Categories of L-Primals, L-Pre-Proximities, and L-Topologies

Abstract

This paper introduces and investigates the fundamental properties of L-primals, a generalization of the primal concept within the framework of L-fuzzy sets and complete lattices. Building upon the established theories of L-topological spaces and L-pre-proximity spaces, this research explores the interrelations among these three generalized topological structures. The study establishes novel categorical links, demonstrating the existence of concrete functors between categories of L-primal spaces and L-pre-proximity spaces, as well as between categories of L-pre-proximity spaces and stratified L-primal spaces. Furthermore, the paper clarifies the existence of a concrete functor between the category of stratified L-primal spaces and the category of L-topological spaces, and vice versa, thereby establishing Galois correspondences between these categories. Theoretical findings are supported by illustrative examples, including applications within the contexts of information systems and medicine, demonstrating the practical aspects of the developed theory.

1. Introduction

Lattice structures, such as L-primals, L-pre-proximities, and L-topologies, form a rich categorical framework generalizing classical topological and proximity theories into the fuzzy- and lattice-enriched domain. These structures offer a foundation for modeling uncertainty, graded membership, and nearness in fields ranging from data science to logic. Various topological spaces play a fundamental role in fuzzy set theory and its applications. Numerous fuzzy structures, such as fuzzy topological spaces, fuzzy rough sets, and fuzzy approximation spaces, are constructed via different modifications of these operators. To analyze the connections among such structures, categorical methods offer a powerful and unifying framework, utilizing morphisms between objects and functors between categories. An early contribution in this direction was made by Goguen [1], who introduced the concept of the image of a fuzzy set under a fuzzy relation, which can be interpreted categorically as a functor between appropriate fuzzy relational categories.

Following the introduction of Goguen’s L-fuzzy sets [1] as an improvement of Zadeh’s fuzzy sets [2], the theory of L-topological spaces has advanced swiftly and has emerged as a prominent area of research. Numerous scholars have conducted theoretical studies of L-topological spaces from different perspectives. Höhle [3] and Kubiak [4] developed the generalized theory of L-topological spaces and analyzed various fundamental properties. Fang [5] developed the notion of L-order convergence structures and strong L-topological spaces. Later on, Li and Jin [6] introduced more properties of stratified L-convergence spaces and L-pre-topologies. Recently, Perfilieva et al. [7] investigated the relation between Čech L-closure spaces and L-co-topological spaces. Some more extensive research on L-topological spaces can be found in [5,8,9].

A proximity structure is a topological construct which has various applications in pattern recognition, digital image classification, and many other fields. In the case of fuzzy structures, there are at least two notions of fuzzy proximity, the first one is given by Katsaras [10,11] as a subset of $I^{X\times X}$ with some axioms, and the other one is given by Artico fuzzy proximity structures [12] as a mapping $I^{X\times X}\to I.$ Many scholars have extended Artico’s fuzzy proximity structures on complete lattices, and they have conducted some research on L-pre-proximity spaces from different perspectives. In particular, Čimoka and Šostak [13] developed the idea of L-proximities on complete residuated lattices, and they studied their relation with L-topological spaces. Kim and Oh [14,15] introduced (Alexandrov) L-fuzzy pre-proximities on complete residuated lattices and investigated their interrelationships with Alexandrov L-fuzzy pre-proximities, Alexandrov L-fuzzy topologies, and L-fuzzy approximate operators. Močkoř [16] provided a systematic study of the relationships among categories of various fuzzy topological structures, in which the morphisms were characterized by L-valued fuzzy relations. He established the existence of functors connecting these categories. Jose and Mathew [17] investigated the relations between L-filters and L-pre-proximity spaces. Recently, Ramadan et al. [18,19,20] studied relations among L-pre-proximity spaces, L-filters, L-ideals, and L-topological spaces.

The theory of primal topological spaces offers effective and powerful tool for constructing topological structures, since it is related to the theory of proximity spaces and topological spaces [21,22,23,24]. It is essential to note that a primal is the dual of a grill. Subsequently, Acharjee et al. presented the concept of primal topological spaces [25], which evolved in many directions. Recently, Al-Omari et al. [26,27] studied the relation between primal structures, closure operators, and topological spaces. More recently, Al-Omari et al. [28] introduced a new approach of proximity structures based on primal spaces.

The target of this manuscript is to present the study of L-primals and some of its basic properties, along with exploring the interrelations between L-primal, L-pre-proximity, and L-topological spaces. Categorical relations among L-topologies and L-pre-proximities induced by L-primals are also established. The existence of a concrete functor between categories of L-primal and L-pre-proximity spaces as well as between categories of L-pre-proximity and stratified L-primal spaces is clarified. Some examples are also given to support this study.

The structure of the paper is as follows: Section 2 presents fundamental concepts and some results that serve as a foundation for the entire paper. Section 3 establishes the links between L-primals and L-neighborhoods and provides some properties of these spaces. In Section 4, we study the categorical aspects of the relationships between L-primal, L-pre-proximity, and L-topological spaces. We show that there are concrete functors between these spaces and discuss the existence of a Galois correspondence among them. Additionally, we present real-world applications in the contexts of information systems and medicine to demonstrate the effectiveness of the proposed approach.

2. Preliminaries

In this section, we recall some basic concepts about complete lattices, L-subsets, L-topological, and L-pre-proximity spaces.

2.1. Complete Lattices

Throughout this research paper, X is a nonempty set, and $L^X$ is the set of all fuzzy subsets of X. $(L,\wedge ,\vee ,\to ,\perp ,\top )$ denotes a frame or Heyting algebra. In other words, $(L,\le )$ is a complete lattice with ⊥ and ⊤ as the bottom and top elements of L, respectively, and for all $\pi ,{\varphi }_j\in L$, and $j\in \Lambda$, it holds that

$$\begin{array}{c}\pi \wedge ({\displaystyle \underset{j\in \Lambda }{\bigvee }}{\varphi }_j)={\displaystyle \underset{j\in \Lambda }{\bigvee }}(\pi \wedge {\varphi }_j).\end{array}$$

The residual implication operation is defined as $\pi \wedge \varphi \le \theta$ iff $\varphi \le \pi \to \theta$ for every $\pi ,\varphi ,\theta \in L$. Note that $(L,\le ,\wedge {,}^{*})$ represents a frame with $*$ as an order-reversing involution defined by $\pi \vee \varphi ={\pi }^{*}\to \varphi ,{\pi }^{*}=\pi \to \perp$, where $\pi$ satisfies the double negation law ${\left({\pi }^{*}\right)}^{*}=\pi$. In the next lemma, we list some of its properties.

Lemma 1 

([3,29]). Let $(L,\le ,\wedge {,}^{*})$ be a frame. For each $\pi ,\varphi ,\theta ,{\pi }_j,{\varphi }_j,\sigma \in L$, and $j\in \Lambda$, the upcoming properties hold

(1) $\pi \to \varphi =\bigvee \{\theta \in L|\pi \wedge \theta \le \varphi \}$;

(2) $\top \to \varphi =\varphi$ and $\pi \le \varphi$ iff $\pi \to \varphi =\top$;

(3) If $\varphi \le \theta$, then $\pi \to \varphi \le \pi \to \theta$ and $\theta \to \pi \le \varphi \to \pi$;

(4) ${\left({\displaystyle \underset{j}{\bigwedge }}{\varphi }_j\right)}^{*}={\displaystyle \underset{j}{\bigvee }}{\varphi }_j^{*},{\left({\displaystyle \underset{j}{\bigvee }}{\varphi }_j\right)}^{*}={\displaystyle \underset{j}{\bigwedge }}{\varphi }_j^{*}$;

(5) $\pi \to \left({\displaystyle \underset{j}{\bigwedge }}{\varphi }_j\right)={\displaystyle \underset{j}{\bigwedge }}(\pi \to {\varphi }_j)$, and $\left({\displaystyle \underset{j}{\bigvee }}{\pi }_j\right)\to \varphi ={\displaystyle \underset{j}{\bigwedge }}({\pi }_j\to \varphi )$;

(6) $(\pi \wedge \varphi )\to \theta =\pi \to (\varphi \to \theta )=\varphi \to (\pi \to \theta )$;

(7) $\pi \wedge \varphi ={(\pi \to {\varphi }^{*})}^{*}$, and $\pi \to \varphi ={\varphi }^{*}\to {\pi }^{*}$;

(8) $\pi \to \varphi \le (\pi \wedge \theta )\to (\varphi \wedge \theta )$, and $(\pi \to \varphi )\wedge (\varphi \to \theta )\le \pi \to t$;

(9) $(\pi \to \varphi )\wedge (\theta \to \sigma )\le (\pi \wedge \theta )\to (\varphi \wedge \sigma )$;

(10) $(\pi \to \varphi )\wedge (\theta \to \sigma )\le (\pi \vee \theta )\to (\varphi \vee \sigma )$;

(11) $(\pi \to \varphi )\vee (\theta \to \sigma )\le (\pi \wedge \theta )\to (\varphi \vee \sigma )$;

(12) $\pi \wedge (\pi \to \varphi )\le \varphi$, and $\varphi \le \pi \to (\pi \wedge \varphi )$;

(13) $\pi \le \varphi \to \theta$ iff $\varphi \le \pi \to \theta$ and $\pi \wedge \theta \le \varphi$ iff $\pi \to \varphi \ge \theta$;

(14) ${\displaystyle \underset{j}{\bigvee }}{\pi }_j\to {\displaystyle \underset{j}{\bigvee }}{\varphi }_j\ge {\displaystyle \underset{j}{\bigwedge }}({\pi }_j\to {\varphi }_j),{\displaystyle \underset{j}{\bigwedge }}{\pi }_j\to {\displaystyle \underset{j}{\bigwedge }}{\varphi }_j\ge {\displaystyle \underset{j}{\bigwedge }}({\pi }_j\to {\varphi }_j)$;

(15) $\varphi \to \pi \le (\pi \to \theta )\to (\varphi \to \theta )$ and $\varphi \to \pi \le (\theta \to \varphi )\to (\theta \to \pi )$.

2.2. L-Subsets and L-Topological Structures

An L-subset [1,2] on a set X is defined by a mapping $\zeta :X\to L.$ For $\pi \in L$, the constant map $\pi :X\to L$ is given by $\pi \left(z\right)=\pi$, for all $z\in X$. The algebraic operations on L can be naturally extended to $L^X$.

Definition 1 

([5,29]). An L-relation θ on X is an L-subset in $X\times X.$ Moreover, θ is said to be

(1) Reflexive if $\theta (x,x)=\top$, for all $x\in X;$

(2) Transitive if $\theta (x,y)\wedge \theta (y,z)\le \theta (x,z)$, for all $x,y,z\in X.$

A reflexive and transitive L-relation on X is called an L-pre-order on X.

If θ is an L-relation, then the pair $(X,\theta )$ is referred as an L-approximation space and the following operators $\underline{\theta },\overline{\theta }:L^X\to L^X$ are defined by

$\underline{\theta }\left(\zeta \right)\left(x\right)={\bigwedge }_{y\in X}\theta (x,y)\to \zeta \left(y\right)$ and $\overline{\theta }\left(\zeta \right)\left(x\right)={\bigvee }_{y\in X}\theta (x,y)\wedge \zeta \left(y\right),\forall \zeta \in L^X,\forall x\in X.$

Note that $\underline{\theta }$ and $\overline{\theta }$ are called lower and upper L-approximation operators, respectively. If θ is reflexive, then $\overline{\theta }\left(\zeta \right)\ge \zeta$ and $\underline{\theta }\left(\zeta \right)\le \zeta .$

Lemma 2 

([5,29]). Let the subsethood degree of two L-subsets η and ζ be given by a mapping $\mathcal{SD}:L^X\times L^X\to L$ defined by $\mathcal{SD}(\eta ,\zeta )={\displaystyle \underset{z\in X}{\bigwedge }}(\eta \left(z\right)\to \zeta \left(z\right))$. Then, for every $\eta ,\zeta ,\nu ,\rho \in L^X$ and $\pi \in L$, the following properties hold:

(1) $\eta \le \zeta$ iff $\mathcal{SD}(\eta ,\zeta )\ge \top$;

(2) $\mathcal{SD}(\eta ,\zeta )\wedge \mathcal{SD}(\zeta ,\nu )\le \mathcal{SD}(\eta ,\rho )$;

(3) If $\eta \le \zeta$, then $\mathcal{SD}(\nu ,\eta )\le \mathcal{SD}(\nu ,\zeta )$ and $\mathcal{SD}(\eta ,\nu )\ge \mathcal{SD}(\zeta ,\nu )$;

(4) $\mathcal{SD}(\eta ,\zeta )\wedge \mathcal{SD}(\nu ,\rho )\le \mathcal{SD}(\eta \vee \nu ,\zeta \vee \rho )$ and $\mathcal{SD}(\eta ,\zeta )\wedge \mathcal{SD}(\rho ,\nu )\le \mathcal{SD}(\eta \wedge \rho ,\zeta \wedge \nu )$;

(5) $\mathcal{SD}(\eta ,\zeta )\le \mathcal{SD}(\rho ,\eta )\to \mathcal{SD}(\rho ,\zeta )$ and $\mathcal{SD}(\eta ,\zeta )\le \mathcal{SD}(\zeta ,\rho )\to \mathcal{SD}(\eta ,\rho )$;

(6) $\mathcal{SD}(\eta ,\zeta )\wedge \eta \le \zeta$ and $\mathcal{SD}(\eta ,\pi \wedge \zeta )\ge \pi \wedge \mathcal{SD}(\eta ,\zeta )$;

(7) $\mathcal{SD}({\eta }^{*},{\zeta }^{*})=\mathcal{SD}(\zeta ,\eta )$ and $\eta \le \mathcal{SD}(\eta ,\zeta )\to \zeta$;

(8) If $\psi :X\to Y$ is a mapping, then for every $\varrho ,\omega \in L^Y$, we have

$\mathcal{SD}(\varrho ,\omega )\le \mathcal{SD}({\psi }^{\leftarrow }\left(\varrho \right),{\psi }^{\leftarrow }\left(\omega \right))$, where ${\psi }^{\leftarrow }\left(\varrho \right)\left(z\right)=\varrho \left(\psi \left(z\right)\right).$

The terminologies of category theory used in this paper can be found in [30].

Definition 2 

([3,4]). The pair $(X,\mathcal{TP})$ is called an L-topological space, where $\mathcal{TP}:L^X\to L$ is a mapping on X, with the following properties:

(T1) $\mathcal{TP}\left({\perp }_X\right)=\mathcal{TP}\left({\top }_X\right)=\top$;

(T2) $\mathcal{TP}(\eta \wedge \lambda )\ge \mathcal{TP}\left(\eta \right)\wedge \mathcal{TP}\left(\lambda \right),\forall \eta ,\lambda \in L^X$;

(T3) $\mathcal{TP}\left({\displaystyle \underset{j\in \Lambda }{\bigvee }}{\eta }_j\right)\ge {\displaystyle \underset{j\in \Lambda }{\bigwedge }}\mathcal{TP}\left({\eta }_j\right),$ for all ${\eta }_j\in L^X,j\in \Lambda$.

Additionally, the L-topological space $(X,\mathcal{TP})$ is called

(AL) Alexandrov if $\mathcal{TP}\left({\displaystyle \underset{j\in \Lambda }{\bigwedge }}{\eta }_j\right)\ge {\displaystyle \underset{j\in \Lambda }{\bigwedge }}\mathcal{TP}\left({\eta }_j\right)$, for each ${\eta }_j\in L^X,j\in \Lambda$;

(R) Enriched if $\mathcal{TP}(\pi \wedge \eta )\ge \mathcal{TP}\left(\eta \right),\forall \eta \in L^X,\pi \in L$.

Note that $\psi :(X,{\mathcal{TP}}_X)\to (Y,{\mathcal{TP}}_Y)$ is a continuous mapping if ${\mathcal{TP}}_Y\left(\varrho \right)\le {\mathcal{TP}}_X\left({\psi }^{\leftarrow }\left(\varrho \right)\right)$, for each $\varrho \in L^Y$.

We assign $\mathit{L}\mathbf{\text{-}}\mathit{TOP}$ to the category of L-topological spaces with continuous maps as morphisms.

Definition 3 

([3]). An L-neighborhood system on X refers to a collection of mappings $\mathcal{N}=\{{\mathcal{N}}^x:L^X\to L\mid x\in X\}$ that satisfy the following properties for all $\eta ,\zeta \in L^X$:

(N1) ${\mathcal{N}}^x\left({\top }_X\right)=\top$;

(N2) ${\mathcal{N}}^x\left(\eta \right)\le \eta \left(x\right)$;

(N3) If $\eta \le \zeta$ then ${\mathcal{N}}^x\left(\eta \right)\le {\mathcal{N}}^x\left(\zeta \right)$;

(N4) ${\mathcal{N}}^x(\eta \wedge \zeta )\ge {\mathcal{N}}^x\left(\eta \right)\wedge {\mathcal{N}}^x\left(\zeta \right)$.

The pair $(X,\mathcal{N})$ is called an L-neighborhood space, and it is called Alexandrov, if

$$\mathcal{N}\left(\underset{j\in \Lambda }{\bigwedge }{\eta }_j\right)=\underset{j\in \Lambda }{\bigwedge }\mathcal{N}\left({\eta }_j\right),\forall {\eta }_j\in L^X,j\in \Lambda .$$

Let $(X,{\mathcal{N}}_X)$ and $(Y,{\mathcal{N}}_Y)$ be two L-neighborhood spaces. A mapping $\psi :X\to Y$ is said to be continuous at $x\in X$ if ${\mathcal{N}}^{\psi \left(x\right)}\left(\varrho \right)\le {\mathcal{N}}^x\left({\psi }^{\leftarrow }\left(\varrho \right)\right)$, for each $\varrho \in L^Y$. ψ is called continuous, if it is continuous at every $x\in X,$ where ${\mathcal{N}}^x\in {\mathcal{N}}_X$ and ${\mathcal{N}}^{\psi \left(x\right)}\in {\mathcal{N}}_Y.$

We assign $\mathbf{L}\mathbf{\text{-}}\mathbf{NB}$ to the category of L-neighborhood spaces with continuous maps as morphisms.

Definition 4 

([19,20]). The pair $(X,\mathcal{PRO})$ is called an L-pre-proximity space, where $\mathcal{PRO}:L^X\times L^X\to L$ is a mapping on X with the following properties for all $\zeta ,\eta ,\varrho ,{\zeta }_1,{\zeta }_2,{\eta }_1,{\eta }_2\in L^X$:

(LP1) $\mathcal{PRO}({\perp }_X,{\top }_X)=\perp ,$ and $\mathcal{PRO}({\top }_X,{\perp }_X)=\perp ;$

(LP2) $\mathcal{PRO}(\zeta ,\eta )\ge {\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge \eta \left(z\right);$

(LP3) $\mathcal{SD}(\zeta ,\eta )\le \mathcal{PRO}(\zeta ,\varrho )\to \mathcal{PRO}(\eta ,\varrho )$ and $\mathcal{SD}(\zeta ,\eta )\le \mathcal{PRO}(\varrho ,\zeta )\to \mathcal{PRO}(\varrho ,\eta );$

(LP4) $\mathcal{PRO}({\zeta }_1\wedge {\zeta }_2,{\eta }_1\vee {\eta }_2)\le \mathcal{PRO}({\zeta }_1,{\eta }_1)\vee \mathcal{PRO}({\zeta }_2,{\eta }_2)$ or

$\mathcal{PRO}({\zeta }_1\vee {\zeta }_2,{\eta }_1\wedge {\eta }_2)\le \mathcal{PRO}({\zeta }_1,{\eta }_1)\vee \mathcal{PRO}({\zeta }_2,{\eta }_2)$.

Additionally, the L-pre-proximity space $(X,\mathcal{PRO})$ is called

(St) Stratified if $\mathcal{PRO}(\pi \wedge \zeta ,\pi \to \eta )\le \mathcal{PRO}(\zeta ,\eta )$ or $\mathcal{PRO}(\pi \to \zeta ,\pi \wedge \eta )\le \mathcal{PRO}(\zeta ,\eta );$

(AL) Alexandrov if $\mathcal{PRO}(\zeta ,{\displaystyle \underset{i\in \Lambda }{\bigvee }}{\eta }_i)={\displaystyle \underset{i\in \Lambda }{\bigvee }}\mathcal{PRO}(\zeta ,{\eta }_i)$ and $\mathcal{PRO}({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i,\eta )={\displaystyle \underset{i\in \Lambda }{\bigvee }}\mathcal{PRO}({\zeta }_i,\eta )$, for all $\{{\zeta }_i,{\eta }_i:i\in \Lambda \}\subseteq L^X$.

Moreover, $\psi :(X,{\mathcal{PRO}}_X)\to (Y,{\mathcal{PRO}}_Y)$ is called an L-proximity map if

$${\mathcal{PRO}}_Y(\zeta ,\eta )\ge {\mathcal{PRO}}_X\left({\psi }^{\leftarrow }\left(\zeta \right),{\psi }^{\leftarrow }\left(\eta \right)\right),\forall \zeta ,\eta \in L^Y.$$

We assign $\mathbf{L}\mathbf{\text{-}}\mathbf{PPROX}$ to the category of L-pre-proximity spaces with L-proximity maps as morphisms.

3. $\mathit{L}$-Primals and $\mathit{L}$-Neighborhoods

In this section, we introduce the notion of an L-primal space and present some of its basic properties. We also establish the links between L-primal spaces and L-neighborhood spaces.

Definition 5. 

The pair $(X,\mathcal{PM})$ is called an L-primal space, where $\mathcal{PM}:L^X\to L$ is a mapping on X with the following properties for all $\eta ,\zeta \in L^X$:

(P1) $\mathcal{PM}\left({\perp }_X\right)=\top$ and $\mathcal{PM}\left({\top }_X\right)=\perp$;

(P2) $\mathcal{SD}(\eta ,\zeta )\le \mathcal{PM}\left(\zeta \right)\to \mathcal{PM}\left(\eta \right)$;

(P3) $\mathcal{PM}(\eta \wedge \zeta )\le \mathcal{PM}\left(\eta \right)\vee \mathcal{PM}\left(\zeta \right)$.

Additionally, the L-primal space $(X,\mathcal{PM})$ is called

(AL) Alexandrov if $\mathcal{PM}\left({\displaystyle \underset{j\in \Lambda }{\bigwedge }}{\eta }_j\right)={\displaystyle \underset{j\in \Lambda }{\bigvee }}\mathcal{PM}\left({\eta }_j\right)$, for each ${\eta }_j\in L^X,j\in \Lambda$;

(St) Stratified if $\mathcal{PM}(\pi \wedge \eta )\le \pi \to \mathcal{PM}\left(\eta \right)$, for each $\pi \in L$.

Note that a mapping $\psi :(X,{\mathcal{PM}}_X)\to (Y,{\mathcal{PM}}_Y)$ is an L-primal map if ${\mathcal{PM}}_X\left({\psi }^{\leftarrow }\left(\varrho \right)\right)\le {\mathcal{PM}}_Y\left(\varrho \right)$, for each $\varrho \in L^Y$.

We assign $\mathit{L}\mathbf{\text{-}}\mathit{PRM}$ to the category of L-primal spaces with L-primal maps as morphisms and $\mathit{SL}\mathbf{\text{-}}\mathit{PRM}$ to the category of stratified L-primal spaces.

One may notice that if $\mathcal{PM}$ is an L-primal map on X and $\eta \le \zeta$, then $\mathcal{PM}\left(\eta \right)\ge \mathcal{PM}\left(\zeta \right)$.

Lemma 3. 

For all $\zeta ,\eta \in L^X,\pi \in L$, the following inequalities are equivalent:

(1) $\mathcal{PM}(\pi \to \zeta )\ge \pi \wedge \mathcal{PM}\left(\zeta \right)$;

(2) $\mathcal{SD}(\zeta ,\eta )\le \mathcal{PM}\left(\eta \right)\to \mathcal{PM}\left(\zeta \right)$;

(3) $\mathcal{PM}(\pi \wedge \zeta )\le \pi \to \mathcal{PM}\left(\zeta \right)$.

Proof. 

(1) ⇒ (2) $\zeta \le \mathcal{SD}(\zeta ,\eta )\to \eta$ and

$$\begin{array}{c}\mathcal{PM}\left(\zeta \right)\ge \mathcal{PM}\left(\mathcal{SD}\right(\zeta ,\eta )\to \eta )\ge \mathcal{SD}(\zeta ,\eta )\wedge \mathcal{PM}\left(\eta \right).\end{array}$$

Therefore, $\mathcal{SD}(\zeta ,\eta )\le \mathcal{PM}\left(\eta \right)\to \mathcal{PM}\left(\zeta \right)$.

(2) ⇒ (3) Taking $\eta =\pi \wedge \zeta$ in (2), then $\pi \le \mathcal{SD}(\zeta ,\pi \wedge \zeta )\le \mathcal{PM}(\pi \wedge \zeta )\to \mathcal{PM}\left(\zeta \right)$. Hence, $\mathcal{PM}(\pi \wedge \zeta )\le \pi \to \mathcal{PM}\left(\zeta \right)$.

(3) ⇒ (1) Since $\pi \wedge (\pi \to \zeta )\le \zeta$, by (3), we get $\mathcal{PM}\left(\zeta \right)\le \mathcal{PM}(\pi \wedge (\pi \to \zeta \left)\right)\le \pi \to \mathcal{PM}(\pi \to \zeta )$. Thus, $\mathcal{PM}(\pi \to \zeta )\ge \pi \wedge \mathcal{PM}\left(\zeta \right)$. □

Theorem 1. 

Assume that ${\left\{{\mathcal{PM}}_i\right\}}_{i\in \Lambda }$ is a family of L-primals on X. Then, the mapping $\mathcal{PM}:L^X\to L$ defined by $\mathcal{PM}\left(\zeta \right)={\displaystyle \underset{i\in \Lambda }{\bigvee }}{\mathcal{PM}}_i\left(\zeta \right)$ is an L-primal map on X.

Proof. 

The proofs of (P1) and (P2) are straightforward and thus omitted.

(P3) $\forall \zeta ,\eta \in L^X$, we have

$$\begin{array}{cc}\mathcal{PM}\left(\zeta \right)\vee \mathcal{PM}\left(\eta \right)& ={\displaystyle \underset{i\in \Lambda }{\bigvee }}{\mathcal{PM}}_i\left(\zeta \right)\vee {\displaystyle \underset{i\in \Lambda }{\bigvee }}{\mathcal{PM}}_i\left(\eta \right)\hfill \\ & ={\displaystyle \underset{i\in \Lambda }{\bigvee }}({\mathcal{PM}}_i\left(\zeta \right)\vee {\mathcal{PM}}_i\left(\eta \right))\ge {\displaystyle \underset{i\in \Lambda }{\bigvee }}{\mathcal{PM}}_i(\zeta \wedge \eta )=\mathcal{PM}(\zeta \wedge \eta ).\hfill \end{array}$$

This completes the proof. □

Theorem 2. 

Let $\mathcal{PM}$ be a stratified L-primal on X and $\psi :X\to Y$ be a mapping. Then, $\psi \left(\mathcal{PM}\right):L^Y\to L$ given by $\psi \left(\mathcal{PM}\right)\left(\zeta \right)=\mathcal{PM}\left({\psi }^{\leftarrow }\left(\zeta \right)\right),\forall \zeta \in L^Y$ is a stratified L-primal map on Y.

Proof. 

(P1) $\psi \left(\mathcal{PM}\right)\left({\top }_Y\right)=\mathcal{PM}\left({\psi }^{\leftarrow }\left({\top }_Y\right)\right)=\mathcal{PM}\left({\top }_X\right)=\perp ,$ and $\psi \left(\mathcal{PM}\right)\left({\perp }_Y\right)=\mathcal{PM}\left({\psi }^{\leftarrow }\left({\perp }_Y\right)\right)=\mathcal{PM}\left({\perp }_X\right)=\top .$

(P2) $\forall \zeta ,\eta \in L^X$, we have

$$\begin{array}{cc}& \psi \left(\mathcal{PM}\right)\left(\eta \right)\to \psi \left(\mathcal{PM}\right)\left(\zeta \right)=\mathcal{PM}\left({\phi }^{\leftarrow }\left(\eta \right)\right)\to \mathcal{PM}\left({\psi }^{\leftarrow }\left(\zeta \right)\right)\hfill \\ & \ge \mathcal{SD}({\psi }^{\leftarrow }\left(\zeta \right),{\psi }^{\leftarrow }\left(\eta \right))\ge \mathcal{SD}(\zeta ,\eta ).\hfill \end{array}$$

(P3) $\forall \zeta ,\eta \in L^X$, we have

$$\begin{array}{cc}\psi \left(\mathcal{PM}\right)(\zeta \wedge \eta )& =\mathcal{PM}\left({\psi }^{\leftarrow }(\zeta \wedge \eta )\right)=\mathcal{PM}({\psi }^{\leftarrow }\left(\zeta \right)\wedge {\psi }^{\leftarrow }\left(\eta \right))\hfill \\ & \le \mathcal{PM}\left({\psi }^{\leftarrow }\left(\zeta \right)\right)\vee \mathcal{PM}\left({\psi }^{\leftarrow }\left(\eta \right)\right)=\psi \left(\mathcal{PM}\right)\left(\zeta \right)\vee \psi \left(\mathcal{PM}\right)\left(\eta \right).\hfill \end{array}$$

(St) Since ${\psi }^{\leftarrow }\left(\pi \right)=\pi ,$ we get

$$\begin{array}{cc}\psi \left(\mathcal{PM}\right)(\pi \wedge \eta )& =\mathcal{PM}\left({\psi }^{\leftarrow }(\pi \wedge \eta )\right)=\mathcal{PM}(\pi \wedge {\psi }^{\leftarrow }\left(\eta \right))\hfill \\ & \le \pi \to \mathcal{PM}\left({\psi }^{\leftarrow }\left(\eta \right)\right)=\pi \to \psi \left(\mathcal{PM}\right)\left(\eta \right).\hfill \end{array}$$

This completes the proof. □

Theorem 3. 

If $\phi :(X,{\mathcal{PM}}_X)\to (Y,{\mathcal{PM}}_Y)$ and $\psi :(Y,{\mathcal{PM}}_Y)\to (Z,{\mathcal{PM}}_Z)$ are L-primal mappings, then $\psi \circ \phi :(X,{\mathcal{PM}}_X)\to (Z,{\mathcal{PM}}_Z)$ is an L-primal mapping.

Proof. 

The proof is straightforward and thus omitted. □

Theorem 4. 

Let $(X,\mathcal{PM})$ be an L-primal space and ${\mathcal{N}}_{\mathcal{PM}}={\mathcal{N}}_{\mathcal{PM}}^x:L^X\to L$ be a mapping defined by

$$\begin{array}{c}{\mathcal{N}}_{\mathcal{PM}}^x\left(\eta \right)=\eta \left(x\right)\wedge {\mathcal{PM}}^{*}\left(\eta \right),\forall x\in X,\eta \in L^X.\end{array}$$

Then,

(1) $(X,{\mathcal{N}}_{\mathcal{PM}})$ is an L-neighborhood space. Moreover, ${\mathcal{N}}_{\mathcal{PM}}$ is Alexandrov if $\mathcal{PM}$ is Alexandrov.

(2) If $\psi :(X,{\mathcal{PM}}_X)\to (Y,{\mathcal{PM}}_Y)$ is an L-primal map, then $\psi :(X,{\mathcal{N}}_{{\mathcal{PM}}_X})\to (X,{\mathcal{N}}_{{\mathcal{PM}}_Y})$ is a continuous map.

Proof. 

(1) (N1) ${\mathcal{N}}_{\mathcal{PM}}^x\left({\top }_X\right)={\top }_X\left(x\right)\wedge {\mathcal{PM}}^{*}\left({\top }_X\right)={\top }_X\left(x\right)\wedge \top =\top$,

(N2) ${\mathcal{N}}_{\mathcal{PM}}^x\left(\eta \right)\le \eta \left(x\right).$

(N3)

$$\begin{array}{cc}\mathcal{SD}\left({\mathcal{N}}_{\mathcal{PM}}^x\left(\zeta \right),{\mathcal{N}}_{\mathcal{PM}}^x\left(\eta \right)\right)& ={\displaystyle \underset{z\in X}{\bigwedge }}\left({\mathcal{N}}_{\mathcal{PM}}^x\left(\zeta \right)\to {\mathcal{N}}_{\mathcal{PM}}^x\left(\eta \right)\right)\hfill \\ & ={\displaystyle \underset{xz\in X}{\bigwedge }}\left(\left(\zeta \left(x\right)\wedge {\mathcal{PM}}^{*}\left(\zeta \right)\right)\to \left(\eta \left(x\right)\wedge {\mathcal{PM}}^{*}\left(\eta \right)\right)\right)\hfill \\ & \ge \left({\displaystyle \underset{z\in X}{\bigwedge }}\left(\zeta \left(x\right)\to \eta \left(x\right)\right)\right)\wedge \left({\mathcal{PM}}^{*}\left(\zeta \right)\to {\mathcal{PM}}^{*}\left(\eta \right)\right)\hfill \\ & =\mathcal{SD}(\zeta ,\eta )\wedge \left(\mathcal{PM}\left(\eta \right)\to \mathcal{PM}\left(\zeta \right)\right)\hfill \\ & \ge \mathcal{SD}(\zeta ,\eta )\wedge \mathcal{SD}(\zeta ,\eta )=\mathcal{SD}(\zeta ,\eta ).\hfill \end{array}$$

(N4)

$$\begin{array}{cc}{\mathcal{N}}_{\mathcal{PM}}^x(\zeta \wedge \eta )& =(\zeta \wedge \eta )\left(x\right)\wedge {\mathcal{PM}}^{*}(\zeta \wedge \eta )\hfill \\ & \ge (\zeta \wedge \eta )\left(x\right)\wedge \left({\mathcal{PM}}^{*}\left(\zeta \right)\wedge {\mathcal{PM}}^{*}\left(\eta \right)\right)\hfill \\ & =\left(\zeta \left(x\right)\wedge {\mathcal{PM}}^{*}\left(\zeta \right)\right)\wedge \left(\eta \left(x\right)\wedge {\mathcal{PM}}^{*}\left(\eta \right)\right)\hfill \\ & ={\mathcal{N}}_{\mathcal{PM}}^x\left(\zeta \right)\wedge {\mathcal{N}}_{\mathcal{PM}}^x\left(\eta \right).\hfill \end{array}$$

Hence, $(X,{\mathcal{N}}_{\mathcal{PM}})$ is an L-neighborhood space.

(2) For all $\zeta \in L^Y,$ we have

$$\begin{array}{cc}& {\mathcal{N}}_{\mathcal{PM}}^x\left({\psi }^{\leftarrow }\left(\zeta \right)\right)={\psi }^{\leftarrow }\left(\zeta \right)\left(x\right)\wedge {\mathcal{PM}}_X^{*}\left({\psi }^{\leftarrow }\left(\zeta \right)\right)\hfill \\ & \ge \zeta \left(\psi \left(x\right)\right)\wedge {\mathcal{PM}}_Y^{*}\left(\zeta \right)={\mathcal{N}}_{\mathcal{PM}}^{\psi \left(x\right)}\left(\zeta \right).\hfill \end{array}$$

This completes the proof. □

Thus, there is a concrete functor $\Psi :\mathbf{L}\mathbf{\text{-}}\mathbf{PRM}\to \mathbf{L}\mathbf{\text{-}}\mathbf{NB}$ defined as $\Psi (X,\mathcal{PM})=(X,{\mathcal{N}}_{\mathcal{PM}})$.

Theorem 5. 

Let $(X,\mathcal{N})$ be an L-neighborhood space and ${\mathcal{PM}}_{\mathcal{N}}={\mathcal{PM}}_{{\mathcal{N}}^x}:L^X\to L$ be a mapping defined by

$$\begin{array}{c}{\mathcal{PM}}_{{\mathcal{N}}^x}\left(\eta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}^x\right)}^{*}\left(\eta \right),\forall \eta \in L^X.\end{array}$$

Then, we have the following:

(1) $(X,{\mathcal{PM}}_{\mathcal{N}})$ is an L-primal space. Moreover, ${\mathcal{PM}}_{\mathcal{N}}$ is Alexandrov if $\mathcal{N}$ is Alexandrov.

(2) If $\psi :(X,{\mathcal{N}}_X)\to (Y,{\mathcal{N}}_Y)$ is a continuous map, then $\psi :(X,{\mathcal{PM}}_{{\mathcal{N}}_X})\to (X,{\mathcal{PM}}_{{\mathcal{N}}_Y})$ is an L-primal map.

(3) ${\mathcal{PM}}_{{\mathcal{N}}_{\mathcal{PM}}}\ge \mathcal{PM}$ and ${\mathcal{N}}_{{\mathcal{PM}}_{\mathcal{N}}}\le \mathcal{N}$.

Proof. 

We prove only (P3) and (3)

(P3)

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{N}}(\zeta \wedge \eta )={\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}^x\right)}^{*}(\zeta \wedge \eta )\le {\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}^x\right)}^{*}\left(\zeta \right)\vee {\left({\mathcal{N}}^x\right)}^{*}\left(\eta \right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}^x\right)}^{*}\left(\zeta \right)\vee {\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}^x\right)}^{*}\left(\eta \right)={\mathcal{PM}}_{\mathcal{N}}\left(\zeta \right)\vee {\mathcal{PM}}_{\mathcal{N}}\left(\eta \right).\hfill \end{array}$$

(3) Let $\zeta \in L^X$, then

$$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{N}}_{\mathcal{PM}}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\left({\mathcal{N}}_{\mathcal{PM}}^x\right)}^{*}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\zeta }^{*}\left(x\right)\vee \mathcal{PM}\left(\zeta \right)\ge \mathcal{PM}\left(\zeta \right),\hfill \end{array}$$

and

$$\begin{array}{cc}& {\mathcal{N}}_{{\mathcal{PM}}_{\mathcal{N}}}\left(\zeta \right)=\zeta \left(x\right)\wedge {\left({\mathcal{PM}}_{{\mathcal{N}}^x}\right)}^{*}\left(\zeta \right)\hfill \\ & =\zeta \left(x\right)\wedge {\displaystyle \underset{z\in X}{\bigwedge }}{\mathcal{N}}^x\left(\zeta \right)\le {\displaystyle \underset{z\in X}{\bigwedge }}\mathcal{N}\left(\zeta \right)\le \mathcal{N}\left(\zeta \right).\hfill \end{array}$$

This completes the proof. □

4. $\mathit{L}$-Primal, $\mathit{L}$-Pre-Proximity, and $\mathit{L}$-Topological Spaces

In this section, we focus on the categorical aspects of the relationships between L-primal, L-pre-proximity, and L-topological spaces. Some applications and examples are also provided.

4.1. On the Adjunction $(\Omega ,\Psi ):\mathit{SL}\mathit{\text{-}}\mathit{PRM}\rightharpoonup \mathit{L}\mathit{\text{-}}\mathit{PPROX}$

The following theorem establishes a concrete functor from the category of L-primal spaces to the category of L-pre-proximity spaces.

Theorem 6. 

Let $(X,\mathcal{PM})$ be an L-primal space and ${\mathcal{PRO}}_{\mathcal{PM}}:L^X\times L^X\to L$ be a mapping defined by:

$${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left(\eta \left(z\right)\vee \mathcal{PM}\left({\eta }^{*}\right)\right)\right),\forall \zeta ,\eta \in L^X.$$

Then, we have the following:

(1) $(X,{\mathcal{PRO}}_{\mathcal{PM}})$ is a pre-proximity space. Moreover, if $\mathcal{PM}$ is Alexandrov, then so is ${\mathcal{PRO}}_{\mathcal{PM}}$.

(2) If $\mathcal{PM}$ is stratified, then ${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\pi \wedge \eta )\ge \pi \wedge {\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta )$ and ${\mathcal{PRO}}_{\mathcal{PM}}$ is stratified.

(3) $\psi :(X,{\mathcal{PRO}}_{{\mathcal{PM}}_X})\to (Y,{\mathcal{PRO}}_{{\mathcal{PM}}_Y})$ is an L-proximity map if $\psi :(X,{\mathcal{PM}}_X)\to (Y,{\mathcal{PM}}_Y)$ is an L-primal map.

Proof. 

(1) (LP1)

$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}({\top }_X,{\perp }_X)={\displaystyle \underset{z\in X}{\bigvee }}\left({\top }_X\left(z\right)\wedge ({\perp }_X\left(z\right)\vee \mathcal{PM}\left({\top }_X\right))\right)=\top \wedge (\perp \vee \perp )=\perp ,\hfill \end{array}$

$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}({\perp }_X,{\top }_X)={\displaystyle \underset{z\in X}{\bigvee }}\left({\perp }_X\left(z\right)\wedge ({\top }_X\left(z\right)\vee \mathcal{PM}\left({\perp }_X\right))\right)=\perp \wedge (\top \vee \top )=\perp .\hfill \end{array}$

(LP2) ${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge (\eta \left(z\right)\vee \mathcal{PM}\left({\eta }^{*}\right))\right)\ge {\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge \eta \left(z\right).$

(LP3) $\forall \zeta ,\eta \in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\varrho )\to {\mathcal{PRO}}_{\mathcal{PM}}(\eta ,\varrho )\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left(\rho \left(z\right)\vee \mathcal{PM}\left({\varrho }^{*}\right)\right)\right)\to {\displaystyle \underset{z\in X}{\bigvee }}\left(\eta \left(z\right)\wedge (\varrho \left(z\right)\vee \mathcal{PM}\left({\varrho }^{*}\right))\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}(\zeta \left(z\right)\to \eta \left(z\right))=\mathcal{SD}(\zeta ,\eta ),\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}& {\mathcal{PR}}_{\mathcal{PM}}(\rho ,\zeta )\to {\mathcal{PR}}_{\mathcal{PM}}(\varrho ,\eta )\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\varrho \left(z\right)\wedge \left(\zeta \left(z\right)\vee \mathcal{PM}\left({\zeta }^{*}\right)\right)\right)\to {\displaystyle \underset{z\in X}{\bigvee }}\left(\varrho \left(z\right)\wedge \left(\eta \left(z\right)\vee \mathcal{PM}\left({\eta }^{*}\right)\right)\right)\hfill \\ & \ge \mathcal{PM}\left({\zeta }^{*}\right)\to \mathcal{PM}\left({\eta }^{*}\right)\ge \mathcal{SD}({\eta }^{*},{\zeta }^{*})=\mathcal{SD}(\zeta ,\eta ).\hfill \end{array}$$

(LP4) $\forall {\zeta }_1,{\zeta }_2,{\eta }_1,{\eta }_2\in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}({\zeta }_1,{\eta }_1)\vee {\mathcal{PRO}}_{\mathcal{PM}}({\zeta }_2,{\eta }_2)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left({\zeta }_1\left(z\right)\wedge \left({\eta }_1\left(z\right)\vee \mathcal{PM}\left({\eta }_1^{*}\right)\right)\right)\vee {\displaystyle \underset{z\in X}{\bigvee }}\left({\zeta }_2\left(z\right)\wedge \left({\eta }_2\left(z\right)\vee \mathcal{PM}\left({\eta }_2^{*}\right)\right)\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}\left({\zeta }_1\left(z\right)\wedge ({\eta }_1\left(z\right)\vee \left(\mathcal{PM}\left({\eta }_1^{*}\right)\right)\vee ({\zeta }_2\left(z\right)\wedge ({\eta }_2\left(z\right)\vee \mathcal{PM}\left({\eta }_2^{*}\right)))\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}({\zeta }_1\left(z\right)\wedge {\zeta }_2\left(z\right))\wedge \left(({\eta }_1\left(z\right)\vee \mathcal{PM}\left({\eta }_1^{*}\right))\vee ({\eta }_2^{*}\left(z\right)\vee \mathcal{PM}\left({\eta }_2^{*}\right))\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}({\zeta }_1\left(z\right)\wedge {\zeta }_2\left(z\right))\wedge \left(({\eta }_1\left(z\right)\vee {\eta }_2\left(z\right))\vee \mathcal{PM}({\eta }_1^{*}\wedge {\eta }_2^{*})\right)\hfill \\ & ={\displaystyle \underset{x\in X}{\bigvee }}\left(({\zeta }_1\wedge {\zeta }_2)\left(z\right)\right)\wedge \left({\eta }_1\vee {\eta }_2)\left(z\right)\vee \mathcal{PM}\left({({\eta }_1\vee {\eta }_2)}^{*}\right)\right)\hfill \\ & ={\mathcal{PRO}}_{\mathcal{PM}}({\zeta }_1\wedge {\zeta }_2,{\eta }_1\vee {\eta }_2).\hfill \end{array}$$

(AL) If $\mathcal{PM}$ is Alexandrov, then

$$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,{\displaystyle \underset{i\in \Gamma }{\bigvee }}{\eta }_i)={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\eta }_i\left(z\right)\vee \mathcal{PM}\left({\displaystyle \underset{i\in \Gamma }{\bigwedge }}{\eta }_i^{*}\right)\right)\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\eta }_i\left(z\right)\vee {\displaystyle \underset{i\in \Gamma }{\bigvee }}\mathcal{PM}\left({\eta }_i^{*}\right)\right)\right)\hfill \\ & ={\displaystyle \underset{i\in \Gamma }{\bigvee }}{\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left({\eta }_i\left(z\right)\vee \mathcal{PM}\left({\eta }_i^{*}\right)\right)\right)\hfill \\ & ={\displaystyle \underset{i\in \Gamma }{\bigvee }}{\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,{\eta }_i).\hfill \end{array}$$

(2) Since $\mathcal{PM}$ is stratified, then it is easy to see that ${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\pi \wedge \eta )\ge \pi \wedge {\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta ).$ Thus,

$$\begin{array}{cc}& {\mathcal{PRO}}_{\mathcal{PM}}(\pi \wedge \zeta ,\pi \to \eta )={\displaystyle \underset{z\in X}{\bigvee }}\left((\pi \wedge \zeta \left(z\right))\wedge ((\pi \to \eta \left(z\right))\vee \mathcal{PM}(\pi \wedge {\eta }^{*}))\right)\hfill \\ & \le {\displaystyle \underset{z\in X}{\bigvee }}\left[(\pi \wedge \zeta \left(z\right))\wedge \left((\pi \to \eta \left(z\right))\vee (\pi \to \mathcal{PM}\left({\eta }^{*}\right))\right)\right]\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left[(\pi \wedge \zeta \left(z\right))\wedge {\left({(\pi \to \eta \left(z\right))}^{*}\wedge {(\pi \to \mathcal{PM}\left({\eta }^{*}\right))}^{*}\right)}^{*}\right]\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left[(\pi \wedge \zeta \left(z\right))\wedge {\left((\pi \wedge {\eta }^{*}\left(z\right))\wedge (\pi \wedge {\mathcal{PM}}^{*}\left({\eta }^{*}\right))\right)}^{*}\right]\hfill \\ & \le {\displaystyle \underset{z\in X}{\bigvee }}\left[(\pi \wedge \zeta \left(z\right))\wedge \left(\pi \to {({\eta }^{*}\left(z\right)\wedge {\mathcal{PM}}^{*}\left({\eta }^{*}\right))}^{*}\right)\right]\hfill \\ & \le {\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge {\left({\eta }^{*}\left(z\right)\wedge {\mathcal{PM}}^{*}\left({\eta }^{*}\right))\right)}^{*}\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right))\wedge \left(\eta \left(z\right)\right)\vee \mathcal{PM}\left({\eta }^{*}\right))\right)\hfill \\ & ={\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta ).\hfill \end{array}$$

Therefore, ${\mathcal{PRO}}_{\mathcal{PM}}$ is stratified.

(3) $\forall \zeta ,\eta \in L^Y,$ we get

$$\begin{array}{cc}{\mathcal{PRO}}_{{\mathcal{PM}}_X}({\psi }^{\leftarrow }\left(\zeta \right),{\psi }^{\leftarrow }\left(\eta \right))& ={\displaystyle \underset{y\in X}{\bigvee }}\left({\psi }^{\leftarrow }\left(\zeta \right)\left(y\right)\wedge ({\psi }^{\leftarrow }\left(\eta \right)\left(z\right)\vee {\mathcal{PM}}_X\left({\psi }^{\leftarrow }\left({\eta }^{*}\right)\right)\right)\hfill \\ & \le {\displaystyle \underset{y\in X}{\bigvee }}\left(\zeta \left(\psi \left(y\right)\right)\wedge (\eta \left(\psi \left(y\right)\right)\vee {\mathcal{PM}}_Y\left({\eta }^{*}\right))\right)\hfill \\ & \le {\displaystyle \underset{z\in Y}{\bigvee }}\left(\zeta \left(z\right)\wedge (\eta \left(z\right)\vee {\mathcal{PM}}_Y\left({\eta }^{*}\right))\right)={\mathcal{PRO}}_{{\mathcal{PM}}_Y}(\zeta ,\eta ).\hfill \end{array}$$

This completes the proof. □

Corollary 1. 

Given $(X,\mathcal{PM})$ as an L-primal space, define a mapping ${\mathcal{PRO}}_{\mathcal{PM}}:L^X\times L^X\to L$ by

$${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\left(\eta \left(z\right)\wedge (\zeta \left(z\right)\vee \mathcal{PM}\left({\zeta }^{*}\right))\right),\forall \zeta ,\eta \in L^X.$$

Then, the following holds:

(1) $(X,{\mathcal{PRO}}_{\mathcal{PM}})$ is an L-pre-proximity space. Moreover, if $\mathcal{PM}$ is Alexandrov, then so is ${\mathcal{PRO}}_{\mathcal{PM}}$.

(2) If $\mathcal{PM}$ is stratified, then ${\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\pi \wedge \eta )\ge \pi \wedge {\mathcal{PRO}}_{\mathcal{PM}}(\zeta ,\eta )$ and ${\mathcal{PRO}}_{\mathcal{PM}}$ is stratified.

Thus, we obtain a concrete functor $\Psi :\mathbf{L}\mathbf{\text{-}}\mathbf{PRM}\to \mathbf{L}\mathbf{\text{-}}\mathbf{PPROX}$ defined by

$$\Psi (X,{\mathcal{PM}}_X)=(X,{\mathcal{PRO}}_{{\mathcal{PM}}_X}),\Psi \left(\psi \right)=\psi .$$

The following theorem establishes a concrete functor from the category of L-pre-proximity spaces to the category of stratified L-primal spaces.

Theorem 7. 

Let $(X,\mathcal{PRO})$ be an L-pre-proximity space. Define a mapping ${\mathcal{PM}}_{\mathcal{PRO}}:L^X⟶L$ as follows:

$${\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}\left({\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left({\mathcal{PRO}}^{*}({\eta }^{*},\eta )\wedge \mathcal{SD}(\eta ,\zeta )\right)\to {\eta }^{*}\right)\left(z\right),\forall \zeta \in L^X.$$

Then, we have the following:

(1) $(X,{\mathcal{PM}}_{\mathcal{PRO}})$ is a stratified L-primal space.

(2) If $\psi :(X,{\mathcal{PRO}}_X)\to (Y,{\mathcal{PRO}}_Y)$ is an L-proximity map, then $\psi :(X,{\mathcal{PM}}_{{\mathcal{PRO}}_X})\to (Y,{\mathcal{PM}}_{{\mathcal{PRO}}_Y})$ is an L-primal map.

Proof. 

(1) (P1)

$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{PRO}}\left({\top }_X\right)=\left({\mathcal{PRO}}^{*}({\perp }_X,{\top }_X)\wedge \mathcal{SD}({\top }_X,{\top }_X)\right)\to {\perp }_X\left(z\right)=\perp ,\hfill \end{array}$

$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{PRO}}\left({\perp }_X\right)=\left({\mathcal{PRO}}^{*}({\top }_X,{\perp }_X)\wedge \mathcal{SD}({\perp }_X,{\perp }_X)\right)\to {\top }_X\left(z\right)=\top .\hfill \end{array}$

(P2) $\forall \zeta ,\eta \in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)\to {\mathcal{PM}}_{\mathcal{PRO}}\left(\eta \right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\right)\left(z\right)\hfill \\ & \to {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\rho }^{*},\rho )\wedge \mathcal{SD}({\eta }^{*},{\rho }^{*}))\to {\rho }^{*}\right)\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}[{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\left(z\right)\right)\hfill \\ & \to {\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\rho }^{*},\rho )\wedge \mathcal{SD}({\eta }^{*},{\rho }^{*}))\to {\rho }^{*}\left(z\right)\right)]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}[\left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\left(z\right)\right)\hfill \\ & \to \left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\eta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\left(z\right)\right)]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left(\mathcal{SD}({\eta }^{*},{\varrho }^{*})\to \mathcal{SD}({\zeta }^{*},{\varrho }^{*})\right)\ge \mathcal{SD}({\zeta }^{*},{\eta }^{*})=\mathcal{SD}(\eta ,\zeta ).\hfill \end{array}$$

(P3) $\forall \zeta ,\eta \in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)\vee {\mathcal{PM}}_{\mathcal{PRO}}\left(\eta \right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\right)\left(z\right)\hfill \\ & \vee {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left(({\mathcal{PRO}}^{*}({\rho }^{*},\rho )\wedge \mathcal{SD}({\eta }^{*},{\rho }^{*}))\to {\rho }^{*}\right)\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}[\left(({\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*}))\to {\varrho }^{*}\right)\left(z\right)\hfill \\ & \vee \left(({\mathcal{PRO}}^{*}({\rho }^{*},\rho )\wedge \mathcal{SD}({\eta }^{*},{\rho }^{*}))\to {\rho }^{*}\right)\left(z\right)]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[({\mathcal{PRO}}^{*}({\rho }^{*},\rho )\wedge {\mathcal{PRO}}^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}({\zeta }^{*},{\varrho }^{*})\wedge \mathcal{SD}({\eta }^{*},{\rho }^{*}))\to {\varrho }^{*}\vee {\rho }^{*})\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[\left({\mathcal{PRO}}^{*}({\varrho }^{*}\vee {\rho }^{*},\varrho \wedge \rho )\right)\wedge \mathcal{SD}({\zeta }^{*}\vee {\eta }^{*},{\varrho }^{*}\vee {\rho }^{*}))\to {\varrho }^{*}\vee {\rho }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[({\mathcal{PRO}}^{*}({(\varrho \wedge \rho )}^{*},\varrho \wedge \rho )\wedge \mathcal{SD}({(\zeta \wedge \eta )}^{*},{(\varrho \wedge \rho )}^{*}))\to {(\varrho \wedge \rho )}^{*}\right]\left(z\right)\hfill \\ & ={\mathcal{PM}}_{\mathcal{PRO}}(\zeta \wedge \eta ).\hfill \end{array}$$

Hence, ${\mathcal{PM}}_{\mathcal{PRO}}(\zeta \wedge \eta )\le {\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)\vee {\mathcal{PM}}_{\mathcal{PRO}}\left(\eta \right).$

(2) $\forall \eta \in L^Y,$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{PRO}}_Y}\left(\eta \right)={\displaystyle \underset{y\in Y}{\bigvee }}{\displaystyle \underset{R\in L^Y}{\bigwedge }}\left(({\mathcal{PRO}}_Y^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}(\varrho ,\eta ))\to {\varrho }^{*}\left(y\right)\right)\hfill \\ & \ge {\displaystyle \underset{\psi \left(z\right)\in Y}{\bigvee }}{\displaystyle \underset{\varrho \in L^Y}{\bigwedge }}\left(({\mathcal{PRO}}_Y^{*}({\varrho }^{*},\varrho )\wedge \mathcal{SD}(\varrho ,\eta ))\to {\varrho }^{*}\left(\psi \left(z\right)\right)\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{{\psi }^{\leftarrow }\left(\varrho \right)\in L^X}{\bigwedge }}\left({\mathcal{PRO}}_X^{*}({\psi }^{\leftarrow }\left({\varrho }^{*}\right),{\psi }^{\leftarrow }\left(\varrho \right))\wedge \mathcal{SD}({\psi }^{\leftarrow }\left(\varrho \right),{\psi }^{\leftarrow }\left(\eta \right))\to {\psi }^{\leftarrow }\left({\varrho }^{*}\right)\left(z\right)\right)\hfill \\ & \ge {\mathcal{PM}}_{{\mathcal{PRO}}_X}\left({\psi }^{\leftarrow }\left(\eta \right)\right).\hfill \end{array}$$

This completes the proof. □

Thus, we obtain a concrete functor $\Omega :\mathbf{L}\mathbf{\text{-}}\mathbf{PPROX}\to \mathbf{SL}\mathbf{\text{-}}\mathbf{PRM}$ defined by

$$\Omega (X,{\mathcal{PRO}}_X)=(X,{\mathcal{PM}}_{{\mathcal{PRO}}_X}),\Omega \left(\psi \right)=\psi .$$

Proposition 1. 

If $(X,\mathcal{PRO})$ is an L-pre-proximity space, then ${\mathcal{PRO}}_{{\mathcal{PM}}_{\mathcal{PRO}}}\le \mathcal{PRO}.$

Proof. 

By Theorem 7, we get ${\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)\le {\displaystyle \underset{z\in X}{\bigvee }}{\zeta }^{*}\left(z\right).$ Thus,

$$\begin{array}{cc}& {\mathcal{PRO}}_{{\mathcal{PM}}_{\mathcal{PRO}}}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge (\eta \left(z\right)\vee {\mathcal{PM}}_{\mathcal{PRO}}\left({\eta }^{*}\right))\right)\le {\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge (\eta \left(z\right)\vee {\displaystyle \underset{z\in X}{\bigvee }}\eta \left(z\right))\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge {\displaystyle \underset{z\in X}{\bigvee }}\eta \left(z\right)\right)={\displaystyle \underset{z\in X}{\bigvee }}(\zeta \left(z\right)\wedge \eta \left(z\right))\le \mathcal{PRO}(\zeta ,\eta ).\hfill \end{array}$$

This completes the proof. □

Proposition 2. 

If $(X,\mathcal{PM})$ is a stratified L-primal space, then ${\mathcal{PM}}_{{\mathcal{PRO}}_{\mathcal{PM}}}\ge \mathcal{PM}.$

Proof. 

Since ${\mathcal{PRO}}_{\mathcal{PM}}^{*}({\eta }^{*},\eta )\le \mathcal{SD}\left(\eta ,{\mathcal{PM}}^{*}\left(\eta \right)\right),$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{PRO}}_{\mathcal{PM}}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left({\mathcal{PRO}}_{\mathcal{PM}}^{*}({\eta }^{*},\eta )\wedge \mathcal{SD}(\eta ,\zeta )\to {\eta }^{*}\right)\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[(\mathcal{SD}(\eta ,{\mathcal{PM}}^{*}\left(\eta \right))\wedge \mathcal{SD}(\eta ,\zeta ))\to {\eta }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[(\mathcal{SD}(\eta ,{\mathcal{PM}}^{*}\left(\eta \right))\wedge \mathcal{SD}(\mathcal{PM}\left(\zeta \right),\mathcal{PM}\left(\eta \right)))\to {\eta }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[(\mathcal{SD}(\eta ,{\mathcal{PM}}^{*}\left(\eta \right))\wedge \mathcal{SD}({\mathcal{PM}}^{*}\left(\eta \right),{\mathcal{PM}}^{*}\left(\zeta \right)))\to {\eta }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[\mathcal{SD}(\eta ,{\mathcal{PM}}^{*}\left(\zeta \right))\to {\eta }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[\mathcal{SD}(\mathcal{PM}\left(\zeta \right),{\eta }^{*})\to {\eta }^{*}\right]\left(z\right)\hfill \\ & \ge \mathcal{PM}\left(\zeta \right).\hfill \end{array}$$

This completes the proof. □

Theorem 8. 

$(\Omega ,\Psi )$ is a Galois correspondence.

Proof. 

Assume that $(X,\mathcal{PM})$ is a stratified L-primal space. By Proposition 2, ${\mathcal{PM}}_{{\mathcal{PRO}}_{\mathcal{PM}}}\ge \mathcal{PM}.$ Therefore, the identity map $i{d}_X:(X,\mathcal{PM})\to (X,{\mathcal{PM}}_{{\mathcal{PRO}}_{\mathcal{PM}}})$ is an L-primal map. If $\mathcal{PRO}$ is an L-pre-proximity on X, then, by Proposition 1, ${\mathcal{PRO}}_{{\mathcal{PM}}_{\mathcal{PRO}}}\le \mathcal{PRO}.$ Therefore, the identity $i{d}_Y:(X,{\mathcal{PRO}}_{{\mathcal{PM}}_{\mathcal{PRO}}})\to (X,\mathcal{PRO})$ is an L-proximity map. Therefore, $(\Omega ,\Psi )$ is a Galois correspondence. □

Example 1. 

(1) Define ${\mathcal{PM}}_1:L^X\to L$ as ${\mathcal{PM}}_1\left(\eta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\eta }^{*}\left(z\right)$. Thus, ${\mathcal{PM}}_1$ is an L-primal on X. By Theorem 6, we have

$$\begin{array}{c}{\mathcal{PRO}}_{{\mathcal{PM}}_1}(\zeta ,\eta )={\bigvee }_{z\in X}\zeta \left(z\right)\wedge (\eta \left(z\right)\vee {\mathcal{PM}}_1\left({\eta }^{*}\right))={\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge {\displaystyle \underset{z\in X}{\bigvee }}\eta \left(z\right).\end{array}$$

(2) Define ${\mathcal{PM}}_2:L^X\to L$ as ${\mathcal{PM}}_2\left(\eta \right)={\eta }^{*}\left(z\right)$. Hence, ${\mathcal{PM}}_2$ is an L-primal on X. By Theorem 6, we have

$$\begin{array}{c}{\mathcal{PRO}}_{{\mathcal{PM}}_2}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge (\eta \left(z\right)\vee {\mathcal{PM}}_2\left({\eta }^{*}\right))={\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge \eta \left(z\right).\end{array}$$

(3) Define $\mathcal{PRO}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge \eta \left(z\right).$ By Theorem 7, we have

$${\mathcal{PM}}_{\mathcal{PRO}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\mathcal{SD}(\eta ,\zeta )\to {\eta }^{*}\left(z\right).$$

(4) Define ${\mathcal{PRO}}_2(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\zeta \left(z\right)\wedge {\displaystyle \underset{z\in X}{\bigvee }}\eta \left(z\right).$ By Theorem 7, we have

$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{PRO}}_2}\left(\zeta \right)\ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[{\displaystyle \underset{z\in X}{\bigwedge }}\eta \left(z\right)\to (\mathcal{SD}({\zeta }^{*},{\eta }^{*})\to {\eta }^{*}\left(z\right))\right]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left[{\displaystyle \underset{z\in X}{\bigwedge }}\eta \left(z\right)\to {\zeta }^{*}\left(z\right)\right]\ge {\displaystyle \underset{z\in X}{\bigvee }}{\zeta }^{*}\left(z\right).\hfill \end{array}$

4.2. On the Adjunction $(\Lambda ,{\rm Y}):\mathit{L}\mathbf{\text{-}}\mathit{PRM}\rightharpoonup \mathit{L}\mathbf{\text{-}}\mathit{TOP}$

The following theorem suggests a concrete functor from the category of L-primal spaces to the category of L-topological spaces.

Theorem 9. 

Let $(X,\mathcal{PM})$ be an L-primal space and ${\mathcal{TP}}_{\mathcal{PM}}:L^X\to L$ be a mapping given by ${\mathcal{TP}}_{\mathcal{PM}}\left(\zeta \right)=\mathcal{SD}(\zeta ,\zeta \wedge {\mathcal{PM}}^{*}\left(\zeta \right)),\forall \zeta \in L^X.$ Then, we have the following:

(1) $(X,{\mathcal{TP}}_{\mathcal{PM}})$ is an L-topological space;

(2) If $\mathcal{PM}$ is stratified, then ${\mathcal{TP}}_{\mathcal{PM}}$ is enriched;

(3) If $\mathcal{PM}$ is Alexandrov, then so is ${\mathcal{TP}}_{\mathcal{PM}}$;

(4) If $\psi :(X,{\mathcal{PM}}_X)\to (Y,{\mathcal{PM}}_Y)$ is an L-primal map, then $\psi :(X,{\mathcal{TP}}_{{\mathcal{PM}}_X})\to (X,{\mathcal{TP}}_{{\mathcal{PM}}_Y})$ is a continuous map.

Proof. 

(1) (T1)

$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left({\top }_X\right)=\mathcal{SD}({\top }_X,{\top }_X\wedge {\mathcal{PM}}^{*}\left({\top }_X\right))=\mathcal{SD}({\top }_X,{\top }_X)=\top ,\hfill \end{array}$

$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left({\perp }_X\right)=\mathcal{SD}({\perp }_X,{\perp }_X\wedge {\mathcal{PM}}^{*}\left({\perp }_X\right))=\mathcal{SD}({\perp }_X,{\perp }_X)=\top .\hfill \end{array}$

(T2)

$$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left(\zeta \right)\wedge {\mathcal{TP}}_{\mathcal{PM}}\left(\eta \right)=\mathcal{SD}(\zeta ,\zeta \wedge {\mathcal{PM}}^{*}\left(\zeta \right))\wedge \mathcal{SD}(\eta ,\eta \wedge {\mathcal{PM}}^{*}\left(\eta \right))\hfill \\ & \le \mathcal{SD}(\zeta \wedge \eta ,\zeta \wedge \eta \wedge {\mathcal{PM}}^{*}\left(\zeta \right)\wedge {\mathcal{PM}}^{*}\left(\eta \right))\hfill \\ & \le \mathcal{SD}(\zeta \wedge \eta ,\zeta \wedge \eta \wedge {\mathcal{PM}}^{*}(\zeta \wedge \eta ))={\mathcal{TP}}_{\mathcal{PM}}(\zeta \wedge \eta ).\hfill \end{array}$$

(T3)

$$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i\right)=\mathcal{SD}({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i,{\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i\wedge {\mathcal{PM}}^{*}\left({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i\right))\hfill \\ & \ge \mathcal{SD}({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i,{\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i\wedge {\mathcal{PM}}^{*}\left({\zeta }_i\right))=\mathcal{SD}({\displaystyle \underset{i\in \Gamma }{\bigvee }}{\zeta }_i,{\displaystyle \underset{i\in \Gamma }{\bigvee }}({\zeta }_i\wedge {\mathcal{PM}}^{*}\left({\zeta }_i\right)))\hfill \\ & \ge {\displaystyle \underset{i\in \Gamma }{\bigwedge }}\mathcal{SD}({\zeta }_i,{\zeta }_i\wedge {\mathcal{PM}}^{*}\left({\zeta }_i\right))={\displaystyle \underset{i\in \Gamma }{\bigwedge }}{\mathcal{TP}}_{\mathcal{PM}}\left(\zeta i\right).\hfill \end{array}$$

(2)

$$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left(\eta \right)=\mathcal{SD}(\eta ,\eta \wedge {\mathcal{PM}}^{*}\left(\eta \right))\hfill \\ & \le \mathcal{SD}(\pi \wedge \eta ,\pi \wedge \eta \wedge \left({\mathcal{PM}}^{*}\left(\eta \right)\right))\le \mathcal{SD}(\pi \wedge \eta ,\pi \wedge \eta \wedge \pi \wedge {\mathcal{PM}}^{*}\left(\eta \right))\hfill \\ & =\mathcal{SD}(\pi \wedge \eta ,\pi \wedge \eta \wedge {\mathcal{PM}}^{*}(\pi \wedge \eta ))={\mathcal{TP}}_{\mathcal{PM}}(\pi \wedge \eta ).\hfill \end{array}$$

(3)

$$\begin{array}{cc}& {\mathcal{TP}}_{\mathcal{PM}}\left({\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i\right)=\mathcal{SD}({\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i,{\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i\wedge {\mathcal{PM}}^{*}\left({\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i\right))\hfill \\ & =\mathcal{SD}({\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i,{\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\eta }_i\wedge {\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\mathcal{PM}}^{*}\left({\eta }_i\right))\ge {\displaystyle \underset{i\in \Lambda }{\bigwedge }}\mathcal{SD}({\eta }_i,{\eta }_i\wedge {\mathcal{PM}}^{*}\left({\eta }_i\right))=\hfill & {\displaystyle \underset{i\in \Lambda }{\bigwedge }}{\mathcal{TP}}_{\mathcal{PM}}\left({\eta }_i\right).\end{array}$$

(4) $\forall \eta \in L^Y,$ we get

$$\begin{array}{cc}& {\mathcal{TP}}_{{\mathcal{PM}}_X}\left({\psi }^{\leftarrow }\left(\eta \right)\right)=\mathcal{SD}({\psi }^{\leftarrow }\left(\eta \right),{\psi }^{\leftarrow }\left(\eta \right)\wedge {\mathcal{PM}}_X^{*}\left({\psi }^{\leftarrow }\left(\eta \right)\right)\hfill \\ & \ge \mathcal{SD}({\psi }^{\leftarrow }\left(\eta \right),{\psi }^{\leftarrow }\left(\eta \right)\wedge {\mathcal{PM}}_Y^{*}\left(\eta \right))\hfill \\ & \ge \mathcal{SD}(\eta ,\eta \wedge {\mathcal{PM}}_Y^{*}\left(\eta \right))\hfill \\ & ={\mathcal{TP}}_{{\mathcal{PM}}_Y}\left(\eta \right).\hfill \end{array}$$

This completes the proof. □

Thus, we obtain a concrete functor $\Lambda :\mathbf{L}\mathbf{\text{-}}\mathbf{PRM}\to \mathbf{L}\mathbf{\text{-}}\mathbf{TOP}$ given by

$$\Lambda (X,\mathcal{PM})=(X,{\mathcal{TP}}_{\mathcal{PM}}).$$

The following theorem suggests a concrete functor from the category of L-topological spaces to the category of stratified L-primal spaces.

Theorem 10. 

Let $(X,\mathcal{TP})$ be an L-topological space and ${\mathcal{PM}}_{\mathcal{TP}}:L^X\to L$ be a mapping defined by

$${\mathcal{PM}}_{\mathcal{TP}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\eta \right)\wedge \mathcal{SD}(\eta ,\zeta ))\to {\eta }^{*}\right)\left(z\right),\forall \zeta \in L^X.$$

Then, the following holds:

(1) $(X,{\mathcal{PM}}_{\mathcal{TP}})$ is a stratified L-primal space;

(2) If $\psi :(X,{\mathcal{TP}}_X)\to (Y,{\mathcal{TP}}_Y)$ is a continuous map, then $\psi :(X,{\mathcal{PM}}_{{\mathcal{TP}}_X})\to (X,{\mathcal{PM}}_{{\mathcal{TP}}_Y})$ is an L-primal map.

Proof. 

(1) (P1)

$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{TP}}\left({\top }_X\right)=\left(\mathcal{TP}\left({\top }_X\right)\wedge \mathcal{SD}({\top }_X,{\top }_X)\to {\perp }_X\right)\left(z\right)=\top \wedge \top \to \perp =\perp ,\hfill \end{array}$

$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{TP}}\left({\perp }_X\right)=\left(\mathcal{TP}\left({\perp }_X\right)\wedge \mathcal{SD}({\perp }_X,{\perp }_X)\to {\top }_X\right)\left(z\right)=\top \wedge \top \to \top =\top .\hfill \end{array}$

(P2)

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{TP}}\left(\zeta \right)\to {\mathcal{PM}}_{\mathcal{TP}}\left(\eta \right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\right)\left(z\right)\to {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\rho \right)\wedge \mathcal{SD}(\rho ,\eta ))\to {\rho }^{*}\right)\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\left(z\right)\right)\to {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\rho \right)\wedge \mathcal{SD}(\rho ,\eta ))\to {\rho }^{*}\left(z\right)\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}\left[{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\left(z\right)\right)\to {\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left((\mathcal{TP}\left(\rho \right)\wedge \mathcal{SD}(\rho ,\eta ))\to {\rho }^{*}\left(z\right)\right)\right]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\left(z\right)\right)\to \left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\eta ))\to {\varrho }^{*}\left(z\right)\right)\right]\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left(\mathcal{SD}(\varrho ,\eta )\to \mathcal{SD}(\varrho ,\zeta )\right)\ge \mathcal{SD}(\eta ,\zeta ).\hfill \end{array}$$

(P3)

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{TP}}\left(\zeta \right)\vee {\mathcal{PM}}_{\mathcal{TP}}\left(\eta \right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[(\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\right]\left(z\right)\vee {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\rho \in L^X}{\bigwedge }}\left[(\mathcal{TP}\left(\rho \right)\wedge \mathcal{SD}(\rho ,\eta ))\to {\rho }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[\left((\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta ))\to {\varrho }^{*}\right)\vee \left((\mathcal{TP}\left(\rho \right)\wedge \mathcal{SD}(\rho ,\eta ))\to {\rho }^{*}\right)\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[(\mathcal{TP}\left(\varrho \right)\wedge \mathcal{TP}\left(\rho \right))\wedge (\mathcal{SD}(\varrho ,\zeta )\wedge \mathcal{SD}(\rho ,\eta ))\to {\varrho }^{*}\vee {\rho }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[(\mathcal{TP}(\varrho \wedge \rho )\wedge \mathcal{SD}(\varrho \wedge \rho ,\zeta \wedge \eta ))\to {\varrho }^{*}\vee {\rho }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho ,\rho \in L^X}{\bigwedge }}\left[(\mathcal{TP}(\varrho \wedge \rho )\wedge \mathcal{SD}(\varrho \wedge \rho ,\zeta \wedge \eta ))\to {(\varrho \wedge \rho )}^{*}\right]\left(z\right)\hfill \\ & ={\mathcal{PM}}_{\mathcal{TP}}(\zeta \wedge \eta ).\hfill \end{array}$$

Hence, ${\mathcal{PM}}_{\mathcal{TP}}(\zeta \wedge \eta )\le {\mathcal{PM}}_{\mathcal{TP}}\left(\zeta \right)\vee {\mathcal{PM}}_{\mathcal{TP}}\left(\eta \right).$

(2) $\forall \eta \in L^Y,$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{TP}}_Y}^{*}\left(\eta \right)={\displaystyle \underset{\varrho \in L^Y}{\bigvee }}\left({\mathcal{TP}}_Y\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\eta )\wedge {\displaystyle \underset{y\in Y}{\bigwedge }}\varrho \left(y\right)\right)\hfill \\ & \le {\displaystyle \underset{\varrho \in L^Y}{\bigvee }}\left({\mathcal{TP}}_X\left({\psi }^{\leftarrow }\left(\varrho \right)\right)\wedge \mathcal{SD}({\psi }^{\leftarrow }\left(\varrho \right),{\phi }^{\leftarrow }\left(\eta \right))\wedge {\displaystyle \underset{\psi \left(y\right)\in Y}{\bigwedge }}\varrho \left(\psi \left(y\right)\right)\right)\hfill \\ & \le {\displaystyle \underset{\varrho \in L^X}{\bigvee }}\left({\mathcal{TP}}_X\left({\psi }^{\leftarrow }\left(\eta \right)\right)\wedge \mathcal{SD}({\psi }^{\leftarrow }\left(\varrho \right),{\psi }^{\leftarrow }\left(\eta \right))\wedge {\displaystyle \underset{z\in X}{\bigwedge }}{\psi }^{\leftarrow }\left(\varrho \right)\left(z\right)\right)\hfill \\ & ={\mathcal{PM}}_{{\mathcal{TP}}_X}^{*}\left({\psi }^{\leftarrow }\left(\eta \right)\right).\hfill \end{array}$$

Hence, ${\mathcal{PM}}_{{\mathcal{TP}}_X}\left({\psi }^{\leftarrow }\left(\eta \right)\right)\le {\mathcal{PM}}_{{\mathcal{TP}}_Y}\left(\eta \right).$ □

Thus, we obtain a concrete functor ${\rm Y}:\mathbf{L}\mathbf{\text{-}}\mathbf{TOP}\to \mathbf{SL}\mathbf{\text{-}}\mathbf{PRM}$ given by:

$${\rm Y}(X,\mathcal{TP})=(X,{\mathcal{PM}}_{\mathcal{TP}}).$$

Proposition 3. 

If $(X,\mathcal{PM})$ is a stratified L-primal space, then ${\mathcal{PM}}_{{\mathcal{TP}}_{\mathcal{PM}}}\ge \mathcal{PM}.$

Proof. 

$\forall \eta \in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{PM}}_{{\mathcal{TP}}_{\mathcal{PM}}}\left(\eta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left({\mathcal{TP}}_{\mathcal{PM}}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\eta )\right)\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left(\mathcal{SD}(\varrho ,\varrho \wedge {\mathcal{PM}}^{*}\left(\varrho \right))\wedge \mathcal{SD}(\varrho ,\eta )\right)\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left(\mathcal{SD}(\varrho ,{\mathcal{PM}}^{*}\left(\varrho \right))\wedge \mathcal{SD}(\varrho ,\eta )\right)\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left(\mathcal{SD}(\varrho ,{\mathcal{PM}}^{*}\left(\varrho \right))\wedge \mathcal{SD}({\mathcal{PM}}^{*}\left(\varrho \right),{\mathcal{PM}}^{*}\left(\eta \right))\right)\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\left(\mathcal{SD}(\varrho ,{\mathcal{PM}}^{*}\left(\eta \right))\right)\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\varrho \in L^X}{\bigwedge }}\left[\mathcal{SD}(\mathcal{PM}\left(\eta \right),{\varrho }^{*})\to {\varrho }^{*}\right]\left(z\right)\hfill \\ & \ge \mathcal{PM}\left(\eta \right).\hfill \end{array}$$

Hence, ${\mathcal{PM}}_{{\mathcal{TP}}_{\mathcal{PM}}}\ge \mathcal{PM}.$ □

Proposition 4. 

If $(X,\mathcal{TP})$ is an L-topological space, then ${\mathcal{TP}}_{{\mathcal{PM}}_{\mathcal{TP}}}\ge \mathcal{TP}.$

Proof. 

$\forall \zeta \in L^X,$ we get

$$\begin{array}{cc}& {\mathcal{TP}}_{{\mathcal{PM}}_{\mathcal{TP}}}\left(\zeta \right)=\mathcal{SD}(\zeta ,\zeta \wedge {\mathcal{PM}}_{\mathcal{TP}}^{*}\left(\zeta \right))\hfill \\ & =\mathcal{SD}\left(\zeta ,\zeta \wedge {\displaystyle \underset{\varrho \in L^X}{\bigvee }}\left(\mathcal{TP}\left(\varrho \right)\wedge \mathcal{SD}(\varrho ,\zeta )\wedge {\displaystyle \underset{z\in X}{\bigwedge }}\varrho \left(z\right)\right)\right)\hfill \\ & \ge \mathcal{SD}\left(\zeta ,\zeta \wedge {\displaystyle \underset{z\in X}{\bigwedge }}(\mathcal{TP}\left(\zeta \right)\wedge \zeta \left(z\right))\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigwedge }}\left(\zeta \left(z\right)\to {\displaystyle \underset{z\in X}{\bigwedge }}(\mathcal{TP}\left(\zeta \right)\wedge \zeta \left(z\right))\wedge \zeta \left(z\right)\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigwedge }}\left(\zeta \left(z\right)\to {\displaystyle \underset{z\in X}{\bigwedge }}(\mathcal{TP}\left(\zeta \right)\wedge \zeta \left(z\right))\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigwedge }}\left(\zeta \left(z\right)\to \left(\mathcal{TP}\right(\zeta )\wedge \zeta (z\left)\right)\right)\ge \mathcal{TP}\left(\zeta \right).\hfill \end{array}$$

Therefore, ${\mathcal{TP}}_{{\mathcal{PM}}_{\mathcal{TP}}}\ge \mathcal{TP}.$ □

Corollary 2. 

$(\Lambda ,{\rm Y})$ is a Galois correspondence.

4.3. Applications

In this subsection, we present real-world applications that illustrate practical systems and demonstrate the effectiveness of the proposed approach.

(1) Information system

As an information system and an extension of Pawlak’s rough set [31] and fuzzy rough sets [32], we provide the following examples of L-primal, L-pre-proximity, and L-topology. Let X be a non-empty set and let $\theta :X\times X\to L$ be a reflexive L-relation on X, that is, $\theta (z,z)=\top$ for all $z\in X$.

(i) Define a mapping ${\mathcal{PM}}_{\theta }^z:L^X\to L$ as follows:

$${\mathcal{PM}}_{\theta }^z\left(\eta \right)={\displaystyle \underset{y\in X}{\bigvee }}\left(\theta (z,y)\wedge {\eta }^{*}\left(y\right)\right)=\overline{\theta }\left({\eta }^{*}\right)\left(z\right),\forall z\in X,\eta \in L^X.$$

We can easily show that ${\mathcal{PM}}_{\theta }^z$ is an Alexandrov L-primal on X.

By Theorem 6, we obtain that

$$\begin{array}{cc}& {\mathcal{PRO}}_{{\mathcal{PM}}_{\theta }^z}(\zeta ,\eta )={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left(\eta \left(z\right)\vee {\mathcal{PM}}_{\theta }^z\left({\eta }^{*}\right)\right)\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \left(\eta \left(z\right)\vee \overline{\theta }\left(\eta \right)\left(z\right)\right)\right)={\displaystyle \underset{z\in X}{\bigvee }}\left(\zeta \left(z\right)\wedge \overline{\theta }\left(\eta \right)\left(z\right)\right)\hfill \\ & ={\displaystyle \underset{z,y\in X}{\bigvee }}\left(\theta (z,y)\wedge \left(\zeta \left(z\right)\wedge \eta \left(y\right)\right)\right).\hfill \end{array}$$

By Theorem 9, we obtain that

$$\begin{array}{cc}& {\mathcal{TP}}_{{\mathcal{PM}}_{\theta }^z}\left(\eta \right)=\mathcal{SD}\left(\eta ,\eta \wedge {\left({\mathcal{PM}}_{\theta }^z\right)}^{*}\left(\eta \right)\right)=\mathcal{SD}\left(\eta ,\eta \wedge {\overline{\theta }}^{*}\left({\eta }^{*}\right)\left(z\right)\right)\hfill \\ & ={\displaystyle \underset{z\in X}{\bigwedge }}\left(\eta \left(z\right)\to \left(\eta \left(z\right)\wedge {\displaystyle \underline{\theta }}\left(\eta \right)\left(z\right)\right)\right)\hfill \\ & ={\displaystyle \underset{z,y\in X}{\bigwedge }}\left(\theta (z,y)\to \left(\eta \right(z)\to \eta (y\left)\right)\right).\hfill \end{array}$$

By Theorem 10, we obtain that

$$\begin{array}{cc}& {\mathcal{PM}}_{\mathcal{TP}}\left(\zeta \right)={\displaystyle \underset{z\in X}{\bigvee }}{\displaystyle \underset{\eta \in L^X}{\bigwedge }}\left(\mathcal{TP}\left(\eta \right)\to \left(\mathcal{SD}(\eta ,\zeta )\to {\eta }^{*}\right)\right)\left(z\right)\hfill \\ & \ge {\displaystyle \underset{z\in X}{\bigvee }}\left({\displaystyle \underset{\eta \in L^X}{\bigwedge }}\mathcal{TP}\left(\eta \right)\to {\zeta }^{*}\left(z\right)\right)\ge {\displaystyle \underset{z\in X}{\bigvee }}{\zeta }^{*}\left(z\right).\hfill \end{array}$$

By Theorem 4, we obtain that

$$\begin{array}{cc}& {\mathcal{N}}_{{\mathcal{PM}}_{\theta }^x}^x\left(\zeta \right)=\zeta \left(x\right)\wedge {\displaystyle \underline{\theta }}\left(\zeta \right)\left(x\right)={\displaystyle \underline{\theta }}\left(\zeta \right)\left(x\right).\hfill \end{array}$$

(ii) Let $([0,1],\wedge ,\to {,}^{*},0,1)$ be a frame given by

$$\begin{array}{c}x\wedge y=max\{0,x+y-1\},x\to y=min\{1-x+y,1\},x^{*}=1-x.\end{array}$$

Let $X=\{x_1,x_2,x_3\}$ be a set of three houses, and let $e,b:X\to [0,1]$ be two graded functions defined by

$$e\left(x_1\right)=1,e\left(x_2\right)=0.8,e\left(x_3\right)=0.5$$

and

$$b\left(x_1\right)=0.8,b\left(x_2\right)=0.9,b\left(x_3\right)=0.2,$$

Here, e and b represent the degrees to which the houses are expensive and beautiful, respectively. Then, $(X,AT=\{e,b\},{\mu }_d)$ is an information system, where $AT$ is a set of attributes with ${\mu }_d:X\to [0,1]$ for $d\in AT$, and is given by

$$\begin{array}{ccc}& e& b\\ x_1& 1& 0.8\\ x_2& 0.8& 0.9\\ x_3& 0.5& 0.2\end{array}$$

Define a reflexive relation $\theta \in {[0,1]}^{X\times X}$ as

$$\theta (x_i,x_j)=(e\left(x_i\right)\to e\left(x_j\right))\wedge (b\left(x_i\right)\to b\left(x_j\right)).$$

Then, $\theta$ is a $[0,1]$-preorder given by

$$\theta =\left(\begin{array}{ccc}1& 0.8& 0.4\\ 0.9& 1& 0.3\\ 1& 1& 1\end{array}\right).$$

The $[0,1]$-preorder $\theta (x_i,x_j)$ can be interpreted as how much better $x_j$ is than $x_i.$ We obtain that:

$${\mathcal{PM}}_{\theta }^{x_i}\left(\zeta \right)={\displaystyle \underset{x_j\in X}{\bigvee }}\theta (x_i,x_j)\wedge {\zeta }^{*}\left(x_j\right).$$

From Theorem 6, we obtain that ${\mathcal{PRO}}_{{\mathcal{PM}}_{\theta }^{x_i}}$ is given by

$$\begin{array}{cc}& {\mathcal{PRO}}_{{\mathcal{PM}}_{\theta }^{x_i}}(\zeta ,\eta )={\displaystyle \underset{x_i,x_j\in X}{\bigvee }}\theta (x_i,x_j)\wedge \zeta \left(x_i\right)\wedge \eta \left(x_j\right).\hfill \end{array}$$

From Theorem 9, we obtain that ${\mathcal{TP}}_{{\mathcal{PM}}_{\theta }^{x_i}}$ is given by

$$\begin{array}{cc}& {\mathcal{TP}}_{{\mathcal{PM}}_{\theta }^{x_i}}\left(\zeta \right)={\displaystyle \underset{x_i,x_j\in X}{\bigwedge }}\theta (x_i,x_j)\to (\zeta \left(x_i\right)\to \zeta \left(x_j\right)).\hfill \end{array}$$

(2) Medicine

In medical practice, it is common to combine multiple types of drugs to treat a disease. Let $X=\{x_i\mid i=1,\dots ,6\}$ be the universe of six distinct types of medicines, and let $Y=\{y_j\mid j=1,\dots ,6\}$ denotes the set of six symptoms associated with a disease (e.g., fever, cough, dizziness, etc.). For each pair $(x_i,y_j)$, let ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_i}$ represent the efficacy value of medicine $x_i$ in treating symptom $y_j$. Moreover, let $L=([0,1],\wedge ,\to {,}^{*},0,1)$ be a frame with implication $x\wedge y=min\{x,y\}$, $x\to y=max\{1-x,y\}$ and $x^{*}=1-x$.

(i) Let $\theta \in {[0,1]}^{X\times Y}$ be a $[0,1]$-relation, where $\theta (x_i,y_j)$ represents the degree to which medicine $x_i$ is effective for symptom $y_j$, as shown in

Table 1. Definition of $[0,1]$-relation $\theta$.

$\mathit{\theta }$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$	${\mathit{y}}_5$	${\mathit{y}}_6$
$x_1$	0.6	0.3	0.2	0.4	0.3	0.2
$x_2$	0.1	0.5	0.4	0.5	0.3	0.3
$x_3$	0.1	0.2	0.5	0.3	0.3	0.1
$x_4$	0.1	0.5	0.4	0.5	0.3	0.2
$x_5$	0.3	0.2	0.2	0.3	0.6	0.1
$x_6$	0.3	0.3	0.2	0.3	0.7	0.7

.

Note that $\theta (x_i,y_j)=1$ means medicine $x_i$ is fully effective for symptom $y_j$ and $\theta (x_i,y_j)=0$ means medicine $x_i$ has no effect on symptom $y_j$. By (1) above, we obtain that ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_i}\left(\zeta \right)={\displaystyle \underline{\theta }}\left(\zeta \right)\left(x_i\right).$

Input: A $[0,1]$-relation $\theta$ and a $[0,1]$-subset $\zeta .$

Output: Degree of the $[0,1]$-neighborhood $\zeta$ of $x_i$.

Step1: Calculating ${\displaystyle \underline{\theta }}\left(\zeta \right)\left(x_i\right)$, for all $x_i\in X$.

Step2: ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_i}\left(\zeta \right)={\displaystyle \underline{\theta }}\left(\zeta \right)\left(x_i\right)$.

Given that $\zeta =(0.6,0.4,0.3,0.5,0.7,0.4)$, which denotes the ability of all medicines in X to cure the disease (based on numerous experiments), then due to the inaccuracy of $\zeta$, we can consider its approximate evaluation using a $[0,1]$-neighborhood.

Firstly, calculating ${\displaystyle \underline{\theta }}\left(\zeta \right)\left(x_i\right)$, for all $x_i\in X$, we get ${\displaystyle \underline{\theta }}\left(\zeta \right)=(0.6,0.5,0.5,0.5,0.7,0.4)$.

Secondly, ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_i}\left(\zeta \right)={\displaystyle \underline{\theta }}\left(\zeta \right)\left(x_i\right)$, for all $x_i\in X$. Thus, we obtain the following values:

${\mathcal{N}}_{\theta }^{x_1}\left(\zeta \right)=0.6$, ${\mathcal{N}}_{\theta }^{x_2}\left(\zeta \right))={\mathcal{N}}_{\theta }^{x_3}\left(\zeta \right)={\mathcal{N}}_{\theta }^{x_4}\left(\zeta \right)=0.5$, ${\mathcal{N}}_{\theta }^{x_5}\left(\zeta \right)=0.7$ and ${\mathcal{N}}_{\theta }^{x_6}\left(\zeta \right)=0.4$.

Since $x_1$ and $x_5$ have the highest neighborhood values, the fuzzy set $\zeta$ can be interpreted as a strong neighborhood for these medicines. This indicates that $x_1$ and $x_5$ are considered the most effective or important medicines for treating the disease. Moreover, since $x_2$, $x_3$, and $x_4$ share moderately high values, they are considered moderately effective medicines. Finally, as $x_6$ has the lowest neighborhood value, the fuzzy set $\zeta$ can be interpreted as a weak neighborhood for this medicine, indicating that $x_6$ appears to be the least significant among the considered medicines for disease treatment.

(ii) Let $X=\{x_1,x_2,x_2\}$ and ${\theta }_X^{\{y_2,y_3\}}\in {[0,1]}^{X\times Y}$ be a $[0,1]$-relation on X defined by

$$\begin{array}{cc}{\theta }_X^{\{y_2,y_3\}}(x_i,x_j)& ={\displaystyle \underset{y\in \{y_2,y_3\}}{\bigwedge }}\left(\theta (x_i,y)\to \theta (x_j,y)\right).\hfill \end{array}$$

Then, ${\theta }_X^{\{y_2,y_3\}}$ is a pre-order relation, given by

$${\theta }_X^{\{y_2,y_3\}}=\left(\begin{array}{ccc}1& 0.5& 0.5\\ 0.7& 1& 0.5\\ 0.7& 0.5& 1\end{array}\right).$$

Given that $\zeta =(0.6,0.4,0.3)$, which denotes the ability of all medicines in X to cure the disease, then the efficacy values ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_1}\left(\zeta \right)=0.3,$${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_2}\left(\zeta \right)=0.4$, and ${\mathcal{N}}_{{\mathcal{PM}}_{\theta }}^{x_3}\left(\zeta \right)=0.3$.

Since $x_2$ has the highest neighborhood value, the fuzzy set $\zeta$ can be regarded as a strong neighborhood for this medicine, indicating that $x_2$ is the most effective or important option for treating the disease. In contrast, $x_1$ and $x_3$ having the lowest neighborhood values, form a weak neighborhood in $\zeta$, suggesting they are the least significant among the considered medicines.

5. Conclusions

The theory of L-structures, such as L-primals, L-pre-proximities, and L-topologies, form a rich categorical framework generalizing classical topological and proximity theories into the fuzzy- and lattice-enriched domain. These structures offer a foundation for modeling uncertainty, graded membership, and nearness in fields ranging from data science to logic. Various topological spaces play a fundamental role in fuzzy set theory and its applications. To analyze the connections among such structures, categorical methods offer a powerful and unifying framework, utilizing morphisms between fuzzy sets and functors between fuzzy categories. An early contribution in this direction was made by Goguen [1], who introduced the concept of the image of a fuzzy set under a fuzzy relation, which can be interpreted categorically as a functor between appropriate fuzzy relational categories. These L-structures have developed rapidly and have become a significant area of research. Numerous scholars have explored L-structures from various theoretical perspectives.

In this research paper, we introduced and thoroughly investigated the concept of L-primals, indicating its basic properties within the context of L-fuzzy sets on complete lattices. A primary objective was to explore the interrelations among L-primal spaces, L-pre-proximity spaces, and L-topological spaces. We achieved this by establishing new L-topological and L-pre-proximity spaces that were directly induced by L-primal spaces.

Essentially, our study delved into the categorical aspects of these interrelationships. We demonstrated the existence of a concrete functor that mapped between categories of L-primal spaces and L-pre-proximity spaces, and similarly between categories of L-pre-proximity spaces and stratified L-primal spaces. These findings are significant as they indicate the presence of a Galois correspondence between these respective categories, presenting a deep structural connection. Furthermore, we showed the existence of a concrete functor between the category of stratified L-primal spaces and the category of L-topological spaces, and vice versa, thereby confirming another Galois correspondence. The theoretical developments were supported by several illustrative examples, including a practical application within an information system and in medicine, which underscored the utility and relevance of the developed theory.

This work lays a foundation for future research, opening several avenues for further investigation. These include

• Exploring the category of L-primal-proximities based on the L-primal notion and its relationship with the category of L-topological structures.
• Investigating the connections between the category of L-topogenous structures and the category of L-primal spaces.
• Developing L-neighborhoods induced L-pre-proximities and exploring their potential applications.
• Investigating the connections between the category of L-primal approximation spaces by L-neighborhoods with possible applications.

These future directions aim to further enrich the theoretical framework and expand the practical applicability of L-primals and their related topological structures in addressing some problems in fuzzy systems and decision-making.