Rough Approximation Operators on a Complete Orthomodular Lattice

Abstract

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

1. Introduction

In 1982, Pawlak [1] proposed rough set theory as an excellent tool for incomplete information processing. Since then, a series of works describe rough sets relying on the mathematical structure of the set, such as logics [2,3,4], algebraic and topological structures [5,6]. Rough sets over different mathematical structures have different properties. In particular, rough sets over Boolean algebra [7,8], lattice effect algebra [9], residuated lattices [10,11,12,13,14,15,16] and other lattices [17,18,19] have been studied.

In 1936, Birkhoff and von Neumann [20] considered the orthomodular lattice as quantum logic for studying the algebraic structure of quantum mechanics. There are many extensions of Pawlak’s rough set, such as fuzzy rough sets [21,22], covering based rough sets [23,24,25], probabilistic rough sets [26,27], soft rough sets [28,29,30,31], Diophantine fuzzy rough sets [32,33,34,35], multi-granulation rough sets [36,37,38], hesitant fuzzy rough set [39] and so on. However, as far as I know, there is only a little literature addressing the rough sets and quantum logics together. In 2017, Hassan [40] considered a rough set classification method via quantum logic. In our previous work [41], we proposed rough set via join and meet on a complete orthomodular lattice (COL). Since orthomodular lattices are different from Boolean algebras, in particular, the distributive law not hold in orthomodular lattices, then we find many basic properties of rough sets rely on the distributive law. Obviously, the underlying laws of logic play an important part in the concept of rough approximations. However, sometimes the underlying laws of logic were taken for granted in rough approximations based on logics. In order to enhance the importance of the underlying laws of logic in rough set theory, a useful method is set up based on the equivalence between the underlying laws of logic and the basic properties of rough approximations. Note that Pawlak’s rough sets also rely on the equivalence relation. In this paper, we studied the relations among the distributive law, rough approximations and lattice-valued relation. Moreover, some topological structures of orthomodular lattice-valued rough approximations are investigated.

The paper is organized as follows: In Section 2, we recall definitions of orthomodular lattices and orthomodular lattice-valued rough approximations. In Section 3, we study the relationship among the distributive law, rough approximations and lattice-valued relations. In Section 4 is the conclusion.

2. Preliminaries

2.1. Orthomodular Lattices

First, we recall definitions of orthomodular lattices, for details see [42,43,44,45,46,47,48].

A COL $L=\mathbf{L},\le ,\wedge ,\vee ,\perp ,0,1$ is a complete bounded lattice and unary operator ⊥ has the following properties: for all $\eta ,\xi \in \mathbf{L}$

(1)

${\eta }^{\perp }\vee \eta =1$, ${\eta }^{\perp }\wedge \eta =0$;

(2)

${\eta }^{\perp \perp }=\eta$;

(3)

$\eta \le \xi \Rightarrow {\xi }^{\perp }\le {\eta }^{\perp }$;

(4)

$\eta \ge \xi \Rightarrow \eta \wedge ({\eta }^{\perp }\vee \xi )=\xi$.

The property (4) is the orthomodular law, denoted by (OL). It is weaker than the distributive law, denoted by DL, which is not valid in orthomodular lattice.

An orthomodular lattice-valued set (l-valued set for short) is a mapping $E:U\to L$, where U is a finite universe. For any $a\in L$, $\widehat{a}$ is the constant l-valued set, i.e., $\widehat{a}\left(x\right)=a$, $\forall x\in U$. Similarly, an orthomodular lattice-valued relation (l-valued relation for short) on U is a mapping $E:U\times U\to L$. We say R is serial, if ${\vee }_{y\in U}R(x,y)=1$ for all $x\in U$. We say R is $\wedge -$transitive, if $R(\alpha ,\gamma )\ge {\bigvee }_{\beta \in U}R(\alpha ,\beta )\wedge R(\beta ,\gamma )$ for all $\alpha ,\beta ,\gamma \in U$.

2.2. Rough Approximations on a COL

Then, we recall the rough approximations on a COL [41].

Definition 1

([41]). Let L be a COL and R be an L-valued relation on a finite universe U. With each L-valued set E on U, the lower approximation operator (LAO) and the upper approximation operator (UAO) of E are defined, respectively, as follows:

$$\begin{array}{c}\hfill L_R\left(E\right)\left(\mu \right)=\underset{\nu \in U}{\bigwedge }\left(R{(\mu ,\nu )}^{\perp }\vee E\left(\nu \right)\right),\mu \in U\end{array}$$

(1)

and

$$\begin{array}{c}\hfill T_R\left(E\right)\left(\mu \right)=\underset{\nu \in U}{\bigvee }\left(R(\mu ,\nu )\wedge E\left(\nu \right)\right),\mu \in U.\end{array}$$

(2)

The pair $\left(L_R\left(E\right),T_R\left(E\right)\right)$ is a $L-$valued rough set of E relative to COL L.

Example 1.

Let C be the set of complex numbers. In the complex Hilbert space ${⨂}^2{C}^2$, $|00\rangle ,|01\rangle ,$$|10\rangle$ and $|11\rangle$ represent its orthonormal base. ${\rho }_{ij}=span\left\{\right|ij\rangle \}$ is denoted the subspace spanned by $|ij\rangle$, $i,j=0,1$. For any closed subspace G and H of ${⨂}^2{C}^2$, $G\le H$ if the subspace G is included in H, ∧ is intersection of subspaces, ∨ is union of subspaces, $G^{\perp }$ is the orthogonal space of G, 0 is the zero subspace and 1 is ${⨂}^2{C}^2$. Then $L_2=L_2,\le ,\wedge ,\vee ,\perp ,0,1$ is a COL, where $L_2$ is the set of all closed subspaces of ${⨂}^2{C}^2$, more details see [43,47].

We define an COL-valued rough approximation whose truth value is a closed subspace of ${⨂}^2{C}^2$. Let $U=\{p,q\}$, $R(p,p)=R(q,q)=1$ and $R(p,q)=R(q,p)=v_{01}$. Moreover, define $E:U\to L_2$ as $E\left(p\right)=v_{00},E\left(q\right)=v_{11}$. Then

• $T_R\left(E\right)\left(p\right)=(R(p,p)\wedge E\left(p\right))\vee (R(p,q)\wedge E\left(q\right))=(1\wedge v_{00})\vee (v_{01}\wedge v_{11})=v_{00}$
• $T_R\left(E\right)\left(q\right)=(R(q,p)\wedge E\left(p\right))\vee (R(q,q)\wedge E\left(q\right))=(v_{01}\wedge v_{00})\vee (1\wedge v_{11})=v_{11}$
• $L_R\left(E\right)\left(p\right)=(R{(p,p)}^{\perp }\vee E\left(p\right))\wedge (R{(p,q)}^{\perp }\vee E\left(q\right))=(1^{\perp }\vee v_{00})\wedge (v_{01}^{\perp }\vee v_{11})=0$
• $L_R\left(E\right)\left(q\right)=(R{(q,p)}^{\perp }\vee E\left(p\right))\wedge (R{(q,q)}^{\perp }\vee E\left(q\right))=(v_{01}^{\perp }\vee v_{00})\wedge (1^{\perp }\vee v_{11})=0$

Example 2.

Consider the smallest OL which is not a Boolean algebra, called $MO2$ [49], as shown in Figure 1. Let the universe $U=\{u_1,u_2,u_3\}$. Define a L-valued set

$$\begin{array}{c}\hfill E={\displaystyle \frac{a}{u_1}}+{\displaystyle \frac{a^{\perp }}{u_2}}+{\displaystyle \frac{b}{u_3}}\end{array}$$

(3)

and a L-valued relation R on $MO2$ in

Table 1. The L-valued relation R in Example 2.

R	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$
$u_1$	1	0	0
$u_2$	0	a	0
$u_3$	0	0	b

. Thus, we have

$$\begin{array}{cc}\hfill L_R\left(E\right)\left({\mu }_1\right)& =\left(R{({\mu }_1,{\mu }_1)}^{\perp }\vee E\left({\mu }_1\right)\right)\bigwedge \left(R{({\mu }_1,{\mu }_2)}^{\perp }\vee E\left({\mu }_2\right)\right)\bigwedge \left(R{({\mu }_1,{\mu }_3)}^{\perp }\vee E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(1^{\perp }\vee a\right)\bigwedge \left(0^{\perp }\vee a^{\perp })\right)\bigwedge \left(0^{\perp }\vee b\right)\hfill \\ \hfill & =a\bigwedge 1\bigwedge 1=a\hfill \\ \hfill L_R\left(E\right)\left({\mu }_2\right)& =\left(R{({\mu }_2,{\mu }_1)}^{\perp }\vee E\left({\mu }_1\right)\right)\bigwedge \left(R{({\mu }_2,{\mu }_2)}^{\perp }\vee E\left({\mu }_2\right)\right)\bigwedge \left(R{({\mu }_2,{\mu }_3)}^{\perp }\vee E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(0^{\perp }\vee a\right)\bigwedge \left(a^{\perp }\vee a^{\perp })\right)\bigwedge \left(0^{\perp }\vee b\right)\hfill \\ \hfill & =1\bigwedge a^{\perp }\bigwedge 1=a^{\perp }\hfill \\ \hfill L_R\left(E\right)\left({\mu }_3\right)& =\left(R{({\mu }_3,{\mu }_1)}^{\perp }\vee E\left({\mu }_1\right)\right)\bigwedge \left(R{({\mu }_3,{\mu }_2)}^{\perp }\vee E\left({\mu }_2\right)\right)\bigwedge \left(R{({\mu }_3,{\mu }_3)}^{\perp }\vee E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(0^{\perp }\vee a\right)\bigwedge \left(0^{\perp }\vee a^{\perp })\right)\bigwedge \left(b^{\perp }\vee b\right)\hfill \\ \hfill & =1\bigwedge 1\bigwedge 1=1\hfill \\ \hfill T_R\left(E\right)\left({\mu }_1\right)& =\left(R({\mu }_1,{\mu }_1)\wedge E\left({\mu }_1\right)\right)\bigvee \left(R({\mu }_1,{\mu }_2)\vee E\left({\mu }_2\right)\right)\bigvee \left(R({\mu }_1,{\mu }_3)\vee E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(1\wedge a\right)\bigvee \left(0\wedge a^{\perp })\right)\bigvee \left(0\wedge b\right)\hfill \\ \hfill & =a\bigwedge 0\bigwedge 0=a\hfill \\ \hfill T_R\left(E\right)\left({\mu }_2\right)& =\left(R({\mu }_2,{\mu }_1)\wedge E\left({\mu }_1\right)\right)\bigvee \left(R({\mu }_2,{\mu }_2)\wedge E\left({\mu }_2\right)\right)\bigvee \left(R({\mu }_2,{\mu }_3)\wedge E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(0\wedge a\right)\bigvee \left(a\wedge a^{\perp })\right)\bigvee \left(0\wedge b\right)\hfill \\ \hfill & =0\bigwedge 0\bigwedge 0=0\hfill \\ \hfill T_R\left(E\right)\left({\mu }_3\right)& =\left(R({\mu }_3,{\mu }_1)\wedge E\left({\mu }_1\right)\right)\bigvee \left(R({\mu }_3,{\mu }_2)\wedge E\left({\mu }_2\right)\right)\bigvee \left(R({\mu }_3,{\mu }_3)\wedge E\left({\mu }_3\right)\right)\hfill \\ \hfill & =\left(0\wedge a\right)\bigvee \left(0\wedge a^{\perp })\right)\bigvee \left(b\wedge b\right)\hfill \\ \hfill & =0\bigwedge 0\bigwedge b=b.\hfill \end{array}$$

3. Relation among the Distributive Law, Rough Approximations and Lattice-Valued Relations

First, we give some relation between distributive law and rough approximations.

Proposition 1.

The following three statements are equivalent:

(1) 

L satisfies DL.

(2) 

$T_R(E\cup F)\equiv (T_R\left(E\right)\cup T_R\left(F\right))$.

(3) 

$L_R(E\cap F)\equiv (L_R\left(E\right)\cap L_R\left(F\right))$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$: By using the distributive law of meet over join, we have

$$\begin{array}{ccc}\hfill T_R(E\vee F)\left(x\right)& =& \underset{y\in U}{\bigvee }(R(x,y)\wedge (E\vee F)\left(y\right))\hfill \\ & =& \underset{y\in U}{\bigvee }(R(x,y)\wedge (E\left(y\right)\vee F\left(y\right)))\hfill \\ & =& \underset{y\in U}{\bigvee }((R(x,y)\wedge E\left(y\right))\vee (R(x,y)\wedge F\left(y\right)))\hfill \\ & =& (T_R\left(E\right)\vee T_R\left(F\right))\left(x\right).\hfill \end{array}$$

(4)

$\left(2\right)\Rightarrow \left(1\right)$: Our purpose is to show $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$, $\forall a,b,c\in L$. Put $R(x,y_1)=a$, and $R(x,y)=0$ for other $y\in U$; $E\left(y_1\right)=b$, $F\left(y_1\right)=c$, and $E\left(y\right)=F\left(y\right)=0$ for other $y\in U$. Then, we have

$$\begin{array}{ccc}\hfill T_R(E\vee F)\left(x\right)& =& \underset{y\in U}{\bigvee }(R(x,y)\wedge (E\vee F)\left(y\right))\hfill \\ & =& R(x,y_1)\wedge (E\vee F)\left(y_1\right)\hfill \\ & =& a\wedge (b\vee c).\hfill \end{array}$$

(5)

and

$$\begin{array}{ccc}\hfill (T_R\left(E\right)\vee T_R\left(F\right))\left(x\right)& =& T_R\left(E\right)\left(x\right)\vee T_R\left(F\right)\left(x\right)\hfill \\ & =& \left(\underset{y\in U}{\bigvee }(R(x,y)\wedge E\left(y\right))\right)\vee \left(\underset{y\in U}{\bigvee }(R(x,y)\wedge F\left(y\right))\right)\hfill \\ & =& \left(R(x,y_1)\wedge E\left(y_1\right)\right)\vee \left(R(x,y_1)\wedge F\left(y_1\right)\right)\hfill \\ & =& (a\wedge b)\vee (a\wedge c)\hfill \end{array}$$

(6)

Therefore, we obtain $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$ from (2).

$\left(1\right)\Rightarrow \left(3\right)$: By using the distributive law of join over meet,

$$\begin{array}{ccc}\hfill L_R(F\wedge F)\left(x\right)& =& \underset{y\in U}{\bigwedge }\left(R{(x,y)}^{\perp }\vee (E\wedge F)\left(y\right)\right)\hfill \\ & =& \underset{y\in U}{\bigwedge }(R{(x,y)}^{\perp }\vee (E\left(y\right)\wedge F\left(y\right)))\hfill \\ & =& \underset{y\in U}{\bigwedge }\left((R{(x,y)}^{\perp }\vee E\left(y\right)\right)\wedge \left(R{(x,y)}^{\perp }\vee F\left(y\right))\right)\hfill \\ & =& (L_R\left(E\right)\wedge L_R\left(F\right))\left(x\right).\hfill \end{array}$$

(7)

$\left(3\right)\Rightarrow \left(1\right)$: Similarly, our purpose is to show $a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$, $\forall a,b,c\in L$. Let $R{(x,y_1)}^{\perp }=a$, and $R{(x,y)}^{\perp }=1$ for other $y\in U$; $E\left(y_1\right)=b$, $F\left(y_1\right)=c$, and $E\left(y\right)=F\left(y\right)=1$ for other $y\in U$. Then,

$$\begin{array}{ccc}\hfill L_R(E\wedge F)\left(x\right)& =& \underset{y\in U}{\bigwedge }(R{(x,y)}^{\perp }\vee (E\wedge F)\left(y\right))\hfill \\ & =& R{(x,y_1)}^{\perp }\vee (E\left(y_1\right)\wedge F\left(y_1\right))\hfill \\ & =& a\vee (b\wedge c).\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill (L_R\left(E\right)\wedge L_R\left(F\right))\left(x\right)& =& L_R\left(E\right)\left(x\right)\wedge L_R\left(F\right)\left(x\right)\hfill \\ & =& \left(\underset{y\in U}{\bigwedge }(R{(x,y)}^{\perp }\vee E\left(y\right))\right)\wedge \left(\underset{y\in U}{\bigwedge }(R{(x,y)}^{\perp }\vee F\left(y\right))\right)\hfill \\ & =& \left(R{(x,y_1)}^{\perp }\vee E\left(y_1\right)\right)\wedge \left(R{(x,y_1)}^{\perp }\vee F\left(y_1\right)\right)\hfill \\ & =& (a\vee b)\wedge (a\vee c)\hfill \end{array}$$

(9)

Therefore, we obtain $a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$ from (3). □

The following results are also based on the distributive law of join over meet.

Proposition 2.

If L satisfies DL, then the following three statements are equivalent

(1) 

R is serial, i.e., ${\vee }_{y\in U}R(x,y)=1$ for all $x\in U$.

(2) 

$T_R\left(\widehat{a}\right)\equiv \widehat{a}$, for any $a\in L$.

(3) 

$L_R\left(\widehat{a}\right)\equiv \widehat{a}$, for any $a\in L$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$: By using the distributive law of join over meet, we have

$$\begin{array}{ccc}\hfill T_R\left(\widehat{a}\right)\left(x\right)& =& \underset{y\in U}{\bigvee }\left(R(x,y)\wedge \widehat{a}\left(y\right)\right)\hfill \\ & =& \underset{y\in U}{\bigvee }\left(R(x,y)\wedge a\right)\hfill \\ & =& a\wedge \underset{y\in U}{\bigvee }R(x,y)\hfill \\ & =& a\wedge 1\hfill \\ & =& a.\hfill \end{array}$$

(10)

$\left(2\right)\Rightarrow \left(1\right)$: Take $a=1$; then it follows from the proof of necessity and $T_R\left(\widehat{1}\right)\left(x\right)=1$ for every $x\in X$ that ${\bigvee }_{y\in U}R(x,y)=1$ holds for every $x\in U$. Hence R is serial.

Similarly, we can prove $\left(1\right)\iff \left(3\right)$. □

Now, we study the relationship among the distributive law, rough approximation and COL-valued relation.

Proposition 3.

If two of the following statements hold, then the third statement holds:

(1) 

L satisfies DL.

(2) 

R is $\wedge -$transitive, i.e., $R(\alpha ,\gamma )\ge {\bigvee }_{\beta \in U}R(\alpha ,\beta )\wedge R(\beta ,\gamma )$ holds for all $\alpha ,\beta ,\gamma \in U$.

(3) 

$T_R\left(T_R\left(E\right)\right)\subseteq T_R\left(E\right)$.

Proof. 

$\left(1\right)+\left(2\right)\Rightarrow \left(3\right)$: By using the distributive law of join over meet, we have

$$\begin{array}{ccc}\hfill T_R\left(T_R\left(E\right)\right)\left(x\right)& =& \underset{y\in U}{\bigvee }(R(x,y)\wedge T_R\left(E\right)\left(y\right))\hfill \\ & =& \underset{y\in U}{\bigvee }(R(x,y)\wedge \left(\underset{z\in U}{\bigvee }(R(y,z)\wedge E\left(z\right))\right))\hfill \\ & =& \underset{z\in U}{\bigvee }(\underset{y\in U}{\bigvee }(R(x,y)\wedge R(y,z))\wedge E\left(z\right))\hfill \\ & \le & \underset{z\in U}{\bigvee }(R(x,z)\wedge E\left(z\right))\hfill \\ & =& T_R\left(E\right)\left(x\right).\hfill \end{array}$$

(11)

$\left(1\right)+\left(3\right)\Rightarrow \left(2\right)$: Assume that R is not $\wedge -$transitive. It follows than that for some $x_0,z_0\in U$, ${\bigvee }_{y\in U}(R(x_0,y)\wedge R(y,z_0))\le R(x_0,z_0)$ is not hold.

Let $E\left(z_0\right)=1$ and $E\left(y\right)=0$ for other $y\in U$. Then, we have

$$\begin{array}{c}\hfill T_R\left(T_R\left(E\right)\right)\left(x_0\right)=\underset{y\in U}{\bigvee }(R(x_0,y)\wedge R(y,z_0))\end{array}$$

(12)

and

$$\begin{array}{c}\hfill T_R\left(E\right)\left(x_0\right)=R(x_0,z_0).\end{array}$$

(13)

Therefore, it follows from (3) that ${\bigvee }_{y\in U}(R(x_0,y)\wedge R(y,z_0))\le R(x_0,z_0)$.

$\left(2\right)+\left(3\right)\Rightarrow \left(1\right)$: Given $a,b,c\in L$, let $U=x,y,z$, $R\in L^{U\times U}$ which $R(x,y)=R(x,z)=a$, $R(y,z)=R(z,z)=b,R(z,y)=R(y,y)=c$ and others are o, and $E\in L^U$ which $E\left(y\right)=c$, $E\left(z\right)=b$, $E\left(x\right)=0$. It easy to see R is $\wedge -$transitive. Then, we have

$$\begin{array}{c}\hfill T_R\left(E\right)\left(x\right)=(R(x,y)\wedge E\left(y\right))\vee (R(x,z)\wedge E\left(z\right))=(a\wedge c)\vee (a\wedge b),\end{array}$$

(14)

$$\begin{array}{c}\hfill T_R\left(E\right)\left(y\right)=(R(y,y)\wedge E\left(y\right))\vee (R(y,z)\wedge E\left(z\right))=b\vee c,\end{array}$$

(15)

$$\begin{array}{c}\hfill T_R\left(E\right)\left(z\right)=(R(z,z)\wedge E\left(z\right))\vee (R(z,y)\wedge E\left(y\right))=b\vee c,\end{array}$$

(16)

and

$$\begin{array}{ccc}\hfill T_R\left(T_R\left(E\right)\right)\left(x\right)& =& (R(x,y)\wedge T_R\left(E\right)\left(y\right))\vee (R(x,z)\wedge T_R\left(E\right)\left(z\right))\hfill \\ & =& (a\wedge (b\vee c\left)\right)\vee (a\wedge (b\vee c\left)\right)\hfill \\ & =& a\wedge (b\vee c).\hfill \end{array}$$

(17)

Therefore, by $T_R\left(T_R\left(E\right)\right)\left(x\right)\le T_R\left(E\right)\left(x\right)$ we obtain $(a\wedge (b\vee c\left)\right)\le \left(\right(a\wedge c)\vee (a\wedge b\left)\right)$.

Since $(a\wedge (b\vee c\left)\right)\ge \left(\right(a\wedge b)\vee (a\wedge c\left)\right)$ always holds. Thus $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$.

 □

Proposition 4.

If two of the following statements hold, then the third statement holds:

(1) 

L satisfies DL.

(2) 

R is $\wedge -$transitive.

(3) 

$L_R\left(E\right)\subseteq L_R\left(L_R\left(E\right)\right)$.

Proof. 

Similar to Proposition 3. □

Propositions 3 and 4 give some basic properties of rough approximations that do not only rely on the binary relation but also on the distributive law.

Definition 2

([50]). Let U be a non-empty set, a function $in$: $L^U\to L^U$ is an $l-$valued interior operator if and only if (iff) for all $E,F\in L^U$ it satisfies:

(1) $in\left(\widehat{a}\right)=\widehat{a}$;

(2) $in\left(E\right)\subseteq E$;

(3) $in(E\cap F)=in\left(E\right)\cap in\left(F\right)$;

(4) $in\left(in\right(E\left)\right)=in\left(E\right)$;

Definition 3

([50]). Let U be a non-empty set, a function $cl$: $L^U\to L^U$ is an $l-$valued closure operator iff for all $E,F\in L^U$ it satisfies:

(1) $cl\left(\widehat{a}\right)=\widehat{a}$;

(2) $E\subseteq cl\left(E\right)$;

(3) $cl(E\cup F)=cl\left(E\right)\cup cl\left(F\right)$;

(4) $cl\left(cl\right(E\left)\right)=cl\left(E\right)$;

Proposition 5.

If two of the following statements hold, then the third statement holds:

(1) L satisfies DL.

(2) R is serial and $\wedge -$transitive.

(3) $L_R$ is an $l-$valued interior operator.

Proof. 

Immediate from Proposition 1 of [41] and Propositions 1, 2 and 4. □

Note that DL is also a condition of that lower rough approximation is an interior operator.

Proposition 6.

If two of the following statements hold, then the third statement holds:

(1) L satisfies DL.

(2) R is serial and $\wedge -$transitive.

(3) $T_R$ is an $l-$valued closure operator.

Proof. 

Immediate from Proposition 1 of [41] and Propositions 1, 2 and 3. □

Similar to Propositions 3 and 4, Propositions 5 and 6 also show that some topology properties of rough approximations that do not only rely on the binary relation but also on the distributive law.

Example 3.

Consider a L-valued relation R on $MO2$ in

Table 2. The L-valued relation R in Example 3.

R	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$
$u_1$	1	0	0
$u_2$	0	1	0
$u_3$	0	0	1

.

Then $L_R$ and $T_R$, respectively, are $l-$valued interior operator and $l-$valued closure operator for any l-valued set $E:U\to MO2$.

Moreover, if we use the following definitions of $l-$valued interior operator and $l-$valued closure operator which are weaker than Definitions 2 and 3, respectively.

Definition 4.

Let U be a non-empty set, a function $in$: $L^U\to L^U$ is an $l-$valued interior operator iff for all $E,F\in L^U$ it satisfies:

(1) $in\left(\widehat{0}\right)=\widehat{0}$;

(2) $in\left(E\right)\subseteq E$;

(3) $in(E\cap F)=in\left(E\right)\cap in\left(F\right)$;

(4) $in\left(in\right(E\left)\right)=in\left(E\right)$;

Definition 5.

Let U be a non-empty set, a function $cl$: $L^U\to L^U$ is an $l-$valued closure operator iff for all $E,F\in L^U$ it satisfies:

(1) $cl\left(\widehat{1}\right)=\widehat{1}$;

(2) $E\subseteq cl\left(E\right)$;

(3) $cl(E\cup F)=cl\left(E\right)\cup cl\left(F\right)$;

(4) $cl\left(cl\right(E\left)\right)=cl\left(E\right)$;

Then, we have the following results.

Proposition 7.

If two of the following statements hold, then the third statement holds:

(1) L satisfies DL.

(2) R is $\wedge -$transitive.

(3) $L_R$ is an $l-$valued interior operator.

Proof. 

Immediate from Proposition 1 of [41] and Propositions 1 and 4. □

Proposition 8.

If two of the following statements hold, then the third statement holds:

(1) L satisfies DL.

(2) R is $\wedge -$transitive.

(3) $T_R$ is an $l-$valued closure operator.

Proof. 

Immediate from Proposition 1 of [41] and Propositions 1 and 3. □

Example 4.

Consider a L-valued relation R on $MO2$ in

Table 3. The L-valued relation R in Example 4.

R	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$
$u_1$	a	0	0
$u_2$	0	a	0
$u_3$	0	0	a

.

Then $L_R$ is an $l-$valued interior operator of Defintion 4 but is not an $l-$valued interior operator of Defintion 2, and $T_R$ is an $l-$valued closure operator of Defintion 5 but is not an $l-$valued closure operator of Defintion 3.

4. Conclusions

In this paper, we studied COL-valued rough approximation. Some properties of rough approximations rely on DL of ∨ over ∧ and binary relation. Obviously, the distributive law plays an important part in the operation of rough approximations. Many basic properties of rough approximations do not only rely on the binary relation but also on the distributive law (see Propositions 2–4). Moreover, some topology properties of rough approximations do not only rely on the binary relation but also on the distributive law (see Propositions 5–8).

Orthomodular lattices can be viewed as a sharp quantum structure. There is another concrete or standard quantum logic, called unsharp quantum logics which do not satisfy the non-contradiction principle [49]. Proceeding from this angle, we can study the rough approximations based on unsharp quantum logics as future work.