\hfill \end{array}$$

It follows that $i{d}_X:(X,\overline{\mathsf{\Delta }}\circ \mathsf{\Gamma }\left({\mathcal{F}}_X\right))\to (X,{\mathcal{F}}_X)$ is an $LF$-continuous map.

Let $(X,R_X)$ be a reflexive L-fuzzy approximation space. Then, it is not difficult to check that ${\mathcal{C}}_{R_X}={\overline{R}}_X$ is an Alexandrov L-fuzzy Čech closure space on X, and then one can define a concrete functor $\overline{\mathsf{\Theta }}:\mathbf{L}-\mathbf{FAPP}\to \mathbf{AL}-\mathbf{FCCs}$ by $\overline{\mathsf{\Theta }}(X,R_X)=(X,{\mathcal{C}}_{R_X}),\overline{\mathsf{\Theta }}\left(\phi \right)=\phi .$

Conversely, let ${\mathcal{F}}_X$ be an Alexandrov L-fuzzy pre-co-topology on X, then it is easily seen that $R_{{\mathcal{F}}_X}(x,y)=\mathcal{F}\left({\top }_y\right)$ is a reflexive L-fuzzy relation. □

Proposition 2.

Let $\phi :(X,R_X)\to (Y,R_Y)$ be a mapping between two L-fuzzy approximation spaces based on a reflexive L-fuzzy relation, then $D_R\left(\phi \right)\le D_{R_{\mathcal{F}}}\left(\phi \right)$.

Proof. 

Similar to the proof of Theorem 8.

The above proposition shows that the correspondence $(X,{\mathcal{F}}_X)\mapsto (X,R_{{\mathcal{F}}_X})$ induces a concrete functor $\overline{\mathsf{\Phi }}:\mathbf{AL}-\mathbf{FCTop}\to \mathbf{L}-\mathbf{FAPP}$ by $\overline{\mathsf{\Phi }}(X,{\mathcal{F}}_X)=(X,R_{{\mathcal{F}}_X}),\overline{\mathsf{\Phi }}\left(\phi \right)=\phi$. □

Theorem 10. 

$(\overline{\mathsf{\Theta }},\overline{\mathsf{\Phi }})$ forms a Galois connection between the categoryL-FAPPand the categoryAL-FCCs. Moreover, $\overline{\mathsf{\Phi }}$ is a left inverse of $\overline{\mathsf{\Theta }}$, i.e., $\overline{\mathsf{\Phi }}\circ \overline{\mathsf{\Theta }}(X,R_X)=(X,R_X)$ for any $(X,R_X)\in \mathbf{L}-\mathbf{FAPP}$.

Proof. 

Firstly, we show that $D_R\left(i{d}_X\right)=\top ,$ where $i{d}_X:(X,R_X)\to (X,\overline{\mathsf{\Phi }}\circ \overline{\mathsf{\Theta }}\left(R_X\right)).$

For any $(X,R_X)\in \mathbf{L}-\mathbf{FAPP}$,

$$\begin{array}{cc}{D}_R\left(i{d}_X\right)& ={\bigwedge }_{x,y\in X}\left(R_{{\mathcal{C}}_{R_X}}(x,y)\to R_X(x,y)\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left({\mathcal{C}}_{R_X}\left({\top }_y\right)\to R_X(x,y)\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left({\overline{R}}_X\left({\top }_y\right)\left(x\right)\to R_X(x,y)\right)\hfill \\ & ={\bigwedge }_{x,y\in X}\left({\bigvee }_{z\in X}{R}_X(x,z)\wedge {\top }_y\left(z\right)\to R_X(x,y)\right)\hfill \\ & \ge {\bigwedge }_{x,y\in X}\left(R_X(x,y)\to R_X(x,y)\right)=\top .\hfill \end{array}.$$

It follows that $\overline{\mathsf{\Phi }}\left(\overline{\mathsf{\Theta }}(X,R_X)\right)=(X,R_X)$ and then $i{d}_X:(X,R_X)\to (X,\overline{\mathsf{\Phi }}\circ \overline{\mathsf{\Theta }}\left(R_X\right))$ is an $LF$-approximation map.

Second, we show that $D_{\mathcal{C}}\left(i{d}_X\right)=\top ,$ where $i{d}_X:(X,\overline{\mathsf{\Theta }}\circ \overline{\mathsf{\Phi }}\left({\mathcal{C}}_X\right))\to (X,{\mathcal{C}}_X)$.

For any $(X,{\mathcal{C}}_X)\in \mathbf{AL}-\mathbf{FCCs}$, we have

$$\begin{array}{cc}{D}_{\mathcal{C}}\left(i{d}_X\right)& ={\bigwedge }_{\lambda \in L^X}{\bigwedge }_{x\in X}\left({\mathcal{C}}_{R_{{\mathcal{C}}_X}}\left(\lambda \right)\to {\mathcal{C}}_X\left(\lambda \right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}{\bigwedge }_{x\in X}\left({\overline{R}}_{{\mathcal{C}}_X}\left(\lambda \right)\left(x\right)\to {\mathcal{C}}_X\left({\bigvee }_{y\in X}({\top }_y\wedge \lambda \left(y\right))\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}{\bigwedge }_{x\in X}\left({\bigvee }_{y\in X}\left(R_{{\mathcal{C}}_X}(x,y)\wedge \lambda \left(y\right)\right)\left(x\right)\to {\bigvee }_{y\in X}\left({\mathcal{C}}_X\left({\top }_y\right)\wedge \lambda \left(y\right)\right)\right)\hfill \\ & ={\bigwedge }_{\lambda \in L^X}{\bigwedge }_{x\in X}\left({\bigvee }_{y\in X}\left({\mathcal{C}}_X\left({\top }_y\right)\wedge \lambda \left(y\right)\right)\to {\bigvee }_{y\in X}\left({\mathcal{C}}_X\left({\top }_y\right)\wedge \lambda \left(y\right)\right)\right)\hfill \\ & \ge {\bigwedge }_{x,y\in X}\left(({\mathcal{C}}_X\left({\top }_y\right)\to {\mathcal{C}}_X\left({\top }_y\right))\wedge (\lambda \left(y\right)\to \lambda \left(y\right))\right)=\top .\hfill \end{array}$$

It follows that $i{d}_X:(X,\overline{\mathsf{\Theta }}\circ \overline{\mathsf{\Phi }}\left({\mathcal{C}}_X\right))\to (X,{\mathcal{C}}_X)$ is an $LF$-closure map.

Let $(X,R)$ be a reflexive L-fuzzy approximation space. Then it is easily seen that the $F_{\overline{R}}:P\left(X\right)\to L$ defined by $F_{\overline{R}}\left(A\right)={\bigwedge }_{x\notin A}\overline{R}{\left(A\right)}^{*}\left(x\right)$ is an Alexandrov L-fuzzifying co-topology on X (see [39]). Moreover, if $c{l}_R\left(A\right)\left(x\right)=\overline{R}\left(A\right)\left(x\right),$ then $(X,c{l}_R)$ is an Alexandrov L-fuzzifying Čech closure space. □

Proposition 3.

Let $(X,cl)$ be an Alexandrov L-fuzzifying Čech closure space, then for any $x\in X$, we define $R_{cl}(x,y)=cl\left(\left\{y\right\}\right)$. Hence, $(X,R_{cl})$ is an L-fuzzifying approximation space with a reflexive L-fuzzy relation $R_{cl}$.

Corollary 6. 

(1) If $\phi :(X,R_X)\to (Y,R_Y)$ is a mapping between L-fuzzifying approximation spaces, then $D_R\left(\phi \right)\le D_{c{l}_R}\left(\phi \right)$,

(2) If $\phi :(X,c{l}_X)\to (Y,c{l}_Y)$ is a mapping between L-fuzzifying Čech closure spaces, then $D_{c{l}_R}\left(\phi \right)\le D_{{\overline{R}}_{cl}}\left(\phi \right).$

The above corollary gives a concrete functors $\mathsf{\Theta }:\mathbf{L}-\mathbf{FYCCs}\to \mathbf{L}-\mathbf{FYAPP}$ between the category of L-fuzzifying Čech closure spaces and that L-fuzzifying approximation spaces and $\mathsf{\Phi }:\mathbf{L}-\mathbf{FYAPP}\to \mathbf{L}-\mathbf{FYCCs}$ between the category of L-fuzzifying approximation spaces and that L-fuzzifying Čech closure spaces.

Corollary 7. 

$(\mathsf{\Theta },\mathsf{\Phi })$ forms a Galois connection between the categoryL-FYAPPand the categoryL-FYCCs. Moreover, Θ is a left inverse of Φ.

5. Conclusions

In this paper, we established relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces, and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. In addition, we used the concept of degrees of $LF$-continuity to study the relationships between the categories of all given spaces. In particular, we obtained some interesting adjunctions between the considered categories.