Rough set model on strong L-fuzzy Alexandrov spaces

Abstract

For a commutative unital quantale L and a strong L-fuzzy topological space X, we define the open-hood $O_x$ for every point $x\in X$. Considering $\{O_x\mid x\in X\}$ as the family of basic granules, we define the upper and lower approximation operators $\varphi ,\psi$. It is shown that $\varphi ,\psi$ are a kind of a closure operator and an interior operator respectively, and they form an L-Galois adjunction on $L^X$. By introducing notions of one zooming-in operator and two zooming out operators, we study the composition and decomposition problems of $\varphi ,\psi$.

Introduction

We live in an information explosion world since a large amount of information appears in our everyday life and the rapid development of information science, computer science with the maturity of network technology. We always deal with information and design related algorithms from given massive data by using theory and methods of information science, computer science and applied mathematics. Among them, information granulation and granular computing are very effective methods to various challenges, and have received much attention from a lot of fields, such as computational intelligence, clustering analysis, Dempster–Shafer theory, machine learning, rough set theory, etc.

Information granulation and granular computing were presented by Zadeh (1979, 1997) by making use of fuzzy set theory. He thought that human cognition was mainly composed of three concepts: granulation, organization and causality. Yager and Filev (1998) suggested to develop a general method to granular computing as one of the main tasks in this field. Various granular computing models can be obtained by reformulating re-interpreting and combining from rough sets (Ma et al. 2007; Pawlak 1982, 1991; Yao 2004), quotient spaces (Zhang and Zhang 2005), belief functions (Shafer 1976), and word computing (Wang and Lin 1998; Wang 2001; Zadeh 1996), etc.

Rough set theory was initially introduced and studied by Pawlak (1982, 1991), providing a systematic approach and method to object classification via different kinds of binary relations and has wide applications in a lot of engineering fields. Mathematically speaking, rough sets were taken its rise from the classification problem with equivalence relations and partition as the fundamental structures. Nevertheless, the traditional rough set theory can no longer handle certain kinds of granular problems since it is hard for us to get a very reasonable equivalence relation from a data set. We then need some of general binary relation (Skowron and Stepaniuk 1996; Zheng et al. 2005; Slowinski and Vanderpooten 2000) or fuzzy types of binary relations (Dubois and Prade 1990; Liu and Sai 2010; Mi and Zhang 2004; Morsi and Yakout 1998; Wu et al. 2005, 2016) to build rough set models. While except for binary relations, the notion of neighborhood operators/systems and that of coverings is other useful tools to construct rough set models (Yao 1998, 1999; Zhu 2009), which can be considered as different generalizations of granule-based models. It should be mentioned that various distance functions are a further kind of structures to construct and study roughness by means of fuzzy set theory (Yao et al. 2019, 2023).

In a broad sense, all of relation-based, neighborhood operators/systems-based, covering-based and even distance function-based rough set models are indeed topological models, since the former three are based on powersets and certain subfamilies, and the latter one is strongly related with metric topology. By this regard, in Yao and Han (2023) we introduced a topological model of granular computing by defining the notions of open/closed-hoods and zooming-in/out operators for Alexandrov spaces (Alexandroff 1937; Miller et al. 2010). This idea in fact has already been stated showing close relations among ordered structures, rough set structures and topological structures in Steiner (1966), Skowron (1988), Wiweger (1988), Pawlak (1991), SP, Srivastava AK (2013), Perfilieva et al. (2018), Perfilieva et al. (2020), Perfilieva (2022) and Sostak and Uljane (2023).

The topological model in Yao and Han (2023) can be considered as a common framework of the preordered relation model, the Alexandrov neighborhood operator model and the covering model. In this paper we would like to follow the step of Yao and Han (2023) to establish a rough set model of lattice-valued fuzzy Alexandrov topology, and wish to supply a common framework of the lattice-valued fuzzy preordered relation model, certain types of lattice-valued fuzzy neighborhood operator model and certain types of lattice-valued fuzzy covering model.

The structure of this paper is arranged as follows. For a commutative unital quantale L and a strong L-fuzzy topological space X, we define the open-hood $O_x$ for every point $x\in X$. Considering $\{O_x\mid x\in X\}$ as the family of basic granules, we define the upper and lower approximation operators $\varphi ,\psi$. It is shown that $\varphi ,\psi$ are a kind of a closure operator and an interior operator respectively, and form an L-Galois connection on $L^X$. By introducing notions of one zooming-in operator and two zooming out operators, we study the composition and decomposition problems of $\varphi ,\psi$.

Preliminaries

In this section, we recall some basic concepts and results on quantales, lattice-valued topology and fuzzy posets.

Definition 1

Rosenthal (1990) A commutative unital quantale is a triple $(L,\ast ,e)$ such that

(Q1) L is a complete lattice with 0, 1 as the bottom and the top element respectively;

(Q2) $(L,\ast )$ is a commutative monoid with e as the unit;

(Q3) the operation $\ast$ is distributive over joins of arbitrary subsets of Q, that is,$$\begin{array}{c}\hfill a\ast \left(\bigvee ,S\right)=\underset{s\in S}{\bigvee }a\ast s(\forall a\in L,\forall S\subseteq L).\end{array}$$

By lattice theory, the operation $\ast$ can induce an implication operation $\to :L\times L⟶L$ satisfying that $a\ast b\le c⟺a\le b\to c(a,b,c\in L)$.

Example 1

(1)

Let $L=[0,\infty ]$ denote the extended interval of all non-negative real numbers with the same ordering as the real numbers. Let $\times$ be the usual multiplication on real numbers extended to cope with the infinity such that $x\times \infty =\infty$ for every $x\in [0,+\infty ]$ and $0\times \infty =0$. Then $(L,\times ,1)$ is a commutative unital quantale.

(2)

Let $L={[0,\infty ]}^{\mathit{op}}$ denote the extended interval of all non-negative real numbers with the opposite ordering as real numbers (so 0 is the greatest element). Let $+$ be the usual addition on real numbers extended to cope with infinity, such that $x+\infty =\infty$ for every $x\in L$. Then $(L,+,0)$ is a commutative unital quantale.

(3)

The unit interval [0, 1] equipped with a left continuous t-norm (e.g., the usual $\times$, $\wedge$ and $min$) becomes a commutative unital quantale.

Proposition 1

(Rosenthal 1990) Suppose that $(L,\ast ,e)$ is a commutative unital quantale. Then (1)

$e\le a\to b⟺a\le b$;

(2)

$e\to a=a$;

(3)

$a\ast (a\to b)\le b$;

(4)

$b\le a\to (a\ast b)$;

(5)

$(a\to b)\ast (b\to c)\le a\to c$;

(6)

$a\to b\le (a\ast c)\to (b\ast c)$;

(7)

$a\to b\le (c\to a)\to (c\to b)$;

(8)

$({\bigvee }_i{p}_i)\to q={\bigwedge }_i(p_i\to q)$;

(9)

$p\to ({\bigwedge }_i{q}_i)={\bigwedge }_i(p\to q_i)$;

(10)

$(r\to p)\to (r\to q)\ge p\to q$;

(11)

$(p\to r)\to (q\to r)\ge q\to p$;

(12)

$p\to (q\to r)=(p\ast q)\to r$;

(13)

$a\to (b\ast c)\ge (a\to b)\ast c$;

(14)

${\bigwedge }_i{a}_i\to b_i\le ({\bigvee }_i{a}_i)\to ({\bigvee }_i{b}_i)$;

(15)

${\bigwedge }_i{a}_i\to b_i\le ({\bigwedge }_i{a}_i)\to ({\bigwedge }_i{b}_i)$.

Definition 2

Yao and Lu (2009) Let X be a nonempty set and let $p:X\times X⟶L$ be a mapping. The pair (X, p) or just X, is called an L-preordered set if (P1)

Self-reflexivity: $p(a,a)\ge e(\forall a\in X)$;

(P2)

Transitivity: $p(a,b)\ast p(b,c)\le p(a,c)(\forall a,b,c\in X).$

Example 2

(1)

Let $e_L:L\times L⟶L$ by $e(a,b)=a\to b$. Then $(L,e_L)$ is an L-preordered set.

(2)

Define ${\text{Sub}}_X:L^X\times L^X⟶L$ by ${\text{Sub}}_X(A,B)={\bigwedge }_{x\in X}A(x)\to B(x)$. Then ${\text{Sub}}_X$ is an L-preorder on $L^X$.

An L-subset A of an L-preordered set (X, p) is called a lower set (resp., an upper set if $p(b,a)\ast A(a)\le A(b)(\forall a,b\in X)$ (resp., $p(a,b)\ast A(a)\le A(b)(\forall a,b\in X)$). Denote by Low(X) (resp., Upp(X)) the family of all lower sets (resp., upper sets) of X.

Definition 3

Yao and Lu (2009) Suppose that $f:(P,p)⟶(Q,q)$ and $g:(Q,q)⟶(P,p)$ are two monotone mappings between two L-preordered sets. The pair (f, g) is called an L-Galois adjunction from P to Q if$$\begin{array}{c}\hfill q(f(a),b)=p(a,g(b))(\forall a\in P,\forall b\in Q).\end{array}$$

Example 3

Let $f:X⟶Y$ be a mapping. Then $(f_L^{\to },f_L^{\leftarrow })$ is an L-Galois adjunction from $(L^X,{\text{Sub}}_X)$ to $(L^Y,{\text{Sub}}_Y)$, where $f_L^{\to }:L^X⟶L^Y$ and $f_L^{\leftarrow }:L^Y⟶L^X$ are respectively defined by$$\begin{array}{cc}\hfill f_L^{\to }(A)(y)=& \underset{f(x)=y}{\bigvee }A(x),\hfill \\ \hfill f_L^{\leftarrow }(B)=& B\circ f(\forall A\in L^X,\forall B\in L^Y).\hfill \end{array}$$

Definition 4

Chen and Zhang (2010) Suppose that X is a nonempty set. A strong L-fuzzy Alexandrov topology on X is a family $\delta$ of L-subsets of X satisfying that (A1)

$0_X,1_X\in \delta$;

(A2)

${\bigwedge }_i{A}_i,{\bigvee }_i{A}_i\in \delta (\forall \{A_i\mid i\in I\}\subseteq \delta )$;

(A3)

$\lambda \ast A,\lambda \to A\in \delta (\forall \lambda \in L,\forall A\in \delta )$.

The pair $(X,\delta )$ is called a strongL-fuzzy Alexandrov space.

For every L-preordered set X, it is easy to show that both Low(X) and Upp(X) are strong L-fuzzy Alexandrov topologies on X.

Rough approximation operators of strong L-fuzzy Alexandrov spaces

In this section, the pair $(X,\delta )$ always denotes a strong L-fuzzy Alexandrov space.

For each $x\in X$, define $O_x:X⟶L$ by$$\begin{array}{c}\hfill O_x(y)=\underset{A\in \delta }{\bigwedge }A(x)\to A(y)(\forall y\in X).\end{array}$$

Theorem 2

For every $x\in X$, $O_x$ is the minimal open set $B\in \delta$ such that $B(x)\ge e$, called the open-hood of x.

Proof

By (A2, A3), it is easy to show that $O_x\in \delta$ and $O_x(x)\ge e$. Suppose there exists $B\in \delta$ such that $B(x)\ge e$. Then for every $y\in X$, it holds that $O_x(y)\le B(x)\to B(y)\le e\to B(y)=B(y)$, and therefore $O_x\le B$. $\square$

Define two operators $\varphi ,\psi :L^X⟶L^X$ by: $\forall A\in L^X,\forall x\in X$,$$\begin{array}{cc}\hfill \varphi (A)(x)=& \underset{y\in X}{\bigvee }{O}_y(x)\ast A(y);\hfill \\ \hfill \psi (A)(x)=& \underset{y\in X}{\bigwedge }{O}_x(y)\to A(y),\hfill \end{array}$$called the upper approximation operator and the lower approximation operator of X respectively.

Remark 1

(1)

By Theorem 2, for every $x\in X$, the subset $O_x$ is the minimal open set containing x. This feature is also possessed by the equivalence classes in Pawlak approximation space, namely for an equivalence relation on a set X and $x\in X$, the equivalence class [x] is the minimal definable set containing x. In this sense, the family $\{O_x\mid x\in X\}$ in a strong L-fuzzy Alexandrov space can be considered as a kind of fuzzy topological types of approximation spaces.

(2)

In the non-fuzzy setting, for an Alexandrov topology X and $A\subseteq X$, the subset $\varphi (A)$ can be written as ${\bigcup }_{y\in A}{O}_y$ and the subset $\psi (A)$ can be written as $\{x\in X\mid O_x\subseteq A\}$. By (1), $\varphi (A)$ and $\psi (A)$ are just the upper and lower approximations of A respectively. In this sense, the pair $(\varphi ,\psi )$ is a kind of fuzzy topological generalizations of the classical approximation operators.

In the following, we will study the properties of the operators $\varphi ,\psi$. Firstly, let us prove a very useful lemma.

Lemma 1

${\text{Sub}}_X(O_x,O_y)=O_y(x)={\bigvee }_{z\in X}{O}_y(z)\ast O_z(x)(\forall x,y\in X).$

Proof

Firstly, since $O_y(x)\ast O_x(z)\le O_y(z)$, we have $O_y(x)\le O_x(z)\to O_y(z)$, and then $O_y(x)\le {\bigwedge }_{z\in X}{O}_x(z)\to O_y(z)$, which means $O_y(x)\le {\text{Sub}}_X(O_x,O_y)$. Conversely,$$\begin{array}{cc}\hfill \underset{z\in X}{\bigwedge }{O}_x(z)\to O_y(z)\le & O_x(x)\to O_y(x)\hfill \\ \hfill \le & e\to O_y(x)\le O_y(x),\hfill \end{array}$$and thus ${\text{Sub}}_X(O_x,O_y)\le O_y(x)$. Hence, ${\text{Sub}}_X(O_x,O_y)=O_y(x)$.

Secondly, on one hand we have$$\begin{array}{cc}& O_y(z)\ast O_z(x)\hfill \\ \hfill & =\left(\underset{A\in \delta }{\bigwedge },A,(y),\to ,A,(z)\right)\ast \left(\underset{B\in \delta }{\bigwedge },B,(z),\to ,B,(x)\right)\hfill \\ \hfill & \le \underset{A,B\in \delta }{\bigwedge }(A(y)\to A(z))\ast (B(z)\to B(x))\hfill \\ \hfill & \le \underset{A\in \delta }{\bigwedge }(A(y)\to A(z))\ast (A(z)\to A(x))\hfill \\ \hfill & \le \underset{A\in \delta }{\bigwedge }A(y)\to A(x)\hfill \\ \hfill & =O_y(x),\hfill \end{array}$$and then ${\bigvee }_{z\in X}{O}_y(z)\ast O_z(x)\le O_y(x)$. On the other hand,$$\begin{array}{c}\hfill \underset{z\in X}{\bigvee }{O}_y(z)\ast O_z(x)\ge O_y(x)\ast O_x(x)\ge O_y(x)\ast e=O_y(x).\end{array}$$Hence, ${\bigvee }_{z\in X}{O}_y(z)\ast O_z(x)=O_y(x)$. $\square$

Theorem 3

The pair $(\varphi ,\psi )$ is an L-Galois adjunction on $(L^X,{\text{Sub}}_X)$.

Proof

For all $A,B\in L^X$, it holds that$$\begin{array}{cc}& {\text{Sub}}_X(\varphi (A),B)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\varphi (A)(x)\to B(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\left(\underset{y\in X}{\bigvee },A,(y),,\ast ,O_y,(x)\right)\to B(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }(A(y)\ast O_y(x))\to B(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }(A(x)\ast O_x(y))\to B(y)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }A(x)\to (O_x(y)\to B(y))\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }A(x)\to \left(\underset{y\in X}{\bigwedge },O_x,(y),\to ,B,(y)\right)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }A(x)\to \psi (B)(x)\hfill \\ \hfill & ={\text{Sub}}_X(A,\psi (B)).\hfill \end{array}$$$\square$

Theorem 4

Let $(X,\delta )$ be a strong L-fuzzy Alexandrov space. Then (Cl1)

$\varphi (0_X)=0_X$;

(Cl2)

$A\le \varphi (A)(\forall A\in L^X)$;

(Cl3)

$\varphi ({\bigvee }_i{A}_i)={\bigvee }_i\varphi (A_i)(\forall \{A_i\in L^X\mid i\in I\})$;

(Cl4)

$\varphi (\varphi (A))=\varphi (A)(\forall A_i\in L^X)$;

(Cl5)

$\varphi (\lambda \ast A)=\lambda \ast \varphi (A)(\forall \lambda \in L,\forall A\in L^X)$;

(Cl6)

${\text{Sub}}_X(A,B)\le {\text{Sub}}_X(\varphi (A),\varphi (B))(\forall A,B\in L^X)$.

Proof

(Cl1) Clearly $\varphi (0_X)=0_X$. For every $x\in X$ and every $a\in L$,$$\begin{array}{c}\hfill \varphi (a_X)(x)=\underset{y\in X}{\bigvee }{O}_y(x)\ast a_X(y)=\underset{y\in X}{\bigvee }{O}_y(x)\ast a\ge e\ast a=a.\end{array}$$Then $\varphi (a_X)\ge a_X$.

(Cl2) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}\hfill \varphi (A)(x)=& \underset{y\in X}{\bigvee }{O}_y(x)\ast A(y)\ge O_x(x)\ast A(x)\hfill \\ \hfill =& e\ast A(x)=A(x).\hfill \end{array}$$Then $\varphi (A)\ge A$.

(Cl3) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \varphi \left(\underset{i}{\bigvee },A_i\right)(x)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \left(\underset{i}{\bigvee },A_i\right)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }\underset{i}{\bigvee }{O}_y(x)\ast A_i(y)\hfill \\ \hfill & =\underset{i}{\bigvee }\underset{y\in X}{\bigvee }{O}_y(x)\ast A_i(y)\hfill \\ \hfill & =\underset{i}{\bigvee }\varphi (A_i)(x).\hfill \end{array}$$Hence, $\varphi ({\bigvee }_i{A}_i)={\bigvee }_i\varphi (A_i)$.

(Cl4) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \varphi (\varphi (A))(x)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \varphi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \left(\underset{z\in X}{\bigvee },O_z,(y),,\ast ,A,(z)\right)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }\underset{z\in X}{\bigvee }{O}_y(x)\ast O_z(y)\ast A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigvee }{O}_z(x)\ast A(z)\hfill \\ \hfill & =\varphi (A)(x).\hfill \end{array}$$Hence, $\varphi (\varphi (A))=\varphi (A)$.

(Cl5) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \varphi (\lambda \ast A)(x)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast (\lambda \ast A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \lambda \ast A(y)\hfill \\ \hfill & =\lambda \ast \underset{y\in X}{\bigvee }{O}_y(x)\ast A(y)\hfill \\ \hfill & =\lambda \ast \varphi (A)(x).\hfill \end{array}$$Hence, $\varphi (\lambda \ast A)=\lambda \ast \varphi (A)$.

(Cl6) For all $A,B\in L^X$, we have$$\begin{array}{cc}& {\text{Sub}}_X(\varphi (A),\varphi (B))\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\varphi (A)(x)\to \varphi (B)(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\left[\underset{y\in X}{\bigvee },O_y,(x),,\ast ,A,(y)\right]\to \left[\underset{y\in X}{\bigvee },O_y,(x),,\ast ,B,(y)\right]\hfill \\ \hfill & \ge \underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }[O_y(x)\ast A(y)]\to [O_y(x)\ast B(y)]\hfill \\ \hfill & \ge \underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }A(y)\to B(y)\hfill \\ \hfill & ={\text{Sub}}_X(A,B).\hfill \end{array}$$$\square$

Theorem 5

Let $(X,\delta )$ be a strong L-fuzzy Alexandrov space. Then (Int1)

$\psi (1_X)=1_X$;

(Int2)

$\psi (A)\le A(\forall A\in L^X)$;

(Int3)

$\psi ({\bigwedge }_i{A}_i)={\bigwedge }_i\psi (A_i)(\forall \{A_i\in L^X\mid i\in I\})$;

(Int4)

$\psi (\psi (A))=\psi (A)(\forall A_i\in L^X)$.

(Int5)

$\psi (\lambda \to A)=\lambda \to \psi (A)(\forall \lambda \in L,\forall A\in L^X)$;

(Int6)

${\text{Sub}}_X(A,B)\le {\text{Sub}}_X(\psi (A),\psi (B))(\forall A,B\in L^X)$.

Proof

(Int1) Clearly $\psi (1_X)=1_X$. For every $x\in X$ and every $a\in L$,$$\begin{array}{cc}\hfill \psi (a_X)(x)=& \underset{y\in X}{\bigwedge }{O}_x(y)\to a_X(y)=\underset{y\in X}{\bigwedge }{O}_x(y)\to a\hfill \\ \hfill \le & O_x(x)\to a\le e\to a=a.\hfill \end{array}$$Then $\psi (a_X)\le a_X$.

(Int2) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}\hfill \psi (A)(x)=& \underset{y\in X}{\bigwedge }{O}_x(y)\to A(y)\hfill \\ \hfill \le & O_x(x)\to A(x)=e\to A(x)=A(x).\hfill \end{array}$$(Int3) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \psi \left(\underset{i}{\bigwedge },A_i\right)(x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \left(\underset{i}{\bigwedge },A_i\right)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{i}{\bigwedge }{O}_x(y)\to A_i(y)\hfill \\ \hfill & =\underset{i}{\bigwedge }\underset{y\in X}{\bigwedge }{O}_x(y)\to A_i(y)\hfill \\ \hfill & =\underset{i}{\bigwedge }\psi (A_i)(x).\hfill \end{array}$$Hence, $\psi ({\bigwedge }_i{A}_i)={\bigwedge }_i\psi (A_i).$ (Int4) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \psi (\psi (A))(x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \psi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \left(\underset{z\in X}{\bigwedge },O_z,(y),\to ,A,(z)\right)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{z\in X}{\bigwedge }{O}_x(y)\to (O_y(z)\to A(z))\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{z\in X}{\bigwedge }(O_x(y)\ast O_y(z))\to A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigwedge }{O}_x(z)\to A(z)\hfill \\ \hfill & =\psi (A)(x).\hfill \end{array}$$Hence, $\psi (\psi (A))=\psi (A)$.

(Int5) For every $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \psi (\lambda \to A)(x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_y(x)\to (\lambda \to A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_y(x)\to (\lambda \to A(y))\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }(O_y(x)\ast \lambda )\to A(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\lambda \to (O_y(x)\to A(y))\hfill \\ \hfill & =\lambda \to \left(\underset{y\in X}{\bigwedge },O_y,(x),\to ,A,(y)\right)\hfill \\ \hfill & =\lambda \to \psi (A)(x).\hfill \end{array}$$Hence, $\psi (\lambda \to A)=\lambda \to \psi (A)$.

(Int6) For all $A,B\in L^X$, we have$$\begin{array}{cc}& {\text{Sub}}_X(\psi (A),\psi (B))\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\psi (A)(x)\to \psi (B)(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\left[\underset{y\in X}{\bigwedge },O_x,(y),\to ,A,(y)\right]\to \left[\underset{y\in X}{\bigwedge },O_x,(y),\to ,B,(y)\right]\hfill \\ \hfill & \ge \underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }[O_x(y)\to A(y)]\to [O_x(y)\to B(y)]\hfill \\ \hfill & \ge \underset{x\in X}{\bigwedge }\underset{y\in X}{\bigwedge }A(y)\to B(y)\hfill \\ \hfill & ={\text{Sub}}_X(A,B).\hfill \end{array}$$$\square$

Theorem 6

Let $(X,\delta )$ be a strong L-fuzzy Alexandrov space. (1)

$\varphi \circ \varphi =\varphi =\psi \circ \varphi$;

(2)

$\psi \circ \psi =\psi =\varphi \circ \psi$.

Proof

The first equality in each of (1–2) is just (Cl4) and (Int4) respectively. For the second equality, we give the following proof.

(1) By (Int2), we have $\varphi \ge \psi \circ \varphi$. Thus we only need to prove $\varphi \le \psi \circ \varphi$. In fact, for any $A\in L^X$ and $x\in X$, we have$$\begin{array}{cc}& \psi (\varphi (A))(x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \varphi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \left(\underset{z\in X}{\bigvee },O_z,(y),,\ast ,A,(z)\right)\hfill \\ \hfill & \ge \underset{y\in X}{\bigwedge }\underset{z\in X}{\bigvee }{O}_x(y)\to (O_z(y)\ast A(z))\hfill \\ \hfill & \ge \underset{y\in X}{\bigwedge }\underset{z\in X}{\bigvee }(O_x(y)\to O_z(y))\ast A(z)\hfill \\ \hfill & \ge \underset{y\in X}{\bigwedge }\underset{z\in X}{\bigvee }{\text{Sub}}_X(O_x,O_z)\ast A(z)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{z\in X}{\bigvee }{O}_z(x)\ast A(z)\hfill \\ \hfill & =\varphi (A)(x).\hfill \end{array}$$Hence, $\varphi =\psi \circ \varphi .$

(2) By (Cl2), we have $\psi \le \varphi \circ \psi$. Thus we only need to prove $\psi \ge \varphi \circ \psi$.

In fact, for any $A\in L^X,x\in X,$ we have$$\begin{array}{cc}& \varphi (\psi (A))(x)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \psi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \left(\underset{z\in X}{\bigwedge },O_y,(z),\to ,A,(z)\right)\hfill \\ \hfill & \le \underset{y\in X}{\bigvee }\underset{z\in X}{\bigwedge }{O}_y(x)\ast (O_y(z)\to A(z))\hfill \\ \hfill & =\underset{y\in X}{\bigvee }\underset{z\in X}{\bigwedge }{\text{Sub}}_X(O_x,O_y)\ast (O_y(z)\to A(z))\hfill \\ \hfill & \le \underset{y\in X}{\bigvee }\underset{z\in X}{\bigwedge }(O_x(z)\to O_y(z))\ast (O_y(z)\to A(z))\hfill \\ \hfill & \le \underset{y\in X}{\bigvee }\underset{z\in X}{\bigwedge }{O}_x(z)\to A(z)\hfill \\ \hfill & =\psi (A)(x).\hfill \end{array}$$Hence we have $\psi =\varphi \circ \psi .$$\square$

Theorem 7

The following statements are equivalent: (1)

$A\in \delta$;

(2)

$\psi (A)=A$;

(3)

$\varphi (A)=A$;

(4)

$A(x)={\text{Sub}}_X(O_x,A)(\forall x\in X)$.

Proof

By the definition of $\psi$, (2) is equivalent to (4).

(1) $⟹$ (2) Firstly, by (Int2), it is obviously $\psi (A)\le A$. Secondly, for every $x\in X$, we have$$\begin{array}{cc}& \psi (A)(x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to A(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\left(\underset{B\in \delta }{\bigwedge },B,(x),\to ,B,(y)\right)\to A(y)\hfill \\ \hfill & \ge \underset{y\in X}{\bigwedge }(A(x)\to A(y))\to A(y)\hfill \\ \hfill & \ge A(x).\hfill \end{array}$$Thus $\psi (A)=A$.

(2) $⟹$ (3) By $\psi =\varphi \circ \psi$ and $\psi (A)=A$, we have $\varphi (A)=A$.

(3) $⟹$ (1) For every $y\in X$, we have $O_y\in \delta$ and then $A(y)\ast O_y\in \delta$. By (A2), $A=\varphi (A)={\bigvee }_{y\in X}{O}_y\ast A(y)={\bigvee }_{y\in X}A(y)\ast O_y\in \delta$. $\square$

Remark 2

For an L-fuzzy topological space $(X,\delta )$, the related interior operator is computed as $\text{Int}(A)=\bigvee \{B\in \delta \mid B\le A\}(\forall A\in L^X)$ (Höhle and Sostak 1999). We can show, for a strong L-fuzzy Alexandrov topological space $(X,\delta )$, $\psi =\text{Int}$.

Proof

For every $A\in L^X$ and every $x\in X$,$$\begin{array}{cc}& \text{Int}(A)(x)\hfill \\ \hfill & =\bigvee \{B(x)\mid B\in \delta ,B\le A\}\hfill \\ \hfill & \ge ({\text{Sub}}_X(O_x,A)\ast O_x)(x)\hfill \\ \hfill & \ge {\text{Sub}}_X(O_x,A)\ast e\hfill \\ \hfill & ={\text{Sub}}_X(O_x,A)\hfill \\ \hfill & =\psi (A)(x).\hfill \end{array}$$Conversely, for every $B\in \delta$ with $B\le A$, we have$$\begin{array}{cc}\hfill B(x)=& {\text{Sub}}_X(O_x,B)\le {\text{Sub}}_X(O_x,B)\ast {\text{Sub}}_X(B,A)\hfill \\ \hfill \le & {\text{Sub}}_X(O_x,A)=\psi (A)(x).\hfill \end{array}$$By the arbitrariness of B, $\text{Int}(A)\le \psi (A)$. Hence, $\psi =\text{Int}$.

$\square$

It has been explained in the last section of Yao and Han (2023) that: for a fuzzy Alexandrov topology $\delta$, it would be hard to get some similar results as those in Yao and Han (2023) unless $a\to b=b^{\prime }\to a^{\prime }$. We now will apply this condition on the commutative unital quantale L to get a dual result of Remark 2.

A commutative unital quantale called Girard if there is an element $d\in L$ such that $(\lambda \to d)\to d=\lambda$ holds for all $\lambda \in L$. The element d is called a dual cyclic element.

Proposition 8

Let L be a Girard commutative unital quantale with d as a dual cyclic element of L. Denote ${\lambda }^{\prime }=\lambda \to d(\forall d\in L)$. Then for all $a,b\in L$ and all $\{a_i\mid i\in I\}\subseteq L$, it holds that: (G1)

$a\to b=b^{\prime }\to a^{\prime }$;

(G2)

$a\to b={(a\ast b^{\prime })}^{\prime }$;

(G3)

${({\bigwedge }_i{a}_i)}^{\prime }={\bigvee }_i{a}_i^{\prime }$.

Proof

Since L is Girard and d is a dual cyclic element, we have ${\lambda }^{\prime \prime }=\lambda (\forall \lambda \in L)$.

1. $b^{\prime }\to a^{\prime }=a\to b^{\prime \prime }=a\to b$.

2. ${(a\ast b^{\prime })}^{\prime }=(a\ast b^{\prime })\to d=a\to b^{\prime \prime }=a\to b$.

3. ${({\bigwedge }_i{a}_i)}^{\prime }={({\bigwedge }_i{a}_i^{\prime \prime })}^{\prime }={[{({\bigvee }_i{a}_i^{\prime })}^{\prime }]}^{\prime }={\bigvee }_i{a}_i^{\prime }$.

$\square$

Remark 3

For an L-fuzzy cotopological space $(X,\delta )$, the related closure operator is computed as $\text{Cl}(A)=\bigwedge \{B\in \delta \mid$$A\le B\}(\forall A\in L^X)$. We can show, for a strong L-fuzzy Alexandrov topological space $(X,\delta )$, $\varphi =\text{Cl}$.

Proof

For all $D_1,D_2\in L^X$, denote $\text{Mt}(D_1,D_2)={\bigvee }_{y\in X}{D}_1(y)\ast D_2(y)$. For every $x\in X$, it easily seen that $A\le C_x\to \text{Mt}(C_x,A)$ and then we would like to show that $C_x\to \text{Mt}(C_x,A)\in \delta$. In fact, for every $y\in X$, we have$$\begin{array}{cc}\hfill C_x(y)\to & \text{Mt}(C_x,A)\hfill \\ \hfill =& {[\text{Mt}(C_x,A)]}^{\prime }\to C_x{(y)}^{\prime }\hfill \\ \hfill =& {[\text{Mt}(C_x,A)]}^{\prime }\to O_y{(x)}^{\prime }\hfill \\ \hfill =& {[\text{Mt}(C_x,A)]}^{\prime }\to {\left[\underset{D\in \delta }{\bigwedge },D,(y),\to ,D,(x)\right]}^{\prime }\hfill \\ \hfill =& {[\text{Mt}(C_x,A)]}^{\prime }\to \underset{D\in \delta }{\bigvee }{(D(y)\to D(x))}^{\prime }\hfill \\ \hfill =& {[\text{Mt}(C_x,A)]}^{\prime }\to \underset{D\in \delta }{\bigvee }D{(x)}^{\prime }\ast D(y).\hfill \end{array}$$Thus $C_x\to \text{Mt}(C_x,A)={[\text{Mt}(C_x,A)]}^{\prime }\to \underset{D\in \delta }{\bigvee }D{(x)}^{\prime }\ast D\in \delta$, as desired.

Then$$\begin{array}{cc}& \text{Cl}(A)(x)\hfill \\ \hfill & =\bigwedge \{B\in \delta \mid A\le B\}(\forall A\in L^X)\hfill \\ \hfill & \le [C_x\to \text{Mt}(C_x,A)](x)\hfill \\ \hfill & \le e\to \text{Mt}(C_x,A)\hfill \\ \hfill & =\text{Mt}(C_x,A)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{C}_x(y)\ast A(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast A(y)\hfill \\ \hfill & =\varphi (A)(x).\hfill \end{array}$$Conversely, for every $B\in \delta$ with $A\le B$,$$\begin{array}{cc}\hfill \varphi (A)(x)=& \underset{y\in X}{\bigvee }{O}_y(x)\ast A(y)\le \underset{y\in X}{\bigvee }{O}_y(x)\ast B(y)\hfill \\ \hfill \le & \underset{y\in X}{\bigvee }[B(y)\to B(x)]\ast B(y)\le B(x).\hfill \end{array}$$By the arbitrariness of $B\in \delta$, we have $\varphi (A)(x)\le \text{Cl}(A)(x)$. Hence, $\varphi =\text{Cl}$. $\square$

Composition and decomposition of rough approximation operators

As is well known, the upper and lower approximations of classical sets are also subsets of the same universes. From the definitions of zooming-in and zooming-out operations, we can combine them to derive some new operators in the same universes. In fact, not only we can obtain approximation operators of a reflexive approximation space, but we can also obtain approximation operators of a granulated universe of discourse. In this section, we will study composition and decomposition problems of rough approximation operators of strong L-fuzzy Alexandrov spaces.

Let $(X,\delta )$ be a strong L-fuzzy Alexandrov space and let$$\begin{array}{c}\hfill \mathcal{O}(X)=\{O_p\mid p\in X\}.\end{array}$$Then $\mathcal{O}(X)$ is preordered set under the set-inclusion.

Define a zooming-in operator$\omega :L^{\mathcal{O}(X)}⟶L^X$ by$$\begin{array}{c}\hfill \omega (P)(x)=P(O_x).\end{array}$$Define two zooming-out operators$\overline{\mathit{apr}},\underline{\mathit{apr}}:L^X⟶L^{\mathcal{O}(X)}$ by$$\begin{array}{cc}\hfill \overline{\mathit{apr}}(A)(O_x)=& \underset{y\in X}{\bigvee }{O}_y(x)\ast A(y)(\forall x,y\in X),\hfill \\ \hfill \underline{\mathit{apr}}(A)(O_x)=& \underset{y\in X}{\bigwedge }{O}_x(y)\to A(y)(\forall x,y\in X).\hfill \end{array}$$

Theorem 9

Let $(X,\delta )$ be a strong L-fuzzy Alexandrov space. Then (1)

$\omega \circ \overline{\mathit{apr}}=\varphi$;

(2)

$\omega \circ \underline{\mathit{apr}}=\psi$;

(3)

$\overline{\mathit{apr}}\circ \varphi =\overline{\mathit{apr}}$;

(4)

$\underline{\mathit{apr}}\circ \psi =\underline{\mathit{apr}}$.

Proof

Let $A\in \delta$, $x\in X$.

1. $\omega (\overline{\mathit{apr}}(A))(x)=\overline{\mathit{apr}}(A)(O_x)={\bigvee }_{y\in X}{O}_y(x)\ast A(y)=\varphi (A)(x).$

2. $\omega (\underline{\mathit{apr}}(A))(x)=\underline{\mathit{apr}}(A)(O_x)={\bigwedge }_{y\in X}{O}_x(y)\to A(y)=\psi (A)(x).$

3. $$\begin{array}{cc}& \overline{\mathit{apr}}(\varphi (A))(O_x)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast \varphi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigvee }{O}_y(x)\ast (\underset{z\in X}{\bigvee }{O}_z(y)\ast A(z))\hfill \\ \hfill & =\underset{y\in X}{\bigvee }\underset{z\in X}{\bigvee }(O_y(x)\ast O_z(y))\ast A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigvee }(\underset{y\in X}{\bigvee }{O}_y(x)\ast O_z(y))\ast A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigvee }{O}_z(x)\ast A(z)\hfill \\ \hfill & =\overline{\mathit{apr}}(A)(O_x).\hfill \end{array}$$

4. $$\begin{array}{cc}& \underline{\mathit{apr}}(\psi (A))(O_x)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to \psi (A)(y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }{O}_x(y)\to (\underset{z\in X}{\bigwedge }{O}_y(z)\to A(z))\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{z\in X}{\bigwedge }{O}_x(y)\to (O_y(z)\to A(z))\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }\underset{z\in X}{\bigwedge }(O_x(y)\ast O_y(z))\to A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigwedge }(\underset{y\in X}{\bigvee }{O}_x(y)\ast O_y(z))\to A(z)\hfill \\ \hfill & =\underset{z\in X}{\bigwedge }{O}_x(z)\to A(z)\hfill \\ \hfill & =\psi (A)(O_x).\hfill \end{array}$$

$\square$

Theorem 10

Let $A\in L^X$. Then $A\in \delta$ if and only if that there is a lower set P of $\mathcal{O}(X)$ such that $A=\omega (P)$.

Proof

The necessity: Let $A=\omega (P)$. We have$$\begin{array}{cc}& {\text{Sub}}_X(O_y,O_x)\ast P(O_x)\hfill \\ \hfill & ={\text{Sub}}_X(O_y,O_x)\ast A(x)\hfill \\ \hfill & =O_x(y)\ast A(x)\hfill \\ \hfill & =\left(\underset{B\in \delta }{\bigwedge },B,(x),\to ,B,(y)\right)\ast A(x)\hfill \\ \hfill & \le (A(x)\to A(y))\ast A(x)\hfill \\ \hfill & \le A(y)=P(O_y),\hfill \end{array}$$which indicates P is a lower set of $\mathcal{O}(X)$.

The sufficiency: By Theorem 7, we only need to prove $A(x)={\text{Sub}}_X(O_x,A)(\forall x\in X)$. On one hand,$$\begin{array}{c}\hfill {\text{Sub}}_X(O_x,A)\le O_x(x)\to A(x)\le e\to A(x)=A(x).\end{array}$$On the other hand, since P is a lower set of $\mathcal{O}(X)$ and $A=\omega (P)$, we have ${\text{Sub}}_X(O_y,O_x)\ast P(O_x)\le P(O_y)$ and then $P(O_x)\le {\text{Sub}}_X(O_y,O_x)\to P(O_y)$. Thus$$\begin{array}{cc}\hfill {\text{Sub}}_X(O_x,A)=& \underset{y\in X}{\bigwedge }{O}_x(y)\to A(y)\hfill \\ \hfill =& \underset{y\in X}{\bigwedge }\text{Sub}(O_y,O_x)\to P(O_y)\ge P(O_x).\hfill \end{array}$$Hence, $A(x)={\text{Sub}}_X(O_x,A)$. $\square$

Lemma 2

Let (X, p) be an L-preordered set and A is a lower set. Then for every $x\in X$,$$\begin{array}{c}\hfill \underset{y\in X}{\bigvee }A(y)\ast p(x,y)=A(x)=\underset{y\in X}{\bigwedge }p(y,x)\to A(y).\end{array}$$

By Lemma 2, for a strong L-fuzzy Alexandrov space $(X,\delta )$ and for every lower set A of $(\delta ,{\text{Sub}}_X)$, we have$$\begin{array}{cc}& \underset{O_y\in \mathcal{O}(X)}{\bigvee }P(O_y)\ast O_y(x)\hfill \\ \hfill & =P(O_x)=\underset{O_y\in \mathcal{O}(X)}{\bigwedge }{O}_x(y)\to P(O_y)(\forall x\in X).\hfill \end{array}$$

Theorem 11

The operators $\overline{\mathit{apr}},\underline{\mathit{apr}}$ are respectively the left and right adjoint of $\omega$ for $L^X$ and $Low(\mathcal{O}(X))$.

Proof

Let P be a lower set of $\mathcal{O}(X)$ and $A\in L^X$. Then$$\begin{array}{cc}& {\text{Sub}}_{\mathcal{O}(\mathcal{X})}(\overline{\mathit{apr}}(A),P)\hfill \\ \hfill & =\underset{O_x\in \mathcal{O}(X)}{\bigwedge }\overline{\mathit{apr}}(A)(O_x)\to P(O_x)\hfill \\ \hfill & =\underset{O_x\in \mathcal{O}(X)}{\bigwedge }\left(\underset{y\in X}{\bigvee },O_y,(x),,\ast ,A,(y),\to ,P,(O_x)\right)\hfill \\ \hfill & =\underset{O_x\in \mathcal{O}(X)}{\bigwedge }\underset{y\in X}{\bigwedge }(O_y(x)\ast A(y))\to P(O_x)\hfill \\ \hfill & =\underset{O_x\in \mathcal{O}(X)}{\bigwedge }\underset{y\in X}{\bigwedge }A(y)\to (O_y(x)\to P(O_x))\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }A(y)\to \left(\underset{O_x\in \mathcal{O}(X)}{\bigwedge },O_y,(x),\to ,P,(O_x)\right)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }A(y)\to P(O_y)\hfill \\ \hfill & =\underset{y\in X}{\bigwedge }A(y)\to \omega (P)(y)\hfill \\ \hfill & ={\text{Sub}}_X(A,\omega (P))\hfill \end{array}$$and$$\begin{array}{cc}& {\text{Sub}}_{\mathcal{O}(X)}(P,\underline{\mathit{apr}}(A))\hfill \\ \hfill & =\underset{O_y\in \mathcal{O}(X)}{\bigwedge }P(O_y)\to \underline{\mathit{apr}}(A)(O_y)\hfill \\ \hfill & =\underset{O_y\in \mathcal{O}(X)}{\bigwedge }P(O_y)\to (\underset{x\in X}{\bigwedge }{O}_y(x)\to A(x))\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\underset{O_y\in \mathcal{O}(X)}{\bigwedge }(P(O_y)\ast O_y(x))\to A(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }(\underset{O_y\in \mathcal{O}(X)}{\bigvee }P(O_y)\ast O_y(x))\to A(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }P(O_x)\to A(x)\hfill \\ \hfill & =\underset{x\in X}{\bigwedge }\omega (P)(x)\to A(x)\hfill \\ \hfill & ={\text{Sub}}_X(\omega (P),A).\hfill \end{array}$$$\square$

Conclusions and further work

For a commutative unital quantale L and a strong L-fuzzy Alexandrov topological space X, we defined the open-hood $O_x$ for every point $x\in X$. Considering $\{O_x\mid x\in X\}$ as the family of basic granules, we defined the upper and lower approximation operators $\varphi ,\psi$. It is shown that $\varphi ,\psi$ are a kind of a closure operator and an interior operator respectively, and form an L-Galois connection on $L^X$. By introducing notions of one zooming-in operator and two zooming out operators, we studied the composition and decomposition problems of $\varphi ,\psi$.

We have some further work:

1. In Yao and Han (2023), by an Alexandrov space and by using the open-hoods and closed-hoods, we define four rough approximation operators, while in this paper, on two rough approximation operators related to the open-hoods have been defined for strong L-fuzzy Alexandrov topological space. We can further define the closed-hood $C_x$ for every point, define another two rough approximation operators and then study the mutual interactions among them.

2. It is shown in Yao and Han (2023), rough set model of classical Alexandrov topology can be considered as a common framework of the preordered relation model, the neighborhood operator model and the covering model. We can further study this problem, to show that the rough set model of strong L-fuzzy Alexandrov topology can be considered as a common framework of the L-preordered relation model, certain types of neighborhood operator model and certain types of the covering model.

Acknowledgements

This paper is supported by the NNSF of China (1237146, 12231007), Jiangsu Provincial Innovative and Entrepreneurial Talent Support Plan (JSSCRC2021521).

Author Contributions

All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Funding

Funding was supported by National Natural Science Foundation of China (1237146, 12231007), Jiangsu Provincial Innovative and Entrepreneurial Talent Support Plan (JSSCRC2021521).

Data availability

All data, models, or code generated or used during the study are available from the corresponding author by request.

Declarations

Conflict of interest

The authors have no conflict of interest to declare that are relevant to the content of this article.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Not applicable.

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Not applicable.

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Informed consent was obtained from all individual participants included in the study.

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