Alexandrov L-Fuzzy Pre-Proximities

Abstract

In this paper, we introduce the concepts of Alexandrov L-fuzzy pre-proximities on complete residuated lattices. Moreover, we investigate their relations among Alexandrov L-fuzzy pre-proximities, Alexandrov L-fuzzy topologies, L-fuzzy upper approximate operators, and L-fuzzy lower approximate operators. We give their examples.

1. Introduction

Pawlak [1,2] introduced the concept of rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Ward et al. [3] introduced the concept of the complete residuated lattice, which is an algebraic structure for many-valued logic. It is an important mathematical tool for studying algebraic structure. By using lower and upper approximation operators, information systems and decision rules were investigated in complete residuated lattices [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Bělohlávek [4] developed the notion of fuzzy contexts using Galois connections with $R\in L^{X\times Y}$ on a complete residuated lattice. El-Dardery [6] introduced L-fuzzy pre-proximity in view points of Sostak’s fuzzy topology [9] and Kim’s L-fuzzy proximities [13] on strictly two-sided, commutative quantales. Kim [10,11,12,13,14,15] investigated the properties of Alexandrov L-fuzzy topologies, Alexandrov L-fuzzy quasi-uniformities, and L-fuzzy approximate operators in complete residuated lattices.

In this paper, we introduce the concepts of Alexandrov L-fuzzy pre-proximities on complete residuated lattices, which are a unified approach to the three spaces: Alexandrov L-fuzzy topologies, L-fuzzy lower approximate operators, and L-fuzzy lower approximate operators as an extension of Pawlak’s rough sets. Moreover, we investigate their relations among Alexandrov L-fuzzy pre-proximities, Alexandrov L-fuzzy topologies, L-fuzzy lower approximate operators, and L-fuzzy lower approximate operators. We give their examples.

2. Preliminaries

Definition 1

([4,8,9,10]). An algebra $(L,\wedge ,\vee ,\odot ,\to ,\perp ,\top )$ is a complete residuated lattice if:

(L1)

$(L,\le ,\vee ,\wedge ,\perp ,\top )$ is a complete lattice with the greatest element ⊤ and the least element ⊥;

(L2)

$(L,\odot ,\top )$ is a commutative monoid;

(L3)

$x\odot y\le z$ if and only if $x\le y\to z$ for all $x,y,z\in L$.

In this paper, we always assume that $(L,\le ,\odot ,\to ,\oplus {,}^{*})$ is a complete residuated lattice with an order-reversing involution ${}^{*}$, which is defined by:

$$x\oplus y={(x^{*}\odot y^{*})}^{*},x^{*}=x\to \perp$$

unless otherwise specified. For all $\alpha \in L$,

$(\alpha \to f)\left(x\right)=\alpha \to f\left(x\right),(\alpha \odot f)\left(x\right)=\alpha \odot f\left(x\right),{\alpha }_X\left(x\right)=\alpha$,

$${\top }_x\left(y\right)=\left\{\begin{array}{cc}\top & \mathrm{if}y=x,\\ \perp ,& \mathrm{otherwise}\end{array}\right.\mathrm{and}{\top }_x^{*}\left(y\right)=\left\{\begin{array}{cc}\perp & \mathrm{if}y=x,\\ \top ,& \mathrm{otherwise}.\end{array}\right.$$

Lemma 1

([4,7,8]). Let $x,y,z,x_i,y_i,w\in L$. Then, the following hold.

(1)

$\top \to x=x$ and $\perp \odot x=\perp$.

(2)

If $y\le z$, then $x\odot y\le x\odot z$, $x\oplus y\le x\oplus z$, $x\to y\le x\to z$, and $z\to x\le y\to x$.

(3)

$x\le y$ if and only if $x\to y=\top$.

(4)

${\left({\bigwedge }_i{y}_i\right)}^{*}={\bigvee }_i{y}_i^{*}$ and ${\left({\bigvee }_i{y}_i\right)}^{*}={\bigwedge }_i{y}_i^{*}$.

(5)

$x\to \left({\bigwedge }_i{y}_i\right)={\bigwedge }_i(x\to y_i)$.

(6)

$\left({\bigvee }_i{x}_i\right)\to y={\bigwedge }_i(x_i\to y)$.

(7)

$x\odot \left({\bigvee }_i{y}_i\right)={\bigvee }_i(x\odot y_i)$.

(8)

$\left({\bigwedge }_i{x}_i\right)\oplus y={\bigwedge }_i(x_i\oplus y)$.

(9)

$(x\odot y)\to z=x\to (y\to z)=y\to (x\to z)$.

(10)

$x\odot y={(x\to y^{*})}^{*}$ and $x\oplus y=x^{*}\to y$.

(11)

$(x\to y)\odot (z\to w)\le (x\odot z)\to (y\odot w)$.

(12)

$x\to y\le (x\odot z)\to (y\odot z)$ and $(x\to y)\odot (y\to z)\le x\to z$.

(13)

$(x\to y)\odot (z\to w)\le (x\oplus z)\to (y\oplus w)$.

(14)

$x\to y=y^{*}\to x^{*}$.

(15)

$(x\vee y)\odot (z\vee w)\le (x\vee z)\vee (y\odot w)\le (x\oplus z)\vee (y\odot w).$

(16)

${\bigvee }_i{x}_i\to {\bigvee }_i{y}_i\ge {\bigwedge }_i(x_i\to y_i)$ and ${\bigwedge }_i{x}_i\to {\bigwedge }_i{y}_i\ge {\bigwedge }_i(x_i\to y_i)$.

(17)

$(x\odot y)\odot (z\oplus w)\le (x\odot z)\oplus (y\odot w)$.

(18)

$x\to y\le (y\to z)\to (x\to z)$ and $x\to y\le (z\to x)\to (z\to y)$.

Definition 2

([4]). Let X be a set. A mapping $R:X\times X\to L$ is an L-partial order if:

(E1)

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

(E2)

$R(x,y)\odot R(y,z)\le R(x,z)$ for all $x,y,z\in X$ (transitive);

(E3)

if $R(x,y)=R(y,x)=\top$, then $x=y$ (antisymmetric).

Definition 3

([4]). Let X be a set. Define a mapping $S:L^X\times L^X\to L$ by:

$$S(f,g)=\underset{x\in X}{\bigwedge }(f\left(x\right)\to g\left(x\right))forallf,g\in L^X.$$

Lemma 2 

([4]). Let $f,g,h,k\in L^X,$ and $\alpha \in L$. Then, the following hold.

(1)

S is an L-partial order on $L^X$.

(2)

$f\le g$ if and only if $S(f,g)\ge \top$.

(3)

If $f\le g$, then $S(h,f)\le S(h,g)$ and $S(f,h)\ge S(g,h)$.

(4)

$S(f,g)\odot S(k,h)\le S(f\oplus k,g\oplus h)$ and $S(f,g)\odot S(k,h)\le S(f\odot k,g\odot h).$

(5)

$S(g,h)\le S(f,g)\to S(f,h).$

(6)

$S(f,h)={\bigvee }_{g\in L^X}(S(f,g)\odot S(g,h)).$

Definition 4

([10]). A mapping $\mathcal{J}:L^X\to L^X$ is an L-lower approximation operator on X if:

(J1)

$\mathcal{J}\left({\top }_X\right)={\top }_X$ where ${\top }_X\left(x\right)=\top$ for all $x\in X$;

(J2)

$\mathcal{J}\left(f\right)\le f$ for all $f\in L^X$;

(J3)

$\mathcal{J}\left({\bigwedge }_{i\in \Gamma }{f}_i\right)={\bigwedge }_{i\in \Gamma }\mathcal{J}\left(f_i\right)$ for all ${\left\{f_i\right\}}_{i\in \Gamma }\subseteq L^X$;

(J4)

$\mathcal{J}(\alpha \to f)=\alpha \to \mathcal{J}\left(f\right)$.

The pair $(X,\mathcal{J})$ is called an L-lower approximation space. An L-lower approximation space is called topological if:

(T)

$\mathcal{J}\left(\mathcal{J}\right(f\left)\right)=\mathcal{J}\left(f\right)$ for all $f\in L^X$.

Definition 5

([10]). A mapping $\mathcal{H}:L^X\to L^X$ is an L-upper approximation operator on X if:

(H1)

$\mathcal{H}\left({\perp }_X\right)={\perp }_X$ where ${\perp }_X\left(x\right)=\perp$ for all $x\in X$;

(H2)

$\mathcal{H}\left(f\right)\ge f$ for all $f\in L^X$;

(H3)

$\mathcal{H}\left({\bigvee }_{i\in \Gamma }{f}_i\right)={\bigvee }_{i\in \Gamma }\mathcal{H}\left(f_i\right)$ for all ${\left\{f_i\right\}}_{i\in \Gamma }\subseteq L^X$;

(H4)

$\mathcal{H}(\alpha \odot f)=\alpha \odot \mathcal{H}\left(f\right)$.

The pair $(X,\mathcal{H})$ is called an L-upper approximation space. An L-upper approximation space is called topological if:

(T)

$\mathcal{H}\left(\mathcal{H}\right(f\left)\right)=\mathcal{H}\left(f\right)$ for all $f\in L^X$.

Definition 6

([10,11,12]). Let τ be a subset of $L^X$. τ is an Alexandrov L-topology on X if:

(O1)

${\perp }_X,{\top }_X\in \tau$;

(O2)

If $A_i\in \tau$ for all $i\in I$, then ${\bigwedge }_{i\in I}{A}_i,{\bigvee }_{i\in I}\in \tau$;

(O3)

If $A\in \tau$ and $\alpha \in L$, then $\alpha \odot A,\alpha \to A\in \tau$.

Definition 7

([10]). A mapping $\mathcal{T}:L^X\to L$ is an Alexandrov L-fuzzy topology on X if:

(AT1)

$\mathcal{T}\left({\perp }_X\right)=\mathcal{T}\left({\top }_X\right)=\top$;

(AT2)

$\mathcal{T}\left({\bigwedge }_i{f}_i\right)\ge {\bigwedge }_i\mathcal{T}\left(f_i\right)$ and $\mathcal{T}\left({\bigvee }_i{f}_i\right)\ge {\bigwedge }_i\mathcal{T}\left(f_i\right)$ for all ${\left\{f_i\right\}}_{i\in \Gamma }\subseteq L^X$;

(AT3)

$\mathcal{T}(\alpha \odot f)\ge \mathcal{T}\left(f\right)$ and $\mathcal{T}(\alpha \to f)\ge \mathcal{T}\left(f\right)$ for all $\alpha \in L$ and $f\in L^X$.

The pair $(X,\mathcal{T})$ is called an L-fuzzy topological space.

Theorem 1 

([10,11,12]).

(1)

Let $\mathcal{J}:L^X\to L^X$ be an L-lower approximation operator. Define ${\mathcal{H}}_{\mathcal{J}}:L^X\to L^X$ by ${\mathcal{H}}_{\mathcal{J}}\left(f\right)={\mathcal{J}}^{*}\left(f^{*}\right)$. Then, ${\mathcal{H}}_{\mathcal{J}}$ is an L-upper approximation operator.

(2)

Let $\mathcal{H}:L^X\to L^X$ be an L-upper approximation operator. Define ${\mathcal{J}}_{\mathcal{H}}:L^X\to L^X$ by ${\mathcal{J}}_{\mathcal{H}}\left(f\right)={\mathcal{H}}^{*}\left(f^{*}\right)$. Then, ${\mathcal{J}}_{\mathcal{H}}$ is an L-lower approximation operator.

(3)

Let $\mathcal{T}:L^X\to L$ be an Alexandrov L-fuzzy topology. Define ${\mathcal{T}}^{*}:L^X\to L$ by ${\mathcal{T}}^{*}\left(f\right)=\mathcal{T}\left(f^{*}\right)$. Then, ${\mathcal{T}}^{*}$ is an Alexandrov L-fuzzy topology.

(4)

Let $\tau \subset L^X$ be an Alexandrov L-topology. Define ${\tau }^{*}=\{f\mid f^{*}\in \tau \}$. Then, ${\tau }^{*}$ is an Alexandrov L-topology.

Theorem 2

([10]). Let $(X,\mathcal{H})$ be an L-upper approximation space. Define a mapping ${\mathcal{T}}_{\mathcal{H}}:L^X\to L$ by ${\mathcal{T}}_{\mathcal{H}}\left(f\right)=S(\mathcal{H}\left(f\right),f).$ Then, ${\mathcal{T}}_{\mathcal{H}}$ is an Alexandrov L-fuzzy topology on X with ${\mathcal{T}}_{\mathcal{H}}^{*}\left(f\right)=S(f,{\mathcal{J}}_{\mathcal{H}}\left(f\right))$ where ${\mathcal{J}}_{\mathcal{H}}\left(f\right)={\mathcal{H}}^{*}\left(f^{*}\right)$ for all $f\in L^X$.

Theorem 3

([10]). Let $(X,\mathcal{J})$ be an L-lower approximation space. Define a map ${\mathcal{T}}_{\mathcal{J}}:L^X\to L$ by ${\mathcal{T}}_{\mathcal{J}}\left(f\right)=S(f,\mathcal{J}\left(f\right)).$ Then, ${\mathcal{T}}_{\mathcal{J}}$ is an Alexandrov L-fuzzy topology on X.

3. The Relationships between Alexandrov L-Fuzzy Pre-Proximities and Alexandrov Topological Structures

Definition 8.

A mapping $\delta :L^X\times L^X\to L$ is an Alexandrov L-fuzzy pre-proximity on X if:

(P1)

$\delta ({\perp }_X,{\top }_X)=\delta ({\top }_X,{\perp }_X)=\perp$;

(P2)

$\delta (f,g)\ge {\bigvee }_{x\in X}(f\left(x\right)\odot g\left(x\right))$;

(P3)

If $f\le f_1$ and $g\le g_1$, then $\delta (f,g)\le \delta (f_1,g_1)$;

(P4)

For all $f_i,f,g_i,g\in L^X$, $\delta ({\bigvee }_{i\in \Gamma }{f}_i,g)\le {\bigvee }_{i\in \Gamma }\delta (f_i,g)$ and $\delta (f,{\bigvee }_{i\in \Gamma }{g}_i)\le {\bigvee }_{i\in \Gamma }\delta (f,g_i)$;

(P5)

For all $\alpha \in L$ and $f,g\in L^X$, $\delta (\alpha \odot f,g)=\alpha \odot \delta (f,g)=\delta (f,\alpha \odot g)$.

An Alexandrov L-fuzzy pre-proximity δ on X is called an Alexandrov L-fuzzy quasi-proximity if:

(P)

$\delta (f,g)\ge {\bigwedge }_{h\in L^X}\delta (f,h)\oplus \delta (h^{*},g)$.

Let ${\delta }_1$ and ${\delta }_2$ be two Alexandrov L-fuzzy pre-proximities on X. ${\delta }_1$ is finer than ${\delta }_2$ if ${\delta }_2(f,g)\ge {\delta }_1(f,g)$ for all $f,g\in L^X$.

Example 1.

Let $R\in L^{X\times X}$. Define a mapping $\delta :L^X\times L^X\to L$ by $\delta (f,g)={\bigvee }_{x,y\in X}(R(x,y)\odot f\left(x\right)\odot g\left(y\right)).$

(1)

Assume that R is reflexive. Then:

(P1)

$\delta ({\perp }_X,{\top }_X)=\delta ({\top }_X,{\perp }_X)=\perp$;

(P2)

$\delta (f,g)\ge {\bigvee }_{x\in X}(R(x,x)\odot f\left(x\right)\odot g\left(x\right))={\bigvee }_{x\in X}(f\left(x\right)\odot g\left(x\right))$;

(P3)

If $f\le f_1$ and $g\le g_1$, then $\delta (f,g)\le \delta (f_1,g_1)$;

(P4)

For all $f_i,f,g_i,g\in L^X$, $\delta ({\bigvee }_{i\in \Gamma }{f}_i,g)={\bigvee }_{i\in \Gamma }\delta (f_i,g)$ and $\delta (f,{\bigvee }_{i\in \Gamma }{g}_i)={\bigvee }_{i\in \Gamma }\delta (f,g_i)$.

(P5)

For all $\alpha \in L$ and $f,g\in L^X$,

$$\begin{array}{ll}\delta (\alpha \odot f,g)& ={\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot (\alpha \odot f\left(x\right)\odot g\left(y\right)))\\ & =\alpha \odot {\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot (f\left(x\right)\odot g\left(y\right)))\\ & =\alpha \odot \delta (f,g).\end{array}$$

Hence, δ is an Alexandrov L-fuzzy pre-proximity on X.

(2)

Assume that R is reflexive and transitive. Then, ${\bigvee }_{y\in X}(R(y,z)\odot R(x,y))=R(x,z).$ For all $f,g,h\in L^X$, we have by Lemma 1 (17) that:

$$\begin{array}{ll}\delta (f,h)\oplus \delta (h^{*},g)& =\left({\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot h\left(y\right))\right)\oplus \left({\displaystyle \underset{y,z\in X}{\bigvee }}(R(y,z)\odot h^{*}\left(y\right)\odot g\left(z\right))\right)\\ & \ge \left({\displaystyle \underset{x,y,z\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot h\left(y\right))\oplus (R(y,z)\odot h^{*}\left(y\right)\odot g\left(z\right))\right)\\ & \ge \left({\displaystyle \underset{x,y,z\in X}{\bigvee }}(R(x,y)\odot R(y,z)\odot f\left(x\right)\odot g\left(z\right))\odot (h\left(y\right)\oplus h^{*}\left(y\right))\right)\\ & ={\displaystyle \underset{x,y,z\in X}{\bigvee }}(R(x,y)\odot R(y,z)\odot f\left(x\right)\odot g\left(z\right))\\ & ={\displaystyle \underset{x,z\in X}{\bigvee }}(R(x,z)\odot f\left(x\right)\odot g\left(z\right))=\delta (f,g).\end{array}$$

Thus, $\delta (f,g)\le {\bigwedge }_{h\in L^X}(\delta (f,h)\oplus \delta (h^{*},g))$.

Let $h\left(y\right)={\left({\bigvee }_{x\in X}(R(x,y)\odot f\left(x\right))\right)}^{*}$. Then:

$$\begin{array}{l}{\displaystyle \underset{h\in L^X}{\bigwedge }}(\delta (f,h)\oplus \delta (h^{*},g))\\ ={\displaystyle \underset{h\in L^X}{\bigwedge }}(\left({\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot h\left(y\right))\right)\oplus \left({\displaystyle \underset{y,z\in X}{\bigvee }}(R(y,z)\odot h^{*}\left(y\right)\odot g\left(z\right))\right))\\ \le \left({\displaystyle \underset{y\in X}{\bigvee }}(h^{*}\left(y\right)\odot h\left(y\right))\right)\oplus \left({\displaystyle \underset{y,z\in X}{\bigvee }}(R(y,z)\odot {\displaystyle \underset{x\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot g\left(z\right)))\right)\\ =\perp \oplus \left({\displaystyle \underset{x,z\in X}{\bigvee }}({\displaystyle \underset{y\in X}{\bigvee }}(R(y,z)\odot R(x,y))\odot f\left(x\right)\odot g\left(z\right))\right)\\ ={\displaystyle \underset{x,z\in X}{\bigvee }}(R(x,z)\odot f\left(x\right)\odot g\left(z\right))=\delta (f,g).\end{array}$$

Hence, δ is an Alexandrov L-fuzzy quasi-proximity on X.

By taking $R(x,y)={\top }_{X\times X}$, let:

$$\begin{array}{ll}{\delta }_1(f,g)& ={\displaystyle \underset{x,y\in X}{\bigvee }}({\top }_{X\times X}(x,y)\odot f\left(x\right)\odot g\left(y\right))={\displaystyle \underset{x,y\in X}{\bigvee }}(f\left(x\right)\odot g\left(y\right)).\end{array}$$

Define ${\Delta }_{X\times X}\in L^{X\times X}$ by:

$${\Delta }_{X\times X}(x,y)=\left\{\begin{array}{cc}\top & \mathit{if}x=y,\\ \perp & \mathit{otherwise}.\end{array}\right.$$

By taking $R(x,y)={\Delta }_{X\times X}$, let:

$${\delta }_2(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}({\Delta }_{X\times X}(x,y)\odot (f\left(x\right)\odot g\left(y\right)))={\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot g\left(x\right)).$$

Then, ${\delta }_2(f,g)\le \delta (f,g)\le {\delta }_1(f,g)$ for all $f,g\in L^X$.

Lemma 3.

Let δ be an Alexandrov L-fuzzy pre-proximity on X. For all $\alpha \in L$ and $f,g,f_i,g_i\in L^X$, the following hold.

(1)

$\delta ({\bigvee }_{i\in \Gamma }{f}_i,g)={\bigvee }_{i\in \Gamma }\delta (f_i,g)$ and $\delta (f,{\bigvee }_{i\in \Gamma }{g}_i)={\bigvee }_{i\in \Gamma }\delta (f,g_i)$.

(2)

$\delta (\alpha \odot f,\alpha \to g)\le \delta (f,g)$ and $\delta (\alpha \to f,\alpha \odot g)\le \delta (f,g)$.

Proof. 

(1)

It follows from (P3) and (P4).

(2)

It follows from $\delta (\alpha \odot f,\alpha \to g)=\alpha \odot \delta (f,\alpha \to g)=\delta (f,\alpha \odot (\alpha \to g\left)\right)\le \delta (f,g)$.

 □

Theorem 4.

Let δ be an Alexandrov L-fuzzy pre-proximity on X. Define a mapping ${\delta }^s:L^X\times L^X\to L$ by ${\delta }^s(f,g)=\delta (g,f)$. Then, the following hold.

(1)

${\delta }^s$ is an Alexandrov L-fuzzy pre-proximity on X.

(2)

$\delta (f,g)={\bigvee }_{x,y\in X}(\delta ({\top }_x,{\top }_y)\odot (f\left(x\right)\odot g\left(y\right)).$

(3)

There exists a reflexive L-fuzzy relation $R_{\delta }\in L^{X\times X}$ such that:

$$\delta (f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(R_{\delta }(x,y)\odot (f\left(x\right)\odot g\left(y\right))).$$

(4)

There exists a reflexive L-fuzzy relation $R_{{\delta }^s}=R_{\delta }^{-1}\in L^{X\times X}$ such that:

$${\delta }^s(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(R_{\delta }^{-1}(x,y)\odot (f\left(x\right)\odot g\left(y\right))).$$

Proof. 

(1) It is easily proven.

(2) Since $f={\bigvee }_{x\in X}(f\left(x\right)\odot {\top }_x)$ and $g={\bigvee }_{y\in X}(g\left(y\right)\odot {\top }_y)$, we have:

$$\begin{array}{ll}\delta (f,g)& =\delta ({\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot {\top }_x),{\displaystyle \underset{y\in X}{\bigvee }}(g\left(y\right)\odot {\top }_y))\\ & ={\displaystyle \underset{x\in X}{\bigvee }}\left(f\left(x\right)\odot \delta ({\top }_x,{\displaystyle \underset{y\in X}{\bigvee }}(g\left(y\right)\odot {\top }_y))\right)\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}\left(f\left(x\right)\odot g\left(y\right)\odot \delta ({\top }_x,{\top }_y)\right).\end{array}$$

(3) Let $R_{\delta }(x,y)=\delta ({\top }_x,{\top }_y)$ in the equation in (2). By (P2),

$$R_{\delta }(x,x)=\delta ({\top }_x,{\top }_x)\ge {\displaystyle \underset{x\in X}{\bigvee }}({\top }_x\left(x\right)\odot {\top }_x\left(x\right))=\top .$$

Moreover, $\delta (f,g)={\bigvee }_{x,y\in X}(R_{\delta }(x,y)\odot f\left(x\right)\odot g\left(y\right)).$

(4) Since $R_{{\delta }^s}(x,y)={\delta }^s({\top }_x,{\top }_y)=\delta ({\top }_y,{\top }_x)=R_{\delta }^{-1}(x,y)$ by (2), we have:

$$\begin{array}{ll}{\delta }^s(f,g)& =\delta (g,f)\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}\left(R_{\delta }(x,y)\odot (g\left(x\right)\odot f\left(y\right))\right)\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}\left(R_{\delta }^{-1}(y,x)\odot (f\left(y\right)\odot g\left(x\right))\right).\end{array}$$

 □

Theorem 5.

Let δ be an Alexandrov L-fuzzy pre-proximity on X. Define a mapping ${\mathcal{T}}_{\delta }:L^X\to L$ by ${\mathcal{T}}_{\delta }\left(f\right)={\delta }^{*}(f,f^{*})$. Then, ${\mathcal{T}}_{\delta }$ is an Alexandrov L-fuzzy topology on X such that ${\mathcal{T}}_{\delta }^{*}={\mathcal{T}}_{{\delta }^s}$. If ${\delta }_1\le {\delta }_2$, then ${\mathcal{T}}_{{\delta }_1}\ge {\mathcal{T}}_{{\delta }_2}$.

Proof. 

(AT1) ${\mathcal{T}}_{\delta }\left({\top }_X\right)={\delta }^{*}({\top }_X,{\top }_X^{*})=\top$ and ${\mathcal{T}}_{\delta }\left({\perp }_X\right)={\delta }^{*}({\perp }_X,{\perp }_X^{*})=\top$.

(AT2) By (P3) and (P4), we have:

$${\mathcal{T}}_{\delta }\left({\displaystyle \underset{i}{\bigwedge }}{f}_i\right)={\delta }^{*}({\displaystyle \underset{i}{\bigwedge }}{f}_i,{\displaystyle \underset{i}{\bigvee }}{f}_i^{*})\ge {\delta }^{*}(f_i,{\displaystyle \underset{i}{\bigvee }}{f}_i^{*})={\displaystyle \underset{i}{\bigwedge }}{\delta }^{*}(f_i,f_i^{*})={\displaystyle \underset{i}{\bigwedge }}{\mathcal{T}}_{\delta }\left(f_i\right)$$

and:

$${\mathcal{T}}_{\delta }\left({\displaystyle \underset{i}{\bigvee }}{f}_i\right)={\delta }^{*}({\displaystyle \underset{i}{\bigvee }}{f}_i,{\displaystyle \underset{i}{\bigwedge }}{f}_i^{*})\ge {\delta }^{*}({\displaystyle \underset{i}{\bigvee }}{f}_i,f_i^{*})={\displaystyle \underset{i}{\bigwedge }}{\delta }^{*}(f_i,f_i^{*})={\displaystyle \underset{i}{\bigwedge }}{\mathcal{T}}_{\delta }\left(f_i\right).$$

(AT3) By Lemma 3 (2), we have:

$$\begin{array}{l}{\mathcal{T}}_{\delta }(\alpha \odot f)={\delta }^{*}(\alpha \odot f,\alpha \to f^{*})=\alpha \to {\delta }^{*}(f,\alpha \to f^{*})={\delta }^{*}(f,\alpha \odot (\alpha \to f^{*}))\\ \ge {\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right),{\mathcal{T}}_{\delta }(\alpha \to f)={\delta }^{*}(\alpha \to f,\alpha \odot f^{*})\ge {\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right).\end{array}$$

Then, ${\mathcal{T}}_{\delta }$ is an Alexandrov L-fuzzy topology on X. Moreover,

$$\begin{array}{l}{\mathcal{T}}_{\delta }^{*}\left(f\right)={\mathcal{T}}_{\delta }\left(f^{*}\right)={\delta }^{*}(f^{*},f)={\delta }^{s*}(f,f^{*})={\mathcal{T}}_{{\delta }^s}\left(f\right).\end{array}$$

 □

Example 2.

Let $R\in L^{X\times X}$ be a reflexive fuzzy relation. Define a mapping $\delta :L^X\times L^X\to L$ by $\delta (f,g)={\bigvee }_{x,y\in X}(R(x,y)\odot f\left(x\right)\odot g\left(y\right)).$ Then:

$$\begin{array}{l}{\mathcal{T}}_{\delta }\left(f\right)={\delta }^{*}(f,f^{*})={\left({\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot f^{*}\left(y\right))\right)}^{*}\\ ={\displaystyle \underset{x,y\in X}{\bigwedge }}(R(x,y)\to (f\left(x\right)\to f\left(y\right)).\end{array}$$

If $R={\top }_{X\times X}$, then ${\mathcal{T}}_{\delta }\left(f\right)={\bigwedge }_{x,y\in X}(f\left(x\right)\to f\left(y\right))$.

If $R={\Delta }_{X\times X}$, then ${\mathcal{T}}_{\delta }\left(f\right)={\bigwedge }_{x\in X}(f\left(x\right)\to f\left(x\right))=\top .$

From the following two theorems, we obtain the L-lower approximation operator and the L-lower approximation operator induced by an Alexandrov L-fuzzy pre-proximity.

Theorem 6.

Let δ be an Alexandrov L-fuzzy pre-proximity on X. Define a mapping ${\mathcal{H}}_{\delta }:L^X\to L^X$ by ${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)=\delta (f,{\top }_x).$ Then, the following hold.

(1)

${\mathcal{H}}_{\delta }$ is an L-upper approximation operator on X.

(2)

$\delta ({\top }_x,{\top }_x)=\top$.

(3)

There exists a reflexive L-fuzzy relation $R_{\delta }\in L^{X\times X}$ such that:

$${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigvee }}(R_{\delta }(y,x)\odot f\left(y\right)).$$

Moreover, there exists a reflexive L-fuzzy relation $R_{{\delta }^s}=R_{\delta }^{-1}\in L^{X\times X}$ such that:

$${\mathcal{H}}_{{\delta }^s}\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigvee }}(R_{\delta }(x,y)\odot f\left(y\right)).$$

(4)

${\bigvee }_{y\in X}(\delta ({\top }_x,{\top }_y)\odot \delta ({\top }_y,{\top }_z))\le \delta ({\top }_x,{\top }_z)$ if and only if ${\mathcal{H}}_{\delta }$ is a topological L-upper approximation operator on X.

(5)

${\mathcal{T}}_{{\mathcal{H}}_{\delta }}\left(f\right)={\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right)$ for all $f\in L^X$.

(6)

$\delta (f,g)={\bigvee }_{x\in X}({\mathcal{H}}_{\delta }\left(f\right)\left(x\right)\odot g\left(x\right))$ for all $f,g\in L^X$.

Proof. 

(1)

(H1) Since $\delta ({\perp }_X,{\top }_x)\le \delta ({\perp }_X,{\top }_X)=\perp$, we have ${\mathcal{H}}_{\delta }\left({\perp }_X\right)\left(x\right)=\delta ({\perp }_X,{\top }_x)=\perp .$

(H2)

${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)=\delta (f,{\top }_x)\ge {\bigvee }_{x\in X}(f\left(x\right)\odot {\top }_x\left(x\right))=f\left(x\right).$

(H3)

From Lemma 3, we obtain:

$$\begin{array}{ll}{\mathcal{H}}_{\delta }\left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{f}_i\right)\left(x\right)& =\delta ({\displaystyle \underset{i\in \Gamma }{\bigvee }}{f}_i,{\top }_x)={\displaystyle \underset{i\in \Gamma }{\bigvee }}\delta (f_i,{\top }_x)\\ & ={\displaystyle \underset{i\in \Gamma }{\bigvee }}{\mathcal{H}}_{\delta }\left(f_i\right)\left(x\right).\end{array}$$

(H4)

By (P4), ${\mathcal{H}}_{\delta }(\alpha \odot f)\left(x\right)=\delta (\alpha \odot f,{\top }_x)=\alpha \odot \delta (f,{\top }_x)=\alpha \odot {\mathcal{H}}_{\delta }\left(f\right).$ Hence, ${\mathcal{H}}_{\delta }$ is an L-upper approximation operator on X.

(2)

$\delta ({\top }_x,{\top }_x)\ge {\bigvee }_{x\in X}({\top }_x\left(x\right)\odot {\top }_x\left(x\right))=\top$.

(3)

We obtain ${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)=\delta (f,{\top }_x)=\delta ({\bigvee }_{y\in X}(f\left(y\right)\odot {\top }_y),{\top }_x)={\bigvee }_{y\in X}(f\left(y\right)\odot \delta ({\top }_y,{\top }_x))$. Put $R_{\delta }(x,y)=\delta ({\top }_x,{\top }_y)$. By (2), $R_{\delta }$ is reflexive. Then, ${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)={\bigvee }_{y\in X}(f\left(y\right)\odot R_{\delta }(y,x))$. Moreover, $R_{{\delta }^s}(x,y)={\delta }^s({\top }_x,{\top }_y)=\delta ({\top }_y,{\top }_x)=R_{\delta }(y,x)=R_{\delta }^{-1}(x,y)$ such that:

$$\begin{array}{ll}{\mathcal{H}}_{{\delta }^s}\left(f\right)\left(x\right)& ={\displaystyle \underset{y\in X}{\bigvee }}(f\left(y\right)\odot {\delta }^{s*}({\top }_y,{\top }_x))\\ & ={\displaystyle \underset{y\in X}{\bigvee }}(f\left(y\right)\odot \delta ({\top }_x,{\top }_y))={\displaystyle \underset{y\in X}{\bigvee }}(f\left(y\right)\odot R_{\delta }(x,y)).\end{array}$$

(4)

Since ${\mathcal{H}}_{\delta }\left(f\right)={\bigvee }_{y\in X}({\mathcal{H}}_{\delta }\left(f\right)\left(y\right)\odot {\top }_y)$, we have:

$$\begin{array}{ll}{\mathcal{H}}_{\delta }\left({\mathcal{H}}_{\delta }\left(f\right)\right)\left(x\right)& =\delta ({\mathcal{H}}_{\delta }\left(f\right),{\top }_x)=\delta ({\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left(f\right)\left(y\right)\odot {\top }_y),{\top }_x)\\ & ={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left(f\right)\left(y\right)\odot \delta ({\top }_y,{\top }_x))={\displaystyle \underset{y\in X}{\bigvee }}(\delta (f,{\top }_y)\odot \delta ({\top }_y,{\top }_x))\\ & ={\displaystyle \underset{y\in X}{\bigvee }}(\delta ({\displaystyle \underset{z\in X}{\bigvee }}(f\left(z\right)\odot {\top }_z),{\top }_y)\odot \delta ({\top }_y,{\top }_x))\\ & ={\displaystyle \underset{y\in X}{\bigvee }}({\displaystyle \underset{z\in X}{\bigvee }}(f\left(z\right)\odot \delta ({\top }_z,{\top }_y))\odot \delta ({\top }_y,{\top }_x))\\ & ={\displaystyle \underset{z\in X}{\bigvee }}(f\left(z\right)\odot {\displaystyle \underset{y\in X}{\bigvee }}(\delta ({\top }_z,{\top }_y)\odot \delta ({\top }_y,{\top }_x)))\\ & \le {\displaystyle \underset{z\in X}{\bigvee }}(f\left(z\right)\odot \delta ({\top }_z,{\top }_x))=\delta ({\displaystyle \underset{z\in X}{\bigvee }}(f\left(z\right)\odot {\top }_z),{\top }_x)\\ & =\delta (f,{\top }_x)={\mathcal{H}}_{\delta }\left(f\right)\left(x\right).\end{array}$$

Conversely, since ${\mathcal{H}}_{\delta }\left({\mathcal{H}}_{\delta }\left({\top }_z\right)\right)\left(x\right)\le {\mathcal{H}}_{\delta }\left({\top }_z\right)\left(x\right)$, for ${\mathcal{H}}_{\delta }\left({\top }_z\right)={\bigvee }_{y\in X}({\mathcal{H}}_{\delta }\left({\top }_z\right)\left(y\right)\odot {\top }_y)$, we have:

$$\begin{array}{ll}{\mathcal{H}}_{\delta }\left({\mathcal{H}}_{\delta }\left({\top }_z\right)\right)\left(x\right)& ={\mathcal{H}}_{\delta }\left({\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left({\top }_z\right)\left(y\right)\odot {\top }_y)\right)\left(x\right)\\ & ={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left({\top }_z\right)\left(y\right)\odot {\mathcal{H}}_{\delta }\left({\top }_y\right)\left(x\right))\le {\mathcal{H}}_{\delta }\left({\top }_z\right)\left(x\right).\end{array}$$

(5)

For all $f\in L^X$, we have:

$$\begin{array}{ll}{\mathcal{T}}_{{\mathcal{H}}_{\delta }}\left(f\right)& =S({\mathcal{H}}_{\delta }\left(f\right),f)={\displaystyle \underset{x\in X}{\bigwedge }}({\mathcal{H}}_{\delta }\left(f\right)\left(x\right)\to f\left(x\right))\\ & ={\displaystyle \underset{x\in X}{\bigwedge }}(\delta (f,{\top }_x)\to f\left(x\right))={\displaystyle \underset{x\in X}{\bigwedge }}(f^{*}\left(x\right)\to {\delta }^{*}(f,{\top }_x))\\ & ={\displaystyle \underset{x\in X}{\bigwedge }}{\delta }^{*}(f,f^{*}\left(x\right)\odot {\top }_x)={\delta }^{*}(f,{\displaystyle \underset{x\in X}{\bigvee }}(f^{*}\left(x\right)\odot {\top }_x))\\ & ={\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right).\end{array}$$

(6)

$$\begin{array}{ll}{\displaystyle \underset{x\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left(f\right)\left(x\right)\odot g\left(x\right))& & ={\displaystyle \underset{x\in X}{\bigvee }}(\delta (f,{\top }_x)\odot g\left(x\right))\\ & =\delta (f,{\displaystyle \underset{x\in X}{\bigvee }}({\top }_x\odot g\left(x\right)))=\delta (f,g).\end{array}$$

 □

Theorem 7.

Let δ be an Alexandrov L-fuzzy pre-proximity on X. Define a mapping ${\mathcal{J}}_{\delta }:L^X\to L^X$ by ${\mathcal{J}}_{\delta }\left(f\right)\left(x\right)={\delta }^{*}({\top }_x,f^{*}).$ Then, the following hold.

(1)

${\mathcal{J}}_{\delta }$ is an L-lower approximation operator on X.

(2)

There exists a reflexive L-fuzzy relation $R_{\delta }\in L^{X\times X}$ such that:

$${\mathcal{J}}_{\delta }\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigwedge }}(R_{\delta }(x,y)\to f\left(y\right)).$$

Moreover, there exists a reflexive L-fuzzy relation $R_{{\delta }^s}=R_{\delta }^{-1}\in L^{X\times X}$ such that:

$${\mathcal{J}}_{{\delta }^s}\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigwedge }}(R_{\delta }(y,x)\to f\left(y\right)).$$

(3)

For all $f\in L^X$, ${\bigvee }_{y\in X}(\delta ({\top }_x,{\top }_y)\odot \delta ({\top }_y,{\top }_z))\le \delta ({\top }_x,{\top }_z)$ if and only if ${\mathcal{J}}_{\delta }\left({\mathcal{J}}_{\delta }\left(f\right)\right)\ge {\mathcal{J}}_{\delta }\left(f\right)$.

(4)

${\mathcal{T}}_{{\mathcal{J}}_{\delta }}\left(f\right)={\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right)$ for all $f\in L^X$.

(5)

${\mathcal{J}}_{{\delta }^s}\left(f\right)=\delta (f^{*},{\top }_x^{*})={\mathcal{H}}_{\delta }^{*}\left(f^{*}\right)$ for all $f\in L^X$ and ${\mathcal{T}}_{{\mathcal{J}}_{\delta }}^{*}={\mathcal{T}}_{{\delta }^s}={\mathcal{T}}_{{\mathcal{J}}_{{\delta }^s}}$.

(6)

$\delta (f,g)=S(f,{\mathcal{J}}_{\delta }\left(g\right))$ for all $f,g\in L^X$.

Proof. 

(1)

(J1) Since ${\delta }^{*}({\top }_x,{\top }_X^{*})\ge {\delta }^{*}({\top }_X,{\perp }_X)=\top$, we have ${\mathcal{J}}_{\delta }\left({\top }_X\right)\left(x\right)={\delta }^{*}({\top }_x,{\top }_X^{*})=\top .$

(J2)

Note that:

$${\mathcal{J}}_{\delta }\left(f\right)\left(x\right)={\delta }^{*}({\top }_x,f^{*})\le {\left({\displaystyle \underset{x\in X}{\bigvee }}({\top }_x\left(x\right)\odot f^{*}\left(x\right))\right)}^{*}=f\left(x\right).$$

(J3)

By Lemma 3, we obtain:

$$\begin{array}{l}{\mathcal{J}}_{\delta }\left({\displaystyle \underset{i\in \Gamma }{\bigwedge }}{f}_i\right)\left(x\right)={\delta }^{*}({\top }_x,{\displaystyle \underset{i\in \Gamma }{\bigvee }}{f}_i^{*})={\displaystyle \underset{i\in \Gamma }{\bigwedge }}{\delta }^{*}({\top }_x,f_i^{*})={\displaystyle \underset{i\in \Gamma }{\bigwedge }}{\mathcal{J}}_{\delta }\left(f_i\right)\left(x\right).\end{array}$$

(J4)

By (P4), we have:

$$\begin{array}{l}{\mathcal{J}}_{\delta }(\alpha \to f)\left(x\right)={\delta }^{*}({\top }_x,\alpha \odot f^{*})=\alpha \to {\delta }^{*}({\top }_x,f^{*})=\alpha \to {\mathcal{J}}_{\delta }\left(f\right).\end{array}$$

(2)

For $f^{*}={\bigvee }_{y\in X}(f^{*}\left(y\right)\odot {\top }_y)$, we have:

$$\begin{array}{ll}{\mathcal{J}}_{\delta }\left(f\right)\left(x\right)& ={\delta }^{*}({\top }_x,f^{*})={\delta }^{*}({\top }_x,{\displaystyle \underset{y\in X}{\bigvee }}(f^{*}\left(y\right)\odot {\top }_y))\\ & ={\displaystyle \underset{y\in X}{\bigwedge }}(f^{*}\left(y\right)\to {\delta }^{*}({\top }_x,{\top }_y))={\displaystyle \underset{y\in X}{\bigwedge }}(\delta ({\top }_x,{\top }_y)\to f\left(y\right)).\end{array}$$

Let $R_{\delta }(x,y)=\delta ({\top }_x,{\top }_y)$. By (2), $R_{\delta }$ is reflexive and ${\mathcal{J}}_{\delta }\left(f\right)\left(x\right)={\bigwedge }_{y\in X}(R_{\delta }(x,y)\to f\left(y\right))$. Moreover, $R_{{\delta }^s}(x,y)={\delta }^{s*}({\top }_x,{\top }_y)={\delta }^{*}({\top }_y,{\top }_x)=R_{\delta }(y,x)=R_{\delta }^{-1}(x,y)$ such that:

$$\begin{array}{l}{\mathcal{J}}_{{\delta }^s}\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigwedge }}({\delta }^{s*}({\top }_x,{\top }_y)\to f\left(y\right))={\displaystyle \underset{y\in X}{\bigwedge }}({\delta }^{*}({\top }_y,{\top }_x)\to f\left(y\right))={\displaystyle \underset{y\in X}{\bigwedge }}(R_{\delta }(y,x)\to f\left(y\right)).\end{array}$$

(3)

Since ${\mathcal{J}}_{\delta }\left(f\right)={\bigwedge }_{y\in X}({\mathcal{J}}_{\delta }^{*}\left(f\right)\left(y\right)\to {\top }_y^{*})$, we have:

$$\begin{array}{ll}{\mathcal{J}}_{\delta }\left({\mathcal{J}}_{\delta }\left(f\right)\right)\left(x\right)& ={\delta }^{*}({\top }_x,{\mathcal{J}}_{\delta }^{*}\left(f\right))\\ & ={\delta }^{*}({\top }_x,{\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{J}}_{\delta }^{*}\left(f\right)\left(y\right)\odot {\top }_y))={\displaystyle \underset{y\in X}{\bigwedge }}({\mathcal{J}}_{\delta }^{*}\left(f\right)\left(y\right)\to {\delta }^{*}({\top }_x,{\top }_y))\\ & ={\displaystyle \underset{y\in X}{\bigwedge }}(\delta ({\top }_y,{\displaystyle \underset{z\in X}{\bigvee }}(f^{*}\left(z\right)\odot {\top }_z))\to {\delta }^{*}({\top }_x,{\top }_y))\\ & ={\displaystyle \underset{y\in X}{\bigwedge }}({\displaystyle \underset{z\in X}{\bigvee }}(f^{*}\left(z\right)\odot \delta ({\top }_y,{\top }_z))\to {\delta }^{*}({\top }_x,{\top }_y))\\ & ={\left({\displaystyle \underset{z\in X}{\bigvee }}(f^{*}\left(z\right)\odot {\displaystyle \underset{y\in X}{\bigvee }}(\delta ({\top }_y,{\top }_z)\odot \delta ({\top }_x,{\top }_y)))\right)}^{*}\\ & \ge {\left({\displaystyle \underset{y\in X}{\bigvee }}(f^{*}\left(z\right)\odot \delta ({\top }_x,{\top }_z))\right)}^{*}\\ & ={\left(\delta ({\top }_x,{\displaystyle \underset{y\in X}{\bigvee }}(f^{*}\left(z\right)\odot {\top }_z))\right)}^{*}={\delta }^{*}({\top }_x,f^{*})={\mathcal{J}}_{\delta }\left(f\right)\left(x\right).\end{array}$$

Conversely, since ${\mathcal{J}}_{\delta }\left({\top }_y^{*}\right)\left(x\right)={\delta }^{*}({\top }_x,{\top }_y)$ and ${\mathcal{J}}_{\delta }\left({\top }_z^{*}\right)={\bigwedge }_{y\in X}({\mathcal{J}}_{\delta }^{*}\left({\top }_z^{*}\right)\left(y\right)\to {\top }_y^{*})$, we have that ${\mathcal{J}}_{\delta }\left({\mathcal{J}}_{\delta }\left({\top }_z^{*}\right)\right)\left(x\right)={\bigwedge }_{y\in X}({\mathcal{J}}_{\delta }^{*}\left({\top }_z^{*}\right)\left(y\right)\to {\mathcal{J}}_{\delta }\left({\top }_y^{*}\right)\left(x\right)\ge {\mathcal{J}}_{\delta }\left({\top }_z^{*}\right)\left(x\right)$ if and only if ${\bigvee }_{y\in X}({\mathcal{J}}_{\delta }^{*}\left({\top }_z^{*}\right)\left(y\right)\odot {\mathcal{J}}_{\delta }^{*}\left({\top }_y^{*}\right)\left(x\right)\le {\mathcal{J}}_{\delta }^{*}\left({\top }_z^{*}\right)\left(x\right)$ if and only if ${\bigvee }_{y\in X}(\delta ({\top }_y,{\top }_z)\odot \delta ({\top }_x,{\top }_y)\le \delta ({\top }_x,{\top }_z)$.

(4)

For $f={\bigvee }_{x\in X}(f\left(x\right)\odot {\top }_x)$, we have:

$$\begin{array}{ll}{\mathcal{T}}_{{\mathcal{J}}_{\delta }}\left(f\right)& =S(f,{\mathcal{J}}_{\delta }\left(f\right))={\displaystyle \underset{x\in X}{\bigwedge }}\left(f\left(x\right)\to {\delta }^{*}({\top }_x,f^{*})\right)={\displaystyle \underset{x\in X}{\bigwedge }}{\delta }^{*}(f\left(x\right)\odot {\top }_x,f^{*}))\\ & ={\delta }^{*}({\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot {\top }_x),f^{*})={\delta }^{*}(f,f^{*})={\mathcal{T}}_{\delta }\left(f\right).\end{array}$$

(5)

For all $f\in L^X$, we have:

$$\begin{array}{ll}{\mathcal{J}}_{{\delta }^s}\left(f\right)\left(x\right)& ={\delta }^{s*}({\top }_x,f^{*})={\delta }^{*}(f^{*},{\top }_x)={\mathcal{H}}_{\delta }^{*}\left(f^{*}\right),\\ {\mathcal{T}}_{{\mathcal{J}}_{\delta }}^{*}\left(f\right)& ={\mathcal{T}}_{{\mathcal{J}}_{\delta }}\left(f^{*}\right)={\delta }^{*}(f^{*},f)={\mathcal{T}}_{{\delta }^s}\left(f\right)={\mathcal{T}}_{{\mathcal{J}}_{{\delta }^s}}\left(f\right).\end{array}$$

(6)

For all $f,g\in L^X$, we have:

$$\begin{array}{ll}S(f,{\mathcal{J}}_{\delta }\left(g\right))& ={\displaystyle \underset{x\in X}{\bigwedge }}(f\left(x\right)\to {\mathcal{J}}_{\delta }\left(g\right)))={\displaystyle \underset{x\in X}{\bigwedge }}(f\left(x\right)\to {\delta }^{*}({\top }_x,g^{*}))\\ & ={\delta }^{*}({\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot {\top }_x),g^{*})={\delta }^{*}(f,g^{*}).\end{array}$$

 □

From the following theorem, we obtain the Alexandrov L-fuzzy pre-proximity induced by an L-upper approximation operator.

Theorem 8.

Let $(X,\mathcal{H})$ be an L-upper approximation space. Define a mapping ${\delta }_{\mathcal{H}}:L^X\times L^X\to L$ by:

$${\delta }_{\mathcal{H}}(f,g)={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(y\right)\odot g\left(y\right)).$$

Then, the following hold.

(1)

${\delta }_{\mathcal{H}}$ is an Alexandrov L-fuzzy proximity such that:

$${\delta }_{\mathcal{H}}(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(\mathcal{H}\left({\top }_y\right)\left(x\right)\odot (f\left(y\right)\odot g\left(x\right))).$$

(2)

${\delta }_{\mathcal{H}}(f,g)\le {\bigwedge }_{h\in L^X}({\delta }_{\mathcal{H}}(f,h)\oplus {\delta }_{\mathcal{H}}(h^{*},g))$. Moreover, the equality holds if $\mathcal{H}$ is topological.

(3)

If $\mathcal{H}$ is topological, then ${\delta }_{\mathcal{H}}$ is an Alexandrov L-fuzzy quasi-proximity on X.

(4)

$\mathcal{H}={\mathcal{H}}_{{\delta }_{\mathcal{H}}}$.

(5)

${\mathcal{T}}_{\mathcal{H}}\left(f\right)={\delta }_{\mathcal{H}}(f,f)={\mathcal{T}}_{{\delta }_{\mathcal{H}}}\left(f\right)$ for all $f\in L^X$.

(6)

If δ is an Alexandrov L-fuzzy pre-proximity on X, then ${\delta }_{{\mathcal{H}}_{\delta }}(f,g)=\delta (f,g)$ for all $f,g\in L^X$.

Proof. 

(1)

(P1) Since $\mathcal{H}\left({\perp }_X\right)={\perp }_X$ and $\mathcal{H}\left({\top }_X\right)={\top }_X$, we have:

$$\begin{array}{ll}{\delta }_{\mathcal{H}}({\top }_X,{\perp }_X)& ={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left({\top }_X\right)\left(y\right)\odot {\perp }_X\left(y\right))=\perp ,\\ {\delta }_{\mathcal{H}}({\perp }_X,{\top }_X)& ={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left({\perp }_X\right)\left(y\right)\odot {\top }_X\left(y\right))=\top .\end{array}$$

(P2)

Since $\mathcal{H}\left(f\right)\ge f$, we have:

$$\begin{array}{ll}{\delta }_{\mathcal{H}}(f,g)& ={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(y\right)\odot g\left(y\right))\ge {\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot g\left(x\right)).\end{array}$$

(P3)

If $f\le f_1$ and $g\le g_1$, then $\mathcal{H}\left(f\right)\le \mathcal{H}\left(f_1\right)$. Thus,

$$\begin{array}{ll}{\delta }_{\mathcal{H}}(f,g)& ={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(y\right)\odot g\left(y\right))\le {\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left(f_1\right)\left(y\right)\odot g_1\left(y\right))={\delta }_{\mathcal{H}}(f_1,g_1).\end{array}$$

(P4)

Note that:

$$\begin{array}{ll}{\delta }_{\mathcal{H}}({\displaystyle \underset{i\in \Gamma }{\bigvee }}{f}_i,g)& ={\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{f}_i\right)\left(x\right)\odot g\left(x\right))\\ & ={\displaystyle \underset{x\in X}{\bigvee }}({\displaystyle \underset{i\in \Gamma }{\bigvee }}\mathcal{H}\left(f_i\right)\left(x\right)\odot g\left(x\right))={\displaystyle \underset{i\in \Gamma }{\bigvee }}{\delta }_{\mathcal{H}}(f_i,g),\end{array}$$

$$\begin{array}{ll}{\delta }_{\mathcal{H}}(f,{\displaystyle \underset{i\in \Gamma }{\bigvee }}{g}_i)& ={\displaystyle \underset{x\in X}{\bigvee }}(f\left(x\right)\odot {\displaystyle \underset{i\in \Gamma }{\bigvee }}{g}_i\left(x\right))={\displaystyle \underset{i\in \Gamma }{\bigvee }}{\delta }_{\mathcal{H}}(f,g_i)\end{array}$$

and:

$$\begin{array}{ll}{\delta }_{\mathcal{H}}(\alpha \odot f,g)& ={\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}(\alpha \odot f)\left(x\right)\odot g\left(x\right))\\ & ={\displaystyle \underset{x\in X}{\bigvee }}(\alpha \odot \mathcal{H}\left(f\right)\left(x\right)\odot g\left(x\right))=\alpha \odot {\delta }_{\mathcal{H}}(f,g).\end{array}$$

Hence, ${\delta }_{\mathcal{H}}$ is an Alexandrov L-fuzzy pre-proximity. For $f=\bigvee (f\left(y\right)\odot {\top }_y)$, we have:

$$\begin{array}{ll}{\delta }_{\mathcal{H}}(f,g)& ={\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(x\right)\odot g\left(x\right))={\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}(\bigvee (f\left(y\right)\odot {\top }_y))\left(x\right)\odot g\left(x\right))\\ & ={\displaystyle \underset{x\in X}{\bigvee }}({\displaystyle \underset{y\in X}{\bigvee }}(f\left(y\right)\odot \mathcal{H}\left({\top }_y\right)\left(x\right))\odot g\left(x\right))\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}(\mathcal{H}\left({\top }_y\right)\left(x\right)\odot (f\left(y\right)\odot g\left(x\right))).\end{array}$$

(2)

For each $f,g,h\in L^X$, we have:

$$\begin{array}{l}{\delta }_{\mathcal{H}}(f,h)\oplus {\delta }_{\mathcal{H}}(h^{*},g)\\ =\left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(x\right)\odot h\left(x\right))\right)\oplus \left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(h^{*}\right)\left(x\right)\odot g\left(x\right))\right)\\ \ge {\displaystyle \underset{x\in X}{\bigvee }}\left((\mathcal{H}\left(f\right)\left(x\right)\odot h\left(x\right))\oplus (\mathcal{H}\left(h^{*}\right)\left(x\right)\odot g\left(x\right))\right)\\ \ge {\displaystyle \underset{x\in X}{\bigvee }}\left((\mathcal{H}\left(f\right)\left(x\right)\odot f\left(x\right))\odot (h\left(x\right)\oplus \mathcal{H}\left(h^{*}\right)\left(x\right))\right)\mathrm{by}\mathrm{Lemma}1\left(17\right)\\ ={\displaystyle \underset{x\in X}{\bigvee }}\left((\mathcal{H}\left(f\right)\left(x\right)\odot f\left(x\right))\odot (h^{*}\left(x\right)\to \mathcal{H}\left(h^{*}\right)\left(x\right))\right)\\ ={\delta }_{\mathcal{H}}(f,g).\end{array}$$

Hence, ${\delta }_{\mathcal{H}}(f,g)\le {\bigwedge }_{h\in L^X}({\delta }_{\mathcal{H}}(f,h)\oplus {\delta }_{\mathcal{H}}(h^{*},g))$.

If $\mathcal{H}$ is topological, then:

$$\begin{array}{l}{\displaystyle \underset{h\in L^X}{\bigwedge }}({\delta }_{\mathcal{H}}(f,h)\oplus {\delta }_{\mathcal{H}}(h^{*},g))\\ ={\displaystyle \underset{h\in L^X}{\bigwedge }}(\left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(f\left(x\right)\right)\odot h\left(x\right))\right)\oplus \left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(h^{*}\right)\left(x\right)\odot g\left(x\right))\right))\\ (put{h}^{*}=\mathcal{H}\left(f\right),)\\ \le ({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(f\left(x\right)\right)\odot {\mathcal{H}}^{*}\left(f\left(x\right)\right))\oplus \left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(\mathcal{H}\left(f\right)\right)\left(x\right)\odot g\left(x\right))\right))\\ =\left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(\mathcal{H}\left(f\right)\right)\left(x\right)\odot g\left(x\right))\right))={\delta }_{\mathcal{H}}(f,g).\end{array}$$

(3)

It follows by (2).

(4)

For all $f\in L^X$, we have:

$$\begin{array}{l}{\mathcal{H}}_{{\delta }_{\mathcal{H}}}\left(f\right)={\delta }_{\mathcal{H}}(f,{\top }_x)={\displaystyle \underset{y\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(y\right)\odot {\top }_x\left(y\right))=\mathcal{H}\left(f\right)\left(x\right).\end{array}$$

(5)

For all $f\in L^X$, we have:

$${\mathcal{T}}_{{\delta }_{\mathcal{H}}}\left(f\right)={\delta }_{\mathcal{H}}^{*}(f,f^{*})={\left({\displaystyle \underset{x\in X}{\bigvee }}(\mathcal{H}\left(f\right)\left(x\right)\odot f^{*}\left(x\right))\right)}^{*}={\mathcal{T}}_{\mathcal{H}}\left(f\right).$$

(6)

For all $f,g\in L^X$, we have:

$$\begin{array}{ll}{\delta }_{{\mathcal{H}}_{\delta }}(f,g)& ={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left(f\right)\left(y\right)\odot g\left(y\right))={\displaystyle \underset{y\in X}{\bigvee }}(\delta (f,{\top }_y)\odot g\left(y\right))\\ & =\delta (f,{\displaystyle \underset{y\in X}{\bigvee }}({\top }_y\odot g\left(y\right)))=\delta (f,g).\end{array}$$

 □

By the above theorem, we obtain the Alexandrov L-fuzzy pre-proximity induced by an L-lower approximation operator in a sense ${\mathcal{H}}_{\mathcal{J}}\left(f\right)={\mathcal{J}}^{*}\left(f^{*}\right)$ for all $f\in L^X$.

Corollary 1.

Let $(X,\mathcal{J})$ be an L-lower approximation space. Define a mapping ${\delta }_{\mathcal{J}}:L^X\times L^X\to L$ by:

$${\delta }_{\mathcal{J}}(f,g)={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{J}}^{*}\left(f^{*}\right)\left(y\right)\odot g\left(y\right)).$$

Then, the following hold.

(1)

${\delta }_{\mathcal{J}}$ is an Alexandrov L-fuzzy proximity such that:

$${\delta }_{\mathcal{J}}(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}({\mathcal{J}}^{*}\left({\top }_y^{*}\right)\left(x\right)\odot (f\left(y\right)\odot g\left(x\right))).$$

(2)

${\delta }_{\mathcal{J}}(f,g)\le {\bigwedge }_{h\in L^X}({\delta }_{\mathcal{J}}(f,h)\oplus {\delta }_{\mathcal{J}}(h^{*},g))$. Moreover, the equality holds if $\mathcal{J}$ is topological.

(3)

If $\mathcal{J}$ is topological, then ${\delta }_{\mathcal{J}}$ is an Alexandrov L-fuzzy quasi-proximity on X.

(4)

$\mathcal{J}={\mathcal{J}}_{{\delta }_{\mathcal{J}}}$.

(5)

${\mathcal{T}}_{\mathcal{J}}\left(f\right)={\delta }_{\mathcal{J}}(f,f)={\mathcal{T}}_{{\delta }_{\mathcal{J}}}\left(f\right)$ for all $f\in L^X$.

(6)

If δ is an Alexandrov L-fuzzy pre-proximity on X, then ${\delta }_{{\mathcal{J}}_{\delta }}(f,g)=\delta (f,g)$ for all $f,g\in L^X$.

Example 3.

Let $([0,1],\odot ,\to {,}^{*},0,1)$ be a complete residuated lattice [4,8,9,10] where:

$$x\odot y=max\{0,x+y-1\},x\to y=min\{1-x+y,1\}$$

$$x\oplus y=min\{x+y,1\},x^{*}=1-x.$$

Let $X=\{x,y,z\}$. Consider the reflexive and transitive L-fuzzy relation $R\in {[0,1]}^{X\times X}$ defined by:

$$\left(\begin{array}{ccc}1& 0.7& 0.8\\ 0.5& 1& 0.4\\ 0.6& 0.7& 1\end{array}\right)$$

(1)

By Example 1, we obtain two Alexandrov L-fuzzy quasi-proximities $\delta ,{\delta }^s:{[0,1]}^X\times {[0,1]}^X\to [0,1]$ where:

$$\delta (f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot (f\left(x\right)\odot g\left(y\right)),$$

$${\delta }^s(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(R(y,x)\odot (f\left(x\right)\odot g\left(y\right)).$$

(2)

By Theorem 5, we obtain two Alexandrov L-fuzzy topologies ${\mathcal{T}}_{\delta },{\mathcal{T}}_{{\delta }^s}:{[0,1]}^X\times {[0,1]}^X\to [0,1]$ where:

$$\begin{array}{ll}{\mathcal{T}}_{\delta }\left(f\right)& ={\delta }^{*}(f,f^{*})={\left({\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot (f\left(x\right)\odot f^{*}\left(y\right))\right)}^{*}\\ & ={\displaystyle \underset{x,y\in X}{\bigwedge }}(R(x,y)\to {(f\left(x\right)\odot f^{*}\left(y\right))}^{*})\\ & ={\displaystyle \underset{x,y\in X}{\bigwedge }}(R(x,y)\to (f\left(x\right)\to f\left(y\right)),\end{array}$$

$${\mathcal{T}}_{{\delta }^s}\left(f\right)={\delta }^{*}(f^{*},f)={\displaystyle \underset{x,y\in X}{\bigwedge }}(R(y,x)\to (f\left(x\right)\to f\left(y\right)).$$

(3)

From Theorem 6 (4), since R is a reflexive and transitive L-fuzzy relation, we obtain two topological L-upper approximation operators ${\mathcal{H}}_{\delta },{\mathcal{H}}_{{\delta }^s}:{[0,1]}^X\to {[0,1]}^X$ where:

$${\mathcal{H}}_{\delta }\left(f\right)\left(x\right)=\delta (f,{\top }_x)={\displaystyle \underset{y\in X}{\bigvee }}(R(y,x)\odot f\left(y\right)),$$

$${\mathcal{H}}_{{\delta }^s}\left(f\right)\left(x\right)={\displaystyle \underset{y\in X}{\bigvee }}(R(x,y)\odot f\left(y\right)).$$

(4)

By Theorem 6 (4), we obtain two topological L-lower approximation operators ${\mathcal{J}}_{\delta },{\mathcal{J}}_{{\delta }^s}:{[0,1]}^X\to {[0,1]}^X$ where:

$${\mathcal{J}}_{\delta }\left(f\right)\left(x\right)={\delta }^{*}({\top }_x,f^{*})={\displaystyle \underset{y\in X}{\bigwedge }}(R(x,y)\to f\left(y\right)),$$

$${\mathcal{J}}_{{\delta }^s}\left(f\right)\left(x\right)={\delta }^{*}(f^{*},{\top }_x)={\displaystyle \underset{y\in X}{\bigwedge }}(R(y,x)\to f\left(y\right)).$$

(5)

From Theorem 8, since ${\mathcal{H}}_{\delta }$ and ${\mathcal{H}}_{{\delta }^s}$ are topological L-upper approximation operators, we obtain two Alexandrov L-fuzzy quasi-proximities ${\delta }_{{\mathcal{H}}_{\delta }},{\delta }_{{\mathcal{H}}_{{\delta }^s}}:{[0,1]}^X\times {[0,1]}^X\to [0,1]$ where:

$$\begin{array}{l}{\delta }_{{\mathcal{H}}_{\delta }}(f,g)={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{H}}_{\delta }\left(f\right)\left(y\right)\odot \left(y\right))={\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot f\left(x\right))\odot g\left(y\right))=\delta (f,g).\\ {\delta }_{{\mathcal{H}}_{{\delta }^s}}(f,g)={\displaystyle \underset{x,y\in X}{\bigvee }}(R(y,x)\odot (f\left(x\right)\odot g\left(y\right)))={\delta }^s(f,g).\end{array}$$

(6)

By Corollary 1, since ${\mathcal{J}}_{\delta }$ and ${\mathcal{J}}_{{\delta }^s}$ are topological L-lower approximation operators, we obtain Alexandrov L-fuzzy quasi-proximities ${\delta }_{{\mathcal{J}}_{\delta }},{\delta }_{{\mathcal{J}}_{{\delta }^s}}:{[0,1]}^X\times {[0,1]}^X\to [0,1]$ as:

$$\begin{array}{ll}{\delta }_{{\mathcal{J}}_{\delta }}(f,g)& ={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{J}}_{\delta }^{*}\left(f^{*}\right)\left(y\right)\odot \left(y\right))\\ & ={\displaystyle \underset{y\in X}{\bigvee }}({\left({\displaystyle \underset{x\in X}{\bigwedge }}(R(y,x)\to f^{*}\left(x\right))\right)}^{*}\odot g\left(y\right))\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}(R(y,x)\odot f\left(x\right)\odot g\left(y\right))={\delta }^s(f,g).\end{array}$$

$$\begin{array}{ll}{\delta }_{{\mathcal{J}}_{{\delta }^s}}(f,g)& ={\displaystyle \underset{y\in X}{\bigvee }}({\mathcal{J}}_{{\delta }^s}^{*}\left(f^{*}\right)\left(y\right)\odot \left(y\right))\\ & ={\displaystyle \underset{y\in X}{\bigvee }}({\left({\displaystyle \underset{x\in X}{\bigwedge }}(R(x,y)\to f^{*}\left(x\right))\right)}^{*}\odot g\left(y\right))\\ & ={\displaystyle \underset{x,y\in X}{\bigvee }}(R(x,y)\odot f\left(x\right)\odot g\left(y\right))=\delta (f,g).\end{array}$$