On the Relationship Between L-fuzzy Closure Spaces and L-fuzzy Rough Sets

Abstract

This work is towards the establishment of bijective correspondence between the family of all L-fuzzy reflexive/tolerance approximation spaces and the family of all quasi-discrete L-fuzzy closure spaces satisfying a certain condition.

1 Introduction

Rough sets, firstly introduced by Pawlak [11] has been advanced notably with worthy of attention due to its widespread applications in both mathematics and computer sciences for the study of intelligent systems having insufficient, imprecise, uncertain and incomplete information. The partition or equivalence (indiscernibility) relations were the fundamental and abstract tools of the rough set theory introduced by Pawlak. Researchers have made several generalizations of rough sets using an arbitrary relation in place of an equivalence relation (cf., [4, 7, 21, 22]). Dubois and Prade [3], proposed fuzzy version of rough sets in which fuzzy relations play a key roll instead of crisp relations. The fuzzy rough sets and their relationship with fuzzy topological spaces were described in detail by several authors (e.g., cf., [2, 6, 10, 12,13,14, 16, 17, 19, 20]). Moreover, in [6, 10, 17], the set of all L-fuzzy preorder approximation spaces together with the set of all saturated L-fuzzy topological spaces were center of interest, and it was shown that under a certain extra condition there exists a bijective correspondence between them. The silence on such relationship between the set of other generalized approximation spaces (such as L-fuzzy reflexive approximation space and L-fuzzy tolerance approximation spaces) and the set of some L-fuzzy topological structures, in the cited work, attract our attention and lead us an attempt to establish such relationships by using the concept of L-fuzzy closure spaces. Finally, we have established the similar result for the set of all, L-fuzzy preorder approximation spaces and L-fuzzy closure spaces, respectively.

2 Preliminaries

We begin by recalling the following concept of a residuated lattice from [1].

Definition 1

An algebra $L=(L, \wedge , \vee , *, \rightarrow , 0, 1)$ define a residuated lattice, if $(L, \wedge , \vee , 0, 1)$ is a lattice having 0 and 1 as least and greatest element, respectively, $(L,*, 1)$ is a commutative monoid having unit 1, and $*$ and $\rightarrow$ form an adjoint pair, i.e., $\forall$ $x, y, z\in L$, $x*y\le z$ $\Leftrightarrow$ $x\le y\rightarrow z$. Also, L is said to be a complete residuated lattice if lattice $(L, \vee , \wedge , 0, 1)$ is complete.

Definition 2

The precomplement on L is a map $\rightharpoondown : L\longrightarrow L$ such that $\rightharpoondown x=x\rightarrow 0, \forall x\in {L}$.

Throughout, L denotes the complete residuated lattice. For a nonempty set X, $L^X$ denote the collection of all L-fuzzy sets in X, for $\alpha \in L$, $\bar{\alpha }$ denotes the constant L-fuzzy set.

Definition 3

A complete residuated lattice L is called regular if $\rightharpoondown (\rightharpoondown a)=a$, $\forall a\in L$.

The basic properties of a complete regular residuated lattice, which we use in subsequent sections are listed in following proposition.

Proposition 1

For all $a, b, a_i\in L$, $i\in J$ an index set, we have

• (i) $a *b$ $=\rightharpoondown (a\rightarrow$ $(\rightharpoondown b))$,

• (ii) $a \rightarrow b=\rightharpoondown (a*(\rightharpoondown b))$,

• (iii) $\rightharpoondown (\wedge \{a_i\})=\vee \{\rightharpoondown a_i\}$,

• (iv) $\rightharpoondown (\vee \{a_i\})=\wedge \{\rightharpoondown a_i\}$.

Definition 4

[5]. Let X be a nonempty set, then L-fuzzy relation on X is a map $R: X\times X\rightarrow L$.

For, properties of an L-fuzzy relation we refer to [5, 10, 15]. However, for completeness we emphasize from [10, 15] that an L-fuzzy reflexive and L-fuzzy symmetric relation R is known as L-fuzzy tolerance relation and, if R is L-fuzzy reflexive as well as L-fuzzy transitive then it is called L-fuzzy preorder.

Definition 5

[6, 10, 15, 17]. Let R be an L-fuzzy relation on a nonempty set X, then an L-fuzzy approximation space is a pair (X, R), which is further known as L-fuzzy reflexive/tolerance/preorder approximation space, respectively, according as underlying L-fuzzy relation R is an reflexive, tolerance or preorder.

Throughout, set of all L-fuzzy approximation space over a nonempty set X is denoted by $\Omega$.

Definition 6

[10, 15, 17]. Consider an $(X,R)\in \Omega$ and $A \in L^X$. The lower approximation $\underline{apr}_{R}(A)$ of A and the upper approximation $\overline{apr}_{R}(A)$ of A in (X, R) are respectively defined as follows:

$$\begin{aligned}&\underline{apr}_{R}(A)(x)=\wedge \{R(x, y)\rightarrow A(y): y\in X\}, and\\&\quad \overline{apr}_{R}(A)(x)=\vee \{R(x, y)*A(y): y\in X\}. \end{aligned}$$

For an $(X, R)\in \Omega$ and $A\in L^X$, we called the pair $(\underline{apr}_R(A), \overline{apr}_R(A))$ an L-fuzzy rough set.

Proposition 2

[17]. Consider an $(X,R)\in \Omega$, where L is regular as well, then $forall A\in L^X$,

• (i) $\underline{apr}_{R}(A)=\rightharpoondown \overline{apr}_{R}(\rightharpoondown A)$, and

• (ii) $\overline{apr}_{R}(A)=\rightharpoondown \underline{apr}_{R}(\rightharpoondown A)$.

Proposition 3

[6, 15, 17]. Consider an $(X,R)\in \Omega$, then $\forall A_i \in L^X, i\in J$ and $\alpha \in L$,

• (i) $\overline{apr}_{R}(\vee \{A_i: i\in J\})=\vee \overline{apr}_{R}\{A_i: i\in J\}$,

• (ii) $\underline{apr}_{R}(\wedge \{A_i: i\in J\})= \wedge \underline{apr}_{R}\{A_i: i\in J\}$, and

• (iii) $\overline{apr}_{R}(A*\bar{\alpha })= \overline{apr}_{R}(A)*\bar{\alpha }$.

Proposition 4

[17]. Consider an $(X,R)\in \Omega$, which is reflexive and $A \in L^X$, then

• (i) $\underline{apr}_{R}(A)\le A$, and

• (ii) $A\le \overline{apr}_{R}(A)$.

Proposition 5

[17]. Consider an $(X,R)\in \Omega$ and $A \in L^X$, then R is an L-fuzzy transitive relation on X iff $\overline{apr}_{R}(\overline{apr}_{R}(A))\le \overline{apr}_{R}(A)$.

Proposition 6

Let $(X,R), (X, S)\in \Omega$, then $R\le S$ iff $\overline{apr}_{R}(A)\le \overline{apr}_{S}(A)$, $\forall A\in L^X$.

Proof

Let $\overline{apr}_{R}(A)\le \overline{apr}_{S}(A)$, $\forall A\in L^X$, i.e., $\vee \{R(x, y)$ $*$ $A(y)\}$ $\le$ $\vee \{S(x, y)*A(y)\}$, $\forall A\in L^X$. Thus $R\le S$, $\forall x, y\in X$.

Conversely, let $R \le S$ and $x\in X$. Then $\overline{apr}_{R}(A)(x)=\vee$ $\{R(x, y)$ $*$ A(y) :  $y\in X\}\le \vee \{S(x, y)$ $*$ $A(y): y\in X\}=\overline{apr}_{S}(A)(x)$. Thus $\overline{apr}_{R}(A)\le \overline{apr}_{S}(A)$.

The L-fuzzy topological concepts, we use here, are fairly standard and based on [8].

Definition 7

An L-fuzzy topology $\tau$ over a nonempty set X is a subset of $L^X$ closed under arbitrary suprema and finite infima and which contains all constant L-fuzzy sets.

The pair $(X, \tau )$ is called an L-bffuzzy topological space. As usual, the member of $\tau$ are called L-fuzzy $\tau$-open sets.

Definition 8

A Kuratowski L-fuzzy closure operator over a nonempty set X is a map $k: L^X\rightarrow L^X$, whit property that $\forall A, \in L^X$ and $\forall \alpha \in L$,

• (i) $k(\bar{\alpha })=\bar{\alpha }$,

• (ii) $A\le k ( A )$,

• (iii) $k( A\vee B )=k ( A )\vee k( B )$, and

• (iv) $k( k ( A )) = k ( A )$.

Proposition 7

[6]. Consider an $(X,R)\in \Omega$, where R be an L-fuzzy reflexive relation, then $\tau _{R}=\{A\in L^X: \underline{apr}_{R}(A)=A\}$ is an L-fuzzy topology.

One can easily verify that $\tau _{R}$ is a saturated¹ L-fuzzy topology over X.

Proposition 8

[17]. Let k be as defined in Definition 8, then $\exists$ an L-fuzzy preorder $S_{k}$ over X for which $\overline{apr}_{S_{k}}(A)=k(A)$ iff (i) $\forall i\in J$ an indexed set $k(\vee \{A_i\})=\vee \{k(A_i)\}$, $\forall A_i\in L^X$ and (ii) $k(A*\bar{\alpha })=k(A)*\bar{\alpha }$, $\forall A\in L^X$, $\forall {\alpha }\in L$.

The concept of fuzzy closure spaces was proposed in (cf., [9]). Further, the concepts of subspace of a fuzzy closure space, sum of a family of pairwise disjoint fuzzy closure spaces and product of a family of fuzzy closure spaces were studied in [18]. Now, we introduce here the following concept of an L-fuzzy closure space as a generalization of the concept of a fuzzy closure space studied in [9, 18].

Definition 9

An L-fuzzy closure space over a nonempty set X is a pair (X, c), where the map $c: L^X\rightarrow L^X$ is such that $\forall A, B\in L^X$ and $\forall \alpha \in L$,

• (i) $c ( \bar{\alpha } )= \bar{ \alpha }$,

• (ii) $A \le c( A )$, and

• (iii) $c(A \vee B )=c( A )\vee c( B )$.

Definition 10

An L-fuzzy closure space (X, c) is called

• (i) quasi-discrete if $c\{\vee \{A_i:i\in J\}\}=\vee \{c(A_i): i\in J\}$, $\forall A_i\in L^X$,

• (ii) symmetric if $c(1_y)(x) =c(1_x)(y)$, $\forall x, y\in X$, and

• (iii) $A\in L^X$ is called L-fuzzy closed if $c (A ) = A$.

Proposition 9

Let (X, c) be as in 9, then

• (i) for $A, B\in L^X$ if $A\le B$ then $c(A)\le c(B)$,

• (ii) $c\{\wedge \{A_i: i\in J\}\}\le \wedge \{c(A_i): i\in J\}$, $\forall A_i\in L^X, i\in J$.

Proof

Follows obviously.

Proposition 10

Consider L-fuzzy closure space (X, c), $H\in L^X$ and $\bar{c}: L^X\rightarrow L^X$ be a map such that $\bar{c}(H)=\wedge \{K\in L^X: H\le K$ and $c(K)=K\}$. Then $\bar{c}$ is a Kuratowski L-fuzzy closure operator on X.

Proof

Obviously $\forall {\alpha }\in L, c(\bar{\alpha })=\bar{\alpha }$ and $\forall H\in L^X, H\le \bar{c}(H)$. Now, let $H, K\in X$. Then $\bar{c}(H\vee K)=\wedge \{G\in L^X: (H\vee K)\le G\ and\ c(G)=G\}$. Thus $\bar{c}(H\vee K)=\wedge \{G\in L^X: H\le G, K\le G\ and\ c(G)=G\}=\{\wedge \{G\in L^X: H\le G\ and\ c(G)=G\}\vee \{\wedge (G\in L^X: K\le G\ and\ c(G)=G)\}\}=\bar{c}(H)\vee \bar{c}(K)$. Finally, $\bar{c}(\bar{c}(H))=\bar{c}\{\{\wedge \{K\in L^X: H \le K\ and\ c(K)=K\}\}\le \wedge \{\bar{c}(K): H\le K, c(K)=K\}=\wedge \{\wedge \{G: K\le G, c(G)=G\}: H\le K, c(K)=K\}=\wedge \{G: H\le G, c(G)=G\}=\bar{c}(H)$.

Thus $\bar{c}$ induces an L-fuzzy topology, say, $\tau _{\bar{c}}$ and is given by $\tau _{\bar{c}}=\{H\in L^X:\bar{c}(\rightharpoondown H)=\rightharpoondown H\}$.

Proposition 11

Let (X, c) be an L-fuzzy closure space. Then $\forall H\in L^X$,

• (i) $c(\bar{c}(H))=\bar{c}(H)$, i.e., $\bar{c}(H)$ is L-fuzzy closed.

• (ii) $c(H)\le \bar{c}(H)$,

• (iii) $c(H)=H$ iff $\bar{c}(H)=H$.

Proof

(i) Let $H\in L^X$. Then from Proposition 9, $c(\bar{c}(H))=c(\wedge \{K: H\le K$ and $c(K)=K\})\le \wedge \{c(K): H\le K$ and $c(K)=K\}=\wedge \{K: H\le K$ and $c(K)=K\}=\bar{c}(H)$.

(ii) $H\le \bar{c}(H)\Rightarrow c(H)\le c(\bar{c}(H))=\bar{c}(H)$.

(iii) Let $c(H)=H$, $\forall H\in L^X$. Then H is L-fuzzy closed. Therefore $\bar{c}(H)\le H$ (cf., Proposition 10). This together with $H\le \bar{c}(H)$ shows that $\bar{c}(H)=H$.

Conversely, let $\bar{c}(H)=H$. Then from (ii), $H\le c(H)\le \bar{c}(H)=H$. Thus $\bar{c}(H)=H$, whereby $c(H)=H$.

Proposition 12

Let (X, c) be an L-fuzzy closure space. Then $\forall H\in L^X$, $c(H)=\bar{c}(H)$ iff $c(c(H))=c(H)$.

Proof

Let $c(H)=\bar{c}(H)$, $H\in L^X$. Then $c(c(H))=c(\bar{c}(H))=\bar{c}(H)=c(H)$.

Conversely, let $c(c(H))=c(H)$. Then c(H) is L-fuzzy closed. Hence from Proposition 11 (iii), $c(H)=\bar{c}(H)$.

Proposition 13

Let (X, c) be a quasi-discrete L-fuzzy closure space. Then the L-fuzzy topology $\tau _{\bar{c}}$ on X is a saturated L-fuzzy topology.

Proof

Follows from Definition 10 and Propositions 10 and 12.

3 L-fuzzy Closure Spaces and L-fuzzy Approximation Spaces

The existence of a bijective correspondence between the set of all L-fuzzy reflexive approximation spaces and the set of all quasi-discrete L-fuzzy closure spaces under a certain extra condition is established here. The similar relationship between the set of all L-fuzzy tolerance approximation spaces and the set of all symmetric quasi-discrete L-fuzzy closure spaces satisfying a certain extra condition is also demonstrated.

We begin with the following.

Proposition 14

Consider an $(X, R)\in \Omega$, where R is L-fuzzy reflexive relation then $(X, \overline{apr}_R)$ is a quasi-discrete L-fuzzy closure space such that $\overline{apr}_{R}(A*\bar{\alpha })=\overline{apr}_{R}(A)*\bar{\alpha }$, $\forall A\in L^X$ and $\forall \alpha \in L$.

Proof

Follows from Propositions 3 and 4.

Definition 11

For $y\in X$ and $\alpha \in L$, the L-fuzzy subset $1_y*\bar{\alpha }$ of X is called an L-fuzzy point in X, and is denoted as $y_{\alpha }$.

Proposition 15

Let (X, c) be a quasi-discrete L-fuzzy closure space such that $c(A*\bar{\alpha })=c(A)*\bar{\alpha }$, $\forall A\in L^X$ and $\forall \alpha \in L$. Then $\exists$ a L-fuzzy reflexive relation $R_c$ over X which is unique and satisfy $\overline{apr}_{R_c}(A)=c(A)$, $\forall A\in L^X$.

Proof

Let (X, c) be a quasi-discrete L-fuzzy closure space such that $c(A*\bar{\alpha })=c(A)*\bar{\alpha }$, $\forall A\in L^X$ and $\forall \alpha \in L$. Also, let $R_{c}(x, t)=c(1_t)(x)$, $\forall x, t\in X$. Then $R_c$ is an L-fuzzy relation on X such that $1=1_x(x)\le c(1_x)(x)$. Thus $c(1_x)(x)=1$, whereby $R_c$ is an L-fuzzy reflexive relation over X. Now, let $A\in L^X$, $\alpha \in L$ and $x\in X$. Then

$$\begin{aligned} \overline{apr}_{{R}_{c}}(A)(x)= & {} \overline{apr}_{{R}_{c}}(\vee \{t_{\alpha }: t \in X\} )( x ),~\text {where}~\alpha =A( t )\\= & {} \vee \{\vee \{R_c(x, r)*t_\alpha (r): r\in X\}: t\in X\}\\= & {} \vee \{\vee \{R_c(x, r)*t_\alpha (r): r\in X, r\ne t\},\\&\quad \vee \{R_c(x, t)*\,t_\alpha (r):\,r\in X, \, r=t\}: t\in X\}\\= & {} \vee \{0\vee (R_c(x, t) \,*\,{\alpha }): t\in X\}\\= & {} \vee \{R_c(x, t) \,*\, \alpha : t\in X\}\\= & {} \vee \{c(1_t)( x)\,*\, \alpha : \,t\in X\}\\= & {} \vee \{c\{(1_t)*\bar{\alpha }\}(x): t\in X\}\\= & {} c\{\vee \{1_t *\bar{\alpha }: t\in X\}(x)\}\\= & {} c(A).\\ \end{aligned}$$

Hence $\overline{apr}_{R_c}(A)=c(A)$. To show the uniqueness of L-fuzzy relation $R_c$, let $R'$ be another L-fuzzy reflexive relation on X such that $\overline{apr}_{R'}(A)=c(A)$, $\forall A\in L^X$. Then $R_c(x, t)=c(1_t)(x)$ $=\overline{apr}_{R'}(1_t)(x)$ $=\vee \{R'(x, r)$ $*1_t(r)$ $: r\in X\}$ $=R'(x, t)$. Thus $R_c=R'$. Hence the L-fuzzy relation $R_c$ on X is unique.

Now, Propositions 14 and 15 lead us to the following.

Proposition 16

Let $\mathcal{F}$ be the set of all L-fuzzy reflexive approximation spaces and $\mathcal{T}$ be the set of all quasi-discrete L-fuzzy closure spaces satisfying $c(A*\bar{\alpha })=c(A)*\bar{\alpha }$, $\forall A\in L^X$ and $\forall \alpha \in L$. Then there exists a bijective correspondence between $\mathcal{F}$ and $\mathcal{T}$.

Remark 1

In [6], it has been pointed out that for $A\in L^X$, $\underline{apr}_{R}(A)$ and $\overline{apr}_{R}(A)$ are not dual to each other. Therefore $\tau _{R_{c}}\ne \tau _{\bar{c}}$. The next proposition says that the equality holds if L is regular.

Proposition 17

Let L be regular and (X, c) be a quasi-discrete satisfying $c(A\,*\,\bar{\alpha })=c(A)\,*\,\bar{\alpha }, \forall A\in L^X, \forall \alpha \in L$. Then $\tau _{R_{c}}=\tau _{\bar{c}}$, where $R_{c}$ is an L-fuzzy reflexive relation on X induced by c.

Proof

Let $A\in \tau _{\bar{c}}$. Then $\bar{c}(\rightharpoondown A)=\rightharpoondown A$. As from Proposition 11, $c(A)$ $\,\le \,$ $\bar{c}(A)$, $\forall A\in L^X$, $c(\rightharpoondown A)$ $\le$ $\bar{c}(\rightharpoondown A)$, or that $A\le$ $\rightharpoondown$ $c(\rightharpoondown A)$.

$$\begin{aligned} \text {Now},~\rightharpoondown c (\rightharpoondown A)= & {} \rightharpoondown \overline{apr}_{R_c}(\rightharpoondown A)\\= & {} \rightharpoondown \{\vee \{R_c(w, t)*(\rightharpoondown A(t))\}: t\in X\}\\= & {} \rightharpoondown \{\vee \{\rightharpoondown \{R_c(w, t)\rightarrow (\rightharpoondown (\rightharpoondown A(t)))\}\}: t\in X\}\\= & {} \rightharpoondown \{\vee \{\rightharpoondown \{R_c(w, t)\rightarrow A(t)\}\}: t\in X\}\\= & {} \wedge \{\rightharpoondown \rightharpoondown \{R_c(w, t)\rightarrow A(t)\}: t\in X\}\\= & {} \wedge \{R_c(w, t)\rightarrow A(t)\}\\= & {} \underline{apr}_{R_c}(A). \end{aligned}$$

Thus $A\le \underline{apr}_{R_{c}}(A)$. Also, $\underline{apr}_{R_{c}}\le A$, whereby $\underline{apr}_{R_{c}}=A$. Hence $\tau _{\bar{c}}\le \tau _{R_{c}}$.

Conversely, let $A \in \tau _{R_c}$. Then $\underline{apr}_{R_c}(A)=A$, or that $\wedge \{R_c(w, t)\rightarrow A(t): t\in X\}=A$, i.e., $\wedge \{\rightharpoondown \{R_c(w, t)\,*\, (\rightharpoondown A(t)): t\in X\}\}=A$, or that $\rightharpoondown \{\vee \{R_c(w, t)*(\rightharpoondown A(t)): t\in X\}\}=A$, i.e., $\vee \{R_c(w, t)\,*\,(\rightharpoondown A(t)): t\in X\}=\rightharpoondown A$, or that $\overline{apr}_{R_{c}}(\rightharpoondown A)=\rightharpoondown A$, whereby $c(\rightharpoondown A)=\rightharpoondown A$. Thus from Proposition 11, $\bar{c}(\rightharpoondown A)=\rightharpoondown A$, whereby $A\in \tau _{\bar{c}}$, or that $\tau _{R_{c}}\le \tau _{\bar{c}}$. Hence $\tau _{R_{c}}=\tau _{\bar{c}}$.

For a given quasi-discrete L-fuzzy closure space (X, c) satisfying $c(A*\bar{\alpha })= c(A)*\bar{\alpha }$, $\forall A\in L^X$, $\forall \alpha \in L$ and its associated Kuratowski L-fuzzy closure operator $\bar{c}$, $(X, \bar{c})$ is obviously a quasi-discrete L-fuzzy closure space such that $\bar{c}(A\,*\,\bar{\alpha })=\bar{c}(A)\,*\,\bar{\alpha }$, $\forall A\in L^X$, $\forall \alpha \in L$. Hence from Proposition 15, there exists an L-fuzzy reflexive relation, say, $S_{\bar{c}}$ on X, given by $S_{\bar{c}}(w, t)\!=\!\bar{c}(1_t)(w), \forall w, t\!\in \! X$.

Before stating next, we introduce the following.

Definition 12

Let R and T be two L-fuzzy relations on X. Then T is called L-fuzzy transitive closure of R if T is the smallest L-fuzzy transitive relation containing R.

Now, we have the following.

Proposition 18

Let (X, c) be a quasi-discrete L-fuzzy closure space such that $c(H*\bar{\alpha })=c(H)*\bar{\alpha }$, $\forall H\in L^X$, $\forall \alpha \in L$ and $\bar{c}$ be the associated Kuratowski L-fuzzy closure operator. Then the L-fuzzy relation $S_{\bar{c}}$ is L-fuzzy transitive closure of L-fuzzy relation $R_{c}$.

Proof

Let $S_{\bar{c}}=\bar{c}(1_y)(x), \forall x, y\in X$.Transitivity of $S_{\bar{c}}$ follows from Propositions 5 and 15. Also, $R_{c}\le S_{\bar{c}}$ follows from Proposition 11. To show the relation $S_{\bar{c}}$ is an L-fuzzy transitive closure of L-fuzzy relation $R_{c}$, it only remains to show that $S_{\bar{c}}$ is the smallest L-fuzzy reflexive and transitive relation containing $R_{c}$. For this, let T be another L-fuzzy reflexive and transitive relation on X such that $R_{c}\le T$. Then from the reflexivity of T, $(X, \overline{apr}_T)$ is quasi-discrete L-fuzzy closure space. Now, from transitivity of T and Proposition 12 followed by Proposition 10, we have $\overline{apr}_T(H)=\wedge \{K\in L^X: H\le K, \overline{apr}_T(K)=K\}, \forall H\in L^X$. Also, $S_{\bar{c}}$ being L-fuzzy reflexive and L-fuzzy transitive relation associated with Kuratowski L-fuzzy closure operator $\bar{c}$, from Proposition 8 $\overline{apr}_{S_{\bar{c}}}(H)=\bar{c}(H), \forall H\in L^X$ and $\bar{c}$ being Kuratowski L-fuzzy closure operator associated with quasi-discrete L-fuzzy closure space (X, c), $\forall H\in L^X$, it follows from Proposition 15 that $\bar{c}(H)=\wedge \{K\in L^X: H\le K, c(K)=K\}=\wedge \{K\in L^X: H\le K, \overline{apr}_{R_{c}}(K)=K\}$. Thus from Proposition 6, $\overline{apr}_{S_{\bar{c}}}(H)= \wedge \{K\in L^X: H\le K, \overline{apr}_{R_{c}}(K)=K\}\le \wedge \{K\in L^X: H\le K, \overline{apr}_T(K)=K\}=\overline{apr}_T(H)$, whereby $\overline{apr}_{S_{\bar{c}}}(H)\le \overline{apr}_T(H)$, showing that $S_{\bar{c}}\le T$.

Now, we show that there is a bijective correspondence between the set of all L-fuzzy tolerance approximation spaces and the set of all symmetric quasi-discrete L-fuzzy closure spaces satisfying an extra condition.

Proposition 19

Let (X, R) be an L-fuzzy tolerance approximation space. Then $(X, \overline{apr}_R)$ is a symmetric quasi-discrete L-fuzzy closure space such that $\overline{apr}_{R}(H*\bar{\alpha })=\overline{apr}_{R}(H)*\bar{\alpha }$, $\forall H\in L^X$ and $\forall {\alpha }\in L$.

Proof

From Propositions 3 and 4 it follows that $(X, \overline{apr}_{R})$ is an L-fuzzy closure space and quasi-discrete. Now, $\forall x, y\in X, \overline{apr}_{R}(1_y)(x)=\vee \{R(x, t)*1_y(t): t\in X\}=\vee \{R(y, t)*1_x(t): t\in X\}=\overline{apr}_{R}(1_x)(y)$, showing that $(X, \overline{apr}_{R})$ is symmetric. Also, for all $H\in L^X$ and $\alpha \in L, \overline{apr}_{R}(H*\bar{\alpha })=\overline{apr}_{R}(H)*\bar{\alpha }$ follows from Proposition 3.

Proposition 20

Let (X, c) be a symmetric quasi-discrete L-fuzzy closure space such that $c(H\,*\,\bar{\alpha })=c(H)\,*\,\bar{\alpha }$, $\forall H\in L^X$ and $\forall \alpha \in L$. Then $\exists$ a L-fuzzy tolerance relation $R_{c}$ over X which is unique and satisfy $\overline{apr}_{{R}_{c}}(H)=c(H)$, $\forall H\in L^X$.

Proof

Let (X, c) be a quasi-discrete and such that $c(H*\bar{\alpha })=c(H)*\bar{\alpha }, \forall H\in L^X, \forall \alpha \in L$. Let L-fuzzy relation $R_c$ on X be such that $R_{c}(x, y)=c(1_y)(x), \forall x, y\in X$. Then $1=1_x(x)\le c(1_x)(x)$. Thus $c(1_x)(x)=1$. Hence $R_c$ is an L-fuzzy reflexive relation on X. Also, (X, c) being an L-fuzzy symmetric closure space, the L-fuzzy relation $R_c$ is symmetric and $\overline{apr}_{R_c}(H)=c(H)$ (cf., Proposition 15). To show the uniqueness of L-fuzzy relation $R_{c}$, let $R'$ be another L-fuzzy tolerance relation on X such that $\overline{apr}_{R'}(H)=c(H)$, $\forall H\in L^X$. Then $R_c(x, y)=c(1_y)(x)=\overline{apr}_{R'}(1_y)(x)=\vee \{R'(x,t)*1_y(t): t\in X\}=R'(x, y)$. Thus $R_c=R'$. Hence the L-fuzzy relation $R_c$ on X is unique.

Proposition 21

Let $\mathcal{F}$ be the set of all L-fuzzy tolerance approximation spaces and $\mathcal{T}$ be the set of all symmetric quasi-discrete L-fuzzy closure spaces satisfying $c(H*\bar{\alpha })=c(H)$ $*\,\bar{\alpha }$, $\forall H\in L^X$ and $\forall$ $\alpha \in L$, then $\exists$ a bijective correspondence between $\mathcal{F}$ and $\mathcal{T}$.

Proof

Follows from Propositions 19 and 20.

Proposition 22

Let (X, c) be a symmetric quasi-discrete L-fuzzy closure space such that $c(H*\bar{\alpha })=c(H)*\bar{\alpha }$, $\forall H\in L^X$, $\forall \alpha \in L$ and $\bar{c}$ be the associated Kuratowski L-fuzzy closure operator. Then the L-fuzzy relation $S_{\bar{c}}$ is an L-fuzzy transitive closure of L-fuzzy relation $R_{c}$.

Proof

Similar to that of Proposition 18.

Proposition 23

Let (X, R) be an L-fuzzy preorder approximation space. Then $(X, \overline{apr}_R)$ is a quasi-discrete L-fuzzy closure space such that (i) $\overline{apr}_R(\overline{apr}_R(H))=H$ and (ii) $\overline{apr}_{R}(H*\bar{\alpha })=\overline{apr}_{R}(H)*\bar{\alpha }$, $\forall H\in L^X$, $\forall \alpha \in L$.

Proof

Follows from Propositions 5 and 14.

Proposition 24

Let (X, c) be a quasi-discrete L-fuzzy closure space such that (i) $c(c(H))=c(H)$ and (ii) $c(H*\bar{\alpha })=c(H)*\bar{\alpha }$, $\forall H\in L^X$, $\forall \alpha \in L$. Then there exists an unique L-fuzzy preorder $R_c$ on X such that $\overline{apr}_{R_c}(H)=c(H)$, $\forall H\in L^X$.

Proof

Follows from Propositions 8, 12 and 15.

Finally, the following is an equivalent characterization of the result regarding the bijective correspondence between the set of all L-fuzzy preorder approximation spaces and the set of all saturated L-fuzzy topological spaces observed in [6, 10, 17].

Proposition 25

Let $\mathcal{F}$ be the set of all L-fuzzy preorder approximation spaces and $\mathcal{T}$ be the set of all quasi-discrete L-fuzzy closure spaces satisfying (i) $c(c(H))=c(H)$ and (ii) $c(H*\bar{\alpha })=c(H)*\bar{\alpha }$, $\forall H\in L^X$, $\forall \alpha \in L$. Then there exists a bijective correspondence between $\mathcal{F}$ and $\mathcal{T}$.

Proof

Follows from Propositions 23 and 24.

4 Conclusion

The present paper established an association between L-fuzzy rough sets and L-fuzzy closure spaces. In literature, the bijective correspondence between the set of all L-fuzzy preorder approximation spaces and the set of all L-fuzzy topological spaces of certain type is well known (cf., [6, 10]). But the work done in this paper shows that actual theory for such bijective correspondence begins from the notion of L-fuzzy closure spaces. In future we will try to associate L-fuzzy approximation spaces and L-fuzzy topological spaces in categorical point of view.