On α-Multiplier on Almost Distributive Lattices

Abstract

In this paper, we initiate the concept of $\alpha$-multiplier on almost distributive lattices. We prove some useful results by using the notion of $\alpha$-multiplier and generalize the idea of multiplier on almost distributive lattices.

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1. Introduction

A lattice is an advanced abstract structure that has been studied in abstract algebra during last few decades. Birkhoff introduced the concept of lattice theory in 1940 [1]. Lattice is generalization of Boolean and fuzzy algebras. Latter on Gratzer and Schmidt worked together and showed their interest in the development of lattice theory [2]. In 1955, Helgason introduced the concept of multiplier in Banach Algebra [3]. The idea of multiplier in lattice was given by Larsen [4] in 1971, and Cornish extended this concept of multiplier in distributive lattice [5].

In 1981, the idea of ADLs was initiated by Swamy and Rao [6]. An almost distributive lattice satisfies all the properties of distributive lattice except commutativity of $\wedge$ and $\vee$ and right distributivity of $\vee$ over $\wedge$. Recently, Kim has introduced the idea of a multiplier in ADLs [7] and discussed some fundamental properties of this notion. For detailed study of the subject, we refer to readers [8–10].

Now, we have generalized certain properties of $\alpha$-multiplier. The notion of $\alpha$-multiplier for an almost distributive lattice is introduced, and some related properties are investigated. Moreover, we introduced principle $\alpha$-multiplier and isotone $\alpha$-multiplier on almost distributive lattices.

2. Preliminaries

Definition 1.

(see [6]). An algebra $G,\wedge ,\vee ,0$ is said to be an almost distributive lattice if it satisfies the following:

(i) $e_1\vee 0=e_1$

(ii) $0\wedge e_1=0$

(iii) $e_1\vee h_1\wedge {\omega }_1=e_1\wedge {\omega }_1\vee h_1\wedge {\omega }_1$

(iv) $e_1\wedge h_1\vee {\omega }_1=e_1\wedge h_1\vee e_1\wedge {\omega }_1$

(v) $e_1\vee h_1\wedge {\omega }_1=e_1\vee h_1\wedge e_1\vee {\omega }_1$

(vi) $e_1\vee h_1\wedge h_1=h_1$

(vii) $e_1\vee h_1\wedge e_1=e_1$

(viii) $e_1\vee h_1\wedge e_1=e_1$, $\forall e_1$, $h_1$, ${\omega }_1\in G$

Lemma 1.

(see [6]). Let $G$ be an almost distributive lattice. For any $e_1$, $h_1\in G$, we have

(i) $e_1\vee e_1=e_1$

(ii) $e_1\vee e_1=e_1$

(iii) $e_1\wedge h_1\vee h_1=h_1$

(iv) $e_1\wedge e_1\vee h_1=e_1$

(v) $e_1\vee h_1\wedge e_1=e_1$

(vi) $e_1\vee h_1=e_1\iff e_1\wedge h_1=h_1$

(vii) $e_1\vee h_1=h_1\iff e_1\wedge h_1=e_1$

Definition 2.

(see [6]). For any $e_1$, $h_1\in G$, we say that $e_1\le h_1$ if $e_1\wedge h_1=e_1$ or equivalently, $e_1\vee h_1=h_1$.

Lemma 2.

(see [6]). Let $G$ be an almost distributive lattice. For any $e_1$, $h_1$, ${\omega }_1$, $h_3\in G$, then the following identities hold:

(i) $e_1\wedge h_1\le h_1$ and $e_1\le e_1\vee h_1$

(ii) $e_1\wedge h_1=h_1\wedge e_1$ whenever $e_1\le h_1$

(iii) $e_1\vee h_1\vee {\omega }_1\wedge h_3=e_1\vee h_1\vee {\omega }_1\wedge h_3$

(iv) $e_1\le h_1\Rightarrow e_1\wedge {\omega }_1\le h_1\wedge {\omega }_1,{\omega }_1\wedge e_1\le {\omega }_1\wedge h_1$ and ${\omega }_1\vee e_1\le {\omega }_1\vee h_1$

Definition 3.

(see [6]). Let $G$ be a lattice, and 0 is known as a zero element of a lattice $G$ if $0\wedge e_1=0,\forall e_1\in G$.

Lemma 3.

(see [6]). Let $G$ be an almost distributive lattice. If $G$ has 0, then for any $e_1$, $h_1\in G$, the following identities hold:

(i) $e_1\vee 0=e_1$ and $0\vee e_1=e_1$

(ii) $e_1\wedge 0=0$

(iii) $e_1\wedge h_1=0$ if and only if $h_1\wedge e_1=0$

Definition 4.

(see [6]). Let $I$ be a nonempty subset of $G$ which is called an ideal of $G$ if $h_3\vee {\omega }_3\in I$ and $h_3\wedge e_3\in I$ whenever $h_3$, ${\omega }_3\in I$ and $e_3\in G$.

If $I$ is an ideal of $G$ and $e_3$, $h_3\in G$, then $e_3\wedge h_3\in I$ if and only if $h_3\wedge e_3\in I$.

Lemma 4.

(see [6]). For any $e_1$, $h_1\in G$, we have

(i) $e_1\wedge h_1\vee h_1=h_1$

(ii) $e_1\vee e_1\wedge h_1=e_1=e_1\wedge e_1\vee h_1$

(iii) $e_1\vee h_1\wedge e_1=e_1=e_1\vee h_1\wedge e_1$

Definition 5.

(see [7]). Let $G$ be an almost distributive lattice and ${\zeta }_1\text{\hspace{0.17em}and\hspace{0.17em}}{\zeta }_2$ be two self maps. We define ${\zeta }_1\vee {\zeta }_2:G⟶G$ by ${\zeta }_1\vee {\zeta }_2{e}_1={\zeta }_1{e}_1\vee {\zeta }_2{e}_1,\forall e_1\in G$.

Definition 6.

(see [7]). Let $G_1$ and $G_2$ be two almost distributive lattices. Then, $G_1\times G_2$ is also an $\text{ADL}$ with respect to the pointwise operation given by $e_1,h_1\wedge {\omega }_1,x=e_1\wedge {\omega }_1,h_1\wedge x\text{\hspace{0.17em}and\hspace{0.17em}}{e}_1,h_1\vee {\omega }_1,x=e_1\vee {\omega }_1,h_1\vee x,\forall e_1,h_1\in G_1$, ${\omega }_1,x\in G_2.$

Definition 7.

(see [7]). Let $G$ be an almost distributive lattice and $\zeta$ be a multiplier of $G$. Define a set ${\text{fix}}_{\zeta }G$ by ${\text{fix}}_{\zeta }G=e_1\in G:\zeta e_1=e_1$.

Definition 8.

(see [7]). Let $G,\wedge ,\vee ,0$ be an almost distributive lattice. For any $u\in G$, define ${\mathrm{\Gamma }}_u=e_1,h_1\in G\times G|f_u{e}_1=f_u{h}_1$, where $f_u$ is a principle multiplier induced by $u\in G$.

3. $\alpha$-Multiplier on Almost Distributive Lattices

Definition 9.

Let $G$ be an almost distributive lattice. A function $\zeta :G⟶G$ is called $\alpha$-multiplier if $\zeta e_1\wedge h_1=\zeta e_1\wedge \alpha h_1$$\forall e_1$, $h_1\in G$, where $\alpha$ is a mapping on $G$.

Example 1.

Let $G$ be an almost distributive lattice with $0\in G$. A function $\zeta$ defined by $\zeta e_1=0$$\forall e_1\in G$ is called zero $\alpha$-multiplier.

Proof.

Let $G$ be an almost distributive lattice with $0\in G$; then, we have to prove that $\zeta$ is a zero $\alpha$-multiplier.

Let $e_1,h_1\in G$, $\zeta e_1\wedge h_1=\zeta e_1\wedge \alpha h_1=0\wedge \alpha h_1=0$. Hence, $\zeta$ is a zero $\alpha$-multiplier.

Lemma 5.

Let $\zeta$ be $\alpha$-multiplier of $G$. If $\alpha :G⟶G$ is homomorphism, then following conditions hold:

(i) $\zeta e_1\le \alpha e_1$

(ii) $\zeta e_1\wedge \zeta h_1\le \zeta e_1\wedge h_1$, $\forall e_1,h_1\in G$

Proof.

(i) Since $\zeta e_1=\zeta e_1\wedge e_1=\zeta e_1\wedge \alpha e_1$, it implies that $\zeta e_1\le \alpha e_1$.

(ii) Let $e_1$, $h_1\in G$. We have to show that $\zeta e_1\wedge \zeta h_1\le \zeta e_1\wedge h_1$.$$\begin{array}{c}\text{(1)}& \zeta e_1\le \alpha e_1,\zeta h_1\le \alpha h_1,\end{array}$$

$\zeta e_1\wedge \zeta h_1\le \zeta e_1\wedge \alpha h_1=\zeta e_1\wedge h_1$. It implies $\zeta e_1\wedge \zeta h_1\le \zeta e_1\wedge h_1$.

Definition 10.

Let $G$ be an almost distributive lattice. A function $K_e$ is defined by $K_{e}g=e\wedge ag,\forall g\in G$, where $\alpha :G⟶G$ is a homomorphism, then $K_e$ is $\alpha -$multiplier of $G$, and such $\alpha -$multiplier of $G$ is called a principle $\alpha -$multiplier of $G$.

Definition 11.

Let $G,\wedge ,\vee ,0$ be an almost distributive lattice and $K$ be $\alpha$-multiplier on $G$. For any $u\in G$, define ${\mathrm{\Gamma }}_u=e_1,h_1\in G\times G|k_u{e}_1=k_u{h}_1$, where $K_u$ is a principle $\alpha$-multiplier induced by $u\in G$.

Lemma 6.

Let $G$ be an almost distributive lattice. A function $K_{e_3}$ is defined by $K_{e_3}a=e_3\wedge \alpha a$, where $\alpha :G⟶G$ is a homomorphism, then $K_{e_3}$ is $\alpha$-multiplier of $G$, and such $\alpha$-multiplier of $G$ is called a principle $\alpha$-multiplier of $G$.

Proof.

Let $h_3$, ${\omega }_3$, $e_3\in G$, and $K$ be an $\alpha$-multiplier; then, we have to prove that $K_{e_3}$ is an $\alpha$-multiplier.$$\begin{array}{}\text{(2)}& K_e{h}_3=e\wedge \alpha h_3,\hspace{1em}\forall h_3\in G,\end{array}$$$K_e{h}_3\wedge {\omega }_3=e_3\wedge \alpha h_3\wedge {\omega }_3=e_3\wedge \alpha h_3\wedge \alpha {\omega }_3=e_3\wedge \alpha h_3\wedge \alpha {\omega }_3=K_{e_3}{h}_3\wedge \alpha {\omega }_3$. This implies that $K_{e_3}$ is an $\alpha$-multiplier.

Definition 12.

Let $G$ be an almost distributive lattice and $\zeta$ be $\alpha -$multiplier on $G$, where $\alpha$ is a mapping on $G$. If for $e_1\le h_1$ implies $\zeta e_1\le \zeta h_1$, then $\zeta$ is an isotone $\alpha -$multiplier.

Proposition 1.

Let $G$ be an almost distributive lattice. If $\alpha :G⟶G$ is an increasing homomorphism, then ${\zeta }_a{h}_2=a\wedge \alpha h_2$, and $\forall h_2\in G$ is an isotone $\alpha$-multiplier of $G$.

Proof.

Let $G$ be an $\text{ADL}$ and $h_2,{\omega }_2\in G$, with $h_2\le {\omega }_2$ such that $\alpha h_2\le \alpha {\omega }_2$, then we have ${\zeta }_a{h}_2\wedge {\omega }_2=a\wedge \alpha h_2\wedge \alpha {\omega }_2=a\wedge \alpha h_2\wedge a\wedge \alpha {\omega }_2={\zeta }_a{h}_2\wedge {\zeta }_a{\omega }_2$. It implies ${\zeta }_a{h}_2\le {\zeta }_a{\omega }_2$. Hence, ${\zeta }_a$ is an isotone $\alpha$-multiplier.

Lemma 7.

Let $G$ be an almost distributive lattice and $\zeta$ be an $\alpha$-multiplier of $G$ and $\alpha$ be an increasing homomorphism on $G$. If $h_3\le {\omega }_3$ and $\zeta {\omega }_3=\alpha {\omega }_3$, then $\zeta h_3=\alpha h_3$.

Proof.

Let $h_3$, ${\omega }_3\in G$ for $h_3\le {\omega }_3$. Since $\alpha$ is an increasing homomorphism, so$$\begin{array}{c}\text{(3)}& \alpha h_3\le \alpha {\omega }_3,\zeta {\omega }_3=\alpha {\omega }_3.\end{array}$$

By using equation (3), we have $\zeta h_3=\zeta {\omega }_3\wedge h_3=\zeta {\omega }_3\wedge \alpha h_3=\alpha {\omega }_3\wedge \alpha h_3$ since $\alpha$ is an increasing homomorphism. It implies that $\zeta h_3=\alpha h_3$.

Theorem 1.

Let $G$ be an almost distributive lattice and $\zeta$ be an $\alpha$-multiplier of $G$ and $\alpha$ be an increasing homomorphism on $G$. Then, $\zeta$ is an isotone $\alpha$-multiplier.

Proof.

Suppose $h_3$, ${\omega }_3\in G$. By using Lemma 5 (i), we have$$\begin{array}{c}\text{(4)}& \zeta {\omega }_3\le \alpha {\omega }_3,\text{(5)}& \text{for\hspace{0.17em}}{h}_3\le {\omega }_3,\alpha h_3\le \alpha {\omega }_3.\end{array}$$

Since $h_3\le {\omega }_3$, therefore, we have $\zeta h_3=\zeta {\omega }_3\wedge h_3=\zeta {\omega }_3\wedge \alpha h_3$. By equations (4) and (5), we have $\zeta h_3\le \zeta {\omega }_3\wedge \alpha {\omega }_3=\zeta {\omega }_3$. It implies $\zeta h_3\le \zeta {\omega }_3$. Hence, $\zeta$ is an $\text{isotone\hspace{0.17em}}\alpha$-multiplier.

Proposition 2.

Let $G$ be an almost distributive lattice, $\zeta$ be an $\alpha$-multiplier of $G$, and $\alpha$ be homomorphism on $G$. Then, $\zeta {\omega }_1\vee e_1=\zeta {\omega }_1\vee \zeta e_1$, $\forall {\omega }_1$, $e_1\in G$.

Proof.

Let ${\omega }_1$, $e_1\in G$ and $\zeta$ be an $\alpha$-multiplier of $G$; then, we have to show that $\zeta {\omega }_1\vee e_1=\zeta {\omega }_1\vee \zeta e_1$. By Definition 1, we have $\zeta {\omega }_1\vee \zeta e_1=\zeta {\omega }_1\vee e_1\wedge {\omega }_1\vee \zeta {\omega }_1\vee e_1\wedge e_1=\zeta {\omega }_1\vee e_1\wedge \alpha {\omega }_1\vee \zeta {\omega }_1\vee e_1\wedge \alpha e_1$. By Definition 1, we have $\zeta {\omega }_1\vee \zeta e_1=\zeta {\omega }_1\vee e_1\wedge \alpha {\omega }_1\vee \alpha e_1=\zeta {\omega }_1\vee e_1\wedge \alpha {\omega }_1\vee e_1=\zeta {\omega }_1\vee e_1$. It implies that $\zeta {\omega }_1\vee \zeta e_1=\zeta {\omega }_1\vee e_1$.

Proposition 3.

Let $G$ be an almost distributive lattice and ${\zeta }_1$ and ${\zeta }_2$ be two $\alpha$-multipliers of $G$. Then, ${\zeta }_1\vee {\zeta }_2$ is also an $\alpha$-multiplier of $G$.

Proof.

Let $G$ be an $\text{ADL}$ and ${\zeta }_1$ and ${\zeta }_2$ be $\alpha$-multiplier such that$$\begin{array}{c}\text{(6)}& {\zeta }_1{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge \alpha {\omega }_3,{\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_2{h}_3\wedge \alpha {\omega }_3.\end{array}$$

Let $h_3$, ${\omega }_3\in G$, ${\zeta }_1$, and ${\zeta }_2$ be $\alpha$-multipliers of $G$. Now, by Definition 5, we have ${\zeta }_1\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge {\omega }_3\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge \alpha {\omega }_3\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3$. By Definitions 1 and 5, we have ${\zeta }_1\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3={\zeta }_1\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3$ which along with equation (6) implies that (${\zeta }_1\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3$. Hence, ${\zeta }_1\vee {\zeta }_2$ is called $\alpha$-multiplier of $G$.

Proposition 4.

Let $G_1$ and $G_2$ be two almost distributive lattices with 0. A function $\zeta :G_1\times G_2⟶G_1\times G_2$ defined by $\zeta s,{\omega }_3=0,\alpha {\omega }_3\forall s,{\omega }_3\in G_1\times G_2$ and $\alpha$ is a homomorphism. Then, $\zeta$ is $\alpha$-multiplier of $G_1\times G_2$ with pointwise operation.

Proof.

Let $G_1$ and $G_2$ be two $\text{ADLs}$ with 0. We define a mapping $\zeta :G_1\times G_2⟶G_1\times G_2$ by$$\begin{array}{}\text{(7)}& \zeta s_1,t_1=0,\alpha t_1.\end{array}$$

Then, we have to show that $\zeta$ is an $\alpha$-multiplier with pointwise operation such that$$\begin{array}{}\text{(8)}& \zeta s_1,t_1\wedge s_2,t_2=\zeta s_1,t_1\wedge \alpha s_2,\alpha t_2.\end{array}$$

Let $s_1,t_1,s_2,t_2\in G_1\times G_2$. By Definition 6, $\zeta s_1,t_1\wedge s_2,t_2=\zeta s_1\wedge s_2,t_1\wedge t_2$. By using equation (7) and Definition 1, we have $\zeta s_1,t_1\wedge s_2,t_2=0,\alpha t_1\wedge \alpha t_1=0\wedge \alpha s_2,\alpha t_1\wedge \alpha t_2$. By Definition 6 and equation (7), we have $\zeta s_1,t_1\wedge s_2,t_2=0,\alpha t_1\wedge \alpha s_2,\alpha t_2=\zeta s_1,t_1\wedge \alpha s_2,\alpha t_2$. This implies that $\zeta$ is an $\alpha$-multiplier with pointwise operation on $G_1\times G_2$.

Theorem 2.

Let $G$ be an almost distributive lattice and $BG$ be the set of all $\alpha$-multipliers of $G$. Then, $BG$ under binary operations $\vee$ and $\wedge$ is an almost distributive lattice, where for any ${\zeta }_1$, ${\zeta }_2\in BG$, $h_3\in G$.$$\begin{array}{c}\text{(9)}& {\zeta }_1\wedge {\zeta }_2{h}_3={\zeta }_1{h}_3\wedge {\zeta }_2{h}_3,{\zeta }_1\vee {\zeta }_2{h}_3={\zeta }_1{h}_3\vee {\zeta }_2{h}_3.\end{array}$$

Proof.

Let ${\zeta }_1$, ${\zeta }_2\in BG$. Then, by equation (9), we have (${\zeta }_1\wedge {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge {\omega }_3\wedge {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge \alpha {\omega }_3\wedge {\zeta }_2{h}_3\wedge \alpha {\omega }_3={\zeta }_1{h}_3\wedge {\zeta }_2{h}_3\wedge \alpha {\omega }_3={\zeta }_1\wedge {\zeta }_2{h}_3\wedge \alpha {\omega }_3$. This implies that ${\zeta }_1\wedge {\zeta }_2$ is an $\alpha$-multiplier. Let ${\zeta }_1$, ${\zeta }_2\in BG$, and along with equation (9), we have ${\zeta }_1\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge {\omega }_3\vee {\zeta }_2{h}_3\wedge {\omega }_3={\zeta }_1{h}_3\wedge \alpha {\omega }_3\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3={\zeta }_1{h}_3\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3={\zeta }_1\vee {\zeta }_2{h}_3\wedge \alpha {\omega }_3$. This implies that (${\zeta }_1\vee {\zeta }_2$) is an $\alpha$-multiplier. Hence, ($BG$, $\vee$, $\wedge$) is closed under $\vee$, $\wedge$. Hence, ($BG$, $\vee$, $\wedge$) is an $\text{ADL}$.

Theorem 3.

Let $G$ be an almost distributive lattice and $BG$ be the set of all $\alpha$-multiplier on $G$. Then, set of all principal $\alpha$-multiplier $PG$${\zeta }_x|x\in G$ is distributive lattice with the following operation ${\zeta }_{h_2}\vee {\zeta }_{{\omega }_3}={\zeta }_{h_2\vee {\omega }_3}$ and ${\zeta }_{h_2}\wedge {\zeta }_{{\omega }_3}={\zeta }_{h_2\wedge {\omega }_3}$  for all $h_2$, ${\omega }_3\in G$.

Proof.

Let $h_2$, ${\omega }_3\in G$. Then, ${\zeta }_{h_2}\vee {\zeta }_{{\omega }_3}{e}_1={\zeta }_{h_2}{e}_1\vee {\zeta }_{{\omega }_3}{e}_1=h_2\wedge \alpha e_1\vee {\omega }_3\wedge \alpha e_1=h_2\vee {\omega }_3\wedge \alpha e_1={\zeta }_{h_2\vee {\omega }_3}{e}_1$. For some $e_1\in G$, it implies that ${\zeta }_{h_2}\vee {\zeta }_{{\omega }_3}={\zeta }_{h_2\vee {\omega }_3}\in PG$. Also ${\zeta }_{h_2}\wedge {\zeta }_{{\omega }_3}{e}_1={\zeta }_{h_2}{e}_1\wedge {\zeta }_{{\omega }_3}{e}_1=h_2\wedge \alpha e_1\wedge {\omega }_3\wedge \alpha e_1=h_2\wedge {\omega }_3\wedge \alpha e_1={\zeta }_{h_2\wedge {\omega }_3}{e}_1$. For any $e_1\in G$, it implies ${\zeta }_{h_2}\wedge {\zeta }_{{\omega }_3}={\zeta }_{h_2\wedge {\omega }_3}\in PG$. Hence, ($PG$, $\vee$, $\wedge$) is closed, and so $PG$ is sub almost distributive lattice. Moreover, for any $e_1\in G$, ${\zeta }_{h_2\wedge {\omega }_3}{e}_1=h_2\wedge {\omega }_3\wedge \alpha e_1={\omega }_3\wedge h_2\wedge \alpha e_1={\zeta }_{{\omega }_3}\wedge {\zeta }_{h_2}$. Thus, ${\zeta }_{h_2}\wedge {\zeta }_{{\omega }_3}={\zeta }_{{\omega }_3}\wedge {\zeta }_{h_2}$. Hence, $PG$ is a distributive lattice.

4. Conclusion

In this paper, we have generalized the idea of multiplier to $\alpha$-multiplier in almost distributive lattices and investigated some properties of ADLs. We have also explored some results by using the notion of principal $\alpha$-multiplier and isotone $\alpha$-multiplier. This generalized concept played a vital role in exploring different properties of almost distributive lattices.

Authors’ Contributions

Ying Wang analyzed the results, drafted the final version of the paper, and arranged funding for this paper. Abdul Rauf Khan and Zafar Ullah proved the results. Zahid Karim and Abid Mahmoob approved the results and supervised this work. Mamoona Karim wrote the first version of the paper.

Acknowledgments

This work was supported by the National Key R & D Program of China (no. 2018YFB1005104), the Guangzhou Academician and Expert Workstation (no. 20200115-9), and Key Disciplines of Guizhou Province-Computer Science and Technology (no. ZDXK [2018]007). This work is supported by the Innovation Projects of Universities in Guangdong Province (No. 2020KTSCX215).

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