p-Regularity and p-Regular Modification in ⊤-Convergence Spaces

Abstract

Fuzzy convergence spaces are extensions of convergence spaces. ⊤-convergence spaces are important fuzzy convergence spaces. In this paper, p-regularity (a relative regularity) in ⊤-convergence spaces is discussed by two equivalent approaches. In addition, lower and upper p-regular modifications in ⊤-convergence spaces are further investigated and studied. Particularly, it is shown that lower (resp., upper) p-regular modification and final (resp., initial) structures have good compatibility.

1. Introduction

Convergence spaces [1] are generalizations of topological spaces. Regularity is an important property in convergence spaces. In general, there are two equivalent approaches to characterize regularity. One approach is stated through a diagonal condition of filters [2,3], the other approach is represented through a closure condition of filters [4]. In [5,6], for a pair of convergence structures $p,q$ on the same underlying set, Wilde-Kent-Richardson considered a relative regularity (called p-regularity) both from two equivalent approaches. When $p=q$, p-regularity is nothing but regularity. Wilde-Kent [6] further presented a theory of lower and upper p-regular modifications in convergence spaces. Said precisely, for convergence structures $p,q$ on a set X, the lower (resp., upper) p-regular modification of q is defined as the finest (resp., coarsest) p-regular convergence structure coarser (resp., finer) than q.

Fuzzy convergence spaces are natural extensions of convergence spaces. Quite recently, two types of fuzzy convergence spaces received wide attention: (1) stratified L-generalized convergence spaces (resp., stratified L-convergence spaces) initiated by Jäger [7] (resp., Flores [8]) and then developed by many scholars [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]; and (2) ⊤-convergence spaces introduced by Fang [31] and then discussed by many researchers [32,33,34,35,36]. Regularity in stratified L-generalized convergence spaces (resp., stratified L-convergence spaces) was studied by Jäger [37] (resp., Boustique-Richardson [38,39]), p-regularity and p-regular modifications in stratified L-generalized convergence spaces and that in stratified L-convergence spaces were discussed by Li [40,41]. Regularity in ⊤-convergence spaces by different diagonal conditions of ⊤-filters were researched by Fang [31] and Li [42], respectively. Regularity in ⊤-convergence spaces by closure condition of ⊤-filters were studied by Reid and Richardson [36]. In this paper, we shall discuss p-regularity and p-regular modifications in ⊤-convergence spaces.

The contents are arranged as follows. Section 2 recalls some notions and notations for later use. Section 3 presents p-regularity in ⊤-convergence spaces by a diagonal condition of ⊤-filters and a closure condition of ⊤-filters, respectively. Section 4 mainly discusses p-regular modifications in ⊤-convergence spaces. The lower and upper p-regular modifications in ⊤-convergence spaces are investigated and researched. Especially, it is shown that lower (resp., upper) p-regular modification and final (resp., initial) structures have good compatibility.

2. Preliminaries

In this paper, if not otherwise stated, $L=(L,\le )$ is always a complete lattice with a top element ⊤ and a bottom element ⊥, which satisfies the distributive law $\alpha \wedge \left({\bigvee }_{i\in I}{\beta }_i\right)={\bigvee }_{i\in I}(\alpha \wedge {\beta }_i)$. A lattice with these conditions is called a complete Heyting algebra. The operation $\to :L\times L⟶L$ given by

$$\alpha \to \beta =\bigvee \{\gamma \in L:\alpha \wedge \gamma \le \beta \}.$$

is called the residuation with respect to ∧. We collect here some basic properties of the binary operations ∧ and → [43].

(1)

$a\to b=\top \iff a\le b$;

(2)

$a\wedge b\le c\iff b\le a\to c$;

(3)

$a\wedge (a\to b)\le b$;

(4)

$a\to (b\to c)=(a\wedge b)\to c$;

(5)

$({\bigvee }_{j\in J}{a}_j)\to b={\bigwedge }_{j\in J}(a_j\to b)$;

(6)

$a\to ({\bigwedge }_{j\in J}{b}_j)={\bigwedge }_{j\in J}(a\to b_j)$.

A function $\mu :X\to L$ is said to be an L-fuzzy set in X, and all L-fuzzy sets in X are denoted as $L^X$. The operators $\vee ,\wedge ,\to$ on L can be translated onto $L^X$ pointwisely. Precisely, for any $\mu ,\nu ,{\mu }_t(t\in T)\in L^X$,

$$(\underset{t\in T}{\bigvee }{\mu }_t)\left(x\right)=\underset{t\in T}{\bigvee }{\mu }_t\left(x\right),(\underset{t\in T}{\bigwedge }{\mu }_t)\left(x\right)=\underset{t\in T}{\bigwedge }{\mu }_t\left(x\right),(\mu \to \nu )\left(x\right)=\mu \left(x\right)\to \nu \left(x\right).$$

Let $f:X⟶Y$ be a function. We define $f^{\to }:L^X⟶L^Y$ by $f^{\to }\left(\mu \right)\left(y\right)={\bigvee }_{f\left(x\right)=y}\mu \left(x\right)$ for $\mu \in L^X$ and $y\in Y$, and define $f^{\leftarrow }:L^Y⟶L^X$ by $f^{\leftarrow }\left(\nu \right)\left(x\right)=\nu \left(f\left(x\right)\right)$ for $\nu \in L^Y$ and $x\in X$ [43].

Let $\mu ,\nu$ be L-fuzzy sets in X. The subsethood degree of $\mu ,\nu$, denoted as $S_X(\mu ,\nu )$, is defined by $S_X(\mu ,\nu )=\underset{x\in X}{\wedge }(\mu \left(x\right)\to \nu \left(x\right))$ [44,45,46].

Lemma 1.

[31,42,47] Let $f:X⟶Y$ be a function and ${\mu }_1,{\mu }_2\in L^X$, ${\lambda }_1,{\lambda }_2\in L^Y$. Then

(1) 

$S_X({\mu }_1,{\mu }_2)\le S_Y(f^{\to }\left({\mu }_1\right)$, $f^{\to }\left({\mu }_2\right)),$

(2) 

$S_Y({\lambda }_1,{\lambda }_2)\le S_X(f^{\leftarrow }\left({\lambda }_1\right),f^{\leftarrow }\left({\lambda }_2\right))$,

(3) 

$S_Y(f^{\to }\left({\mu }_1\right),{\lambda }_1)=S_X({\mu }_1,f^{\leftarrow }\left({\lambda }_1\right))$.

⊤-Filters and ⊤-Convergence Spaces

Definition 1.

[43,48] A nonempty subset $\mathbb{F}\subseteq L^X$ is said to be a ⊤-filter on the set X if it satisfies the following three conditions:

(TF1) 

$\forall \lambda \in \mathbb{F},{\displaystyle \underset{x\in X}{\bigvee }}\lambda \left(x\right)=\top$,

(TF2) 

$\forall \lambda ,\mu \in \mathbb{F}$, $\lambda \wedge \mu \in \mathbb{F}$,

(TF3) 

if ${\displaystyle \underset{\mu \in \mathbb{F}}{\bigvee }}{S}_X(\mu ,\lambda )=\top$, then $\lambda \in \mathbb{F}$.

The set of all ⊤-filters on X is denoted by ${\mathbb{F}}_L^{\top }\left(X\right)$.

Definition 2.

[43] A nonempty subset $\mathbb{B}\subseteq L^X$ is referred to be a ⊤-filter base on the set X if it holds that:

(TB1) 

$\forall \lambda \in \mathbb{B}$, ${\displaystyle \underset{x\in X}{\bigvee }}\lambda \left(x\right)=\top$,

(TB2) 

if $\lambda ,\mu \in \mathbb{B}$, then ${\displaystyle \underset{\nu \in \mathbb{B}}{\bigvee }}{S}_X(\nu ,\lambda \wedge \mu )=\top$.

Each ⊤-filter base generates a ⊤-filter ${\mathbb{F}}_{\mathbb{B}}$ by

$${\mathbb{F}}_{\mathbb{B}}:=\{\lambda \in L^X|\underset{\mu \in \mathbb{B}}{\bigvee }{S}_X(\mu ,\lambda )=\top \}.$$

Example 1.

[31,43] Let $f:X⟶Y$ be a function.

(1) 

For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, the family $\left\{f^{\to }\left(\lambda \right)\right|\lambda \in \mathbb{F}\}$ forms a ⊤-filter base on Y. The generated ⊤-filter is denoted as $f^{\Rightarrow }\left(\mathbb{F}\right)$, called the image of $\mathbb{F}$ under f. It is known that $\mu \in f^{\Rightarrow }\left(\mathbb{F}\right)⟺f^{\leftarrow }\left(\mu \right)\in \mathbb{F}$.

(2) 

For any $\mathbb{G}\in {\mathbb{F}}_L^{\top }\left(Y\right)$, the family $\left\{f^{\leftarrow }\left(\mu \right)\right|\mu \in \mathbb{G}\}$ forms a ⊤-filter base on Y iff ${\displaystyle \underset{y\in f\left(X\right)}{\bigvee }}\mu \left(y\right)=\top$ holds for all $\mu \in \mathbb{G}$. The generated ⊤-filter (if exists) is denoted as $f^{\Leftarrow }\left(\mathbb{G}\right)$, called the inverse image of $\mathbb{G}$ under f. It is known that $\mathbb{G}\subseteq f^{\Rightarrow }\left(f^{\Leftarrow }\left(\mathbb{G}\right)\right)$ holds whenever $f^{\Leftarrow }\left(\mathbb{G}\right)$ exists. Furthermore, $f^{\Leftarrow }\left(\mathbb{G}\right)$ always exists and $\mathbb{G}=f^{\Rightarrow }\left(f^{\Leftarrow }\left(\mathbb{G}\right)\right)$ whenever f is surjective.

(3) 

For any $x\in X$, the family ${\left[x\right]}_{\top }=:\{\lambda \in L^X|\lambda \left(x\right)=\top \}$ is a ⊤-filter on X, and $f^{\Rightarrow }\left({\left[x\right]}_{\top }\right)={\left[f\left(x\right)\right]}_{\top }$.

Lemma 2.

Let $f:X⟶Y$ be a function.

(1) 

If $\mathbb{B}$ is a ⊤-filter base of $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, then $\left\{f^{\to }\left(\lambda \right)\right|\lambda \in \mathbb{B}\}$ is a ⊤-filter base of $f^{\Rightarrow }\left(\mathbb{F}\right)$, see Example 2.9 (1) in [31].

(2) 

If $\mathbb{B}$ is a ⊤-filter base of $\mathbb{G}\in {\mathbb{F}}_L^{\top }\left(Y\right)$ and $f^{\Leftarrow }\left(\mathbb{G}\right)$ exists, then $\left\{f^{\leftarrow }\left(\mu \right)\right|\mu \in \mathbb{B}\}$ is a ⊤-filter base of $f^{\Leftarrow }\left(\mathbb{G}\right)$, see Example 2.9 (2) in [31].

(3) 

Let $\mathbb{F},\mathbb{G}\in {\mathbb{F}}_L^{\top }\left(X\right)$ and $\mathbb{B}$ be a ⊤-filter base of $\mathbb{F}$. Then $\mathbb{B}\subseteq \mathbb{G}$ implies that $\mathbb{F}\subseteq \mathbb{G}$, see Lemma 2.5 (1) in [42].

(4) 

Let $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$ and $\mathbb{B}$ be a ⊤-filter base of $\mathbb{F}$. Then ${\displaystyle \underset{\mu \in \mathbb{B}}{\bigvee }}{S}_X(\mu ,\lambda )={\displaystyle \underset{\mu \in \mathbb{F}}{\bigvee }}{S}_X(\mu ,\lambda )$, see Lemma 3.1 in [36].

For each $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, we define $\mathsf{\Lambda }\left(\mathbb{F}\right):L^X⟶L$ as

$$\forall \lambda \in L^X,\mathsf{\Lambda }\left(\mathbb{F}\right)\left(\lambda \right)=\underset{\mu \in \mathbb{F}}{\bigvee }{S}_X(\mu ,\lambda ),$$

then $\mathsf{\Lambda }\left(\mathbb{F}\right)$ is a tightly stratified L-filter on X [47].

In the following, we recall some notions and notations collected in [29].

Definition 3.

[31] Let X be a nonempty set. Then a function $q:{\mathbb{F}}_L^{\top }\left(X\right)⟶2^X$ is said to be a ⊤-convergence structure on X if it satisfies the following two conditions:

(TC1) 

$\forall x\in X$, ${\left[x\right]}_{\top }\stackrel{q}{⟶}x$;

(TC2) 

if $\mathbb{F}\stackrel{q}{⟶}x$ and $\mathbb{F}\subseteq \mathbb{G}$, then $\mathbb{G}\stackrel{q}{⟶}x$.

where $\mathbb{F}\stackrel{q}{⟶}x$ is short for $x\in q\left(\mathbb{F}\right)$. The pair $(X,q)$ is said to be a ⊤-convergence space.

A function $f:X⟶X^{\prime }$ between ⊤-convergence spaces $(X,q)$, $(X^{\prime },q^{\prime })$ is said to be continuous if $f^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q^{\prime }}{⟶}f\left(x\right)$ for any $\mathbb{F}\stackrel{q}{⟶}x$.

We denote the category consisting of ⊤-convergence spaces and continuous functions as ⊤-CS. It has been known that ⊤-CS is topological over SET [31].

For a source ${(X\stackrel{f_i}{⟶}(X_i,q_i))}_{i\in I}$, the initial structure, q on X is defined by

$\mathbb{F}\stackrel{q}{⟶}x⟺\forall i\in I,f_i^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q_i}{⟶}{f}_i\left(x\right)$ [35,49].

For a sink ${((X_i,q_i)\stackrel{f_i}{⟶}X)}_{i\in I}$, the final structure, q on X is defined as

$$\mathbb{F}\stackrel{q}{\to }x⟺\left\{\begin{array}{cc}\mathbb{F}\supseteq {\left[x\right]}_{\top },\hfill & x\notin {\cup }_{i\in I}{f}_i\left(X_i\right);\hfill \\ \mathbb{F}\supseteq f_i^{\Rightarrow }\left({\mathbb{G}}_i\right),\hfill & \exists i\in I,x_i\in X_i,{\mathbb{G}}_i\in {\mathbb{F}}_L^{\top }\left(X_i\right)s.tf\left(x_i\right)=x,{\mathbb{G}}_i\stackrel{q_i}{\to }{x}_i.\hfill \end{array}\right.$$

When $X={\cup }_{i\in I}{f}_i\left(X_i\right)$, the final structure q can be characterized as

$$\mathbb{F}\stackrel{q}{\to }x⟺\mathbb{F}\supseteq f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\mathrm{for}\mathrm{some}{\mathbb{G}}_i\stackrel{q_i}{\to }{x}_i\mathrm{with}f\left(x_i\right)=x.$$

Let $\top \left(X\right)$ denote the set of all ⊤-convergence structures on a set X. For $p,q\in \top \left(X\right)$, we say that q is finer than p (or p is coarser than q), denoted as $p\le q$ for short, if the identity ${\mathrm{id}}_X:(X,q)⟶(X,p)$ is continuous. It has been known that $(\top (X),\le )$ forms a completed lattice. The discrete (resp., indiscrete) structure $\delta$ (resp., $\iota$) is the top (resp., bottom) element of $\left(T\right(X),\le )$, where $\delta$ is defined as $\mathbb{F}\stackrel{\delta }{⟶}x$ iff $\mathbb{F}\supseteq {\left[x\right]}_{\top }$; and $\iota$ is defined as $\mathbb{F}\stackrel{\iota }{⟶}x$ for all $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, $x\in X$ [42].

Remark 1.

When $L=\{\perp ,\top \}$, ⊤-convergence spaces degenerate into convergence spaces. Therefore, ⊤-convergence spaces are natural generalizations of convergence spaces.

3. p-Regularity in ⊤-Convergence Spaces

In this section, we shall discuss the p-regularity in ⊤-convergence spaces. Two equivalent approaches are considered, one approach using diagonal ⊤-filter and the other approach using closure of ⊤-filter. Moreover, it will be proved that p-regularity is preserved under the initial and final structures in the category ⊤-CS.

At first, we recall the notions of diagonal ⊤-filter and closure of ⊤-filter to define p-regularity.

Let $J,X$ be any sets and $\varphi :J⟶{\mathbb{F}}_L^{\top }\left(X\right)$ be any function. Then we define a function $\widehat{\varphi }:L^X\to L^J$ as

$$\forall \lambda \in L^X,\forall j\in J,\widehat{\varphi }\left(\lambda \right)\left(j\right)=\mathsf{\Lambda }\left(\varphi \left(j\right)\right)\left(\lambda \right)=\underset{\mu \in \varphi \left(j\right)}{\bigvee }{S}_X(\mu ,\lambda ).$$

For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(J\right)$, it is known that the subset of $L^X$ defined by

$$k\varphi \mathbb{F}:=\{\lambda \in L^X|\widehat{\varphi }\left(\lambda \right)\in \mathbb{F}\}$$

forms a ⊤-filter on X, called diagonal ⊤-filter of $\mathbb{F}$ under $\varphi$ [31]. It was shown that $S_X(\lambda ,\mu )\le S_J(\widehat{\varphi }\left(\lambda \right),\widehat{\varphi }\left(\mu \right))$ for any $\lambda ,\mu \in L^X$.

Lemma 3.

Let $f:X⟶Y$ and $\varphi :J⟶{\mathbb{F}}_L^{\top }\left(X\right)$ be functions. Then for any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(J\right)$ we have $f^{\Rightarrow }\left(k\varphi \mathbb{F}\right)=k(f^{\Rightarrow }\circ \varphi )\mathbb{F}$.

Proof. 

$f^{\Rightarrow }\left(k\varphi \mathbb{F}\right)\subseteq k(f^{\Rightarrow }\circ \varphi )\mathbb{F}$. By Lemma 2 (3) we need only check that $f^{\to }\left(\lambda \right)\in k(f^{\Rightarrow }\circ \varphi )\mathbb{F}$ for any $\lambda \in k\varphi \mathbb{F}$. Take $\lambda \in k\varphi \mathbb{F}$ then $\widehat{\varphi }\left(\lambda \right)\in \mathbb{F}$. Please note that $\forall j\in J$,

$$\widehat{\varphi }\left(\lambda \right)\left(j\right)=\underset{\mu \in \varphi \left(j\right)}{\bigvee }{S}_X(\mu ,\lambda )\le \underset{\mu \in \varphi \left(j\right)}{\bigvee }{S}_Y(f^{\to }\left(\mu \right),f^{\to }\left(\lambda \right))\le \underset{\nu \in f^{\Rightarrow }\left(\varphi \left(j\right)\right)}{\bigvee }{S}_Y(\nu ,f^{\to }\left(\lambda \right))=\widehat{f^{\Rightarrow }\circ \varphi }\left(f^{\to }\left(\lambda \right)\right)\left(j\right),$$

i.e., $\widehat{\varphi }\left(\lambda \right)\le \widehat{f^{\Rightarrow }\circ \varphi }\left(f^{\to }\left(\lambda \right)\right)$, and so $\widehat{f^{\Rightarrow }\circ \varphi }\left(f^{\to }\left(\lambda \right)\right)\in \mathbb{F}$, i.e., $f^{\to }\left(\lambda \right)\in k(f^{\Rightarrow }\circ \varphi )\mathbb{F}$.

$k(f^{\Rightarrow }\circ \varphi )\mathbb{F}\subseteq f^{\Rightarrow }\left(k\varphi \mathbb{F}\right)$. For any $\lambda \in k(f^{\Rightarrow }\circ \varphi )\mathbb{F}$ we have $\widehat{f^{\Rightarrow }\circ \varphi }\left(\lambda \right)\in \mathbb{F}$. By Lemma 2 (4), $\forall j\in J$,

$$\widehat{f^{\Rightarrow }\circ \varphi }\left(\lambda \right)\left(j\right)=\underset{\nu \in f^{\Rightarrow }\left(\varphi \left(j\right)\right)}{\bigvee }{S}_Y(\nu ,\lambda )=\underset{\mu \in \varphi \left(j\right)}{\bigvee }{S}_Y(f^{\to }\left(\mu \right),\lambda )\le \underset{\mu \in \varphi \left(j\right)}{\bigvee }{S}_X(\mu ,f^{\leftarrow }\left(\lambda \right))=\widehat{\varphi }\left(f^{\leftarrow }\left(\lambda \right)\right)\left(j\right),$$

i.e., $\widehat{f^{\Rightarrow }\circ \varphi }\left(\lambda \right)\le \widehat{\varphi }\left(f^{\leftarrow }\left(\lambda \right)\right)$, and so $\widehat{\varphi }\left(f^{\leftarrow }\left(\lambda \right)\right)\in \mathbb{F}$, i.e., $f^{\leftarrow }\left(\lambda \right)\in k\varphi \mathbb{F}$ then $\lambda \in f^{\Rightarrow }\left(k\varphi \mathbb{F}\right)$. □

Definition 4.

[36] Let $(X,p)$ be a ⊤-convergence space. For each $\lambda \in L^X$, the L-set ${\overline{\lambda }}_p\in L^X$ defined by

$$\forall x\in X,{\overline{\lambda }}_p\left(x\right)=\underset{\mathbb{F}\stackrel{p}{⟶}x}{\bigvee }\mathsf{\Lambda }\left(\mathbb{F}\right)\left(\lambda \right)=\underset{\mathbb{F}\stackrel{p}{⟶}x}{\bigvee }\underset{\mu \in \mathbb{F}}{\bigvee }{S}_X(\mu ,\lambda )$$

is called closure of λ w.r.t p.

For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, the closure of $\mathbb{F}$ regarding p, denoted as $c{l}_p\left(\mathbb{F}\right)$, is defined to be the ⊤-filter generated by $\left\{{\overline{\lambda }}_p\right|\lambda \in \mathbb{F}\}$ as a ⊤-filter base.

Lemma 4.

[36] Let $(X,p)$ be a ⊤-convergence space. Then for all $\lambda ,\mu \in L^X$ we get

(1) 

$\lambda \le {\overline{\lambda }}_p$;

(2) 

$\lambda \le \mu$ implies ${\overline{\lambda }}_p\le {\overline{\mu }}_p$;

(3) 

$S_X(\lambda ,\mu )\le S_X(\overline{\lambda },\overline{\mu })$.

Let $\mathbb{N}$ be the set of natural numbers containing 0 and let $(X,p)$ be a ⊤-convergence space. For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, we define $c{l}_p^0\left(\mathbb{F}\right)=\mathbb{F}$. Furthermore, for any $n\in \mathbb{N}$, we define the $n+1$ th iteration of the closure ⊤-filter of $\mathbb{F}$ as $c{l}_p^{n+1}\left(\mathbb{F}\right)=c{l}_p\left(c{l}_p^n\left(\mathbb{F}\right)\right)$ if $c{l}_p^n\left(\mathbb{F}\right)$ has been defined.

The next proposition collects the properties of closure of ⊤-filters. We omit the obvious proofs.

Proposition 1.

Let $(X,p)$ be a ⊤-convergence space and $\mathbb{F},\mathbb{G}\in {\mathbb{F}}_L^{\top }\left(X\right)$. Then for any $n\in \mathbb{N}$,

(1) 

$c{l}_p^n\left(\mathbb{F}\right)\subseteq \mathbb{F}$,

(2) 

if $\mathbb{F}\subseteq \mathbb{G}$, then $c{l}_p^n\left(\mathbb{F}\right)\subseteq c{l}_p^n\left(\mathbb{G}\right)$,

(3) 

if $p^{\prime }\in T\left(X\right)$ and $p\le p^{\prime }$, then $c{l}_p^n\left(\mathbb{F}\right)\subseteq c{l}_{p^{\prime }}^n\left(\mathbb{F}\right)$.

Definition 5.

A function $f:(X,q)⟶(Y,p)$ between ⊤-convergence spaces is said to be a closure function if ${\overline{f^{\to }\left(\lambda \right)}}_p\le f^{\to }\left(\overline{{\lambda }_q}\right)$ for any $\lambda \in L^X$.

Proposition 2.

Suppose that $f:(X,q)⟶(Y,p)$ is a function between ⊤-convergence spaces and $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, $n\in \mathbb{N}$.

(1) 

If f is a continuous function, then $f^{\Rightarrow }\left(c{l}_q^n\left(\mathbb{F}\right)\right)\supseteq c{l}_p^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$.

(2) 

If f is a closure function, then $f^{\Rightarrow }\left(c{l}_q^n\left(\mathbb{F}\right)\right)\subseteq c{l}_p^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$.

Proof. 

(1) Let’s prove it by mathematical induction.

Firstly, we check ${\overline{f^{\leftarrow }\left(\lambda \right)}}_q\le f^{\leftarrow }\left({\overline{\lambda }}_p\right)$ for any $\lambda \in L^Y$. In fact, for any $x\in X$, by continuity of f we obtain

$$\begin{array}{ccc}\hfill {\overline{f^{\leftarrow }\left(\lambda \right)}}_q\left(x\right)& =& \underset{\mathbb{G}\stackrel{q}{⟶}x}{\bigvee }\underset{\mu \in \mathbb{G}}{\bigvee }{S}_X(\mu ,f^{\leftarrow }\left(\lambda \right))=\underset{\mathbb{G}\stackrel{q}{⟶}x}{\bigvee }\underset{\mu \in \mathbb{G}}{\bigvee }{S}_Y(f^{\to }\left(\mu \right),\lambda )\hfill \\ \hfill & \le & \underset{f^{\Rightarrow }\left(\mathbb{G}\right)\stackrel{p}{⟶}f\left(x\right)}{\bigvee }\underset{f^{\to }\left(\mu \right)\in f^{\Rightarrow }\left(\mathbb{G}\right)}{\bigvee }{S}_Y(f^{\to }\left(\mu \right),\lambda )\le \underset{\mathbb{H}\stackrel{p}{⟶}f\left(x\right)}{\bigvee }\underset{\nu \in \mathbb{H}}{\bigvee }{S}_Y(\nu ,\lambda )=f^{\leftarrow }\left({\overline{\lambda }}_p\right)\left(x\right).\hfill \end{array}$$

Secondly, we prove $f^{\Rightarrow }\left(c{l}_q^n\left(\mathbb{F}\right)\right)\supseteq c{l}_p^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$ when $n=1$. Let $\lambda \in f^{\Rightarrow }\left(\mathbb{F}\right)$, i.e., $f^{\leftarrow }\left(\lambda \right)\in \mathbb{F}$. Then by ${\overline{f^{\leftarrow }\left(\lambda \right)}}_q\le f^{\leftarrow }\left({\overline{\lambda }}_p\right)$ we have $f^{\leftarrow }\left({\overline{\lambda }}_p\right)\in c{l}_q\left(\mathbb{F}\right)$, i.e., ${\overline{\lambda }}_p\in f^{\Rightarrow }\left(c{l}_q\left(\mathbb{F}\right)\right)$. It follows by Lemma 2 (3) that $f^{\Rightarrow }\left(c{l}_q\left(\mathbb{F}\right)\right)\supseteq c{l}_p\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$.

Thirdly, we assume that $f^{\Rightarrow }\left(c{l}_q^n\left(\mathbb{F}\right)\right)\supseteq c{l}_p^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$ when $n=k$. Then we prove $f^{\Rightarrow }\left(c{l}_q^n\left(\mathbb{F}\right)\right)\supseteq c{l}_p^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$ when $n=k+1$. In fact,

$$f^{\Rightarrow }\left(c{l}_q^{k+1}\left(\mathbb{F}\right)\right)=f^{\Rightarrow }\left(c{l}_q\left(c{l}_q^k\left(\mathbb{F}\right)\right)\right)\supseteq c{l}_p\left(f^{\Rightarrow }\left(c{l}_q^k\left(\mathbb{F}\right)\right)\right)\supseteq c{l}_p\left(c{l}_p^k\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)\right)=c{l}_p^{k+1}\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right).$$

(2) We prove only that the inequality holds for $n=1$, and the rest of the proof is similar to (1).

For any $\lambda \in \mathbb{F}$, we have ${\overline{\lambda }}_q\in c{l}_q\left(\mathbb{F}\right)$ and then $f^{\to }\left(\lambda \right)\in f^{\Rightarrow }\left(\mathbb{F}\right)$. From f is a closure function, we conclude that $f^{\to }\left({\overline{\lambda }}_q\right)\ge {\overline{f^{\to }\left(\lambda \right)}}_p\in c{l}_p\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$ and so $f^{\to }\left({\overline{\lambda }}_q\right)\in c{l}_p\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)$. By Lemma 2 (1), (3) we obtain $f^{\Rightarrow }\left(c{l}_q\left(\mathbb{F}\right)\right)\subseteq c{l}_p\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right).$ □

Now, we tend our attention to p-regularity and its equivalent characterization. In the following, we shorten a pair of ⊤-convergence spaces $(X,p)$ and $(X,q)$ as $(X,p,q)$.

Definition 6.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces. Then q is said to be p-regular if the following condition p-(TC) is fulfilled.

p-(TC): $\forall \mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right),\forall x\in X,\mathbb{F}\stackrel{q}{⟶}x⟹c{l}_p\left(\mathbb{F}\right)\stackrel{q}{⟶}x$.

Remark 2.

When $L=\{\perp ,\top \}$, a ⊤-convergence space degenerates into a convergence space, and the condition p-(TC) degenerates into the crisp p-regularity condition in [5]. When $p=q$, the condition p-(TC) is precisely the regular characterization in [36].

We say a pair of ⊤-convergence spaces $(X,p,q)$ fulfill the Fischer ⊤-diagonal condition whenever

p-(TR): Let $J,X$ be any sets, $\psi :J⟶X$, and $\varphi :J⟶{\mathbb{F}}_L^{\top }\left(X\right)$ such that $\varphi \left(j\right)\stackrel{p}{⟶}\psi \left(j\right)$, for each $j\in J$. Then for each $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(J\right)$ and each $x\in X$, $k\varphi \mathbb{F}\stackrel{q}{⟶}x$ implies ${\psi }^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q}{⟶}x$.

Remark 3.

When $L=\{\perp ,\top \}$, a ⊤-convergence space degenerates into a convergence space, and the condition p-(TC) degenerates into the Fischer diagonal condition $R_{p,q}$ in [6]. When $p=q$, the condition p-(TR) is precisely the diagonal condition (TR) in [31].

In the following, we shall show that p-regularity can be described by Fischer ⊤-diagonal condition p-(TR).

Lemma 5.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces and let $J,X,\varphi ,\psi$ be defined as in p-(TR) . Then $S_X({\overline{\mu }}_p,\lambda )\le S_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right))$ for all $\lambda ,\mu \in L^X$.

Proof. 

Let $\lambda ,\mu \in L^X$.

$$\begin{array}{ccc}\hfill S_X({\overline{\mu }}_p,\lambda )& =& \underset{x\in X}{\bigwedge }([\underset{\mathbb{G}\stackrel{p}{⟶}x}{\bigvee }\mathsf{\Lambda }\left(\mathbb{G}\right)\left(\mu \right)]\to \lambda \left(x\right)),\mathrm{by}\psi \left(j\right)\in X,\varphi \left(j\right)\stackrel{p}{⟶}\psi \left(j\right)\hfill \\ \hfill & \le & \underset{j\in J}{\bigwedge }(\mathsf{\Lambda }\left(\varphi \left(j\right)\right)\left(\mu \right)\to \lambda \left(\psi \left(j\right)\right))=\underset{j\in J}{\bigwedge }(\widehat{\varphi }\left(\mu \right)\left(j\right)\to {\psi }^{\leftarrow }\left(\lambda \right)\left(j\right))\hfill \\ \hfill & =& S_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right)).\hfill \end{array}$$

 □

Theorem 1. 

(Theorem 4.8 in [36] for $p=q$) Let $(X,p,q)$ be a pair of ⊤-convergence spaces. Then p-(TC)⟺p-(TR).

Proof. 

p-(TC)⟹p-(TR). Let $J,X,\varphi ,\psi$ be defined as in p-(TR). Assume that $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(J\right)$ and $k\varphi \mathbb{F}\stackrel{q}{⟶}x$. Then it follows by p-(TC) that $c{l}_p\left(k\varphi \mathbb{F}\right)\stackrel{q}{⟶}x$.

Next we prove that $c{l}_p\left(k\varphi \mathbb{F}\right)\subseteq {\psi }^{\Rightarrow }\left(\mathbb{F}\right)$. Indeed, for any $\lambda \in c{l}_p\left(k\varphi \mathbb{F}\right)$, we have

$$\begin{array}{ccc}\hfill \top & =& \underset{\mu \in k\varphi \mathbb{F}}{\bigvee }{S}_X({\overline{\mu }}_p,\lambda )\stackrel{\mathrm{Lemma}5}{\le }\underset{\mu \in k\varphi \mathbb{F}}{\bigvee }{S}_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right))=\underset{\widehat{\varphi }\left(\mu \right)\in \mathbb{F}}{\bigvee }{S}_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right))\le \underset{\nu \in \mathbb{F}}{\bigvee }{S}_J(\nu ,{\psi }^{\leftarrow }\left(\lambda \right)),\hfill \end{array}$$

which means ${\psi }^{\leftarrow }\left(\lambda \right)\in \mathbb{F}$, i.e., $\lambda \in {\psi }^{\Rightarrow }\left(\mathbb{F}\right)$.

Now we have known that $c{l}_p\left(k\varphi \mathbb{F}\right)\stackrel{q}{⟶}x$ and $c{l}_p\left(k\varphi \mathbb{F}\right)\subseteq {\psi }^{\Rightarrow }\left(\mathbb{F}\right)$. Therefore, ${\psi }^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q}{⟶}x$, as desired.

p-(TR)⟹p-(TC). Let

$$J=\{(\mathbb{G},y)\in {\mathbb{F}}_L^{\top }\left(X\right)\times X|\mathbb{G}\stackrel{p}{⟶}y\};\psi :J⟶X,(\mathbb{G},y)\mapsto y;\varphi :J⟶{\mathbb{F}}_L^{\top }\left(X\right),(\mathbb{G},y)\mapsto \mathbb{G}.$$

Then $\forall j\in J$, $\varphi \left(j\right)\stackrel{p}{⟶}\psi \left(j\right)$. Please note that $j=(\mathbb{G},y)\in J⟺\mathbb{G}=\varphi \left(j\right),y=\psi \left(j\right)$.

(1) For any $\lambda ,\mu \in L^X$, $S_X({\overline{\mu }}_p,\lambda )=S_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right))$. Indeed,

$$\begin{array}{ccc}\hfill S_X({\overline{\mu }}_p,\lambda )& =& \underset{y\in X}{\bigwedge }([\underset{\mathbb{G}\stackrel{p}{⟶}y}{\bigvee }\mathsf{\Lambda }\left(\mathbb{G}\right)\left(\mu \right)]\to )=\underset{y\in X}{\bigwedge }\underset{(\mathbb{G},y)\in J}{\bigwedge }(\mathsf{\Lambda }\left(\mathbb{G}\right)\left(\mu \right)\to )\hfill \\ \hfill & =& \underset{j=(\mathbb{G},y)\in J}{\bigwedge }(\mathsf{\Lambda }\left(\varphi \left(j\right)\right)\left(\mu \right)\to \lambda \left(\psi \left(j\right)\right))=\underset{j\in J}{\bigwedge }(\widehat{\varphi }\left(\mu \right)\left(j\right)\to {\psi }^{\leftarrow }\left(\lambda \right)\left(j\right))\hfill \\ \hfill & =& S_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right)).\hfill \end{array}$$

(2) For each $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, the family $\{\widehat{\varphi }\left(\lambda \right)|\lambda \in \mathbb{F}\}$ forms a ⊤-filter base on J. Indeed,

(TB1): For any $\lambda \in \mathbb{F}$, by ${\left[y\right]}_{\top }\stackrel{p}{⟶}y$ for any $y\in X$, we have

$$\underset{j\in J}{\bigvee }\widehat{\varphi }\left(\lambda \right)\left(j\right)=\underset{j\in J}{\bigvee }\mathsf{\Lambda }\left(\varphi \left(j\right)\right)\left(\lambda \right)=\underset{y\in X}{\bigvee }\underset{\mathbb{G}\stackrel{p}{⟶}y}{\bigvee }\mathsf{\Lambda }\left(\mathbb{G}\right)\left(\lambda \right)\ge \underset{y\in X}{\bigvee }\mathsf{\Lambda }\left({\left[y\right]}_{\top }\right)\left(\lambda \right)=\underset{y\in X}{\bigvee }\lambda \left(y\right)=\top .$$

(TB2): For any $\lambda ,\mu \in \mathbb{F}$, note that for any $j\in J$,

$$\begin{array}{ccc}\hfill \widehat{\varphi }\left(\lambda \right)\left(j\right)\wedge \widehat{\varphi }\left(\mu \right)\left(j\right)& =& \underset{{\lambda }_1\in \varphi \left(j\right)}{\bigvee }{S}_X({\lambda }_1,\lambda )\wedge \underset{{\mu }_1\in \varphi \left(j\right)}{\bigvee }{S}_X({\mu }_1,\mu )\hfill \\ \hfill & \le & \underset{{\lambda }_1,{\mu }_1\in \varphi \left(j\right)}{\bigvee }{S}_X({\lambda }_1\wedge {\mu }_1,\lambda \wedge \mu )\hfill \\ \hfill & \le & \underset{\nu \in \varphi \left(j\right)}{\bigvee }{S}_X(\nu ,\lambda \wedge \mu )=\widehat{\varphi }(\lambda \wedge \mu )\left(j\right),\hfill \end{array}$$

i.e., $\widehat{\varphi }\left(\lambda \right)\wedge \widehat{\varphi }\left(\mu \right)\le \widehat{\varphi }(\lambda \wedge \mu )$. It follows easily that (TB2) is satisfied. We denote the ⊤-filter generated by $\{\widehat{\varphi }\left(\lambda \right)|\lambda \in \mathbb{F}\}$ as ${\mathbb{F}}^{\varphi }$.

(3) For each $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, $k\varphi {\mathbb{F}}^{\varphi }\supseteq \mathbb{F}$. Indeed, for any $\lambda \in \mathbb{F}$, we have $\widehat{\varphi }\left(\lambda \right)\in {\mathbb{F}}^{\varphi }$, i.e., $\lambda \in k\varphi {\mathbb{F}}^{\varphi }$.

(4) For each $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$, ${\psi }^{\Rightarrow }\left({\mathbb{F}}^{\varphi }\right)=c{l}_p\left(\mathbb{F}\right)$. Indeed,

$$\begin{array}{ccc}\hfill \lambda \in {\psi }^{\Rightarrow }\left({\mathbb{F}}^{\varphi }\right)& ⟺& {\psi }^{\leftarrow }\left(\lambda \right)\in {\mathbb{F}}^{\varphi }⟺\underset{\mu \in \mathbb{F}}{\bigvee }{S}_J(\widehat{\varphi }\left(\mu \right),{\psi }^{\leftarrow }\left(\lambda \right))=\top \stackrel{\left(1\right)}{⟺}\underset{\mu \in \mathbb{F}}{\bigvee }{S}_X({\overline{\mu }}_p,\lambda )=\top ⟺\lambda \in c{l}_p\left(\mathbb{F}\right).\hfill \end{array}$$

Assume that $\mathbb{F}\stackrel{q}{⟶}x$, then by (3), we have $k\varphi {\mathbb{F}}^{\varphi }\supseteq \mathbb{F}$, and so $k\varphi {\mathbb{F}}^{\varphi }\stackrel{q}{⟶}x$. From p-(TR) and (4), we get that $c{l}_p\left(\mathbb{F}\right)={\psi }^{\Rightarrow }\left({\mathbb{F}}^{\varphi }\right)\stackrel{q}{⟶}x$. Therefore, the condition p-(TC) is satisfied. □

The next theorem shows that p-regularity is preserved under initial structures.

Theorem 2.

Let ${\left\{(X_i,q_i,p_i)\right\}}_{i\in I}$ be pairs of ⊤-convergence spaces such that each $q_i$ is $p_i$-regular. If q (resp., p) is the initial structure on X regarding the source ${(X\stackrel{f_i}{⟶}(X_i,q_i))}_{i\in I}$ (resp., ${(X\stackrel{f_i}{⟶}(X_i,p_i))}_{i\in I}$), then q is also p-regular.

Proof. 

Let $\psi :J⟶X$ and $\varphi :J⟶{\mathbb{F}}_L^{\top }\left(X\right)$ be any function such that $\varphi \left(j\right)\stackrel{p}{⟶}\psi \left(j\right)$ for any $j\in J$. Then

$$\forall i\in I,\forall j\in J,(f_i^{\Rightarrow }\circ \varphi )\left(j\right)=f_i^{\Rightarrow }\left(\varphi \left(j\right)\right)\stackrel{p_i}{⟶}{f}_i\left(\psi \left(j\right)\right)=(f_i\circ \psi )\left(j\right).$$

Let $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(J\right)$ satisfy $k\varphi \mathbb{F}\stackrel{q}{⟶}x$. Then by definition of q and Lemma 3 we have

$$\forall i\in I,k(f_i^{\Rightarrow }\circ \varphi )\mathbb{F}=f_i^{\Rightarrow }\left(k\varphi \mathbb{F}\right)\stackrel{q_i}{⟶}{f}_i\left(x\right).$$

Since $q_i$ is $p_i$-regular we have $f_i^{\Rightarrow }{\psi }^{\Rightarrow }\left(\mathbb{F}\right)={(f_i\circ \psi )}^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q_i}{⟶}{f}_i\left(x\right)$. By definition of q we have ${\psi }^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{q}{⟶}x$. Thus q is p-regular. □

The next theorem shows that p-regularity is preserved under final structures with some additional assumptions.

Theorem 3.

Let ${\left\{(X_i,q_i,p_i)\right\}}_{i\in I}$ be pairs of ⊤-convergence spaces such that each $q_i$ is $p_i$-regular. Let q (resp., p) be the final structure on X relative to the sink ${((X_i,q_i)\stackrel{f_i}{⟶}X)}_{i\in I}$ (resp., ($(X_i,p_i)$$\stackrel{f_i}{⟶}{X)}_{i\in I}$). If $X={\cup }_{i\in I}{f}_i\left(X_i\right)$ and each $f_i:(X_i,p_i)⟶(X,p)$ is a closure function, then q is also p-regular.

Proof. 

Let $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)\stackrel{q}{⟶}x$. Then by definition of q, there exists $i\in I,x_i\in X_i,{\mathbb{G}}_i\in {\mathbb{F}}_L^{\top }\left(X_i\right),f_i\left(x_i\right)=x$ such that ${\mathbb{G}}_i\stackrel{q_i}{⟶}{x}_i$ and $f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\subseteq \mathbb{F}$. Because $q_i$ is $p_i$-regular we get $c{l}_{p_i}\left({\mathbb{G}}_i\right)\stackrel{q_i}{⟶}{x}_i$ and then $f_i^{\Rightarrow }\left(c{l}_{p_i}\left({\mathbb{G}}_i\right)\right)\stackrel{q}{⟶}x$. By $f_i$ is a closure function and Proposition 2 (2) it follows that $c{l}_p\left(f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\right)\stackrel{q}{⟶}x$. Hence $c{l}_p\left(\mathbb{F}\right)\stackrel{q}{⟶}x$ from $c{l}_p\left(f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\right)\subseteq c{l}_p\left(\mathbb{F}\right)$. Thus q is p-regular. □

For any ${\left\{q_i\right\}}_{i\in I}\subseteq \top \left(X\right)$, note that the supremum (resp., infimum) of ${\left\{q_i\right\}}_{i\in I}$ in the lattice $\top \left(X\right)$, denoted as $sup\left\{q_i\right|i\in I\}$ (resp., $inf\left\{q_i\right|i\in I\}$), is precisely the initial structure (resp., final structure) regarding the source ${(X\stackrel{i{d}_X}{⟶}(X,q_i))}_{i\in I}$ (resp., the sink (${(X,q_i)\stackrel{i{d}_X}{⟶}X)}_{i\in I}$). By Theorems 2 and 3, we obtain easily the following corollary. It will show us that p-regularity is preserved under supremum and infimum in the lattice $\top \left(X\right)$.

Corollary 1.

Let $\left\{q_i\right|i\in I\}\subseteq \top \left(X\right)$ and $p\in \top \left(X\right)$ with each $(X,q_i)$ being p-regular. Then both $inf{\left\{q_i\right\}}_{i\in I}$ and $sup{\left\{q_i\right\}}_{i\in I}$ are all p-regular.

4. Lower (Upper) p-Regular Modifications in ⊤-Convergence Spaces

In this section, we shall consider the p-regular modifications in ⊤-convergence spaces.

Lemma 6.

Let $p,q$ be ⊤-convergence structures on X.

(1) 

If q is p-regular, then $\mathbb{F}\stackrel{q}{⟶}x$ implies $c{l}_p^n\left(\mathbb{F}\right)\stackrel{q}{⟶}x$ for any $n\in \mathbb{N}$.

(2) 

If q is p-regular, then q is $p^{\prime }$-regular for any $p\le p^{\prime }$.

(3) 

The indiscrete structure ι is p-regular for any $p\in \top \left(X\right)$.

Proof. 

It is obvious. □

4.1. Lower p-Regular Modification

It has been known that p-regularity is preserved under supremum in the lattice $\top \left(X\right)$ (see Corollary 1), and the indiscrete structure $\iota$ is p-regular for any $p\in \top \left(X\right)$ (see Lemma 6 (3)). So, it follows easily that for a pair of ⊤-convergence spaces $(X,p,q)$, there is a finest p-regular ⊤-convergence structure ${\gamma }_{p}q$ on X which is coarser than q.

Definition 7.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces. Then the ⊤-convergence structure ${\gamma }_{p}q$ on X is said to be the lower p-regular modification of q.

The following theorem gives a characterization on lower p-regular modification.

Theorem 4.

For any $p,q\in \top \left(X\right)$, $\mathbb{F}\stackrel{{\gamma }_{p}q}{⟶}x⟺\exists n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)$.

Proof. 

We define $q^{\prime }$ as $\mathbb{F}\stackrel{q^{\prime }}{⟶}x⟺\exists n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)$, then we prove ${\gamma }_{p}q=q^{\prime }$.

Obviously, $q^{\prime }\in \top \left(X\right)$ and $q^{\prime }\le q$. We check that $q^{\prime }$ is p-regular. In fact, let $\mathbb{F}\stackrel{q^{\prime }}{⟶}x$. Then there exists $n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x$ such that $\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)$. It follows that $c{l}_p\left(\mathbb{F}\right)\supseteq c{l}_p\left(c{l}_p^n\left(\mathbb{G}\right)\right)=c{l}_p^{n+1}\left(\mathbb{G}\right)$, so $c{l}_p\left(\mathbb{F}\right)\stackrel{q^{\prime }}{⟶}x$. Now, we have proved that $q^{\prime }$ is p-regular.

Let r be p-regular with $r\le q$. We prove below $r\le q^{\prime }$. In fact, let $\mathbb{F}\stackrel{q^{\prime }}{⟶}x$. Then there exists $n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x$ such that $\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)$, so $\mathbb{G}\stackrel{r}{⟶}x$ by $q\le r$. Because r is p-regular it follows by Lemma 6 (1) that $\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)\stackrel{r}{⟶}x$. Therefore, $r\le q^{\prime }$. □

Theorem 5.

If $f:(X,q)⟶(X^{\prime },q^{\prime })$ and $f:(X,p)⟶(X^{\prime },p^{\prime })$ are both continuous function between ⊤-convergence spaces then so is $f:(X,{\gamma }_{p}q)⟶(X^{\prime },{\gamma }_{p^{\prime }}{q}^{\prime })$.

Proof. 

For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$ and $x\in X$.

$$\begin{array}{ccc}\hfill \mathbb{F}\stackrel{{\gamma }_{p}q}{⟶}x& ⟹& \exists n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.\mathbb{F}\supseteq c{l}_p^n\left(\mathbb{G}\right)\hfill \\ \hfill & ⟹& \exists n\in \mathbb{N},f^{\Rightarrow }\left(\mathbb{G}\right)\stackrel{q^{\prime }}{⟶}f\left(x\right)\mathrm{s}.\mathrm{t}.f^{\Rightarrow }\left(\mathbb{F}\right)\supseteq f^{\Rightarrow }\left(c{l}_p^n\left(\mathbb{G}\right)\right)\hfill \\ \hfill & ⟹& \exists n\in \mathbb{N},f^{\Rightarrow }\left(\mathbb{G}\right)\stackrel{q^{\prime }}{⟶}f\left(x\right)\mathrm{s}.\mathrm{t}.f^{\Rightarrow }\left(\mathbb{F}\right)\supseteq c{l}_{p^{\prime }}^n\left(f^{\Rightarrow }\left(\mathbb{G}\right)\right)\hfill \\ \hfill & ⟹& f^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{{\gamma }_{p^{\prime }}{q}^{\prime }}{⟶}\left(f\left(x\right)\right),\hfill \end{array}$$

where the second implication uses the continuity of $f:(X,q)⟶(X^{\prime },q^{\prime })$, and the third implication uses the continuity of $f:(X,p)⟶(X^{\prime },p^{\prime })$ and Proposition 2(1). □

The following theorem exhibits us that lower p-regular modification and final structures have good compatibility.

Theorem 6.

Let ${\left\{(X_i,q_i,p_i)\right\}}_{i\in I}$ be pairs of spaces in ⊤-CS and let q be the final structure relative to the sink ${((X_i,q_i)\stackrel{f_i}{⟶}X)}_{i\in I}$ with $X={\cup }_{i\in I}{f}_i\left(X_i\right)$. If $p\in \top \left(X\right)$ such that each $f_i:(X_i,p_i)⟶(X,p)$ is a continuous closure function, then ${\gamma }_{p}q$ is the final structure relative to the sink ${((X_i,{\gamma }_{p_i}{q}_i)\stackrel{f_i}{⟶}X)}_{i\in I}$.

Proof. 

Let s denote the final structure relative to the sink ${((X_i,{\gamma }_{p_i}{q}_i)\stackrel{f_i}{⟶}X)}_{i\in I}$. Then for any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$ and $x\in X$

$$\begin{array}{ccc}\hfill \mathbb{F}\stackrel{s}{⟶}x& ⟹& \exists i\in I,x_i\in X_i,f_i\left(x_i\right)=x,{\mathbb{G}}_i\stackrel{{\gamma }_{p_i}{q}_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\subseteq \mathbb{F},\mathrm{by}\mathrm{Theorem}4\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,n\in \mathbb{N},f_i\left(x_i\right)=x,{\mathbb{H}}_i\stackrel{q_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.c{l}_{p_i}^n\left({\mathbb{H}}_i\right)\subseteq {\mathbb{G}}_i,f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\subseteq \mathbb{F},\mathrm{by}\mathrm{Proposition}2\left(1\right)\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,n\in \mathbb{N},f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.c{l}_p^n\left(f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\right)\subseteq f_i^{\Rightarrow }\left(c{l}_{p_i}^n\left({\mathbb{H}}_i\right)\right)\subseteq f_i^{\Rightarrow }\left({\mathbb{G}}_i\right),f_i^{\Rightarrow }\left({\mathbb{G}}_i\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,n\in \mathbb{N},f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.c{l}_p^n\left(f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \mathbb{F}\stackrel{{\gamma }_{p}q}{⟶}x.\hfill \end{array}$$

Conversely,

$$\begin{array}{ccc}\hfill \mathbb{F}\stackrel{{\gamma }_{p}q}{⟶}x& ⟹& \exists n\in \mathbb{N},\mathbb{G}\stackrel{q}{⟶}x\mathrm{s}.\mathrm{t}.c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,f_i\left(x_i\right)=x,n\in \mathbb{N},{\mathbb{H}}_i\stackrel{q_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\subseteq \mathbb{G},c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,f_i\left(x_i\right)=x,n\in \mathbb{N},{\mathbb{H}}_i\stackrel{q_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.c{l}_p^n\left(f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\right)\subseteq c{l}_p^n\left(\mathbb{G}\right),c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,f_i\left(x_i\right)=x,n\in \mathbb{N},{\mathbb{H}}_i\stackrel{q_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.f_i^{\Rightarrow }\left(c{l}_{p_i}^n\left({\mathbb{H}}_i\right)\right)\subseteq c{l}_p^n\left(f_i^{\Rightarrow }\left({\mathbb{H}}_i\right)\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists i\in I,x_i\in X_i,f_i\left(x_i\right)=x,n\in \mathbb{N},c{l}_{p_i}^n\left({\mathbb{H}}_i\right)\stackrel{{\gamma }_{p_i}{q}_i}{⟶}{x}_i\mathrm{s}.\mathrm{t}.f_i^{\Rightarrow }\left(c{l}_{p_i}^n\left({\mathbb{H}}_i\right)\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \mathbb{F}\stackrel{s}{⟶}x,\hfill \end{array}$$

where the fourth implication follows by Proposition 2(2). □

The following corollary tells us that lower p-regular modification has good compatibility with infimum in the lattice $\top \left(X\right)$.

Corollary 2.

Assume that $\left\{q_i\right|i\in I\}\subseteq \top \left(X\right)$, $p\in \top \left(X\right)$ and $q=inf\left\{q_i\right|i\in I\}$. Then ${\gamma }_{p}q=inf\left\{{\gamma }_p{q}_i\right|i\in I\}$.

4.2. Upper p-Regular Modification

Similar to the crisp case, the discrete structure $\delta$ is not always p-regular for any $p\in \top \left(X\right)$. This shows that for a given $q\in \top \left(X\right)$, there may not exist p-regular ⊤-convergence structure on X finer than q.

Definition 8.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces. If there exists a coarsest p-regular ⊤-convergence structure ${\gamma }^{p}q$ on X finer than q, then it is said to be the upper p-regular modification of q.

It has been known that the existence of ${\gamma }^{p}q$ depends on the existence of a p-regular ⊤-convergence structure finer than q (see Corollary 1), and ${\gamma }_p\delta$ is the finest p-regular ⊤-convergence structure. So, it follows easily that ${\gamma }^{p}q$ exists if and only if $q\le {\gamma }_p\delta$. By Theorem 4, we obtain the following result.

Theorem 7.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces. Then

$${\gamma }^{p}q exists⟺\forall x\in X,\forall n\in \mathbb{N},c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{q}{⟶}x.$$

Proof. 

For any $\mathbb{F}\in {\mathbb{F}}_L^{\top }\left(X\right)$ and any $x\in X$, from Theorem 4 we obtain

$$\mathbb{F}\stackrel{{\gamma }_p\delta }{⟶}x⟺\exists n\in \mathbb{N},\mathbb{G}\stackrel{\delta }{⟶}x\mathrm{s}.\mathrm{t}.c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}.$$

Necessity. Let ${\gamma }^{p}q$ exist. Then $q\le {\gamma }_p\delta$. So, for any $x\in X$, $n\in \mathbb{N}$

$${\left[x\right]}_{\top }\stackrel{\delta }{⟶}x⟹c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{{\gamma }_p\delta }{⟶}x⟹c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{q}{⟶}x.$$

Sufficiency. Let $c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{q}{⟶}x$ for any $x\in X$, $n\in \mathbb{N}$. Then

$$\begin{array}{ccc}\hfill \mathbb{F}\stackrel{{\gamma }_p\delta }{⟶}x& ⟹& \exists n\in \mathbb{N},\mathbb{G}\stackrel{\delta }{⟶}x\mathrm{s}.\mathrm{t}.c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists n\in \mathbb{N},{\left[x\right]}_{\top }\subseteq \mathbb{G}\mathrm{s}.\mathrm{t}.c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & \stackrel{\mathrm{Proposition}1\left(2\right)}{⟹}& \exists n\in \mathbb{N},\subseteq c{l}_p^n\left(\mathbb{G}\right)\mathrm{s}.\mathrm{t}.c{l}_p^n\left(\mathbb{G}\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \exists n\in \mathbb{N}\mathrm{s}.\mathrm{t}.c{l}_p^n\left({\left[x\right]}_{\top }\right)\subseteq \mathbb{F}\hfill \\ \hfill & ⟹& \mathbb{F}\stackrel{q}{⟶}x.\hfill \end{array}$$

It follows that $q\le {\gamma }_p\delta$, so ${\gamma }^{p}q$ exists. □

The following theorem gives a characterization on upper p-regular modification if it exists.

Theorem 8.

Let $(X,p,q)$ be a pair of ⊤-convergence spaces and ${\gamma }^{p}q$ exists. Then

$$\mathbb{F}\stackrel{{\gamma }^{p}q}{⟶}x⟺\forall n\in \mathbb{N}$$

Proof. 

We define $q^{\prime }$ as $\mathbb{F}\stackrel{q^{\prime }}{⟶}x⟺\forall n\in \mathbb{N},c{l}_p^n\left(\mathbb{F}\right)\stackrel{q}{⟶}x$.

(1)

$q^{\prime }\in \top \left(X\right)$. It is obvious.

(2)

$q\le q^{\prime }$. In fact, let $\mathbb{F}\stackrel{q^{\prime }}{⟶}x$ then $\mathbb{F}=c{l}_p^0\left(\mathbb{F}\right)\stackrel{q}{⟶}x$.

(3)

$q^{\prime }$ is p-regular. In fact, let $\mathbb{F}\stackrel{q^{\prime }}{⟶}x$. Then for any $n\in \mathbb{N}$ it holds that $c{l}_p^n\left(c{l}_p\left(\mathbb{F}\right)\right)=c{l}_p^{n+1}\left(\mathbb{F}\right)\stackrel{q}{⟶}x,$ which means $c{l}_p\left(\mathbb{F}\right)\stackrel{q^{\prime }}{⟶}x$. So, $q^{\prime }$ is p-regular.

(4)

Let r be p-regular with $q\le r$. Then $q^{\prime }\le r$. In fact, let $\mathbb{F}\stackrel{r}{⟶}x$ then for any $n\in \mathbb{N}$, by Proposition 6 (1) it holds that $c{l}_p^n\left(\mathbb{F}\right)\stackrel{r}{⟶}x$ and so $c{l}_p^n\left(\mathbb{F}\right)\stackrel{q}{⟶}x$ by $q\le r$. That means $\mathbb{F}\stackrel{q^{\prime }}{⟶}x$.

By (1)–(4) we get that $q^{\prime }$ is the coarsest p-regular ⊤-convergence structure finer than q. Therefore, ${\gamma }^{p}q=q^{\prime }$. □

Theorem 9.

Let $f:(X,q)⟶(X^{\prime },q^{\prime })$ be a continuous function, and $f:(X,p)⟶(X^{\prime },p^{\prime })$ be a closure function between ⊤-convergence spaces. If ${\gamma }^{p}q$ and ${\gamma }^{p^{\prime }}{q}^{\prime }$ exist then $f:(X,{\gamma }^{p}q)⟶(X^{\prime },{\gamma }^{p^{\prime }}{q}^{\prime })$ is continuous.

Proof. 

Let $\mathbb{F}\stackrel{{\gamma }^{p}q}{⟶}x$. Then $\forall n\in \mathbb{N},c{l}_p^n\left(\mathbb{F}\right)\stackrel{q}{⟶}x$. Since $f:(X,q)⟶(X^{\prime },q^{\prime })$ is a continuous function and $f:(X,p)⟶(X^{\prime },p^{\prime })$ is a closure function it holds that

$$\forall n\in \mathbb{N},c{l}_{p^{\prime }}^n\left(f^{\Rightarrow }\left(\mathbb{F}\right)\right)\supseteq f^{\Rightarrow }\left(c{l}_p^n\left(\mathbb{F}\right)\right)\stackrel{q^{\prime }}{⟶}f\left(x\right),$$

so $f^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{{\gamma }^{p^{\prime }}{q}^{\prime }}{⟶}f\left(x\right)$, as desired. □

The following theorem exhibits us that the upper p-regular modification has good compatibility with initial structures.

Theorem 10.

Let ${\left\{(X_i,q_i,p_i)\right\}}_{i\in I}$ be pairs of spaces in ⊤-CS and q be the initial structure relative to the source ${(X\stackrel{f_i}{⟶}(X_i,q_i))}_{i\in I}$. Let $p\in \top \left(X\right)$ such that each $f_i:(X,p)⟶(X_i,p_i)$ is continuous closure function. If ${\gamma }^{p_i}{q}_i$ exists for all $i\in I$ then so does ${\gamma }^{p}q$, and ${\gamma }^{p}q$ is precisely the initial structure relative to the source ${(X\stackrel{f_i}{⟶}(X_i,{\gamma }^{p_i}{q}_i))}_{i\in I}$.

Proof. 

At first, we show the existence of ${\gamma }^{p}q$. By Theorem 7, it suffices to check that $c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{q}{⟶}x$ for any $x\in X,n\in \mathbb{N}$. In fact, by the existence of ${\gamma }^{p_i}{q}_i$ we have $c{l}_{p_i}^n\left(\left[f_i\left(x\right)\right]\right)\stackrel{q_i}{⟶}{f}_i\left(x\right)$ for any $i\in I,x\in X,n\in \mathbb{N}$. Then by each $f_i:(X,p)⟶(X_i,p_i)$ being a continuous closure function it holds that

$$f_i^{\Rightarrow }(c{l}_p^n\left({\left[x\right]}_{\top }\right)=c{l}_{p_i}^n\left(f_i^{\Rightarrow }\left({\left[x\right]}_{\top }\right)\right)=c{l}_{p_i}^n\left({\left[f_i\left(x\right)\right]}_{\top }\right)\stackrel{q_i}{⟶}{f}_i\left(x\right),$$

so $c{l}_p^n\left({\left[x\right]}_{\top }\right)\stackrel{q}{⟶}x$ for any $x\in X,n\in \mathbb{N}$, i.e., ${\gamma }^{p}q$ exists.

Let s denote the initial structure relative to the source ${(X\stackrel{f_i}{⟶}(X_i,{\gamma }^{p_i}{q}_i))}_{i\in I}$. Then

$$\begin{array}{ccc}\hfill \mathbb{F}\stackrel{s}{⟶}x& ⟺& \forall i\in I,f_i^{\Rightarrow }\left(\mathbb{F}\right)\stackrel{{\gamma }^{p_i}{q}_i}{⟶}{f}_i\left(x\right)\stackrel{\mathrm{Theorem}8}{⟺}\forall i\in I,\forall n\in \mathbb{N},c{l}_{p_i}^n\left(f_i^{\Rightarrow }\left(\mathbb{F}\right)\right)\stackrel{q_i}{⟶}{f}_i\left(x\right)\hfill \\ \hfill & \stackrel{\mathrm{Proposition}2}{⟺}& \forall i\in I,\forall n\in \mathbb{N},f_i^{\Rightarrow }\left(c{l}_p^n\left(\mathbb{F}\right)\right)\stackrel{q_i}{⟶}{f}_i\left(x\right)\hfill \\ \hfill & ⟺& \forall n\in \mathbb{N},c{l}_p^n\left(\mathbb{F}\right)\stackrel{q}{⟶}x\stackrel{\mathrm{Theorem}8}{⟺}\mathbb{F}\stackrel{{\gamma }^{p}q}{⟶}x.\hfill \end{array}$$

 □

The following corollary tells us that upper p-regular modification has good compatibility with supremum in the lattice $\top \left(X\right)$.

Corollary 3.

Assume that $\left\{q_i\right|i\in I\}\subseteq \top \left(X\right)$, $p\in \top \left(X\right)$ and $q=sup\left\{q_i\right|i\in I\}$. If ${\gamma }^p{q}_i$ exists for all $i\in I$ then so does ${\gamma }^{p}q$ and ${\gamma }^{p}q=sup\left\{{\gamma }^p{q}_i\right|i\in I\}$.

5. Conclusions

In this paper, we studied p-regularity in ⊤-convergence spaces by a diagonal condition and a closure condition about ⊤-filter, respectively. We proved that p-regularity was preserved under the initial and final constructions in the category ⊤-CS. We then followed as a conclusion that p-regularity was preserved under the infimum and supremum in the lattice $\top \left(X\right)$. Furthermore, we defined and discussed lower (upper) p-regular modifications in ⊤-convergence spaces. In particular, we showed that lower (resp., upper) p-regular modification has good compatibility with final (resp., initial) construction.