mathematics

Article p-Regularity and p-Regular Modification in >-Convergence Spaces

Qiu Jin 1,*, Lingqiang Li 1,2,3,* and Guangming Lang 2,3 1 Department of Mathematics, Liaocheng University, Liaocheng 252059, China 2 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China; [email protected] 3 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China * Correspondence: [email protected] (Q.J.); [email protected] (L.L.); Tel.: +86-150-6355-2700 (Q.J.); +86-152-0650-6635 (L.L.)


Received: 9 March 2019; Accepted: 22 April 2019; Published: 24 April 2019
 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.

Keywords: fuzzy topology; fuzzy convergence; >-convergence space; regularity

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 [8–30]; and (2) >-convergence spaces introduced by Fang [31] and then discussed by many researchers [32–36]. 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. Section2 recalls some notions and notations for later use. Section3 presents p-regularity in >-convergence spaces by a diagonal condition of >-filters and a

Mathematics 2019, 7, 370; doi:10.3390/math7040370 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 370 2 of 14 closure condition of >-filters, respectively. Section4 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, ≤) is always a complete lattice with a top element > W W and a bottom element ⊥, which satisfies the distributive law α ∧ ( i∈I βi) = i∈I (α ∧ βi). A lattice with these conditions is called a complete Heyting algebra. The operation →: L × L −→ L given by

_ α → β = {γ ∈ L : α ∧ γ ≤ β}. is called the residuation with respect to ∧. We collect here some basic properties of the binary operations ∧ and → [43].

(1) a → b = > ⇔ a ≤ b; (2) a ∧ b ≤ c ⇔ b ≤ a → c; (3) a ∧ (a → b) ≤ b; (4) a → (b → c) = (a ∧ b) → c; W V V V (5) ( j∈J aj) → b = j∈J(aj → b); (6) a → ( j∈J bj) = j∈J(a → bj). A function µ : X → L is said to be an L-fuzzy set in X, and all L-fuzzy sets in X are denoted as X X L . The operators ∨, ∧, → on L can be translated onto L pointwisely. Precisely, for any µ, ν, µt(t ∈ T) ∈ LX,

_ _ ^ ^ ( µt)(x) = µt(x), ( µt)(x) = µt(x), (µ → ν)(x) = µ(x) → ν(x). t∈T t∈T t∈T t∈T → X Y → W X Let f : X −→ Y be a function. We define f : L −→ L by f (µ)(y) = f (x)=y µ(x) for µ ∈ L and y ∈ Y, and define f ← : LY −→ LX by f ←(ν)(x) = ν( f (x)) for ν ∈ LY and x ∈ X [43]. Let µ, ν be L-fuzzy sets in X. The subsethood degree of µ, ν, denoted as SX(µ, ν), is defined by SX(µ, ν) = ∧ (µ(x) → ν(x)) [44–46] x∈X

X Y Lemma 1. [31,42,47] Let f : X −→ Y be a function and µ1, µ2 ∈ L , λ1, λ2 ∈ L . Then → → (1)S X(µ1, µ2) ≤ SY( f (µ1), f (µ2)), ← ← (2)S Y(λ1, λ2) ≤ SX( f (λ1), f (λ2)), → ← (3)S Y( f (µ1), λ1) = SX(µ1, f (λ1)).

2.1. >-Filters and >-Convergence Spaces

Definition 1. [43,48] A nonempty subset F ⊆ LX is said to be a >-filter on the set X if it satisfies the following three conditions: W (TF1) ∀λ ∈ F, λ(x) = >, x∈X (TF2) ∀λ, µ ∈ F, λ ∧ µ ∈ F, W (TF3) if SX(µ, λ) = >, then λ ∈ F. µ∈F > The set of all >-filters on X is denoted by FL (X).

Definition 2. [43] A nonempty subset B ⊆ LX is referred to be a >-filter base on the set X if it holds that: W (TB1) ∀λ ∈ B, λ(x) = >, x∈X W (TB2) if λ, µ ∈ B, then SX(ν, λ ∧ µ) = >. ν∈B Mathematics 2019, 7, 370 3 of 14

Each >-filter base generates a >-filter FB by

X _ FB := {λ ∈ L | SX(µ, λ) = >}. µ∈B

Example 1. [31,43] Let f : X −→ Y be a function.

> → (1) For any F ∈ FL (X), the family { f (λ)|λ ∈ F} forms a >-filter base on Y. The generated >-filter is ⇒ ⇒ ← denoted as f (F), called the image of F under f . It is known that µ ∈ f (F) ⇐⇒ f (µ) ∈ F. > ← W (2) For any G ∈ FL (Y), the family { f (µ)|µ ∈ G} forms a >-filter base on Y iff µ(y) = > holds y∈ f (X) ⇐ for all µ ∈ G. The generated >-filter (if exists) is denoted as f (G), called the inverse image of G under ⇒ ⇐ ⇐ ⇐ f . It is known that G ⊆ f ( f (G)) holds whenever f (G) exists. Furthermore, f (G) always exists ⇒ ⇐ and G = f ( f (G)) whenever f is surjective. X ⇒ (3) For any x ∈ X, the family [x]> =: {λ ∈ L |λ(x) = >} is a >-filter on X, andf ([x]>) = [ f (x)]>.

Lemma 2. Let f : X −→ Y be a function.

> → ⇒ (1) If B is a >-filter base of F ∈ FL (X), then { f (λ)|λ ∈ B} is a >-filter base of f (F), see Example 2.9 (1) in [31]. > ⇐ ← ⇐ (2) If B is a >-filter base of G ∈ FL (Y) and f (G) exists, then { f (µ)|µ ∈ B} is a >-filter base of f (G), see Example 2.9 (2) in [31]. > (3) Let F, G ∈ FL (X) and B be a >-filter base of F. Then B ⊆ G implies that F ⊆ G, see Lemma 2.5 (1) in [42]. > W W (4) Let F ∈ FL (X) and B be a >-filter base of F. Then SX(µ, λ) = SX(µ, λ), see Lemma 3.1 in [36]. µ∈B µ∈F

> X For each F ∈ FL (X), we define Λ(F) : L −→ L as

X _ ∀λ ∈ L , Λ(F)(λ) = SX(µ, λ), µ∈F then Λ(F) is a tightly stratified L-filter on X [47]. In the following, we recall some notions and notations collected in [29].

> X Definition 3. [31] Let X be a nonempty set. Then a function q : FL (X) −→ 2 is said to be a >-convergence structure on X if it satisfies the following two conditions:

q (TC1) ∀x ∈ X, [x]> −→ x; q q (TC1) if F −→ x and F ⊆ G, then G −→ x. q where F −→ x is short for x ∈ q(F). The pair (X, q) is said to be a >-convergence space.

A function f : X −→ X0 between >-convergence spaces (X, q), (X0, q0) is said to be continuous if 0 ⇒ q q f (F) −→ f (x) for any F −→ 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]. fi For a source (X −→ (Xi, qi))i∈I, the initial structure, q on X is defined by q ⇒ qi F −→ x ⇐⇒ ∀i ∈ I, fi (F) −→ fi(x) [35,49]. fi For a sink ((Xi, qi) −→ X)i∈I, the final structure, q on X is defined as ( q F ⊇ [x]>, x 6∈ ∪i∈I fi(Xi); → ⇐⇒ F x ⇒ > qi F ⊇ fi (Gi), ∃i ∈ I, xi ∈ Xi, Gi ∈ FL (Xi) s.t f (xi) = x, Gi → xi. Mathematics 2019, 7, 370 4 of 14

When X = ∪i∈I fi(Xi), the final structure q can be characterized as

q ⇒ qi F → x ⇐⇒ F ⊇ fi (Gi) for some Gi → xi with f (xi) = x.

Let >(X) denote the set of all >-convergence structures on a set X. For p, q ∈ >(X), we say that q is finer than p (or p is coarser than q), denoted as p ≤ q for short, if the identity idX : (X, q) −→ (X, p) is continuous. It has been known that (>(X), ≤) forms a completed lattice. The discrete (resp., indiscrete) δ structure δ (resp., ι) is the top (resp., bottom) element of (T(X), ≤), where δ is defined as F −→ x iff ι > F ⊇ [x]>; and ι is defined as F −→ x for all F ∈ FL (X), x ∈ X [42].

Remark 1. When L = {⊥, >}, >-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 φ : J −→ FL (X) be any function. Then we define a function φˆ : LX → LJ as X _ ∀λ ∈ L , ∀j ∈ J, φˆ(λ)(j) = Λ(φ(j))(λ) = SX(µ, λ). µ∈φ(j)

> X For any F ∈ FL (J), it is known that the subset of L defined by

kφF := {λ ∈ LX|φˆ(λ) ∈ F} forms a >-filter on X, called diagonal >-filter of F under φ [31]. It was shown that SX(λ, µ) ≤ X SJ(φˆ(λ), φˆ(µ)) for any λ, µ ∈ L .

> > Lemma 3. Let f : X −→ Y and φ : J −→ FL (X) be functions. Then for any F ∈ FL (J) we have ⇒ ⇒ f (kφF) = k( f ◦ φ)F.

⇒ ⇒ → ⇒ Proof. f (kφF) ⊆ k( f ◦ φ)F. By Lemma2 (3) we need only check that f (λ) ∈ k( f ◦ φ)F for any λ ∈ kφF. Take λ ∈ kφF then φˆ(λ) ∈ F. Please note that ∀j ∈ J,

_ _ → → _ → ⇒ → φˆ(λ)(j) = SX(µ, λ) ≤ SY( f (µ), f (λ)) ≤ SY(ν, f (λ)) = f\◦ φ( f (λ))(j), µ∈φ(j) µ∈φ(j) ν∈ f ⇒(φ(j))

→ → → ⇒ i.e., φˆ(λ) ≤ f\⇒ ◦ φ( f (λ)), and so f\⇒ ◦ φ( f (λ)) ∈ F, i.e., f (λ) ∈ k( f ◦ φ)F. ⇒ ⇒ ⇒ k( f ◦ φ)F ⊆ f (kφF). For any λ ∈ k( f ◦ φ)F we have f\⇒ ◦ φ(λ) ∈ F. By Lemma2 (4), ∀j ∈ J,

⇒ _ _ → _ ← ← f\◦ φ(λ)(j) = SY(ν, λ) = SY( f (µ), λ) ≤ SX(µ, f (λ)) = φˆ( f (λ))(j), ν∈ f ⇒(φ(j)) µ∈φ(j) µ∈φ(j)

← ← ← ⇒ i.e., f\⇒ ◦ φ(λ) ≤ φˆ( f (λ)), and so φˆ( f (λ)) ∈ F, i.e., f (λ) ∈ kφF then λ ∈ f (kφF).

X X Definition 4. [36] Let (X, p) be a >-convergence space. For each λ ∈ L , the L-set λp ∈ L defined by _ _ _ ∀x ∈ X, λp(x) = Λ(F)(λ) = SX(µ, λ) p p µ∈ F−→x F−→x F Mathematics 2019, 7, 370 5 of 14 is called closure of λ w.r.t p. > For any F ∈ FL (X), the closure of F regarding p, denoted as clp(F), is defined to be the >-filter generated by {λp|λ ∈ F} as a >-filter base.

Lemma 4. [36] Let (X, p) be a >-convergence space. Then for all λ, µ ∈ LX we get

(1) λ ≤ λp; (2) λ ≤ µ implies λp ≤ µp; (3)S X(λ, µ) ≤ SX(λ, µ).

Let N be the set of natural numbers containing 0 and let (X, p) be a >-convergence space. For any > 0 F ∈ FL (X), we define clp(F) = F. Furthermore, for any n ∈ N, we define the n + 1 th iteration of the n+1 n n closure >-filter of F as clp (F) = clp(clp (F)) if clp (F) 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 F, G ∈ FL (X). Then for any n ∈ N, n (1) clp (F) ⊆ F, n n (2) if F ⊆ G, then clp (F) ⊆ clp (G), 0 0 n n (3) if p ∈ T(X) and p ≤ p , then clp (F) ⊆ clp0 (F).

Definition 5. A function f : (X, q) −→ (Y, p) between >-convergence spaces is said to be a closure function → → X if f (λ)p ≤ f (λq) for any λ ∈ L .

Proposition 2. Suppose that f : (X, q) −→ (Y, p) is a function between >-convergence spaces and > F ∈ FL (X), n ∈ N. ⇒ n n ⇒ (1) If f is a continuous function, then f (clq (F)) ⊇ clp ( f (F)). ⇒ n n ⇒ (2) If f is a closure function, then f (clq (F)) ⊆ clp ( f (F)).

Proof. (1) Let’s prove it by mathematical induction. ← ← Y Firstly, we check f (λ)q ≤ f (λp) for any λ ∈ L . In fact, for any x ∈ X, by continuity of f we obtain

← _ _ ← _ _ → f (λ)q(x) = SX(µ, f (λ)) = SY( f (µ), λ) q µ∈ q µ∈ G−→x G G−→x G _ _ → _ _ ← ≤ SY( f (µ), λ) ≤ SY(ν, λ) = f (λp)(x). p f →(µ)∈ f ⇒( ) p ν∈ f ⇒(G)−→ f (x) G H−→ f (x) H

⇒ n n ⇒ ⇒ ← Secondly, we prove f (clq (F)) ⊇ clp ( f (F)) when n = 1. Let λ ∈ f (F), i.e., f (λ) ∈ F. ← ← ← ⇒ Then by f (λ)q ≤ f (λp) we have f (λp) ∈ clq(F), i.e., λp ∈ f (clq(F)). It follows by Lemma2 (3) ⇒ ⇒ that f (clq(F)) ⊇ clp( f (F)). ⇒ n n ⇒ ⇒ n Thirdly, we assume that f (clq (F)) ⊇ clp ( f (F)) when n = k. Then we prove f (clq (F)) ⊇ n ⇒ clp ( f (F)) when n = k + 1. In fact,

⇒ k+1 ⇒ k ⇒ k k ⇒ k+1 ⇒ f (clq (F)) = f (clq(clq(F))) ⊇ clp( f (clq(F))) ⊇ clp(clp( f (F))) = clp ( f (F)).

(2) We prove only that the inequality holds for n = 1, and the rest of the proof is similar to (1). → ⇒ For any λ ∈ F, we have λq ∈ clq(F) and then f (λ) ∈ f (F). From f is a closure function, we → → ⇒ → ⇒ conclude that f (λq) ≥ f (λ)p ∈ clp( f (F)) and so f (λq) ∈ clp( f (F)). By Lemma2 (1), (3) we ⇒ ⇒ obtain f (clq(F)) ⊆ clp( f (F)). 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). Mathematics 2019, 7, 370 6 of 14

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. > q q p-(TC): ∀F ∈ FL (X), ∀x ∈ X, F −→ x =⇒ clp(F) −→ x.

Remark 2. When L = {⊥, >}, 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 p-(TR): Let J, X be any sets, ψ : J −→ X, and φ : J −→ FL (X) such that φ(j) −→ ψ(j), for each > q ⇒ q j ∈ J. Then for each F ∈ FL (J) and each x ∈ X, kφF −→ x implies ψ (F) −→ x.

Remark 3. When L = {⊥, >}, a >-convergence space degenerates into a convergence space, and the condition p-(TC) degenerates into the Fischer diagonal condition Rp,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, φ, ψ be defined as in p-(TR). ˆ ← X Then SX(µp, λ) ≤ SJ(φ(µ), ψ (λ)) for all λ, µ ∈ L .

Proof. Let λ, µ ∈ LX.

^ _ p SX(µp, λ) = ([ Λ(G)(µ)] → λ(x)), by ψ(j) ∈ X, φ(j) −→ ψ(j) x∈X p G−→x ^ ^ ≤ (Λ(φ(j))(µ) → λ(ψ(j))) = (φˆ(µ)(j) → ψ←(λ)(j)) j∈J j∈J ← = SJ(φˆ(µ), ψ (λ)).

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).

> q Proof. p-(TC)=⇒p-(TR). Let J, X, φ, ψ be defined as in p-(TR). Assume that F ∈ FL (J) and kφF −→ x. q Then it follows by p-(TC) that clp(kφF) −→ x. ⇒ Next we prove that clp(kφF) ⊆ ψ (F). Indeed, for any λ ∈ clp(kφF), we have

Lemma5 _ _ ˆ ← _ ˆ ← _ ← > = SX(µp, λ) ≤ SJ(φ(µ), ψ (λ)) = SJ(φ(µ), ψ (λ)) ≤ SJ(ν, ψ (λ)), µ∈kφF µ∈kφF φˆ(µ)∈F ν∈F

← ⇒ which means ψ (λ) ∈ F, i.e., λ ∈ ψ (F). q ⇒ ⇒ q Now we have known that clp(kφF) −→ x and clp(kφF) ⊆ ψ (F). Therefore, ψ (F) −→ x, as desired. p-(TR)=⇒p-(TC). Let

> p > J = {(G, y) ∈ FL (X) × X|G −→ y}; ψ : J −→ X, (G, y) 7→ y; φ : J −→ FL (X), (G, y) 7→ G.

p Then ∀j ∈ J, φ(j) −→ ψ(j). Please note that j = (G, y) ∈ J ⇐⇒ G = φ(j), y = ψ(j). Mathematics 2019, 7, 370 7 of 14

X ˆ ← (1) For any λ, µ ∈ L , SX(µp, λ) = SJ(φ(µ), ψ (λ)). Indeed,

^ _ ^ ^ SX(µp, λ) = ([ Λ(G)(µ)] → λ(y))= (Λ(G)(µ) → λ(y)) y∈X p y∈X ( ,y)∈J G−→y G ^ ^ = (Λ(φ(j))(µ) → λ(ψ(j))) = (φˆ(µ)(j) → ψ←(λ)(j)) j=(G,y)∈J j∈J ← = SJ(φˆ(µ), ψ (λ)).

> ˆ (2) For each F ∈ FL (X), the family {φ(λ)|λ ∈ F} forms a >-filter base on J. Indeed, p (TB1): For any λ ∈ F, by [y]> −→ y for any y ∈ X, we have ______φˆ(λ)(j) = Λ(φ(j))(λ) = Λ(G)(λ) ≥ Λ([y]>)(λ) = λ(y) = >. j∈J j∈J y∈X p y∈X y∈X G−→y

(TB2): For any λ, µ ∈ F, note that for any j ∈ J, _ _ φˆ(λ)(j) ∧ φˆ(µ)(j) = SX(λ1, λ) ∧ SX(µ1, µ) λ1∈φ(j) µ1∈φ(j) _ ≤ SX(λ1 ∧ µ1, λ ∧ µ) λ1,µ1∈φ(j) _ ≤ SX(ν, λ ∧ µ) = φˆ(λ ∧ µ)(j), ν∈φ(j) i.e., φˆ(λ) ∧ φˆ(µ) ≤ φˆ(λ ∧ µ). It follows easily that (TB2) is satisfied. We denote the >-filter generated by {φˆ(λ)|λ ∈ F} as Fφ. > φ ˆ φ φ (3) For each F ∈ FL (X), kφF ⊇ F. Indeed, for any λ ∈ F, we have φ(λ) ∈ F , i.e., λ ∈ kφF . > ⇒ φ (4) For each F ∈ FL (X), ψ (F ) = clp(F). Indeed,

( ) ⇒ φ ← φ _ ˆ ← 1 _ λ ∈ ψ (F ) ⇐⇒ ψ (λ) ∈ F ⇐⇒ SJ (φ(µ), ψ (λ)) = > ⇐⇒ SX(µp, λ) = > ⇐⇒ λ ∈ clp(F). µ∈F µ∈F

q q Assume that F −→ x, then by (3), we have kφFφ ⊇ F , and so kφFφ −→ x. From p-(TR) and (4), ⇒ φ q we get that clp(F) = ψ (F ) −→ x. Therefore, the condition p-(TC) is satisfied.

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

Theorem 2. Let {(Xi, qi, pi)}i∈I be pairs of >-convergence spaces such that each qi is pi-regular. If q (resp., p) fi fi is the initial structure on X regarding the source (X −→ (Xi, qi))i∈I (resp., (X −→ (Xi, pi))i∈I), then q is also p-regular.

> p Proof. Let ψ : J −→ X and φ : J −→ FL (X) be any function such that φ(j) −→ ψ(j) for any j ∈ J. Then ⇒ ⇒ pi ∀i ∈ I, ∀j ∈ J, ( fi ◦ φ)(j) = fi (φ(j)) −→ fi(ψ(j)) = ( fi ◦ ψ)(j). > q Let F ∈ FL (J) satisfy kφF −→ x. Then by definition of q and Lemma3 we have

⇒ ⇒ qi ∀i ∈ I, k( fi ◦ φ)F = fi (kφF) −→ fi(x).

⇒ ⇒ ⇒ qi Since qi is pi-regular we have fi ψ (F) = ( fi ◦ ψ) (F) −→ fi(x). By definition of q we have ⇒ q ψ (F) −→ x. Thus q is p-regular. Mathematics 2019, 7, 370 8 of 14

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

Theorem 3. Let {(Xi, qi, pi)}i∈I be pairs of >-convergence spaces such that each qi is pi-regular. Let q fi fi (resp., p) be the final structure on X relative to the sink ((Xi, qi) −→ X)i∈I (resp., ((Xi, pi) −→ X)i∈I). If X = ∪i∈I fi(Xi) and each fi : (Xi, pi) −→ (X, p) is a closure function, then q is also p-regular.

> q > Proof. Let F ∈ FL (X) −→ x. Then by definition of q, there exists i ∈ I, xi ∈ Xi, Gi ∈ FL (Xi), fi(xi) = q q −→i ⇒( ) ⊆ ( ) −→i x such that Gi xi and fi Gi F. Because qi is pi-regular we get clpi Gi xi and then q q ⇒( ( )) −→ ( ⇒( )) −→ fi clpi Gi x. By fi is a closure function and Proposition2 (2) it follows that clp fi Gi x. q ⇒ Hence clp(F) −→ x from clp( fi (Gi)) ⊆ clp(F). Thus q is p-regular.

For any {qi}i∈I ⊆ >(X), note that the supremum (resp., infimum) of {qi}i∈I in the lattice >(X), denoted as sup{qi|i ∈ I} (resp., inf{qi|i ∈ I}), is precisely the initial structure (resp., final structure) idX idX regarding the source (X −→ (X, qi))i∈I (resp., the sink ((X, qi) −→ X)i∈I). By Theorems2 and3, we obtain easily the following corollary. It will show us that p-regularity is preserved under supremum and infimum in the lattice >(X).

Corollary 1. Let {qi|i ∈ I} ⊆ >(X) and p ∈ >(X) with each (X, qi) being p-regular. Then both inf{qi}i∈I and sup{qi}i∈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.

q n q (1) If q is p-regular, then F −→ x implies clp (F) −→ x for any n ∈ N. (2) If q is p-regular, then q is p0-regular for any p ≤ p0. (3) The indiscrete structure ι is p-regular for any p ∈ >(X).

Proof. It is obvious.

4.1. Lower p-Regular Modification It has been known that p-regularity is preserved under supremum in the lattice >(X) (see Corollary1), and the indiscrete structure ι is p-regular for any p ∈ >(X) (see Lemma6 (3)). So, it follows easily that for a pair of >-convergence spaces (X, p, q), there is a finest p-regular >-convergence structure γpq on X which is coarser than q.

Definition 7. Let (X, p, q) be a pair of >-convergence spaces. Then the >-convergence structure γpq on X is said to be the lower p-regular modification of q.

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

γ q p q n Theorem 4. For any p, q ∈ >(X), F −→ x ⇐⇒ ∃n ∈ N, G −→ x s.t. F ⊇ clp (G).

0 0 q q n 0 Proof. We define q as F −→ x ⇐⇒ ∃n ∈ N, G −→ x s.t. F ⊇ clp (G), then we prove γpq = q . 0 0 0 0 q Obviously, q ∈ >(X) and q ≤ q. We check that q is p-regular. In fact, let F −→ x. Then q n n n+1 there exists n ∈ N, G −→ x such that F ⊇ clp (G). It follows that clp(F) ⊇ clp(clp (G)) = clp (G), 0 q 0 so clp(F) −→ x. Now, we have proved that q is p-regular. Mathematics 2019, 7, 370 9 of 14

0 0 q Let r be p-regular with r ≤ q. We prove below r ≤ q . In fact, let F −→ x. Then there exists q n r n ∈ N, G −→ x such that F ⊇ clp (G), so G −→ x by q ≤ r. Because r is p-regular it follows by n r 0 Lemma6 (1) that F ⊇ clp (G) −→ x. Therefore, r ≤ q .

Theorem 5. If f : (X, q) −→ (X0, q0) and f : (X, p) −→ (X0, p0) are both continuous function between 0 0 >-convergence spaces then so is f : (X, γpq) −→ (X , γp0 q ).

> Proof. For any F ∈ FL (X) and x ∈ X.

γpq q n F −→ x =⇒ ∃n ∈ N, G −→ x s.t. F ⊇ clp (G) 0 ⇒ q ⇒ ⇒ n =⇒ ∃n ∈ N, f (G) −→ f (x) s.t. f (F) ⊇ f (clp (G)) 0 ⇒ q ⇒ n ⇒ =⇒ ∃n ∈ N, f (G) −→ f (x) s.t. f (F) ⊇ clp0 ( f (G)) 0 γ 0 q ⇒ p =⇒ f (F) −→ ( f (x)), where the second implication uses the continuity of f : (X, q) −→ (X0, q0), and the third implication uses the continuity of f : (X, p) −→ (X0, p0) and Proposition2(1).

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

Theorem 6. Let {(Xi, qi, pi)}i∈I be pairs of spaces in >-CS and let q be the final structure relative to the fi sink ((Xi, qi) −→ X)i∈I with X = ∪i∈I fi(Xi). If p ∈ >(X) such that each fi : (Xi, pi) −→ (X, p) is a fi continuous closure function, then γpq is the final structure relative to the sink ((Xi, γpi qi) −→ X)i∈I.

fi Proof. Let s denote the final structure relative to the sink ((Xi, γpi qi) −→ X)i∈I. Then for any > F ∈ FL (X) and x ∈ X

γ q s pi i ⇒ F −→ x =⇒ ∃i ∈ I, xi ∈ Xi, fi(xi) = x, Gi −→ xi s.t. fi (Gi) ⊆ F, by Theorem4 q =⇒ ∃ ∈ ∈ ∈ ( ) = −→i n ( ) ⊆ ⇒( ) ⊆ ( ) i I, xi Xi, n N, fi xi x, Hi xi s.t. clpi Hi Gi, fi Gi F, by Proposition2 1 q =⇒ ∃ ∈ ∈ ∈ ⇒( ) −→ n( ⇒( )) ⊆ ⇒( n ( )) ⊆ ⇒( ) ⇒( ) ⊆ i I, xi Xi, n N, fi Hi x s.t. clp fi Hi fi clpi Hi fi Gi , fi Gi F ⇒ q n ⇒ =⇒ ∃i ∈ I, xi ∈ Xi, n ∈ N, fi (Hi) −→ x s.t. clp ( fi (Hi)) ⊆ F

γpq =⇒ F −→ x.

Conversely,

γpq q n F −→ x =⇒ ∃n ∈ N, G −→ x s.t. clp (G) ⊆ F qi ⇒ n =⇒ ∃i ∈ I, xi ∈ Xi, fi(xi) = x, n ∈ N, Hi −→ xi s.t. fi (Hi) ⊆ G, clp (G) ⊆ F qi n ⇒ n n =⇒ ∃i ∈ I, xi ∈ Xi, fi(xi) = x, n ∈ N, Hi −→ xi s.t. clp ( fi (Hi)) ⊆ clp (G), clp (G) ⊆ F q =⇒ ∃i ∈ I x ∈ X f (x ) = x n ∈ −→i x f ⇒(cln ( )) ⊆ cln( f ⇒( )) ⊆ , i i, i i , N, Hi i s.t. i pi Hi p i Hi F γp qi =⇒ ∃i ∈ I x ∈ X f (x ) = x n ∈ cln ( ) −→i x f ⇒(cln ( )) ⊆ , i i, i i , N, pi Hi i s.t. i pi Hi F s =⇒ F −→ x, where the fourth implication follows by Proposition2(2). Mathematics 2019, 7, 370 10 of 14

The following corollary tells us that lower p-regular modification has good compatibility with infimum in the lattice >(X).

Corollary 2. Assume that {qi|i ∈ I} ⊆ >(X), p ∈ >(X) and q = inf{qi|i ∈ I}. Then γpq = inf{γpqi|i ∈ I}.

4.2. Upper p-Regular Modification Similar to the crisp case, the discrete structure δ is not always p-regular for any p ∈ >(X). This shows that for a given q ∈ >(X), 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 γpq 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 γpq depends on the existence of a p-regular >-convergence structure finer than q (see Corollary1), and γpδ is the finest p-regular >-convergence structure. So, p it follows easily that γ q exists if and only if q ≤ γpδ. By Theorem4, we obtain the following result.

Theorem 7. Let (X, p, q) be a pair of >-convergence spaces. Then p n q γ q exists ⇐⇒ ∀x ∈ X, ∀n ∈ N, clp ([x]>) −→ x.

> Proof. For any F ∈ FL (X) and any x ∈ X, from Theorem4 we obtain

γpδ δ n F −→ x ⇐⇒ ∃n ∈ N, G −→ x s.t. clp (G) ⊆ F.

p Necessity. Let γ q exist. Then q ≤ γpδ. So, for any x ∈ X, n ∈ N

δ n γpδ n q [x]> −→ x =⇒ clp ([x]>) −→ x =⇒ clp ([x]>) −→ x.

n q Sufficiency. Let clp ([x]>) −→ x for any x ∈ X, n ∈ N. Then

γpδ δ n F −→ x =⇒ ∃n ∈ N, G −→ x s.t. clp (G) ⊆ F n =⇒ ∃n ∈ N, [x]> ⊆ G s.t. clp (G) ⊆ F

Proposition1 (2) n n n =⇒ ∃n ∈ N, clp ([x]>) ⊆ clp (G) s.t. clp (G) ⊆ F n =⇒ ∃n ∈ N s.t. clp ([x]>) ⊆ F q =⇒ F −→ x.

p It follows that q ≤ γpδ, so γ 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 γpq exists. Then p γ q n q F −→ x ⇐⇒ ∀n ∈ N, clp (F) −→ x.

0 0 q n q Proof. We define q as F −→ x ⇐⇒ ∀n ∈ N, clp (F) −→ x. (1) q0 ∈ >(X). It is obvious. 0 0 q 0 q (2) q ≤ q . In fact, let F −→ x then F = clp(F) −→ x. Mathematics 2019, 7, 370 11 of 14

0 0 q n n+1 q (3) q is p-regular. In fact, let F −→ x. Then for any n ∈ N it holds that clp (clp(F)) = clp (F) −→ x, 0 q 0 which means clp(F) −→ x. So, q is p-regular. 0 r (4) Let r be p-regular with q ≤ r. Then q ≤ r. In fact, let F −→ x then for any n ∈ N, by 0 n r n q q Proposition6 (1) it holds that clp (F) −→ x and so clp (F) −→ x by q ≤ r. That means F −→ x. By (1)–(4) we get that q0 is the coarsest p-regular >-convergence structure finer than q. Therefore, γpq = q0.

Theorem 9. Let f : (X, q) −→ (X0, q0) be a continuous function, and f : (X, p) −→ (X0, p0) be a closure 0 0 function between >-convergence spaces. If γpq and γp q0 exist then f : (X, γpq) −→ (X0, γp q0) is continuous.

p γ q n q 0 0 Proof. Let F −→ x. Then ∀n ∈ N, clp (F) −→ x. Since f : (X, q) −→ (X , q ) is a continuous function and f : (X, p) −→ (X0, p0) is a closure function it holds that

0 n ⇒ ⇒ n q ∀n ∈ N, clp0 ( f (F)) ⊇ f (clp (F)) −→ f (x),

p0 0 ⇒ γ q so f (F) −→ f (x), as desired.

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

Theorem 10. Let {(Xi, qi, pi)}i∈I be pairs of spaces in >-CS and q be the initial structure relative to the source fi (X −→ (Xi, qi))i∈I. Let p ∈ >(X) such that each fi : (X, p) −→ (Xi, pi) is continuous closure function. p p p If γ i qi exists for all i ∈ I then so does γ q, and γ q is precisely the initial structure relative to the source fi p (X −→ (Xi, γ i qi))i∈I.

p n q Proof. At first, we show the existence of γ q. By Theorem7, it suffices to check that clp ([x]>) −→ x qi x ∈ X n ∈ pi q cln ([ f (x)]) −→ f (x) for any , N. In fact, by the existence of γ i we have pi i i for any i ∈ I, x ∈ X, n ∈ N. Then by each fi : (X, p) −→ (Xi, pi) being a continuous closure function it holds that

q f ⇒(cln([x] ) = cln ( f ⇒([x] )) = cln ([ f (x)] ) −→i f (x) i p > pi i > pi i > i ,

n q p so clp ([x]>) −→ x for any x ∈ X, n ∈ N, i.e., γ q exists. fi p Let s denote the initial structure relative to the source (X −→ (Xi, γ i qi))i∈I. Then

p s γ i qi Theorem8 qi −→ x ⇐⇒ ∀i ∈ I f ⇒( ) −→ f (x) ⇐⇒ ∀i ∈ I ∀n ∈ cln ( f ⇒( )) −→ f (x) F , i F i , N, pi i F i Proposition2 ⇒ n qi ⇐⇒ ∀i ∈ I, ∀n ∈ N, fi (clp (F)) −→ fi(x) p n q Theorem8 γ q ⇐⇒ ∀n ∈ N, clp (F) −→ x ⇐⇒ F −→ x.

The following corollary tells us that upper p-regular modification has good compatibility with supremum in the lattice >(X).

p Corollary 3. Assume that {qi|i ∈ I} ⊆ >(X), p ∈ >(X) and q = sup{qi|i ∈ I}. If γ qi exists for all i ∈ I p p p then so does γ q and γ q = sup{γ qi|i ∈ I}. Mathematics 2019, 7, 370 12 of 14

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 >(X). 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.

Author Contributions: Both authors contributed equally in the writing of this article. Funding: This work is supported by National Natural Science Foundation of China (No. 11801248, 11501278) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09). Acknowledgments: The authors thank the reviewer and the editor for their valuable comments and suggestions. Conflicts of Interest: The authors declare no conflict of interest.

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