(L,M)-Fuzzy k-Pseudo Metric Space

Abstract

Recently, the notion of a classical k-metric, which make the triangle inequality to a more general axiom: $d(x,z)\le k\left(d\right(x,y)+d(y,z\left)\right)$, has been presented and is applied in many fields. In this paper, the definitions of an $(L,M)$-fuzzy k-pseudo metric and an $(L,M)$-fuzzy k-remote neighborhood ball system are introduced. It is proved that the category of $(L,M)$-fuzzy k-pseudo metric spaces is isomorphic to the category of $(L,M)$-fuzzy k-remote neighborhood ball spaces. Besides, $(L,M)$-fuzzy topological structures induced by an $(L,M)$-fuzzy k-pseudo metric are presented and their properties are investigated. Finally, the concept of a nest of pointwise k-pseudo metrics is proposed and it is shown that there is a one-to-one correspondence between $(L,M)$-fuzzy k-pseudo metrics and nests of pointwise k-pseudo metrics.

1. Introduction

Fuzzy set theory [1] has been applied to many areas of research, such as fuzzy convex structure [2,3], fuzzy convergence structure [4,5,6], fuzzy topology [7,8,9,10], fuzzy metric and so on. The directions of fuzzy metric are very important in the theory of fuzzy sets. They have been applied in image processing, decision making, optimization, computer programming, engineering, etc. (for more details see [11,12,13,14,15,16,17]).

Many researchers have done spectacular and creative work in the theory of fuzzy metrics, which can be divided into three groups. The first group is formed by those results in which a fuzzy metric on a set X is viewed as a map $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )$, where the $J\left(L^X\right)$ is the set of all L-fuzzy points on X. In this case, it induces an L-topology [18,19,20]. The second group is formed by those results in which the distance between objects is fuzzy, the objects themselves are crisp, in which a fuzzy metric can be regarded as a map $M:X\times X\times [0,\infty )\to [0,1]$ or a map $d:X\times X⟶[0,\infty )\left(M\right)$, where $[0,\infty )\left(M\right)$ denote the set of all non-negative M-fuzzy real numbers. In this case, it induces a topology or a fuzzifying topology [21,22,23,24,25,26,27]. The third group is a further extension of the first and second group, in which a fuzzy metric is viewed as a map $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ satisfying some conditions. In this case, it can induce an $(L,M)$-fuzzy topology [28].

In 1989, Bakhtin [29] introduced a concept of a metric-type space which is a generalization of a metric space. In 1993, Czerwik [30] rediscovered it and named it a b-metric space. In order to describe the concept more vividly, Šostak [31] used a new name of a “k-metric space”, which make the triangle inequality to a more general axiom: $d(x,z)\le k\left(d\right(x,y)+d(y,z\left)\right)$, where $k\ge 1$ is a fixed constant. In 2015 and 2016, Hussain [32] and Nǎdǎban [33] introduced the concept of a fuzzy b-metric and discussed the corresponding fixed point theorems. In 2018, Šostak [31] gave a definition of a fuzzy k-pseudo metric and studied two crisp structures induced by a fuzzy k-pseudo metric: topology and supratopology. However, he did not consider any fuzzy structures induced by a fuzzy k-pseudo metric.

By simply modifying the definition of Šostak’s fuzzy k-pseudo metric, Zhong et al. [34,35] proposed a new definition of an M-fuzzifying k-pseudo metric in 2021. Additionally, many fuzzifying topological structures are constructed by this new concept of a fuzzy k-pseudo metric. From a completely different direction, Zhong et al. [36] later introduced the notion of a pointwise k-pseudo metric and showed that many L-topological structures can be induced by a pointwise k-pseudo metric.

The main aim of this paper is to generalize the definitions and conclusions of fuzzifying k-pseudo metric spaces and pointwise k-pseudo metric spaces to the $(L,M)$-fuzzy case. To be specific, we shall introduce the concepts of an $(L,M)$-fuzzy k-pseudo metric and an $(L,M)$-fuzzy k-remote neighborhood ball system. Furthermore, we shall show that some $(L,M)$-fuzzy topological structures can be induced by an $(L,M)$-fuzzy k-pseudo metric and show that there is a one-to-one correspondence between $(L,M)$-fuzzy k-pseudo metrics and nests of pointwise k-pseudo metrics.

This paper is organized as follows. In Section 2, some necessary definitions and results about M-fuzzifying k-pseudo metric spaces and $(L,M)$-fuzzy topological spaces are recalled. In Section 3, the definitions of an $(L,M)$-fuzzy k-pseudo metric and an $(L,M)$-fuzzy k-remote neighborhood ball system are introduced. Then the relationships between $(L,M)$-fuzzy k-pseudo metrics and $(L,M)$-fuzzy k-remote neighborhood ball systems are discussed. In Section 4, some $(L,M)$-fuzzy topological structures induced by an $(L,M)$-fuzzy pseudo metric are constructed. In Section 5, the definition of a nest of pointwise k-pseudo metrics is presented. And, the relations between $(L,M)$-fuzzy k-pseudo metrics and nests of pointwise k-pseudo metrics are discussed.

2. Preliminaries

Throughout this paper, both L and M denote a completely distributive De Morgan algebra, i.e., a completely distributive lattice with an order-reserving involution ${}^{\prime }$. The smallest element and the largest element in L are denoted by ${\perp }_L$ and ${\top }_L$, respectively. In the sequel, we adopt the convention $\bigvee \varnothing ={\perp }_L$ and $\bigwedge \varnothing ={\top }_L$.

We say that a is wedge-below b in L, in symbols, $a\prec b$, if for every subset $D\subseteq L$, $\bigvee D\ge b$ implies $a\le c$ for some $c\in D$ [37]. The wedge below relation has the interpolation property, $a\prec {\bigwedge }_{i\in I}{b}_i$ implies $a\prec b_i$ for every $i\in I$, and $a\prec {\bigvee }_{i\in I}{b}_i$ implies $a\prec b_i$ for some $i\in I$ [38,39]. The set of non-zero co-prime elements in L is denoted by $J\left(L\right)$.

Let X be a non-empty set. $L^X$ denotes the set of all L-fuzzy subsets on X and $L^X$ is also a completely distributive De Morgan algebra. The smallest element and the largest element in $L^X$ are denoted by ${\perp }_{L^X}$ and ${\top }_{L^X}$, respectively. The set of all non-zero co-prime elements in $L^X$ is denoted by $J\left(L^X\right)$. We know $J\left(L^X\right)$ is exactly the set of all L-fuzzy points $x_{\lambda }$, that is $J\left(L^X\right)=\{x_{\lambda }\in L^X\mid \lambda \in J\left(L\right)\}$, in which $x_{\lambda }$ is an L-fuzzy set such that $x_{\lambda }\left(x\right)=\lambda$ and otherwise $=0$.

Let $f:X⟶Y$ be a mapping. Define $f_L^{\to }:L^X⟶L^Y$ and $f_L^{\leftarrow }:L^Y\leftrightarrow L^X$ as follows: $f_L^{\to }\left(A\right)\left(y\right)={\bigvee }_{f\left(x\right)=y}A\left(x\right)$ for all $A\in L^X$ and $y\in Y$, and $f_L^{\leftarrow }\left(B\right)=B\circ f$ for all $B\in L^Y$.

Firstly, we recall the definition of k-metric as it was introduced in [31].

Definition 1

([31]). A k-pseudo metric on X is a map $d:X\times X\to [0,\infty )$ satisfying the following conditions: $\forall x,y,z\in X$,

(D1) 

$d(x,x)=0$;

(D2) 

$d(x,z)\le k\left(d\right(x,y)+d(y,z\left)\right)$, $k\ge 1$ is a fixed constant.

(D3) 

$d(x,y)=d(y,x)$;

If the axiom (D1) is replaced by a stronger axiom:

(D1)${}^{*}$ $d(x,y)=0\iff x=y$;

then d is called a k-metric and the pair $(X,d)$ is called a k-metric space.

Example 1.

Let $\mathbb{R}$ be the set of real numbers and let $d:X\times X\to [0,\infty )$ be a mapping defined by $\forall x,y\in \mathbb{R}$, $d(x,y)={|x-y|}^2.$ Then d is a 2-metric.

Similarly, let $(X,\parallel \parallel )$ be a normed space. There also exists 2-metric on X defined by $\forall x,y\in X$, $d(x,y)={\parallel x-y\parallel }^2.$

Example 2.

A series of k-metrics can be obtained from a metric by the following construction. Let $k\ge 1$ be a fixed constant and let $\phi :[0,\infty )\to [0,\infty )$ be a continuous increasing mapping such that $\phi \left(0\right)=0$ and $\phi (a+b)\le k\left(\phi \right(a)+\phi (b\left)\right)$ for any $a,b\in [0,\infty )$. And, let $\rho :X\times X\to [0,\infty )$ be a metric. Then the mapping $d_{\rho \phi }:X\times X\to [0,\infty )$ defined by $\forall x,y\in X$, $d_{\rho \phi }(x,y)=(\phi \circ \rho )(x,y),$ is a k-metric.

(1) 

Let $\phi \left(a\right)=a^2$. Obviously ${(a+b)}^2\le 2(a^2+b^2)$. By defining $d_{\rho \phi }(x,y)=\rho {(x,y)}^2$, we obtain a 2-metric. In particular,

$d_{\rho \phi }(f,g)={\left({\int }_a^b|f\left(x\right)-g\left(x\right)|dx\right)}^2$ on the set of Lebesgue measurable functions on $[a,b]$.

(2) 

Let $\phi \left(a\right)=a^{\frac{3}{2}}$ and we know ${(a+b)}^{\frac{3}{2}}\le \sqrt{2}(a^{\frac{3}{2}}+b^{\frac{3}{2}})$. By defining $d_{\rho \phi }(x,y)=\rho {(x,y)}^{\frac{3}{2}}$, we can obtain a $\sqrt{2}$-metric.

In what follows, we recall the definitions of an M-fuzzifying k-pseudo metric and a pointwise k-pseudo metric, which are different aspects of generalizations of a crsip k-pseudo metric.

Before we recall the definition of an M-fuzzifying k-pseudo metric, we need the following definition of an M-fuzzy non-negative real number, which is a generalization of a crisp non-negative real number.

Definition 2

([28]). An M-fuzzy non-negative real number is an equivalence class $\left[\lambda \right]$ of non-increasing maps $\lambda :\mathbb{R}⟶M$ satisfying

$$\lambda (0-)=\underset{t<0}{\bigwedge }\lambda \left(t\right)={\top }_M,\lambda (+\infty )=\underset{t\in \mathbb{R}}{\bigwedge }\lambda \left(t\right)={\perp }_M,$$

where the equivalence identifies maps $\lambda ,\mu$ if and only if $\forall t\in I,\lambda (t-)=\mu (t-)$.

Example 3

([40]). Let M=[0,1]. Define $\lambda :\mathbb{R}⟶[0,1]$ by

$$\begin{array}{c}\hfill \lambda \left(t\right)=\left\{\begin{array}{cc}1,\hfill & t\le \lambda ;\hfill \\ 0,\hfill & t>\lambda ;\hfill \end{array}\right.\end{array}$$

Then λ is exactly the corresponding non-negative real number.

We shall not distinguish an M-fuzzy real number $\left[\lambda \right]$ and its representative function $\lambda$ being left continuous. The set of all M-fuzzy non-negative real numbers denoted by $[0,+\infty )\left(M\right)$.

Definition 3

([35]). An M-fuzzifying k-pseudo metric on X is a map $d:X\times X⟶[0,+\infty )\left(M\right)$ satisfying: $\forall x,y,z\in X$,

(MFKD1) 

$d(x,x)(0+)={\bigvee }_{t>0}d(x,x)\left(t\right)=0$;

(MFKD2) 

$d(x,z)(r+s)\le d(x,y)\left(r\right)\vee d(y,z)\left(s\right)$, $\forall r,s>0$;

(MFKD3) 

$d(x,y)=d(y,x)$.

Example 4

([35]). Let X=[1,4], M=[0,1]. Define d: $X\times X⟶[0,\infty )\left(M\right)$ by

$$\begin{array}{c}\hfill d(x,y)\left(t\right)=\left\{\begin{array}{cc}1,\hfill & t=0;\hfill \\ 0,\hfill & x=y,t>0;\hfill \\ 0,\hfill & x\ne y,t>x·y;\hfill \\ \frac{1}{1+t},\hfill & x\ne y,0<t\le x·y.\hfill \end{array}\right.\end{array}$$

Then d is an M-fuzzifying 2-pseudo metric and not an M-fuzzifying metric.

Definition 4

([36]). A pointwise k-pseudo metric is a map $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )$ which satisfies: $\forall x_{\lambda },y_{\mu },z_{\gamma }\in J\left(L^X\right)$

(LKD1) 

$d(x_{\lambda },x_{\lambda })=0$;

(LKD2) 

$d(x_{\lambda },z_{\gamma })\le k(d(x_{\lambda },y_{\mu })+d(y_{\mu },z_{\gamma }))$;

(LKD3) 

$d(x_{\lambda },y_{\mu })={\bigwedge }_{\nu \prec \mu }d(x_{\lambda },y_{\nu })$;

(LKD4) 

$d(x_{\lambda },y_{\mu })\le d(x_{\omega },y_{\mu })$ whenever $\lambda \le \omega$.

(LKD5) 

${\bigwedge }_{\omega \nleqq {\lambda }^{\prime }}d(x_{\omega },y_{\mu })={\bigwedge }_{\nu \nleqq {\mu }^{\prime }}d(y_{\nu },x_{\lambda })$.

Example 5

([36]). Let X be a non-empty set and $L=[0,1]$.

Define $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )$ as follows: for any $x_{\lambda },y_{\mu }\in J\left(L^X\right)$,

$$d(x_{\lambda },y_{\mu })=\left\{\begin{array}{ccc}{|\lambda -\mu |}^2,\hfill & & \lambda >\mu ;\hfill \\ 0,\hfill & & \lambda \le \mu .\hfill \end{array}\right.$$

Then d is a pointwise 2-pseudo metric and d is not a pointwise pseudo metric.

Next, the concepts of an $(L,M)$-fuzzy topology, an $(L,M)$-fuzzy quasi-neighborhood system, an $(L,M)$-fuzzy closure operator and an $(L,M)$-fuzzy interior operator are recalled.

Definition 5

([8,10]). An $(L,M)$-fuzzy topology on X is a map $\mathcal{T}:L^X⟶M$ which satisfies:

(LMT1) 

$\mathcal{T}\left({\perp }_{L^X}\right)=\mathcal{T}\left({\top }_{L^X}\right)={\top }_M$;

(LMT2) 

$\forall A_1,A_2\in L^X$, $\mathcal{T}\left(A_1\right)\wedge \mathcal{T}\left(A_2\right)\le \mathcal{T}(A_1\wedge A_2)$;

(LMT3) 

$\forall \left\{A_j\right|j\in J\}\subseteq L^X$, ${\bigwedge }_{j\in J}\mathcal{T}\left(A_j\right)\le \mathcal{T}\left({\bigvee }_{j\in J}{A}_j\right)$.

The pair $(X,\mathcal{T})$ is called an $(L,M)$-fuzzy topological space. A continuous map between two $(L,M)$-fuzzy topological spaces $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ is a map $f:X⟶Y$ such that $\forall B\in L^Y$, ${\mathcal{T}}_Y\left(B\right)\le {\mathcal{T}}_X\left(f_L^{\leftarrow }\left(B\right)\right)$.

The category of $(L,M)$-fuzzy topological spaces and their continuous mappings is denoted by $LM$-Top.

We know $x_{\lambda }$ quasi-coincides with A

if $\lambda \nleqq A^{\prime }\left(x\right)$ i.e., $x_{\lambda }\nleqq A^{\prime }$ [38,41].

The following definitions were presented for $(L,L)$-fuzzy cases. They can easily be transformed to $(L,M)$-fuzzy cases as follows.

Definition 6

([42]). An $(L,M)$-fuzzy quasi-neighborhood system on X is a set $\mathcal{Q}=\left\{{\mathcal{Q}}_{x_{\lambda }}\right|x_{\lambda }\in J\left(L^X\right)\}$ of maps ${\mathcal{Q}}_{x_{\lambda }}:L^X⟶M$ satisfying the following conditions:

(LMQ1) 

${\mathcal{Q}}_{x_{\lambda }}\left({\perp }_{L^X}\right)={\perp }_M$, ${\mathcal{Q}}_{x_{\lambda }}\left({\top }_{L^X}\right)={\top }_M$;

(LMQ2) 

${\mathcal{Q}}_{x_{\lambda }}\left(A\right)\ne {\perp }_M\Rightarrow x_{\lambda }\nleqq A^{\prime }$;

(LMQ3) 

${\mathcal{Q}}_{x_{\lambda }}(A\wedge B)={\mathcal{Q}}_{x_{\lambda }}\left(A\right)\wedge {\mathcal{Q}}_{x_{\lambda }}\left(B\right)$;

(LMQ4) 

${\mathcal{Q}}_{x_{\lambda }}\left(A\right)={\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(B^{\prime }\right)$.

The pair $(X,\mathcal{Q})$ is called an $(L,M)$-fuzzy quasi-neighborhood space. A continuous map between two $(L,M)$-fuzzy quasi-neighborhood spaces $(X,{\mathcal{Q}}_X)$ and $(Y,{\mathcal{Q}}_Y)$ is a map $f:X⟶Y$ such that $\forall x_{\lambda }\in J\left(L^X\right),B\in L^Y$, ${\left({\mathcal{Q}}_Y\right)}_{f{\left(x\right)}_{\lambda }}\left(B\right)\le {\left({\mathcal{Q}}_X\right)}_{x_{\lambda }}\left(f_L^{\leftarrow }\left(B\right)\right)$. The category of $(L,M)$-fuzzy quasi-neighborhood spaces and their continuous mappings is denoted by $LM$-QS. Besides, it is proved that $LM$-Top is isomorphic to $LM$-QS in [28,43].

Definition 7

([43,44]). An $(L,M)$-fuzzy closure operator is a map $Cl:L^X⟶M^{J\left(L^X\right)}$ which satisfies

(LMC1) 

$\forall x_{\lambda }\in J\left(L^X\right)$, $Cl\left({\perp }_{L^X}\right)\left(x_{\lambda }\right)={\perp }_M$;

(LMC2) 

$\forall x_{\lambda }\le A$, $Cl\left(A\right)\left(x_{\lambda }\right)={\top }_M$;

(LMC3) 

$Cl(A\vee B)=Cl\left(A\right)\vee Cl\left(B\right)$;

(LMC4) 

$\forall x_{\lambda }\nleqq A$, $Cl\left(A\right)\left(x_{\lambda }\right)={\bigwedge }_{x_{\lambda }\nleqq B\ge A}{\bigvee }_{y_{\mu }\nleqq B}Cl\left(B\right)\left(y_{\mu }\right)$.

The pair $(X,Cl)$ is called an $(L,M)$-fuzzy closure space. A continuous mapping between two $(L,M)$-fuzzy closure spaces is a map $f:X⟶Y$ such that $C{l}_X\left(A\right)\left(x_{\lambda }\right)\le C{l}_Y\left(f_L^{\to }\left(A\right)\right)\left(f{\left(x\right)}_{\lambda }\right)$ for all $x_{\lambda }\in J\left(L^X\right)$ and for all $A\in L^X$. The category of $(L,M)$-fuzzy closure spaces and their continuous mappings is denoted by $LM$-CS. In [28,43], it is shown that $LM$-QS and $LM$-Top are all isomorphic to $LM$-CS.

Definition 8

([43,44]). An $(L,M)$-fuzzy interior operator is a map $Int:L^X⟶M^{J\left(L^X\right)}$ which satisfies

(LMI1) 

$\forall x_{\lambda }\in J\left(L^X\right)$, $Int\left({\top }_{L^X}\right)\left(x_{\lambda }\right)={\top }_M$;

(LMI2) 

$\forall x_{\lambda }\nleqq A$, $Int\left(A\right)\left(x_{\lambda }\right)={\perp }_M$;

(LMI3) 

$Int(A\wedge B)=Int\left(A\right)\vee Int\left(B\right)$;

(LMI4) 

$\forall x_{\lambda }\le A$, $Int\left(A\right)\left(x_{\lambda }\right)={\bigvee }_{x_{\lambda }\le B\le A}{\bigwedge }_{y_{\mu }\prec B}Int\left(B\right)\left(y_{\mu }\right)$.

The pair $(X,Int)$ is called an $(L,M)$-fuzzy interior space. A continuous mapping between two $(L,M)$-fuzzy interior spaces is a map $f:X⟶Y$ such that $In{t}_Y\left(B\right)\left(f{\left(x\right)}_{\lambda }\right)\le In{t}_X\left(f_L^{\leftarrow }\left(B\right)\right)\left(x_{\lambda }\right)$ for all $x_{\lambda }\in J\left(L^X\right)$ and for all $B\in L^Y$. The category of $(L,M)$-fuzzy interior spaces and their continuous mappings is denoted by $LM$-IS. However, $C{l}^{\mathcal{T}}\left(A\right)={\left(In{t}^{\mathcal{T}}\left(A\right)\right)}^{\prime }$ is not true in an $(L,M)$-fuzzy topological space $(X,\mathcal{T})$ (see [43,44]).

3. (L, M)-Fuzzy k-Pseudo Metric

In this section, the definitions of an $(L,M)$-fuzzy k-pseudo metric and an $(L,M)$-fuzzy k-remote neighborhood ball system are introduced. Then we show the category of $(L,M)$-fuzzy k-pseudo metric spaces is isomorphic to the category of $(L,M)$-fuzzy k-remote neighborhood ball spaces.

Firstly, we introduce the definition of an $(L,M)$-fuzzy k-pseudo metric, which is a generalization of both an M-fuzzifying k-pseudo metric and a pointwise k-pseudo metric.

Definition 9.

An$(L,M)$-fuzzy k-pseudo metricon $L^X$ is a map $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ satisfying: $\forall x_{\lambda },y_{\mu },z_{\gamma }\in J\left(L^X\right)$, $\forall t,s\in [0,\infty )$,

(LMKD1) 

$d(x_{\lambda },x_{\lambda })(0+)={\bigvee }_{t>0}d(x_{\lambda },x_{\lambda })\left(t\right)={\perp }_M$;

(LMKD2) 

$d(x_{\lambda },z_{\gamma })\left(k(t+s)\right)\le d(x_{\lambda },y_{\mu })\left(t\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)$;

(LMKD3) 

$d(x_{\lambda },y_{\mu })\left(t\right)={\bigwedge }_{\nu \prec \mu }d(x_{\lambda },y_{\nu })\left(t\right)$;

(LMKD4) 

$d(x_{\lambda },y_{\mu })\left(t\right)\le d(x_{\omega },y_{\mu })\left(t\right)$ whenever $\lambda \le \omega$.

(LMKD5) 

${\bigwedge }_{\omega \nleqq {\lambda }^{\prime }}d(x_{\omega },y_{\mu })\left(t\right)={\bigwedge }_{\nu \nleqq {\mu }^{\prime }}d(y_{\nu },x_{\lambda })\left(t\right)$.

The pair $(X,d)$ is called an$(L,M)$-fuzzy k-pseudo metric space.

Example 6.

Let X be any set and let $L=[0,1]$, $M=\{{\perp }_M,a_0,{\top }_M\}$. Then $J\left(L\right)=(0,1]$ and $J\left(M\right)=\{a_0,{\top }_M\}$. Define $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$, $\forall t\in [0,\infty )$,

$$d(x_{\lambda },y_{\mu })\left(t\right)=\left\{\begin{array}{ccc}{\top }_M,\hfill & & t=0;\hfill \\ a_0,\hfill & & 0<t\le {|\lambda -\mu |}^2,\lambda >\mu ;\hfill \\ {\perp }_M,\hfill & & {t>|\lambda -\mu |}^2,\lambda >\mu ;\hfill \\ {\perp }_M,\hfill & & \lambda \le \mu .\hfill \end{array}\right.$$

Then d is an $(L,M)$-fuzzy 2-pseudo metric and not an $(L,M)$-fuzzy pseudo metric.

Proof. 

Step 1.We check d is well-defined.

At first, it is clear that $d(x_{\lambda },y_{\mu })(·)$ is non-increasing with respect to t. Then, $d(x_{\lambda },y_{\mu })(0-)=d(x_{\lambda },y_{\mu })\left(0\right)={\top }_M$, $d(x_{\lambda },y_{\mu })(+\infty )=li{m}_{t\to +\infty }d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$. This shows $d(x_{\lambda },y_{\mu })\left(t\right)\in [0,\infty )\left(M\right)$.

Step 2.We check d satisfies (LMKD1)-(LMKD5).

(LMKD1) 

$d(x_{\lambda },x_{\lambda })(0+)={\bigvee }_{t>0}d(x_{\lambda },x_{\lambda })\left(t\right)={\perp }_M$ is obvious.

(LMKD2) 

It must check $d(x_{\lambda },z_{\gamma })\left(k(t+s)\right)\le d(x_{\lambda },y_{\mu })\left(t\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)$.

(1) 

If $d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$, i.e., ${t>|\lambda -\mu |}^2,\lambda >\mu$ and $d(y_{\mu },z_{\gamma })\left(s\right)={\perp }_M$, i.e., ${s>|\mu -\gamma |}^2,\mu >\gamma$, then $\lambda >\mu >\gamma$ and ${2(t+s)>2\left(\right|\lambda -\mu |}^2{+|\mu -\gamma |}^2{)\ge (|\lambda -\mu |+|\mu -\gamma |)}^2\ge {|\lambda -\gamma |}^2$, that is $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)={\perp }_M$.

(2) 

If $d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$, i.e., $\lambda \le \mu$ and $d(y_{\mu },z_{\gamma })\left(s\right)={\perp }_M$, i.e., $\mu \le \gamma$, then $\lambda \le \mu \le \gamma$ that is $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)={\perp }_M$.

(3) 

If $d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$, i.e., ${t>|\lambda -\mu |}^2,\lambda >\mu$ and $d(y_{\mu },z_{\gamma })\left(s\right)={\perp }_M$, i.e., $\mu \le \gamma$, then this lead to two situations. Whenever $\gamma \ge \lambda$, we know $d(x_{\lambda },z_{\nu })\left(2(t+s)\right)={\perp }_M$. Whenever $\mu <\gamma <\lambda$, ${2(t+s)>2t>2|\lambda -\mu |}^2{>|\lambda -\mu |}^2>{|\lambda -\gamma |}^2$, which means $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)={\perp }_M$.

Combing (1)(2)(3), we obtain $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)\le d(x_{\lambda },y_{\mu })\left(t\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)$.

(LMKD3)We must examine $d(x_{\lambda },y_{\mu })\left(t\right)={\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)$.

(1) 

Assume $d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$, that is, $\lambda \le \mu$. If $\lambda <\mu$, then there exists ν such that $\lambda <\nu <\mu$. So $d(x_{\lambda },y_{\nu })\left(t\right)={\perp }_M$. If $\lambda =\mu$, then ${\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)={\perp }_M$. Hence ${\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)={\perp }_M$.

(2) 

Assume $d(x_{\lambda },y_{\mu })={\perp }_M$, that is, ${t>|\lambda -\mu |}^2,\lambda >\mu$. Then there exists some $\nu <\mu$ such that ${t>|\lambda -\nu |}^2>{|\lambda -\mu |}^2$ and $\lambda >\nu$, which shows $d(x_{\lambda },y_{\nu })\left(t\right)={\perp }_M$. Hence ${\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)={\perp }_M$.

(3) 

Assume $d(x_{\lambda },y_{\mu })=a_0$, that is, $0<t\le {|\lambda -\mu |}^2,\lambda >\mu$. For any $\nu <\mu$, we have $\lambda >\nu$ and ${0<t<|\lambda -\nu |}^2$. So $d(x_{\lambda },y_{\nu })\left(t\right)=a_0$. Hence ${\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)=a_0$.

Combing (1)(2)(3), we obtain $d(x_{\lambda },y_{\mu })\left(t\right)={\bigwedge }_{\nu <\mu }d(x_{\lambda },y_{\nu })\left(t\right)$.

(LMKD4)The proof is analogous to that of (LMKD3) and omitted here.

(LMKD5)We must check ${\bigwedge }_{\omega >1-\lambda }d(x_{\omega },y_{\mu })\left(t\right)={\bigwedge }_{\nu >1-\mu }d(y_{\nu },x_{\lambda })\left(t\right)$.

(1) 

If $1-\lambda -\mu <0$, then there exist $\omega >1-\lambda$ and $v>1-\mu$ such that $1-\lambda <\omega <\mu$ and $1-\mu <\nu <\lambda$. Hence ${\bigwedge }_{\omega >1-\lambda }d(x_{\omega },y_{\mu })={\perp }_M$ and ${\bigwedge }_{\nu >1-\mu }d(y_{\nu },x_{\lambda })={\perp }_M$.

(2) 

If $1-\lambda -\mu \ge 0$, then $\omega >1-\lambda >\mu$ and $\nu >1-\mu >\lambda$. When ${\bigwedge }_{\omega >1-\lambda }d(x_{\omega },y_{\mu })\left(t\right)=a_0$, that is, ${|\omega -\mu |}^2$ for any $\omega >1-\lambda$. Since $t\le {\bigwedge }_{\omega >1-\lambda }{|\omega -\mu |}^2={|1-\lambda -\mu |}^2={|1-\mu -\lambda |}^2={\bigwedge }_{\nu >1-\mu }{|\nu -\lambda |}^2$, it follows that $t\le {|\nu -\lambda |}^2$ for any $\nu >1-\mu$. Hence ${\bigwedge }_{\nu >1-\mu }d(y_{\mu },x_{\lambda })\left(t\right)=a_0$. When ${\bigwedge }_{\omega >1-\lambda }d(x_{\omega },y_{\mu })\left(t\right)={\perp }_M$, the proof is analogous.

Combing (1) and (2), we get ${\bigwedge }_{\omega >1-\lambda }d(x_{\omega },y_{\mu })\left(t\right)={\bigwedge }_{\nu >1-\mu }d(y_{\nu },x_{\lambda })\left(t\right)$.

Step 3.d is not an $(L,M)$-fuzzy k-pseudo metric.

Let $\lambda =\frac{5}{8}$, $\mu =\frac{3}{8}$, $\nu =\frac{1}{8}$ and let $t=\frac{1}{8}$, $s=\frac{1}{8}$. Then

$${|\lambda -\nu |}^2=|\frac{5}{8}-\frac{1}{8}{|}^2=\frac{1}{4}{,|\lambda -\mu |}^2=|\frac{5}{8}-\frac{3}{8}{|}^2=\frac{1}{16}{,|\mu -\nu |}^2={|\frac{3}{8}-\frac{1}{8}|}^2=\frac{1}{16}.$$

Hence $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)={\perp }_M$, $d(x_{\lambda },z_{\gamma })(t+s)=a_0$, $d(x_{\lambda },y_{\mu })\left(t\right)={\perp }_M$, $d(y_{\mu },z_{\gamma })\left(t\right)$$={\perp }_M$. Therefore $d(x_{\lambda },z_{\gamma })\left(2(t+s)\right)\le d(x_{\lambda },y_{\mu })\left(t\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)$ and $d(x_{\lambda },z_{\gamma })(t+s)\nleqq d(x_{\lambda },y_{\mu })\left(t\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)$.

Remark 1. 

(1) If $M=\{{\perp }_M,{\top }_M\}$, then Definition 9 is equivalent to Definition 3. If $L=\{{\perp }_L,{\top }_L\}$, then Definition 9 is equivalent to Definition 4. Therefore Definition 9 can be viewed as a generalization of both a pointwise k-pseudo metric and an M-fuzzifying pseudo metric.

(2) $d(x_{\lambda },y_{\mu })\left(t\right)$ can be interpreted as the degree to which the distance from $x_{\lambda }$ to $y_{\mu }$ is more than or equal to t.

Definition 10.

A map $f:X⟶Y$ between two $(L,M)$-fuzzy k-pseudo metric spaces $(X,d_X)$ and $(Y,d_Y)$ is called contractive if $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$, $\forall t\in [0,+\infty )$,

$$d_Y\left(f{\left(x\right)}_{\lambda },f{\left(y\right)}_{\mu }\right)\left(t\right)\le d_X(x_{\lambda },y_{\mu })\left(t\right).$$

We can check that $(L,M)$-fuzzy k-pseudo metric spaces and their contractive maps constitute a category, denoted by $LM$-KPMS.

Proposition 1.

Let d be an $(L,M)$-fuzzy k-pseudo metric on $L^X$. Then the following hold.

(LMKD1)^* 

$\forall \lambda \le \mu$, $d(x_{\lambda },x_{\mu })(0+)={\perp }_M$.

(LMKD3)^* 

${}^{*}$$\forall \nu \le \mu$, $d(x_{\lambda },y_{\nu })\left(t\right)\ge d(x_{\lambda },y_{\mu })\left(t\right)$.

The following theorems present some characterizations of an $(L,M)$-fuzzy k-pseudo metric. Their proofs are not difficult to check and are thus omitted here.

Theorem 1.

Let d be an $(L,M)$-fuzzy k-pseudo metric. Define $M:J\left(L^X\right)\times J\left(L^X\right))\times [0,\infty )⟶M$ as follows: for all $x_{\lambda },y_{\mu }\in J\left(L^X\right)$ and for all $t,s\in [0,\infty )$,

$$M(x_{\lambda },y_{\mu },t)={\left(d(x_{\lambda },y_{\mu })\right)}^{\prime }$$

Then M satisfies the following conditions:

(LMKM1) 

$M(x_{\lambda },y_{\mu },0)={\perp }_M$;

(LMKM2) 

$M(x_{\lambda },y_{\mu },t)\wedge M(y_{\mu },z_{\nu },s)\le M(x_{\lambda },z_{\nu },k(t+s))$;

(LMKM3) 

$\forall t>0$, $M(x_{\lambda },x_{\lambda },t)={\top }_M$;

(LMKM4) 

$M(x_{\lambda },y_{\mu },t)={\bigvee }_{s<t}M(x_{\lambda },y_{\mu },s)$;

(LMKM5) 

${\bigvee }_{t>0}M(x_{\lambda },y_{\mu },t)=li{m}_{t\to \infty }M(x_{\lambda },y_{\mu },t)={\top }_M$;

(LMKM6) 

$M(x_{\lambda },y_{\mu },t)={\bigwedge }_{\nu <\mu }M(x_{\lambda },y_{\nu },t)$;

(LMKM7) 

$\lambda \le \gamma \Rightarrow M(x_{\lambda },y_{\mu },t)\le M(x_{\gamma },y_{\mu },t)$.

(LMKM8) 

${\bigvee }_{\omega \nleqq {\lambda }^{\prime }}M(x_{\omega },y_{\mu },t)={\bigvee }_{\nu \nleqq {\mu }^{\prime }}M(y_{\nu },x_{\lambda },t)$.

Theorem 2.

Let $M:J\left(L^X\right)\times J\left(L^X\right))\times [0,\infty )⟶M$ be a map satisfying(LMKM1) − (LMKM8). Define $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$, $\forall t\in [0,\infty )$,

$$d(x_{\lambda },y_{\mu })\left(t\right)={\left(M(x_{\lambda },y_{\mu },t)\right)}^{\prime }.$$

Then d is an $(L,M)$-fuzzy k-pseudo metric.

Let $(X,d)$ be a k-pseudo metric space.

Define the open ball with the center $x\in X$ and the radius $r>0$ by $B(x,r)=\{y\in X\mid d(x,y)<r\}$. And, the remote neighborhood ball is defined by $R(x,r)={\left(B(x,r)\right)}^{{}^{\prime }}=\{y\in X\mid d(x,y)\ge r\}.$

Next, we shall introduce the concept of an $(L,M)$-fuzzy k-remote neighborhood ball system, which is a generalization to the opposite aspect of the crisp spherical neighborhood system.

Definition 11.

An$(L,M)$-fuzzy k-remote neighborhood ball systemon $L^X$ is defined to be a set $\mathcal{R}=\{R_r\mid r\in (0,\infty )\}$ of maps $\{R_r:J\left(L^X\right)⟶M^{L^X}\}$ which satisfies $\forall x_{\lambda },y_{\mu },z_{\nu }\in J\left(L^X\right)$, $\forall r,s\in [0,\infty )$,

(LMKR1) 

${\bigwedge }_{r>0}{R}_r\left(x_{\lambda }\right)\left(y_{\mu }\right)={\perp }_M$;

(LMKR2) 

$R_r\left(x_{\lambda }\right)\left(x_{\lambda }\right)={\perp }_M$;

(LMKR3) 

$R_{k(r+s)}\left(x_{\lambda }\right)\left(z_{\gamma }\right)\le R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)\vee R_s\left(y_{\mu }\right)\left(z_{\gamma }\right)$;

(LMKR4) 

$R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)={\bigwedge }_{s<r}{R}_s\left(x_{\lambda }\right)\left(y_{\mu }\right)$.

(LMKR5) 

$R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)={\bigwedge }_{\nu \prec \mu }{R}_s\left(x_{\lambda }\right)\left(y_{\nu }\right)$.

(LMKR6) 

$R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)\le R_r\left(x_{\omega }\right)\left(y_{\mu }\right)$. whenever $\lambda \le \omega$.

(LMKR6) 

${\bigwedge }_{\omega \nleqq {\lambda }^{{}^{\prime }}}{R}_r\left(x_{\omega }\right)\left(y_{\mu }\right)={\bigwedge }_{\nu \nleqq {\mu }^{{}^{\prime }}}{R}_r\left(y_{\nu }\right)\left(x_{\lambda }\right)$.

The pair $(X,\mathcal{R})$ is called an$(L,M)$-fuzzy k-remote neighborhood ball space.

Proposition 2.

Let $(X,\mathcal{R})$ be an $(L,M)$-fuzzy k-remote neighborhood ball space. Then the following statements hold.

(LMKR4)${}^{*}$$R_s\left(x_{\lambda }\right)\left(y_{\mu }\right)\le R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)$ whenever $r\le s$.

(LMKR5)${}^{*}$$R_r\left(x_{\lambda }\right)\left(y_{\nu }\right)\ge R_r\left(x_{\lambda }\right)\left(y_{\mu }\right)$ whenever $\nu \le \mu$.

Definition 12.

A map $f:X⟶Y$ between $(L,M)$-fuzzy k-remote neighborhood ball spaces $(X,{\mathcal{R}}^X)$ and $(Y,{\mathcal{R}}^Y)$ is called continuous if $\forall r>0$, $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$,

$$R_r^Y\left(f{\left(x\right)}_{\lambda }\right)\left(f{\left(y\right)}_{\mu }\right)\le R_r^X\left(x_{\lambda }\right)\left(y_{\mu })\right).$$

We can check that $(L,M)$-fuzzy k-remote neighborhood ball spaces and their continuous maps constitute a category, denoted by $LM$-KRNS.

In what follows, the relationships between $(L,M)$-fuzzy k-pseudo metrics and $(L,M)$-fuzzy k-remote neighborhood ball systems are discussed. The following theorems are easily obtained from Definition 9 to Definition 12.

Theorem 3.

Let d be an $(L,M)$-fuzzy k-pseudo metric. For all $r\in (0,\infty )$, define a map $R_r^d:J\left(L^X\right)⟶M^{L^X}$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$,

$$R_r^d\left(x_{\lambda }\right)\left(y_{\mu }\right)=d(x_{\lambda },y_{\mu })\left(r\right).$$

Then ${\mathcal{R}}^d=\{R_r^d\mid r\in (0,\infty )\}$ is an $(L,M)$-fuzzy k-remote neighborhood ball system.

Theorem 4.

If a mapping $f:(X,d_X)⟶(Y,d_Y)$ is contractive between $(L,M)$-fuzzy k-pseudo metric spaces, then the mapping $f:(X,{\mathcal{R}}^{d_X})⟶(Y,{\mathcal{R}}^{d_Y})$ is continuous between $(L,M)$-fuzzy k-remote neighborhood ball spaces.

Conversely, we have:

Theorem 5.

Let $\mathcal{R}=\{R_r\mid r\in (0,\infty )\}$ be an $(L,M)$-fuzzy k-remote neighborhood ball system. Define a map $d^{\mathcal{R}}:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$, $\forall r>0$,

$$d^{\mathcal{R}}(x_{\lambda },y_{\mu })\left(r\right)=R_r\left(x_{\lambda }\right)\left(y_{\mu }\right).$$

Also, define $d^{\mathcal{R}}(x_{\lambda },y_{\mu })\left(0\right)={\top }_M$. Then $d^{\mathcal{R}}$ is an $(L,M)$-fuzzy k-pseudo metric.

Theorem 6.

If $f:(X,{\mathcal{R}}^X)⟶(Y,{\mathcal{R}}^Y)$ is continuous between $(L,M)$-fuzzy k-remote neighborhood ball spaces, then $f:(X,d^{{\mathcal{R}}^X})⟶(Y,d^{{\mathcal{R}}^Y})$ is contractive between $(L,M)$-fuzzy k-pseudo metric spaces.

By Theorems 3 and 5, we obtain ${\mathcal{R}}^{d^{\mathcal{R}}}=\mathcal{R}$ and $d^{{\mathcal{R}}^d}=d$.

In the end, we present a diagram illustrating the relationship between $(L,M)$-fuzzy k-pseudo metrics and $(L,M)$-fuzzy k-remote neighborhood ball systems (see Figure 1).

Theorem 7.

The category$LM$-KPMSis isomorphic to the category$LM$-KRNS.

4. (L, M)-Fuzzy Structures Induced by an (L, M)-Fuzzy k-Pseudo Metric

In this section, some $(L,M)$-fuzzy structures induced by an $(L,M)$-fuzzy k-pseudo metric are constructed and their continuous maps are studied, such as an $(L,M)$-fuzzy quasi-neighborhood system, an $(L,M)$-fuzzy closure operator, an $(L,M)$-fuzzy topology.

Before discussing the $(L,M)$-fuzzy quasi-neighborhood system induced by an $(L,M)$-fuzzy k-pseudo metric, we need the following lemma.

Lemma 1.

Let $\mathcal{Q}=\{{\mathcal{Q}}_{x_{\lambda }}\mid x_{\lambda }\in J\left(L^X\right)\}$ of maps ${\mathcal{Q}}_{x_{\lambda }}:L^X⟶M$ satisfying(LMQ1)-(LMQ3). Then the following conditions are equivalent.

(LMQ4)${\mathcal{Q}}_{x_{\lambda }}\left(A\right)={\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(B^{\prime }\right)$.

(LMQ4)${}^{*}$${\mathcal{Q}}_{x_{\lambda }}\left(A\right)={\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge {\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right)$.

Proof. 

$\left(\mathrm{LMQ}4\right)\Rightarrow {\left(\mathrm{LMQ}4\right)}^{*}$ is trivial.

On the other hand, suppose (LMQ4)${}^{*}$ holds. Let $\alpha \in M$ with

$$\alpha \prec \underset{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigvee }\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge \underset{y_{\mu }\nleqq B}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right).$$

Then there exists some $B\in L^X$ such that $x_{\lambda }\nleqq B\ge A^{\prime }$ and

(i)

$\alpha \prec {\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)$ and

(ii)

$\forall y_{\mu }\nleqq B$, $\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(A\right)$.

Let

$$V_{\alpha }=\bigwedge \{B\mid x_{\lambda }\nleqq B\ge A^{\prime },\alpha \prec {\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right),\forall y_{\mu }\nleqq B,\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(A\right)\}.$$

Then $x_{\lambda }\nleqq V_{\alpha }\ge A^{\prime }$, $\alpha \prec {\mathcal{Q}}_{x_{\lambda }}\left(V_{\alpha }^{\prime }\right)$ and $\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(A\right)$ for any $y_{\mu }\nleqq V_{\alpha }$. Note that $\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(A\right)$. Repeat the above derivation, there exists some $W_{y_{\mu }}\in L^X$ such that $y_{\mu }\nleqq W_{y_{\mu }}\ge A^{\prime }$, $\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(W_{y_{\mu }}^{\prime }\right)$ and $\alpha \prec {\mathcal{Q}}_{z_{\nu }}\left(A\right)$ for any $z_{\nu }\nleqq W_{y_{\mu }}$. By the minimality of $V_{\alpha }$, it follows that $V_{\alpha }\le W_{y_{\mu }}$. So $\alpha \prec {\mathcal{Q}}_{y_{\mu }}\left(W_{y_{\mu }}^{\prime }\right)\le {\mathcal{Q}}_{y_{\mu }}\left(V_{\alpha }^{\prime }\right)$ for any $y_{\mu }\nleqq V_{\alpha }$. This implies $\alpha \le {\bigwedge }_{y_{\mu }\nleqq V_{\alpha }}{\mathcal{Q}}_{y_{\mu }}\left(V_{\alpha }^{\prime }\right)$. By the arbitrariness of $\alpha$, we get

$$\begin{array}{ccc}& & {\mathcal{Q}}_{x_{\lambda }}\left(A\right)=\underset{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigvee }\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge \underset{y_{\mu }\nleqq B}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right)\hfill \\ & \le & \underset{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigvee }\underset{y_{\mu }\nleqq B}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}\left(B^{\prime }\right)\le {\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\le {\mathcal{Q}}_{x_{\lambda }}\left(A\right)\hfill \end{array}$$

Hence $Q_{x_{\lambda }}\left(A\right)={\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(B^{\prime }\right)$. □

Theorem 8.

Let $(X,\mathcal{R})$ be an $(L,M)$-fuzzy k-remote neighborhood ball space. For all $x_{\lambda }\in J\left(L^X\right)$, define a map ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}:L^X⟶M$ by $\forall A\in L^X$,

$${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)=\underset{r>0}{\bigvee }\underset{y_{\mu }\le A^{\prime }}{\bigwedge }{\mathcal{R}}_r\left(x_{\lambda }\right)\left(y_{\mu }\right).$$

Then ${\mathcal{Q}}^{\mathcal{R}}={\left\{{\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\right\}}_{x_{\lambda }\in J\left(L^X\right)}$ is an $(L,M)$-fuzzy quasi-neighborhood system.

Proof. 

It need to examine ${\mathcal{Q}}^{\mathcal{R}}$ satisfies (LMQ1)-(LMQ3) and (LMQ4)${}^{*}$.

(LMQ1), (LMQ2) holds obviously.

(LMQ3) From the definition of ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}$, it is easy to know $A\le B$ implies ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\le {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right)$. This means ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}(A\wedge B)\le {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\wedge {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right)$.

What remains is to prove ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\wedge {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right)\le {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}(A\wedge B)$. Take any $\alpha \in M$ with $\alpha \prec {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\wedge {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right).$ Then

$$\alpha \prec {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)=\underset{r>0}{\bigvee }\underset{y_{\mu }\le A^{\prime }}{\bigwedge }{\mathcal{R}}_r\left(x_{\lambda }\right)\left(y_{\mu }\right),\alpha \prec {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right)=\underset{s>0}{\bigvee }\underset{z_{\nu }\le B^{\prime }}{\bigwedge }{\mathcal{R}}_r\left(x_{\lambda }\right)\left(z_{\nu }\right).$$

This implies that there exist some $r>0$ and $s>0$ such that $\forall y_{\mu }\le A^{\prime },\alpha \le {\mathcal{R}}_r\left(x_{\lambda }\right)\left(y_{\mu }\right)$ and $\forall z_{\nu }\le B^{\prime },\alpha \le {\mathcal{R}}_s\left(x_{\lambda }\right)\left(z_{\nu }\right)$.

Let $t=r\wedge s$. By (LMKR4)${}^{*}$, we have $\forall y_{\mu }\le A^{\prime }$, $\alpha \le {\mathcal{R}}_t\left(x_{\lambda }\right)\left(y_{\mu }\right)$ and $\forall z_{\nu }\le B^{\prime }$, $\alpha \le {\mathcal{R}}_t\left(x_{\lambda }\right)\left(z_{\nu }\right)$. For any $w_l\in J\left(L^X\right)$ with $w_l\le {(A\wedge B)}^{\prime }=A^{\prime }\vee B^{\prime }$, we have $w_l\le A^{\prime }$ or $w_l\le B^{\prime }$. So $\alpha \le {\mathcal{R}}_t\left(x_{\lambda }\right)\left(w_l\right)$ for any $w_l\le {(A\wedge B)}^{\prime }$. Hence $\alpha \le {\bigvee }_{t>0}{\bigwedge }_{w_l\le {(A\wedge B)}^{\prime }}{\mathcal{R}}_r\left(x_{\lambda }\right)\left(w_l\right)={\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}(A\wedge B)$. From the arbitrariness of $\alpha$, we obtain ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\wedge {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B\right)\le {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}(A\wedge B)$.

(LMQ4)${}^{*}$${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\ge {\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge {\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right)$ is trivial. The key proof is that

$${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\le \underset{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigvee }\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge \underset{y_{\mu }\nleqq B}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right).$$

Take any $\alpha \in J\left(M\right)$ with $\alpha \prec {\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)={\bigvee }_{r>0}{\bigwedge }_{z_{\nu }\le A^{\prime }}{\mathcal{R}}_r\left(x_{\lambda }\right)\left(z_{\nu }\right)$. Then there exists some $r_0>0$ such that $\alpha \le {\mathcal{R}}_{r_0}\left(x_{\lambda }\right)\left(z_{\nu }\right)$ for any $z_{\nu }\le A^{\prime }$. Let

$$B_0=\bigvee \{y_{\mu }\in J\left(L^X\right)\mid R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge \alpha \}.$$

Step 1. We have that

$$y_{\nu }\le B_0\iff R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge \alpha .$$

In fact, it is easy to see $R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge \alpha$ implies $y_{\mu }\le B_0$. On other hand, take any $\nu \prec \mu$ with $y_{\nu }\prec B_0$. So there exists $y_w\in J\left(L^X\right)$ such that $R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_w\right)\ge \alpha$ and $\nu \le w$. It follows from (LMKR5)${}^{*}$ that $R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\nu }\right)\ge \alpha$. Hence ${\bigwedge }_{\nu \prec \mu }{R}_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\nu }\right)=R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge \alpha$.

Step 2. We show that

$$x_{\lambda }\nleqq B_0\ge A^{\prime }.$$

since $R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(x_{\lambda }\right)={\perp }_M\ngeqq \alpha$, it follows that $x_{\lambda }\nleqq B_0$. For any $z_{\nu }\le A^{\prime }$, $R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(z_{\nu }\right)\ge R_{r_0}\left(x_{\lambda }\right)\left(z_{\nu }\right)\ge \alpha$, then $z_{\nu }\le B_0$. So $B_0\ge A^{\prime }$. This shows $x_{\lambda }\nleqq B_0\ge A^{\prime }$.

Step 3. We must prove

$$\alpha \le \underset{x_{\lambda }\nleqq B\ge A^{\prime }}{\bigvee }\left({\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B^{\prime }\right)\wedge \underset{y_{\mu }\nleqq B}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}^{\mathcal{R}}\left(A\right)\right).$$

(i)

By Step 1, ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B_0^{\prime }\right)={\bigvee }_{r>0}{\bigwedge }_{y_{\mu }\le B_0}{R}_r\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge {\bigwedge }_{y_{\mu }\le B_0}{R}_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\ge \alpha$.

(ii)

What remains is to prove ${\bigwedge }_{y_{\mu }\nleqq B_0}{\mathcal{Q}}_{y_{\mu }}^{\mathcal{R}}\left(A\right)\ge \alpha$. Note that ${\bigwedge }_{y_{\mu }\nleqq B_0}{\mathcal{Q}}_{y_{\mu }}^{\mathcal{R}}\left(A\right)={\bigwedge }_{y_{\mu }\nleqq B_0}{\bigvee }_{r>0}$ ${\bigwedge }_{z_{\nu }\le A^{\prime }}{R}_r\left(y_{\mu }\right)\left(z_{\nu }\right)\ge {\bigwedge }_{y_{\mu }\nleqq B_0}{\bigwedge }_{z_{\nu }\le A^{\prime }}{R}_{\frac{r_0}{2k}}\left(y_{\mu }\right)\left(z_{\nu }\right)$. It suffices to check ${\bigwedge }_{y_{\mu }\nleqq B_0}{\bigwedge }_{z_{\nu }\le A^{\prime }}$ $R_{\frac{r_0}{2k}}\left(y_{\mu }\right)\left(z_{\nu }\right)\ge \alpha$. Since $\alpha \le R_{r_0}\left(x_{\lambda }\right)\left(z_{\nu }\right)$ for any $z_{\nu }\le A^{\prime }$ and $\alpha \nleqq R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)$ for any $y_{\mu }\le B_0$, it follows that

$$\alpha \le R_{r_0}\left(x_{\lambda }\right)\left(z_{\nu }\right)=R_{k(\frac{r_0}{2k}+\frac{r_0}{2k})}\left(x_{\lambda }\right)\left(z_{\nu }\right)\le R_{\frac{r_0}{2k}}\left(x_{\lambda }\right)\left(y_{\mu }\right)\vee R_{\frac{r_0}{2k}}\left(y_{\mu }\right)\left(z_{\nu }\right).$$

This implies $R_{\frac{r_0}{2k}}\left(y_{\mu }\right)\left(z_{\nu }\right)\ge \alpha$. If not, it is a contradiction. Then

$$\alpha \le \underset{y_{\mu }\nleqq B_0}{\bigwedge }\underset{z_{\nu }\le A^{\prime }}{\bigwedge }{R}_{\frac{r_0}{2k}}\left(y_{\mu }\right)\left(z_{\nu }\right)\le \underset{y_{\mu }\nleqq B_0}{\bigwedge }{\mathcal{Q}}_{y_{\mu }}^{\mathcal{R}}\left(A\right).$$

Combining (i) and (ii), $\alpha \le {\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}\left({\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(B^{\prime }\right)\wedge {\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}^{\mathcal{R}}\left(A\right)\right)$.

Therefore, it follows from Step 1–3 and the arbitrariness of $\alpha$ that ${\mathcal{Q}}_{x_{\lambda }}^{\mathcal{R}}\left(A\right)\le {\bigvee }_{x_{\lambda }\nleqq B\ge A^{\prime }}$ $\left({\mathcal{Q}}_{x_{\lambda }}\left(B^{\prime }\right)\wedge {\bigwedge }_{y_{\mu }\nleqq B}{\mathcal{Q}}_{y_{\mu }}\left(A\right)\right)$. □

Theorem 9.

If a mapping $f:(X,{\mathcal{R}}^X)⟶(Y,{\mathcal{R}}^Y)$ is continuous between $(L,M)$-fuzzy k-remote neighborhood ball spaces, then the mapping $f:(X,{\mathcal{Q}}^{{\mathcal{R}}^X})⟶(Y,{\mathcal{Q}}^{{\mathcal{R}}^Y})$ is continuous between $(L,M$)-fuzzy quasi-neighborhood spaces.

Proof. 

We need to check $\forall x_{\lambda }\in J\left(L^X\right)$, $\forall B\in L^Y$, ${\left({\mathcal{Q}}_Y\right)}_{f{\left(x\right)}_{\lambda }}\left(B\right)\le {\left({\mathcal{Q}}_X\right)}_{x_{\lambda }}\left(f_L^{\leftarrow }\left(B\right)\right)$.

Take any $a\in J\left(M\right)$ with $a\prec {\left({\mathcal{Q}}_Y\right)}_{f{\left(x\right)}_{\lambda }}\left(B\right)$. Suppose that ${\left({\mathcal{Q}}_Y\right)}_{f{\left(x\right)}_{\lambda }}\left(B\right)={\bigvee }_{r>0}{\bigwedge }_{z_{\nu }\le B^{\prime }}$ $R_r^Y(f{\left(x\right)}_{\lambda },z_{\nu }).$ Then there exists $r>0$ such that $a\le R_r^Y(f{\left(x\right)}_{\lambda },z_{\nu })$ for any $z_{\nu }\le B^{\prime }$. For any $w_l\ne {\left(f_L^{\leftarrow }\left(B\right)\right)}^{\prime }=f_L^{\leftarrow }\left(B^{\prime }\right)$, there exists $z_{\nu }\le B^{\prime }$ such that $z_{\nu }=f{\left(w\right)}_l$. By the continuity of $(L,M)$-fuzzy k-remote neighborhood ball spaces, $a\le R_r^Y(f{\left(x\right)}_{\lambda },f{\left(w\right)}_l)\le R_r^X(x_{\lambda },w_l)$. Hence $a\le {\bigvee }_{r>0}{\bigwedge }_{w_l\le {\left(f_L^{\leftarrow }\left(B\right)\right)}^{\prime }}{R}_r^X(x_{\lambda },w_l)={\left({\mathcal{Q}}_X\right)}_{x_{\lambda }}\left(f_L^{\leftarrow }\left(B\right)\right)$. By the arbitrariness of a, we obtain ${\left({\mathcal{Q}}_Y\right)}_{f{\left(x\right)}_{\lambda }}\left(B\right)\le {\left({\mathcal{Q}}_X\right)}_{x_{\lambda }}\left(f_L^{\leftarrow }\left(B\right)\right)$. □

Because of the one-to-one correspondence between $(L,M)$-fuzzy k-pseudo metrics and $(L,M)$-fuzzy k-remote neighborhood ball systems, we have the following.

Theorem 10.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric space. For all $x_{\lambda }\in J\left(L^X\right)$, define a map ${\mathcal{Q}}_{x_{\lambda }}^d:L^X⟶M$ by $\forall A\in L^X$,

$${\mathcal{Q}}_{x_{\lambda }}^d\left(A\right)=\underset{r>0}{\bigvee }\underset{y_{\mu }\le A^{\prime }}{\bigwedge }d(x_{\lambda },y_{\mu })\left(r\right).$$

Then ${\mathcal{Q}}^d=\{{\mathcal{Q}}_{x_{\lambda }}^d\mid x_{\lambda }\in J\left(L^X\right)\}$ is an $(L,M)$-fuzzy quasi-neighborhood system.

In [43,44], it is shown the category of $(L,M)$-fuzzy quasi-neighborhood spaces is isomorphic to the category of $(L,M)$-fuzzy closure spaces. Given an $(L,M)$-fuzzy quasi-neighborhood system $\mathcal{Q}$, a map $C{l}^{\mathcal{Q}}:L^X\to M^{J\left(L^X\right)}$ defined by

$$C{l}^{\mathcal{Q}}\left(A\right)\left(x_{\lambda }\right)={\left({\mathcal{Q}}_{x_{\lambda }}\left(A^{\prime }\right)\right)}^{\prime }$$

is an $(L,M)$-fuzzy closure operator.

Based on these and Figure 1, we have the following theorem.

Theorem 11.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric. Define $C{l}^d:L^X⟶M^{L^X}$ by

$$C{l}^d\left(A\right)\left(x_{\lambda }\right)=\underset{r>0}{\bigwedge }\underset{y_{\mu }\le A}{\bigvee }{\left(d(x_{\lambda },y_{\mu })\left(r\right)\right)}^{\prime }={\left(d(x_{\lambda },A)(0+)\right)}^{\prime },$$

in which $d(x_{\lambda },A)\left(r\right)={\bigwedge }_{y_{\mu }\le A}d(x_{\lambda },y_{\mu })\left(r\right)$. Then $C{l}^d$ is an $(L,M)$-fuzzy closure operator.

Finally, we shall give the expressions of an $(L,M)$-fuzzy topology and an $(L,M)$-fuzzy interior operator induced by an $(L,M)$-fuzzy k-pseudo metric. In [43,44], it is also shown that the category of $(L,M)$-fuzzy quasi-neighborhood spaces is isomorphic to the category of $(L,M)$-fuzzy topology and the category of $(L,M)$-fuzzy interior spaces is isomorphic to the category of $(L,M)$-fuzzy topology.

Given an $(L,M)$-fuzzy quasi-neighborhood system $\mathcal{Q}$, a map ${\mathcal{T}}^{\mathcal{Q}}:L^X\to M$ defined by

$${\mathcal{T}}^{\mathcal{Q}}\left(A\right)=\underset{x_{\lambda }\nleqq A^{\prime }}{\bigwedge }{\mathcal{Q}}_{x_{\lambda }}\left(A\right)$$

is an $(L,M)$-fuzzy topology. Based on these and Figure 1, we have the following theorem.

Theorem 12.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric. Define ${\mathcal{T}}^d:L^X⟶M$ by

$${\mathcal{T}}^d\left(A\right)=\underset{x_{\lambda }\nleqq A^{\prime }}{\bigwedge }\underset{r>0}{\bigvee }\underset{y_{\mu }\le A^{\prime }}{\bigwedge }d(x_{\lambda },y_{\mu })\left(r\right).$$

Then ${\mathcal{T}}^d$ is an $(L,M)$-fuzzy topology.

Given an $(L,M)$-fuzzy topology, a map $in{t}^{\mathcal{T}}:L^X⟶M^{J\left(L^X\right)}$ defined by

$$in{t}^{\mathcal{T}}\left(A\right)\left(x_{\lambda }\right)=\underset{x_{\lambda }\le V\le A}{\bigvee }{\mathcal{T}}_{(}V)$$

is an $(L,M)$-fuzzy interior operator. Based on these, we have the following theorem.

Theorem 13.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric. Define $in{t}^d:L^X⟶M^{J\left(L^X\right)}$ by

$$in{t}^d\left(A\right)\left(x_{\lambda }\right)=\underset{x_{\lambda }\le V\le A}{\bigvee }\underset{x_{\lambda }\nleqq V^{\prime }}{\bigwedge }\underset{r>0}{\bigvee }\underset{y_{\mu }\le V^{\prime }}{\bigwedge }d(x_{\lambda },y_{\mu })\left(r\right).$$

Then $in{t}^d$ is an $(L,M)$-fuzzy interior operator.

5. Relationships between Nests of Pointwise k-Pseudo Metric and (L, M)-Fuzzy k-Pseudo Metrics

In this section, we shall propose the concept of a nest of pointwise k-pseudo metrics and discuss its relations with $(L,M)$-fuzzy k-pseudo metrics.

Denote $PKM\left(X\right)$ as the set of all pointwise k-pseudo metrics on X.

Definition 13.

A mapping $\varphi :J\left(M\right)⟶PKM\left(X\right)$ is called a nested mapping of pointwise k-pseudo metrics, if

$$\forall a\in J\left(M\right),\varphi \left(a\right)=\underset{b\prec a}{\bigwedge }\varphi \left(b\right).$$

And $\left\{\varphi \right(a)\mid a\in J(M\left)\right\}$ is calleda nest of pointwise k-pseudo metrics.

From Definition 13, it is clear that $a_1\le a_2$ implies $\varphi \left(a_1\right)\ge \varphi \left(a_2\right)$.

Given an $(L,M)$-fuzzy k-pseudo metric d and define a map ${\varphi }^d:J\left(M\right)⟶PKM\left(X\right)$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right)$, $\forall a\in J\left(M\right)$,

$${\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })=\bigwedge \{t\in [0,\infty )\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}.$$

Before proving $\{{\varphi }^d\left(a\right)\mid a\in J\left(M\right)\}$ is a nest of pointwise k-pseudo metrics, we need the following lemma.

Lemma 2.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric space. Then

$$d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\iff {\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })<t.$$

Proof. 

⇒ Obviously.

⇐ if ${\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })=\bigwedge \{t\in [0,\infty )\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}<t$, then there exists $t_0$ such that $d(x_{\lambda },y_{\mu })\left(t_0\right)\ngeqq a$ and $t_0<t$. Since $d(x_{\lambda },y_{\mu })\left(t\right)\le d(x_{\lambda },y_{\mu })\left(t_0\right)$, we obtain $d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a$. □

Theorem 14.

Let $(X,d)$ be an $(L,M)$-fuzzy k-pseudo metric space. Then $\{{\varphi }^d\left(a\right)\mid a\in J\left(M\right)\}$ is a nest of pointwise k-pseudo metrics, in which ${\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })=\bigwedge \{t\in [0,\infty )\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}$.

Proof. 

Step 1. We check $\forall a\in J\left(M\right),{\varphi }^d\left(a\right)$ is a pointwise k-pseudo metric.

(LKD1)${\varphi }^d\left(a\right)(x_{\lambda },x_{\lambda })=\bigwedge \{t\in [0,\infty )\mid d(x_{\lambda },x_{\lambda })\left(t\right)\ngeqq a\}=\bigwedge [0,\infty )=0$.

(LKD2) For any $x_{\lambda }$, $y_{\mu }$, $z_{\gamma }\in J\left(L^X\right)$, take any $t>0$ with

$$\begin{array}{ccc}& & k\left({\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })+{\varphi }^d\left(a\right)(y_{\mu },z_{\gamma })\right)\hfill \\ & =& k\left(\bigwedge \{r\mid d(x_{\lambda },y_{\mu })\left(r\right)\ngeqq a\}+\bigwedge \{s\mid d(y_{\mu },z_{\gamma })\left(s\right)\ngeqq a\}\right)<t\hfill \end{array}$$

Then there exist $r,s>0$ such that $d(x_{\lambda },y_{\mu })\left(r\right)\ngeqq a$, $d(y_{\mu },z_{\gamma })\left(s\right)\ngeqq a$ and $k(r+s)<t$. By (LMKD2), $d(x_{\lambda },z_{\gamma })\left(k(r+s)\right)\le d(x_{\lambda },y_{\mu })\left(r\right)\vee d(y_{\mu },z_{\gamma })\left(s\right)\ngeqq a$. This implies $d(x_{\lambda },z_{\gamma })$ $\left(k\right(r+s\left)\right)\ngeqq a$, i.e., ${\varphi }^d\left(a\right)(x_{\lambda },z_{\gamma })<k(r+s)<t$. By the arbitrariness of t, ${\varphi }^d\left(a\right)(x_{\lambda },z_{\gamma })\le k\left({\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })+{\varphi }^d\left(a\right)(y_{\mu },z_{\gamma })\right)$.

(LKD3) It holds by (LMKD3) and the following equations

$$\begin{array}{ccc}& & {\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })=\bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}=\bigwedge \{t\mid \underset{\nu \prec \mu }{\bigwedge }d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}\hfill \\ & =& \underset{\nu \prec \mu }{\bigwedge }\bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}=\underset{\nu \prec \mu }{\bigwedge }{\varphi }^d\left(a\right)(x_{\lambda },y_{\nu })\hfill \end{array}$$

(LKD4) For any $\lambda \le \omega$, we have

$${\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })=\bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}\le \bigwedge \{t\mid d(x_{\omega },y_{\mu })\left(t\right)\ngeqq a\}={\varphi }^d\left(a\right)(x_{\omega },y_{\mu }).$$

(LKD5) It holds by (LKMD5) and the following equations

$$\begin{array}{ccc}& & \underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }{\varphi }^d\left(a\right)(x_{\omega },y_{\mu })=\underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }\bigwedge \{t\mid d(x_{\omega },y_{\mu })\left(t\right)\ngeqq a\}\hfill \\ & =& \bigwedge \{t\mid \underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }d(x_{\omega },y_{\mu })\left(t\right)\ngeqq a\}\hfill \\ & =& \bigwedge \{t\mid \underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }d(y_{\nu },x_{\lambda })\left(t\right)\ngeqq a\}\hfill \\ & =& \underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }\bigwedge \{t\mid d(y_{\nu },x_{\lambda })\left(t\right)\ngeqq a\}=\underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }{\varphi }^d\left(a\right)(y_{\nu },x_{\lambda })\hfill \end{array}$$

Step 2. We prove ${\varphi }^d\left(a\right)={\bigwedge }_{b\prec a}{\varphi }^d\left(b\right)$. Since ${\bigvee }_{b\prec a}b=a$, it follows that

$$\begin{array}{ccc}& & \underset{b\prec a}{\bigwedge }{\varphi }^d\left(b\right)=\underset{b\prec a}{\bigwedge }\bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq b\}\hfill \\ & =& \bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq \underset{b\prec a}{\bigvee }b\}=\bigwedge \{t\mid d(x_{\lambda },y_{\mu })\left(t\right)\ngeqq a\}={\varphi }^d\left(a\right).\hfill \end{array}$$

Therefore $\{{\varphi }^d\left(a\right)\mid a\in J\left(M\right)\}$ is a nest of pointwise k-pseudo metrics. □

In the following, we shall consider the converse question: whether an $(L,M)$-fuzzy k-pseudo metric can be induced by a nested mapping of pointwise k-pseudo metrics or not.

Given a nested mapping of pointwise k-pseudo metrics $\varphi :J\left(M\right)⟶PKM\left(X\right)$, define a map $d^{\varphi }:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ by $\forall x_{\lambda },y_{\mu }\in J\left(L^X\right),\forall t\in [0,\infty )$

$$d^{\varphi }(x_{\lambda },y_{\mu })\left(t\right)=\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t\}.$$

Then we have the following theorem.

Theorem 15.

Let $\varphi :J\left(M\right)⟶PKM\left(X\right)$ be a nested mapping of pointwise k-pseudo metrics. Then $d^{\varphi }$ is an $(L,M)$-fuzzy k-pseudo metric.

Proof. 

We need to check d satisfies (LMKD1)- (LMKD5).

(LMKD1)$d^{\varphi }(x_{\lambda },x_{\lambda })\left(t\right)=\bigvee \{a\mid \varphi \left(a\right)(x_{\lambda },x_{\lambda })\ge t\}=\bigvee \varnothing ={\perp }_M$.

(LMKD2) For all $x_{\lambda },y_{\mu },z_{\gamma }\in J\left(L^X\right)$ and $t,s\in [0,\infty )$,

$$\begin{array}{ccc}& & d^{\varphi }(x_{\lambda },z_{\gamma })\left(k(t+s)\right)=\bigvee \left\{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },z_{\gamma })\ge k(t+s)\right\}\hfill \\ & \le & \bigvee \{a\in J\left(M\right)\mid k\left(\varphi \left(a\right)(x_{\lambda },y_{\mu })+\varphi \left(a\right)(y_{\mu },z_{\gamma })\right)\ge k(t+s)\}\hfill \\ & \le & \left(\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\mu })\ge t\}\right)\vee \left(\bigvee \{c\in J\left(M\right)\mid \varphi \left(c\right)(y_{\mu },z_{\gamma })\ge s\}\right)\hfill \\ & =& d^{\varphi }(x_{\lambda },y_{\mu })\left(t\right)\vee d^{\varphi }(y_{\mu },z_{\gamma })\left(s\right).\hfill \end{array}$$

The inequality holds from $\{a\in J\left(M\right)\mid k(\varphi \left(a\right)(x_{\lambda },y_{\mu })+\varphi \left(a\right)(y_{\mu },z_{\gamma }))\ge k(t+s)\}\subseteq \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\mu })\ge t\}\cup \{c\in J\left(M\right)\mid \varphi \left(c\right)(y_{\mu },z_{\gamma })\ge s\}$.

(LMKD3) We need to prove $d^{\varphi }(x_{\lambda },y_{\mu })={\bigwedge }_{\nu \prec \mu }{d}^{\varphi }(x_{\lambda },y_{\nu })$, i.e.,

$$\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t\}=\underset{\nu \prec \mu }{\bigwedge }\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t\}.$$

On one hand, take $a\in J\left(M\right)$ with $\varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t$. For any $\nu \prec \mu$, we have $\varphi \left(a\right)(x_{\lambda },y_{\nu })\ge \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t$. This implies $a\le {\bigwedge }_{\nu \prec \mu }\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t\}$. Hence $\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t\}\le {\bigwedge }_{\nu \prec \mu }\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t\}$.

On the other hand, take any $r\in M$ with $r\prec {\bigwedge }_{\nu \prec \mu }\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t\}$. Then there exists some $b\in J\left(M\right)$ such that $r\prec b$ and $\varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t$ for any $\nu \prec \mu$. By (LKD3), ${\bigwedge }_{\nu \prec \mu }\varphi \left(b\right)(x_{\lambda },y_{\nu })=\varphi \left(b\right)(x_{\lambda },y_{\mu })\ge t$. Further $\varphi \left(r\right)(x_{\lambda },y_{\mu })\ge \varphi \left(b\right)(x_{\lambda },y_{\mu })\ge t$. This means $r\le \bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t\}$. From the arbitrariness of r, we obtain ${\bigwedge }_{\nu \prec \mu }\bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\lambda },y_{\nu })\ge t\}\le \bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\mu })\ge t\}$.

(LMKD4) For any $\lambda \le \omega$, $d^{\varphi }(x_{\lambda },y_{\mu })\left(t\right)=\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\lambda },y_{\nu })\ge t\}\le \bigvee \{b\in J\left(M\right)\mid \varphi \left(b\right)(x_{\omega },y_{\nu })\ge t\}=d^{\varphi }(x_{\omega },y_{\mu })\left(t\right)$.

(LMKD5) By the symmetry of the pointwise k-pseudo metric $\varphi \left(a\right)$, we have

$$\begin{array}{ccc}& & \underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }{d}^{\varphi }(x_{\omega },y_{\mu })\left(t\right)=\underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(x_{\omega },y_{\mu })\ge t\}\hfill \\ & =& \bigvee \{a\in J\left(M\right)\mid \underset{\omega \nleqq {\lambda }^{\prime }}{\bigwedge }\varphi \left(a\right)(x_{\omega },y_{\mu })\ge t\}=\bigvee \{a\in J\left(M\right)\mid \underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }\varphi \left(a\right)(y_{\nu },x_{\lambda })\ge t\}\hfill \\ & =& \underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }\bigvee \{a\in J\left(M\right)\mid \varphi \left(a\right)(y_{\nu },x_{\lambda })\ge t\}=\underset{\nu \nleqq {\mu }^{\prime }}{\bigwedge }{d}^{\varphi }(y_{\nu },x_{\lambda })\left(t\right).\hfill \end{array}$$

□

Let $\mathcal{D}=\{d\mid d\mathrm{is}\mathrm{an}(L,M)-\mathrm{fuzzy}k-\mathrm{pseudo}\mathrm{metric}\mathrm{on}X\}$.

Let $\mathcal{Q}=\{\varphi \mid \varphi \mathrm{is}\mathrm{a}\mathrm{nested}\mathrm{mapping}\mathrm{of}\mathrm{pointwise}k-\mathrm{pseudo}\mathrm{metric}\mathrm{on}X\}$.

By Theorem 14 and Theorem 15, we can construct a bijection $f:\mathcal{D}⟶\mathcal{Q}$ defined by $f\left(d\right)={\varphi }^d$, and a bijection $g:\mathcal{Q}⟶\mathcal{D}$ defined by $g\left(\varphi \right)=d^{\varphi }$.

In the following theorem, we shall show that there is a one-to-one correspondence between $(L,M)$-fuzzy k-pseudo metrics and nested mappings of pointwise k-pseudo metrics.

Theorem 16.

Let $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ be an $(L,M)$-fuzzy k-pseudo metric and $\varphi :J\left(M\right)⟶PKM\left(X\right)$ be a nested mapping of pointwise k-pseudo metric. Then $d^{{\varphi }^d}=d$ and ${\varphi }^{d^{\varphi }}=\varphi$.

Proof. 

$d^{{\varphi }^d}=d$ holds from the following

$$a\le d^{{\varphi }^d}(x_{\lambda },y_{\mu })\left(t\right)\iff {\varphi }^d\left(a\right)(x_{\lambda },y_{\mu })\ge t\iff a\le d(x_{\lambda },y_{\mu })\left(t\right).$$

and ${\varphi }^{d^{\varphi }}=\varphi$ holds from the following

$$t\le {\varphi }^{d^{\varphi }}\left(a\right)(x_{\lambda },y_{\mu })\iff d^{\varphi }(x_{\lambda },y_{\mu })\ge a\iff t\le \varphi \left(a\right)(x_{\lambda },y_{\mu }).$$

□

Finally, we shall discuss the relationships between the $(L,M)$-fuzzy topology directly induced by an $(L,M)$-fuzzy k-pseudo metric and the $(L,M)$-fuzzy topology induced by a nest of pointwise k-pseudo metrics.

Let ${\tau }_{\varphi \left(a\right)}$ be the L-topology induced by $\varphi \left(a\right)$. In [36], we know

$${\tau }_{\varphi \left(a\right)}=\{A\in L^X\mid \forall x_{\lambda }\nleqq A^{\prime },\exists r>0,\forall y_{\mu }\le A^{\prime },\varphi \left(a\right)(x_{\lambda },y_{\mu })\ge r\}.$$

Beside, refer to the corresponding theorems in [45], it is not hard to prove the following lemma and theorems.

Lemma 3.

If $\varphi :J\left(M\right)⟶PKM\left(X\right)$ is a nested mapping of pointwise k-pseudo metrics, then ${\tau }_{\varphi \left(b\right)}\subseteq {\tau }_{\varphi \left(a\right)}$ for any $a\le b$.

Theorem 17.

Let $\varphi :J\left(L\right)⟶PKM\left(X\right)$ be a nested mapping of pointwise k-pseudo metrics. Define ${\mathcal{T}}^{\varphi }:L^X⟶M$ by $\forall A\in L^X$,

$${\mathcal{T}}^{\varphi }\left(A\right)=\bigvee \{a\in J\left(M\right)\mid A\in {\tau }_{\varphi \left(a\right)}\}.$$

Then ${\mathcal{T}}^{\varphi }$ is an $(L,M)$-fuzzy topology.

Theorem 18.

Let $d:J\left(L^X\right)\times J\left(L^X\right)⟶[0,\infty )\left(M\right)$ be an $(L,M)$-fuzzy k-pseudo metric and $\varphi :J\left(M\right)⟶PKM\left(X\right)$ be a nested mapping of pointwise k-pseudo metric. Then ${\mathcal{T}}^{d^{\varphi }}={\mathcal{T}}^{\varphi }$, that is, the $(L,M)$-fuzzy topology induced by d is exactly the $(L,M)$-fuzzy topology induced by ϕ.

At the end of the paper, we present a diagram illustrating the obtained results about $(L,M)$-fuzzy topological structures induced by an $(L,M)$-fuzzy k-pseudo metrics (see Figure 2).

6. Conclusions

In this paper, the definitions of an $(L,M)$-fuzzy k-pseudo metric and an $(L,M)$-fuzzy k-remote neighborhood ball system were introduced. We showed that the category of $(L,M)$-fuzzy k-pseudo metric spaces is isomorphic to the category of $(L,M)$-fuzzy k-remote neighborhood ball spaces. Besides, we discussed some $(L,M)$-fuzzy structures induced by an $(L,M)$-fuzzy k-pseudo metric and investigated their properties.

Research regarding the concept of an $(L,M)$-fuzzy partial k-metric and its induced $(L,M)$-fuzzy structures would be our interest in the future. Furthermore, we plan to analyze the relationship among $(L,M)$-fuzzy k-pseudo metrics, $(L,M)$-fuzzy uniform structures and $(L,M)$-fuzzy topologies.