# Lattice-Valued Convergence Spaces: Weaker Regularity andp-Regularity

Authors: Lingqiang Li; Qiu Jin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

By using some lattice-valued Kowalsky’s dual diagonal conditions, some weaker regularities for Jäger’s generalized stratifiedL-convergence spaces and those for Boustique et al’s stratified L-convergence spaces are defined and studied. Here, the lattice L is a complete Heyting algebra. Some characterizations and properties of weaker regularities are presented. For Jäger’s generalized stratified L-convergence spaces, a notion of closures of stratified L-filters is introduced and then a new p-regularity is defined. At last, the relationships between p-regularities and weaker regularities are established.


---

## Body

## 1. Introduction

In 1954, Kowalsky [1] introduced a diagonal condition (the K-diagonal condition) to characterize whenever a pretopological convergence space is topological. In 1967, Cook and Fischer [2] defined a stronger diagonal condition (the F-diagonal condition) which, as they showed therein, is necessary and sufficient for a convergence space to be topological. Furthermore, a dual version of F (the DF-diagonal condition) is necessary and sufficient for a convergence space to be regular. Regularity can also be characterized by the requirement that, for each filter 𝔽, if 𝔽 converges to x then so does 𝔽¯ (the closure of 𝔽). In [3, 4], by considering a pair of convergence spaces (X,p) and (X,q), Kent and his coauthors introduced a kind of relative topologicalness (resp., regularity) which was called p-topologicalness (resp., p-regularity). They discussed p-topologicalness (resp., p-regularity) both by neighborhood (resp., closure) of filter [5] and generalized F (resp., DF)-diagonal condition. When p=q, p-topologicalness (resp., p-regularity) is precisely topologicalness (resp., regularity). In 1996, Kent and Richardson defined a weaker regularity by using the duality of Kowalsky’s diagonal condition. They also proved that weaker regularity, regularity, and p-regularity were distinct notions but closely related to each other [6].In [7], Jäger investigated a kind of lattice-valued convergence spaces, which were called generalized stratified L-convergence spaces. Later, the theory of these spaces was extensively discussed under different lattice context [8–19]. A supercategory of generalized stratified L-convergence spaces, called levelwise stratified L-convergence spaces in this paper, was researched in [20–24]. Indeed, a generalized stratified L-convergence space is precisely a left-continuous levelwise stratified L-convergence space [22].Lattice-valuedK- and F-diagonal conditions for generalized stratified L-convergence spaces were studied in [11, 12, 17, 18] and those for levelwise stratified L-convergence spaces were discussed in [18, 23]. Both by lattice-valued DF-diagonal condition and α-level closures of stratified L-filters, the lattice-valued regularity for generalized stratified L-convergence spaces was presented in [13] and that for levelwise stratified L-convergence spaces was given in [20, 21]. Later, by α-level closures of stratified L-filters, p-regularity for levelwise generalized stratified L-convergence spaces was studied in [24]. Recently, p-topologicalness and p-regularity for generalized stratified L-convergence spaces and that for level stratified L-convergence spaces were discussed systemically in [25].In this paper, for generalized stratifiedL-convergence spaces and levelwise stratified L-convergence spaces, we will discuss some lattice-valued weaker regularities, p-regularities, and their relationships. The content is arranged as follows. Section 2 recalls some basic notions as preliminary. Section 3 presents the definitions, characterizations, and properties of lattice-valued weaker regularities. Section 4 presents a notion of closures of stratified L-filters and a new lattice-valued p-regularity for stratified generalized L-convergence spaces. Also, the relationships between lattice-valued weaker regularities and lattice-valued p-regularities are established.


## 2. Preliminaries

In this paper, if not otherwise specified,L=(L,≤) is always a complete lattice with a top element 1 and a bottom element 0, which satisfies the distributive law α∧(⋁i∈Iβi)=⋁i∈I(α∧βi). A lattice with these conditions is called a complete Heyting algebra or a frame. The operation →:L×L→L given by α→β=∨{γ∈L:α∧γ≤β} is called the residuation with respect to ∧. A complete Heyting algebra L is said to be a complete Boolean algebra if it obeys the law of double negation:∀α∈L, (α→0)→0=α.For a setX, the set LX of functions from X to L with the pointwise order becomes a complete lattice. Each element of LX is called an L-set (or a fuzzy subset) of X. For any λ∈LX, 𝒦⊆LX, and α∈L, we denote by α∧λ, α→λ, ∨𝒦, and ∧𝒦 the L-sets defined by (α∧λ)(x)=α∧λ(x), (α→λ)(x)=α→λ(x), (∨𝒦)(x)=⋁μ∈𝒦μ(x), and (∧𝒦)(x)=⋀μ∈𝒦μ(x). Also, we make no difference between a constant function and its value since no confusion will arise. For a crisp subset A⊆X, let 1A be the characteristic function; that is 1A(x)=1 if x∈A and 1A(x)=0 if x∉A. Clearly, the characteristic function 1A of a subset A⊆X can be regarded as a function from X to L.LetX be a set. A fuzzy partial order (or an L-partial order) on X [26] is a function R:X×X→L such that (1) R(a,a)=1 for every a∈X (reflexivity); (2) R(a,b)=R(b,a)=1 implies that a=b for all a,b∈X (antisymmetry); (3) R(a,b)∧R(b,c)≤R(a,c) for all a,b,c∈X (transitivity). The pair (X,R) is called an L-partially ordered set.Let[LX]:LX×LX→L be a function defined by [LX](λ,μ)=⋀x∈X(λ(x)→μ(x)); then [LX] is an L-partial order on LX. The value [LX](λ,μ)∈L is interpreted as the degree that λ is contained in μ. In the sequel, we use the symbol [λ,μ] to denote [LX](λ,μ) for simplicity.Letf:X→Y be an ordinary function. We define f→:LX→LY and f←:LY→LX [27] by f→(λ)(y)=⋁f(x)=yλ(x) for λ∈LX and y∈Y, and f←(μ)=μ∘f for μ∈LY.


### 2.1. StratifiedL-(Ultra)filters

A stratifiedL-filter [27] on a set X is a function ℱ:LX→L such that for each λ,μ∈LX and each α∈L, (F1) ℱ(0)=0, ℱ(1)=1; (F2) ℱ(λ)∧ℱ(μ)=ℱ(λ∧μ); (Fs) ℱ(α)≥α. A stratified L-filter ℱ is called tight if ℱ(α)=α for each α∈L [5]. It is proved in [27] that all stratified L-filters are tight if and only if L is a complete Boolean algebra. It is easily seen that for a stratified L-filter ℱ on X, we have ∀λ∈LX, ℱ(λ)=⋁μ∈LX(ℱ(μ)∧[μ,λ]).The setℱLs(X) of all stratified L-filters on X is ordered by ℱ≤𝒢⇔∀λ∈LX, ℱ(λ)≤𝒢(λ). It is shown in [27] that the partially ordered set (ℱLs(X),≤) has maximal elements which are called stratified L-ultrafilters. The set of all stratified L-ultrafilters on X is denoted as 𝒰Ls(X). Let ℱ∈ℱLs(X). Then ℱ is an L-ultrafilter if and only if for all λ∈LX we have ℱ(λ)=ℱ(λ→0)→0. A stratified L-filter ℱ is called a stratified L-prime filter if ℱ(λ∨μ)=ℱ(λ)∨ℱ(μ) for each λ,μ∈LX. And when L is a complete Boolean algebra then ℱ=⋀ℱ≤𝒢∈𝒰Ls(X)𝒢 and ℱ is prime whenever ℱ is maximal [27].For eachℱ∈ℱLs(X), it is easily seen that 𝔽ℱ={A⊆X∣ℱ(1A)=1} is a filter on X. For each λ∈LX, take ιλ={x∈X∣λ(x)>0}. Let 𝔽 be a filter on X. Then, when L is a linearly order frame or 0∈L is prime (α∧β=0 implies α=0 or β=0), the function ℱ𝔽:LX→L, defined by ∀λ∈LX, ℱ𝔽(λ)=1 if ιλ∈𝔽 and ℱ𝔽(λ)=0 if not so, is a stratified L-filter on X [22]. Also, when L is a linearly order frame or 0∈L is prime, a stratified L-ultrafilter takes values in {0,1} only [10].Lemma 1 (Jäger [28] for L=[0,1]).
LetL be a linearly order frame or let 0∈L be prime. Then, for each ℱ∈𝒰Ls(X), 𝔽ℱ is an ultrafilter on X and ℱ=ℱ𝔽ℱ.Proof.
At first, we check that𝔽ℱ is an ultrafilter on X. For each A⊆X, we assume that A∉𝔽ℱ; that is,  ℱ(1A)=0; then ℱ(1X-A)=ℱ(1X-A→0)→0=ℱ(1A)→0=1. That means X-A∈𝔽ℱ. By the arbitrariness of A we get that 𝔽ℱ is an ultrafilter on X. At second, we check ℱ≤ℱ𝔽ℱ. Note that ℱ takes values in {0,1} only; thus, it suffices to prove that if ℱ(λ)=1; then ℱ𝔽ℱ(λ)=1. Indeed, let ℱ(λ)=1; then ℱ(1ιλ)≥ℱ(λ)=1; that is,  ιλ∈𝔽ℱ and so ℱ𝔽ℱ(λ)=1. Therefore, ℱ≤ℱ𝔽ℱ and it follows that ℱ=ℱ𝔽ℱ by the maximality of ℱ.The following examples belong to the folklore; we list them here because the notations are needed.Example 2.
(1) For each pointx in a set X, the function [x]:LX→L, [x](λ)=λ(x) is a stratified L-filter on X. In general, [x] is not a stratified L-ultrafilter. But when L is a complete Boolean algebra, then it is so.
(2) Let{ℱj∣j∈J} be a family of stratified L-filters on X; then ⋀j∈Jℱj, in particular, ℱ0=∧ℱLs(X), is a stratified L-filter on X.
(3) Letf:X→Y be a function. If ℱ∈ℱLs(X), then the function f⇒(ℱ)∈ℱLs(Y), where f⇒(ℱ):LY→L defined by λ↦ℱ(λ∘f). If ℱ∈𝒰Ls(X), then f⇒(ℱ)∈𝒰Ls(Y).There is a natural fuzzy partial order onℱLs(X) inherited from L(LX). Precisely, for all ℱ,𝒢∈ℱLs(X), if we let [ℱLs(X)](ℱ,𝒢)=[LLX](ℱ,𝒢)=⋀λ∈LX(ℱ(λ)→𝒢(λ)), then [ℱLs(X)] is an L-partially order. For simplicity, we use the symbol [ℱ,𝒢] to denote the value [ℱLs(X)](ℱ,𝒢) below.


### 2.2. Lattice-Valued Convergence Spaces

Definition 3.
A generalized stratifiedL-convergence structure [7] on a set X is a function lim:ℱLs(X)→LX satisfying (LC1)∀x∈X, lim[x](x)=1; and (LC2)∀ℱ,𝒢∈ℱLs(X), ℱ≤𝒢⇒limℱ≤lim𝒢. The pair (X,lim) is called a generalized stratified L-convergence space. If lim further satisfies the strong axiom (LC2′)∀ℱ,𝒢∈ℱLs(X),[ℱ,𝒢]∧limℱ≤lim𝒢, then the pair (X,lim) is called a strong stratified L-convergence space [8, 15, 16].A functionf:X→X′ between two generalized stratified L-convergence spaces (X,lim), (X′,lim′) is called continuous if for all ℱ∈ℱLs(X) and all x∈X we have limℱ(x)≤lim′f⇒(ℱ)(f(x)). The category SL-GCS has as objects all generalized stratified L-convergence spaces and as morphisms the continuous functions. This category is topological over SET [7, 10]. For a given source (X→fi(Xi,limi))i∈I, the initial structure, lim on X is defined by ∀ℱ∈ℱLs(X), ∀x∈X, limℱ(x)=⋀i∈Ilimifi⇒(ℱ)(fi(x)).Definition 4.
A collectionq¯=(qα)α∈L, where qα:ℱLs(X)→𝒫(X), is called a levelwise stratified L-convergence structure on X [20] if it satisfies the following:(LL1)
[x]→qαx for each x∈X;(LL2)
𝒢≥ℱ→qαx implies 𝒢→qαx;(LL3)
ℱ→qαx implies ℱ→qβx whenever β≤α.The notation, ℱ→qαx, means that x∈qα(ℱ). The pair (X,q¯) is called a levelwise stratified L-convergence space.A functionf:X→X′ between two levelwise stratified L-convergence spaces (X,q¯), (X′,q′-) is called continuous if for all ℱ∈ℱLs(X) all x∈X, and all α∈L we have ℱ→qαx implies f⇒(ℱ)→qα′f(x). The category SL-LCS has as objects all levelwise stratified L-convergence spaces and as morphisms the continuous functions. This category is topological over SET [20, 21]. For a given source (X→fi(Xi,qi¯))i∈I, the initial structure, q¯ on X is defined by ℱ→qαx⇔∀i∈I, fi⇒(ℱ)→qαifi(x)(ℱ∈ℱLs(X),x∈X,α∈L).


## 2.1. StratifiedL-(Ultra)filters

A stratifiedL-filter [27] on a set X is a function ℱ:LX→L such that for each λ,μ∈LX and each α∈L, (F1) ℱ(0)=0, ℱ(1)=1; (F2) ℱ(λ)∧ℱ(μ)=ℱ(λ∧μ); (Fs) ℱ(α)≥α. A stratified L-filter ℱ is called tight if ℱ(α)=α for each α∈L [5]. It is proved in [27] that all stratified L-filters are tight if and only if L is a complete Boolean algebra. It is easily seen that for a stratified L-filter ℱ on X, we have ∀λ∈LX, ℱ(λ)=⋁μ∈LX(ℱ(μ)∧[μ,λ]).The setℱLs(X) of all stratified L-filters on X is ordered by ℱ≤𝒢⇔∀λ∈LX, ℱ(λ)≤𝒢(λ). It is shown in [27] that the partially ordered set (ℱLs(X),≤) has maximal elements which are called stratified L-ultrafilters. The set of all stratified L-ultrafilters on X is denoted as 𝒰Ls(X). Let ℱ∈ℱLs(X). Then ℱ is an L-ultrafilter if and only if for all λ∈LX we have ℱ(λ)=ℱ(λ→0)→0. A stratified L-filter ℱ is called a stratified L-prime filter if ℱ(λ∨μ)=ℱ(λ)∨ℱ(μ) for each λ,μ∈LX. And when L is a complete Boolean algebra then ℱ=⋀ℱ≤𝒢∈𝒰Ls(X)𝒢 and ℱ is prime whenever ℱ is maximal [27].For eachℱ∈ℱLs(X), it is easily seen that 𝔽ℱ={A⊆X∣ℱ(1A)=1} is a filter on X. For each λ∈LX, take ιλ={x∈X∣λ(x)>0}. Let 𝔽 be a filter on X. Then, when L is a linearly order frame or 0∈L is prime (α∧β=0 implies α=0 or β=0), the function ℱ𝔽:LX→L, defined by ∀λ∈LX, ℱ𝔽(λ)=1 if ιλ∈𝔽 and ℱ𝔽(λ)=0 if not so, is a stratified L-filter on X [22]. Also, when L is a linearly order frame or 0∈L is prime, a stratified L-ultrafilter takes values in {0,1} only [10].Lemma 1 (Jäger [28] for L=[0,1]).
LetL be a linearly order frame or let 0∈L be prime. Then, for each ℱ∈𝒰Ls(X), 𝔽ℱ is an ultrafilter on X and ℱ=ℱ𝔽ℱ.Proof.
At first, we check that𝔽ℱ is an ultrafilter on X. For each A⊆X, we assume that A∉𝔽ℱ; that is,  ℱ(1A)=0; then ℱ(1X-A)=ℱ(1X-A→0)→0=ℱ(1A)→0=1. That means X-A∈𝔽ℱ. By the arbitrariness of A we get that 𝔽ℱ is an ultrafilter on X. At second, we check ℱ≤ℱ𝔽ℱ. Note that ℱ takes values in {0,1} only; thus, it suffices to prove that if ℱ(λ)=1; then ℱ𝔽ℱ(λ)=1. Indeed, let ℱ(λ)=1; then ℱ(1ιλ)≥ℱ(λ)=1; that is,  ιλ∈𝔽ℱ and so ℱ𝔽ℱ(λ)=1. Therefore, ℱ≤ℱ𝔽ℱ and it follows that ℱ=ℱ𝔽ℱ by the maximality of ℱ.The following examples belong to the folklore; we list them here because the notations are needed.Example 2.
(1) For each pointx in a set X, the function [x]:LX→L, [x](λ)=λ(x) is a stratified L-filter on X. In general, [x] is not a stratified L-ultrafilter. But when L is a complete Boolean algebra, then it is so.
(2) Let{ℱj∣j∈J} be a family of stratified L-filters on X; then ⋀j∈Jℱj, in particular, ℱ0=∧ℱLs(X), is a stratified L-filter on X.
(3) Letf:X→Y be a function. If ℱ∈ℱLs(X), then the function f⇒(ℱ)∈ℱLs(Y), where f⇒(ℱ):LY→L defined by λ↦ℱ(λ∘f). If ℱ∈𝒰Ls(X), then f⇒(ℱ)∈𝒰Ls(Y).There is a natural fuzzy partial order onℱLs(X) inherited from L(LX). Precisely, for all ℱ,𝒢∈ℱLs(X), if we let [ℱLs(X)](ℱ,𝒢)=[LLX](ℱ,𝒢)=⋀λ∈LX(ℱ(λ)→𝒢(λ)), then [ℱLs(X)] is an L-partially order. For simplicity, we use the symbol [ℱ,𝒢] to denote the value [ℱLs(X)](ℱ,𝒢) below.


## 2.2. Lattice-Valued Convergence Spaces

Definition 3.
A generalized stratifiedL-convergence structure [7] on a set X is a function lim:ℱLs(X)→LX satisfying (LC1)∀x∈X, lim[x](x)=1; and (LC2)∀ℱ,𝒢∈ℱLs(X), ℱ≤𝒢⇒limℱ≤lim𝒢. The pair (X,lim) is called a generalized stratified L-convergence space. If lim further satisfies the strong axiom (LC2′)∀ℱ,𝒢∈ℱLs(X),[ℱ,𝒢]∧limℱ≤lim𝒢, then the pair (X,lim) is called a strong stratified L-convergence space [8, 15, 16].A functionf:X→X′ between two generalized stratified L-convergence spaces (X,lim), (X′,lim′) is called continuous if for all ℱ∈ℱLs(X) and all x∈X we have limℱ(x)≤lim′f⇒(ℱ)(f(x)). The category SL-GCS has as objects all generalized stratified L-convergence spaces and as morphisms the continuous functions. This category is topological over SET [7, 10]. For a given source (X→fi(Xi,limi))i∈I, the initial structure, lim on X is defined by ∀ℱ∈ℱLs(X), ∀x∈X, limℱ(x)=⋀i∈Ilimifi⇒(ℱ)(fi(x)).Definition 4.
A collectionq¯=(qα)α∈L, where qα:ℱLs(X)→𝒫(X), is called a levelwise stratified L-convergence structure on X [20] if it satisfies the following:(LL1)
[x]→qαx for each x∈X;(LL2)
𝒢≥ℱ→qαx implies 𝒢→qαx;(LL3)
ℱ→qαx implies ℱ→qβx whenever β≤α.The notation, ℱ→qαx, means that x∈qα(ℱ). The pair (X,q¯) is called a levelwise stratified L-convergence space.A functionf:X→X′ between two levelwise stratified L-convergence spaces (X,q¯), (X′,q′-) is called continuous if for all ℱ∈ℱLs(X) all x∈X, and all α∈L we have ℱ→qαx implies f⇒(ℱ)→qα′f(x). The category SL-LCS has as objects all levelwise stratified L-convergence spaces and as morphisms the continuous functions. This category is topological over SET [20, 21]. For a given source (X→fi(Xi,qi¯))i∈I, the initial structure, q¯ on X is defined by ℱ→qαx⇔∀i∈I, fi⇒(ℱ)→qαifi(x)(ℱ∈ℱLs(X),x∈X,α∈L).


## 3. Lattice-Valued Weaker Regularities

In this section, we will present the definitions, characterizations, and properties of lattice-valued weaker regularities.LetX be a set; a function ϕ:X→ℱLs(X) is usually called an L-filter select function on X. We define ϕ^:LX→LX as ϕ^(λ):X→L, x↦ϕ(x)(λ). Let Σ(X) denote the set of all L-filter select functions on X, and let Σ*(X) be the subset consisting of all ϕ∈Σ such that ϕ(y)∈𝒰Ls(X) for all y∈X.Letϕ∈Σ(X). For all ℱ∈ℱLs(X), it can be proved that the function kLϕℱ:LX→L, defined by ∀λ∈LX, kLϕℱ(λ)=ℱ(ϕ^(λ)), is a stratified L-filter, which is called the L-diagonal filter of (ϕ,ℱ) [11, 17]. Then we have the following obvious lemma. It may have appeared in some other places.Lemma 5.
Letϕ,σ∈Σ(X) or Σ*(X). Then(1)
ϕ^(0)=0, ϕ^(1)=1;(2)
for eachλ,μ∈LX, ϕ^(λ∧μ)=ϕ^(λ)∧ϕ^(μ);(3)
σ≤ϕ implies σ^≤ϕ^;(4)
for allℱ,𝒢∈ℱLs(X), then [ℱ,𝒢]≤[kLϕℱ,kLϕ𝒢]. In particular, if ℱ≤𝒢 then kLϕℱ≤kLϕ𝒢.


### 3.1. For Generalized StratifiedL-Convergence Spaces

Let(X,lim) be a generalized stratified L-convergence space. We consider the following axioms.DLK. For each ϕ∈Σ(X), we have
(1)∀ℱ∈ℱLs(X),⋀y∈Xlimϕ(y)(y)≤[limkLϕℱ,limℱ].D
L
K
′
. Taking ϕ as ∀y∈X, limϕ(y)(y)=1 in DLK.ReplacingℱLs(X) by 𝒰Ls(X) in DLK (resp., DLK′), we obtain a weaker axiom in symbol DLK*(resp., DLK′*).Remark 6.
The axiomDLK is the dual axiom of LK which appeared in [11], and the axiom DLK′ is the dual axiom of LK′ which appeared in [17].Definition 7.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is called k-regular (resp., k′-regular, k*-regular, and k′*-regular) if it satisfies the axiom DLK (resp., DLK′, DLK*, and DLK′*).Lemma 8 (Li and Jin [25]).
Letϕ∈Σ(X) and ℱ∈ℱLs(X). We define ℱϕ:LX→L as ℱϕ(λ)=⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ]). Then ℱϕ satisfies (F1), (F2), and (Fs); thus, we say that ℱϕ is nearly a stratified L-filter. If ℱϕ∈ℱLs(X) then kLϕ(ℱϕ)≥ℱ.Lemma 9.
Letϕ∈Σ(X) and ℱ∈ℱLs(X). Then (kLϕℱ)ϕ∈ℱLs(X) and (kLϕℱ)ϕ≤ℱ.Proof.
For eachλ∈LX, we have
(2)(kLϕℱ)ϕ(λ)=⋁μ∈LX(kLϕℱ(μ)∧[ϕ^(μ),λ])=⋁μ∈LX(ℱ(ϕ^(μ))∧[ϕ^(μ),λ])≤ℱ(λ);
that is, (kLϕℱ)ϕ≤ℱ. It follows that (kLϕℱ)ϕ(0)=0. From the above lemma we have that (kLϕℱ)ϕ is a stratified L-filter on X.By the above two lemmas, we get the following characteristic theorem.Theorem 10.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is k-regular (resp.,k*-regular) if and only if, for each ϕ∈Σ(X)(resp., ϕ∈Σ*(X)), ⋀y∈Xlimϕ(y)(y)≤[limℱ,limℱϕ] whenever ℱϕ∈ℱLs(X).Proof.
We prove only fork-regularity. Assume the given condition is satisfied, let ϕ∈Σ(X) and ℱ∈ℱLs(X). By Lemma 9 we have (kLϕℱ)ϕ∈ℱLs(X) and
(3)⋀y∈Xlimϕ(y)(y)≤[limkLϕℱ,lim(kLϕℱ)ϕ]≤[limkLϕℱ,limℱ],
and so DLK holds; that is,  (X,lim) is k-regular.
Conversely, letℱ∈ℱLs(X), ϕ∈Σ(X) with ℱϕ∈ℱLs(X). By Lemma 8, kLϕ(ℱϕ)≥ℱ. It follows by DLK that
(4)[limℱ,limℱϕ]≥[limkLϕ(ℱϕ),limℱϕ]≥⋀y∈Xlimϕ(y)(y).
Thus, the requirement is satisfied.Corollary 11.
A generalized stratifiedL-convergence space (X,lim) is k′-regular (resp., k′*-regular) if and only if for each ϕ∈Σ(X) (resp., ϕ∈Σ*(X)) with limϕ(y)(y)=1 for all y∈X, we have limℱ≤limℱϕ whenever ℱϕ∈ℱLs(X).The following theorem considers lattice-valued weaker regularities w.r.t. the initial structures.Theorem 12.
Let(X,lim) be the initial structure relative to the source (X→fi(Xi,limi))i∈I with each fi:X→Xi being injective. Then if each (Xi,limi) is k-regular (resp., k′-regular), then the same is true of (X,lim).Proof.
We prove only fork-regularity. Let ϕ∈Σ(X). Fix i∈I; define ϕi∈Σ(Xi) as ϕi(y)=[y] if y∉fi(X) and ϕi(y)=fi⇒(ϕ(fi-1(y))) if y∈fi(X). Then for each i∈I, by lim[y](y)=1 it follows that
(5)⋀y∈Xilimiϕi(y)(y)=⋀y∈fi(X)limiϕi(y)(y)=⋀x∈Xlimifi⇒(ϕ(x))(fi(x)).
(In particular, if ∀x∈X, limϕ(x)(x)=1, then ∀y∈Xi, limiϕi(y)(y)=1).
For eachλ∈LXi and each x∈X, it follows that
(6)ϕ^(λ∘fi)(x)=ϕ(x)(λ∘fi)=fi⇒(ϕ(x))(λ)=ϕi(fi(x))(λ)=ϕ^i(λ)(fi(x)).
Hence, ϕ^(λ∘fi)=ϕ^i(λ)∘fi, and then, for each ℱ∈ℱLs(X),
(7)fi⇒(kLϕℱ)(λ)=kLϕℱ(λ∘fi)=ℱ(ϕ^(λ∘fi))=ℱ(ϕ^i(λ)∘fi)=fi⇒(ℱ)(ϕ^i(λ))=kLϕi(fi⇒(ℱ))(λ).
Therefore, fi⇒(kLϕℱ)=kLϕi(fi⇒(ℱ)). Then, for each x∈X,
(8)⋀y∈Xlimϕ(y)(y)∧limkLϕℱ(x)=⋀y∈X⋀i∈Ilimifi⇒(ϕ(y))(fi(y))∧⋀i∈Ilimifi⇒(kLϕℱ)(fi(x))=⋀i∈I⋀zi∈Xilimiϕi(zi)(zi)∧⋀i∈IlimikLϕi(fi⇒(ℱ))(fi(x))≤⋀i∈I(⋀zi∈Xilimiϕi(zi)(zi)∧limikLϕi(fi⇒(ℱ))(fi(x)))≤⋀i∈Ilimifi⇒(ℱ)(fi(x))=limℱ(x).
Here, the last inequality holds because each (Xi,limi) is k-regular. Now, we have proved that (X,lim) is k-regular.The following theorem gives the relationship between types of lattice-valued weaker regularities.Theorem 13.
LetL be a complete Boolean algebra. Then k-regularity ⇔k*-regularity and k′-regularity ⇔k′*-regularity.Proof.
We check only the equivalencek-regularity ⇔k*-regularity. The other equivalence is similar. Obviously, k-regularity ⇒k*-regularity. Conversely, let (X,lim) be k*-regular. Note that when L is a complete Boolean algebra, then for every stratified L-filter there exists a stratified L-ultrafilter containing it. Thus, for each ϕ∈Σ(X), there is some ϕ*∈Σ* such that ϕ(y)≤ϕ*(y) for all y∈X. Assume that ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X). Then it is easily seen that ℱϕ*≤ℱϕ and ℱϕ*∈ℱLs(X). By Theorem 10,
(9)⋀y∈Xlimϕ(y)(y)≤⋀y∈Xlimϕ*(y)(y)≤[limℱ,limℱϕ*]≤[limℱ,limℱϕ].
Thus, (X,lim) is k-regular.As a consequence, we obtain that whenL is a complete Boolean algebra, Theorem 12 holds for k*-regularity and k′*-regularity.Obviously,k-regularity ⇒k′-regularity and k*-regularity ⇒k′*-regularity. The following example shows that the reverse inclusions do not hold generally.Example 14.
LetX={x,y} and L={0,α,β,1} with ordering 0<α, β<1 and α∧β=0, α∨β=1. Then (L,∧) becomes a complete Boolean algebra. Obviously, [x] and [y] are all stratified L-ultrafilters on X. Thus, it is easily seen that the function lim:ℱLs(X)→LX defined by
(10)limℱ(x)={1,ℱ=[x];α,ℱ=[y];0,otherwise,limℱ(y)={1,ℱ=[y];β,ℱ=[x];0,otherwise,
is a generalized stratified L-convergence structure on X.
(1)
(
X
,
lim
)
satisfies
D
L
K
′
(
D
L
K
′
*
). Let ϕ∈Σ(X) with limϕ(x)(x)=limϕ(y)(y)=1. Then ϕ(x)=[x], ϕ(y)=[y]. Thus, for each ℱ∈ℱLs(X), we have kLϕℱ=ℱ. Then the axiom  DLK′, and thus the axiom  DLK′* holds obviously.
(2)
(
X
,
lim
)
does not satisfy
D
L
K
(
D
L
K
*
). Let ϕ∈Σ(X) be defined by ϕ(x)=ϕ(y)=[y]. Then, for each λ∈LX, we have ϕ^(λ)=λ(y). For each ℱ∈ℱLs(X),
(11)kLϕℱ(λ)=ℱ(ϕ^(λ))=ℱ(λ(y))=tightλ(y)=[y](λ);thatis,kLϕℱ=[y].
Taking 𝒢=[x]∧[y], then lim𝒢(x)=lim𝒢(y)=0, and limkLϕ𝒢(x)=lim[y](x)=α, limkLϕ𝒢(y)=lim[y](y)=1. It follows that
(12)α=⋀z∈Xlimϕ(z)(z)≰0=[limkLϕ𝒢,lim𝒢].
It follows that the axiom DLK* and thus the axiom DLK does not hold.


### 3.2.  For Levelwise StratifiedL-Convergence Spaces

Let(X,q¯) be a levelwise stratified L-convergence space. We consider the following axioms:DLLK. For each ϕ∈Σ(X) and each α∈L with ∀z∈X, ϕ(z)→qαz. Then ∀ℱ∈ℱLs(X), ∀x∈X, ℱ→qαx whenever kLϕℱ→qαx.ReplacingℱLs(X) by 𝒰Ls(X) in DLLK, we obtain a weaker axiom in symbol DLLK*.Remark 15.
The axiomDLLK is a special case of the regular axiom (R2) in [23] with J=X and ψ=id.Definition 16.
Let(X,q¯) be a levelwise stratified L-convergence space. Then (X,q¯) is called k-regular (resp., k*-regular) if it satisfies the axiom DLLK (resp., DLLK*).Fork-regularity (k*-regularity), we have the following characteristic theorem.Theorem 17.
Let(X,q¯) be a levelwise stratified L-convergence space. Then (X,q¯) is k-regular (resp., k*-regular) if and only if for each ℱ∈ℱLs(X) and each ϕ∈Σ(X)(resp., ϕ∈Σ*(X)) and each α∈L with ∀z∈X, ϕ(z)→qαz, we have that ℱ→qαx implies ℱϕ→qαx whenever ℱϕ∈ℱLs(X).Proof.
We prove only fork-regularity. Assume the given condition is satisfied; let ϕ∈Σ(X) satisfy the condition in DLLK and kLϕℱ→qαx. By Lemma 9 we have (kLϕℱ)ϕ∈ℱLs(X) and (kLϕℱ)ϕ≤ℱ. By the given condition, we have (kLϕℱ)ϕ→qαx and then ℱ→qαx. So, the axiom DLLK holds; that is,  (X,q¯) is k-regular. Conversely, Let ϕ∈Σ(X) and α∈L with ∀z∈X, ϕ(z)→qαz. Suppose that ℱ→qαx and ℱϕ∈ℱLs(X). By Lemma 8, kLϕ(ℱϕ)≥ℱ, so, kLϕ(ℱϕ)→qαx. It follows by DLLK that ℱϕ→qαx as desired.The following theorem shows thatk-regular is an initial property relative to any family of injection functions.Theorem 18.
Let(X,q¯) be the initial structure relative to the source (X→fi(Xi,qi¯))i∈I with each fi:X→Xi being injective. If each (Xi,qi¯) is k-regular, then the same is true of (X,q¯).Proof.
Letϕ∈Σ(X) and α∈L satisfy ϕ(x)→qαx for all x∈X. Fix i∈I; define ϕi∈Σ(Xi) as ϕi(y)=[y] if y∉fi(X) and ϕi(y)=fi⇒(ϕ(fi-1(y))) if y∈fi(X). Then ϕi(y)→qαy for each y∈Xi. Indeed, if y∉fi(X), then ϕi(y)=[y]→qαy, and if y∈fi(X), then there exists an x∈X such that fi(x)=y and so ϕi(y)=fi⇒(ϕ(x))→qαfi(x)=y. Let kLϕℱ→qαx. Similar to Theorem 12, we have fi⇒(kLϕℱ)=kLϕi(fi⇒(ℱ)) for all i∈I. Because each fi is continuous, thus kLϕi(fi⇒(ℱ))=fi⇒(kLϕℱ)→qαifi(x). Then fi⇒(ℱ)→qαifi(x) since each (X,qi¯) is k-regular. It follows that ℱ→qαx by the definition of initial structure. We have proved that (X,q¯) is k-regular.Theorem 19.
LetL be a complete Boolean algebra. Then k-regularity ⇔k*-regularity.Proof.
The proof is similar to Theorem13 and thus it is omitted.As a consequence, we obtain that whenL is a complete Boolean algebra, then Theorem 18 holds for k*-regularity.The last theorem gives the relationship betweenk-regularity for generalized stratified L-convergence space and k-regularity for levelwise stratified L-convergence space.Let(X,lim) be a generalized stratified L-convergence space. It is proved in [22] that the pair (X,qlim¯), where ℱ→(qlim)αx if and only if limℱ(x)≥α, is a levelwise stratified L-convergence space.Theorem 20.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is k-regular (resp., k*-regular) if and only if (X,qlim¯) is k-regular (resp., k*-regular).Proof.
We prove only fork-regularity. Let (X,lim) be k-regular. Take ϕ∈Σ(X) and α∈L with ∀z∈X, ϕ(z)→(qlim)αz; then we have α≤⋀y∈Xlimϕ(y)(y). Take ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X); then we have ℱ→(qlim)αx; that is,  limℱ(x)≥α. By Theorem 10 we obtain α≤⋀y∈Xlimϕ(y)(y)≤[limℱ,limℱϕ]. Then limℱϕ(x)≥α; that is, ℱϕ→(qlim)αx. It follows by Theorem 17 that (X,qlim¯) is k-regular.
Conversely, assume that(X,qlim¯) is k-regular. Let us take ϕ∈Σ(X) with ⋀y∈Xlimϕ(y)(y)=α and take ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X). Then if limℱ(x)=β for x∈X, we have ϕ(y)→(qlim)α∧βy and ℱ→(qlim)α∧βx. It follows by Theorem 17 that ℱϕ→(qlim)α∧βx; that is,  limℱϕ(x)≥α∧β. By the arbitrariness of x we note that ⋀y∈Xlimϕ(y)(y)=α≤[limℱ,limℱϕ]. It follows by Theorem 10 that (X,lim) is k-regular.


## 3.1. For Generalized StratifiedL-Convergence Spaces

Let(X,lim) be a generalized stratified L-convergence space. We consider the following axioms.DLK. For each ϕ∈Σ(X), we have
(1)∀ℱ∈ℱLs(X),⋀y∈Xlimϕ(y)(y)≤[limkLϕℱ,limℱ].D
L
K
′
. Taking ϕ as ∀y∈X, limϕ(y)(y)=1 in DLK.ReplacingℱLs(X) by 𝒰Ls(X) in DLK (resp., DLK′), we obtain a weaker axiom in symbol DLK*(resp., DLK′*).Remark 6.
The axiomDLK is the dual axiom of LK which appeared in [11], and the axiom DLK′ is the dual axiom of LK′ which appeared in [17].Definition 7.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is called k-regular (resp., k′-regular, k*-regular, and k′*-regular) if it satisfies the axiom DLK (resp., DLK′, DLK*, and DLK′*).Lemma 8 (Li and Jin [25]).
Letϕ∈Σ(X) and ℱ∈ℱLs(X). We define ℱϕ:LX→L as ℱϕ(λ)=⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ]). Then ℱϕ satisfies (F1), (F2), and (Fs); thus, we say that ℱϕ is nearly a stratified L-filter. If ℱϕ∈ℱLs(X) then kLϕ(ℱϕ)≥ℱ.Lemma 9.
Letϕ∈Σ(X) and ℱ∈ℱLs(X). Then (kLϕℱ)ϕ∈ℱLs(X) and (kLϕℱ)ϕ≤ℱ.Proof.
For eachλ∈LX, we have
(2)(kLϕℱ)ϕ(λ)=⋁μ∈LX(kLϕℱ(μ)∧[ϕ^(μ),λ])=⋁μ∈LX(ℱ(ϕ^(μ))∧[ϕ^(μ),λ])≤ℱ(λ);
that is, (kLϕℱ)ϕ≤ℱ. It follows that (kLϕℱ)ϕ(0)=0. From the above lemma we have that (kLϕℱ)ϕ is a stratified L-filter on X.By the above two lemmas, we get the following characteristic theorem.Theorem 10.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is k-regular (resp.,k*-regular) if and only if, for each ϕ∈Σ(X)(resp., ϕ∈Σ*(X)), ⋀y∈Xlimϕ(y)(y)≤[limℱ,limℱϕ] whenever ℱϕ∈ℱLs(X).Proof.
We prove only fork-regularity. Assume the given condition is satisfied, let ϕ∈Σ(X) and ℱ∈ℱLs(X). By Lemma 9 we have (kLϕℱ)ϕ∈ℱLs(X) and
(3)⋀y∈Xlimϕ(y)(y)≤[limkLϕℱ,lim(kLϕℱ)ϕ]≤[limkLϕℱ,limℱ],
and so DLK holds; that is,  (X,lim) is k-regular.
Conversely, letℱ∈ℱLs(X), ϕ∈Σ(X) with ℱϕ∈ℱLs(X). By Lemma 8, kLϕ(ℱϕ)≥ℱ. It follows by DLK that
(4)[limℱ,limℱϕ]≥[limkLϕ(ℱϕ),limℱϕ]≥⋀y∈Xlimϕ(y)(y).
Thus, the requirement is satisfied.Corollary 11.
A generalized stratifiedL-convergence space (X,lim) is k′-regular (resp., k′*-regular) if and only if for each ϕ∈Σ(X) (resp., ϕ∈Σ*(X)) with limϕ(y)(y)=1 for all y∈X, we have limℱ≤limℱϕ whenever ℱϕ∈ℱLs(X).The following theorem considers lattice-valued weaker regularities w.r.t. the initial structures.Theorem 12.
Let(X,lim) be the initial structure relative to the source (X→fi(Xi,limi))i∈I with each fi:X→Xi being injective. Then if each (Xi,limi) is k-regular (resp., k′-regular), then the same is true of (X,lim).Proof.
We prove only fork-regularity. Let ϕ∈Σ(X). Fix i∈I; define ϕi∈Σ(Xi) as ϕi(y)=[y] if y∉fi(X) and ϕi(y)=fi⇒(ϕ(fi-1(y))) if y∈fi(X). Then for each i∈I, by lim[y](y)=1 it follows that
(5)⋀y∈Xilimiϕi(y)(y)=⋀y∈fi(X)limiϕi(y)(y)=⋀x∈Xlimifi⇒(ϕ(x))(fi(x)).
(In particular, if ∀x∈X, limϕ(x)(x)=1, then ∀y∈Xi, limiϕi(y)(y)=1).
For eachλ∈LXi and each x∈X, it follows that
(6)ϕ^(λ∘fi)(x)=ϕ(x)(λ∘fi)=fi⇒(ϕ(x))(λ)=ϕi(fi(x))(λ)=ϕ^i(λ)(fi(x)).
Hence, ϕ^(λ∘fi)=ϕ^i(λ)∘fi, and then, for each ℱ∈ℱLs(X),
(7)fi⇒(kLϕℱ)(λ)=kLϕℱ(λ∘fi)=ℱ(ϕ^(λ∘fi))=ℱ(ϕ^i(λ)∘fi)=fi⇒(ℱ)(ϕ^i(λ))=kLϕi(fi⇒(ℱ))(λ).
Therefore, fi⇒(kLϕℱ)=kLϕi(fi⇒(ℱ)). Then, for each x∈X,
(8)⋀y∈Xlimϕ(y)(y)∧limkLϕℱ(x)=⋀y∈X⋀i∈Ilimifi⇒(ϕ(y))(fi(y))∧⋀i∈Ilimifi⇒(kLϕℱ)(fi(x))=⋀i∈I⋀zi∈Xilimiϕi(zi)(zi)∧⋀i∈IlimikLϕi(fi⇒(ℱ))(fi(x))≤⋀i∈I(⋀zi∈Xilimiϕi(zi)(zi)∧limikLϕi(fi⇒(ℱ))(fi(x)))≤⋀i∈Ilimifi⇒(ℱ)(fi(x))=limℱ(x).
Here, the last inequality holds because each (Xi,limi) is k-regular. Now, we have proved that (X,lim) is k-regular.The following theorem gives the relationship between types of lattice-valued weaker regularities.Theorem 13.
LetL be a complete Boolean algebra. Then k-regularity ⇔k*-regularity and k′-regularity ⇔k′*-regularity.Proof.
We check only the equivalencek-regularity ⇔k*-regularity. The other equivalence is similar. Obviously, k-regularity ⇒k*-regularity. Conversely, let (X,lim) be k*-regular. Note that when L is a complete Boolean algebra, then for every stratified L-filter there exists a stratified L-ultrafilter containing it. Thus, for each ϕ∈Σ(X), there is some ϕ*∈Σ* such that ϕ(y)≤ϕ*(y) for all y∈X. Assume that ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X). Then it is easily seen that ℱϕ*≤ℱϕ and ℱϕ*∈ℱLs(X). By Theorem 10,
(9)⋀y∈Xlimϕ(y)(y)≤⋀y∈Xlimϕ*(y)(y)≤[limℱ,limℱϕ*]≤[limℱ,limℱϕ].
Thus, (X,lim) is k-regular.As a consequence, we obtain that whenL is a complete Boolean algebra, Theorem 12 holds for k*-regularity and k′*-regularity.Obviously,k-regularity ⇒k′-regularity and k*-regularity ⇒k′*-regularity. The following example shows that the reverse inclusions do not hold generally.Example 14.
LetX={x,y} and L={0,α,β,1} with ordering 0<α, β<1 and α∧β=0, α∨β=1. Then (L,∧) becomes a complete Boolean algebra. Obviously, [x] and [y] are all stratified L-ultrafilters on X. Thus, it is easily seen that the function lim:ℱLs(X)→LX defined by
(10)limℱ(x)={1,ℱ=[x];α,ℱ=[y];0,otherwise,limℱ(y)={1,ℱ=[y];β,ℱ=[x];0,otherwise,
is a generalized stratified L-convergence structure on X.
(1)
(
X
,
lim
)
satisfies
D
L
K
′
(
D
L
K
′
*
). Let ϕ∈Σ(X) with limϕ(x)(x)=limϕ(y)(y)=1. Then ϕ(x)=[x], ϕ(y)=[y]. Thus, for each ℱ∈ℱLs(X), we have kLϕℱ=ℱ. Then the axiom  DLK′, and thus the axiom  DLK′* holds obviously.
(2)
(
X
,
lim
)
does not satisfy
D
L
K
(
D
L
K
*
). Let ϕ∈Σ(X) be defined by ϕ(x)=ϕ(y)=[y]. Then, for each λ∈LX, we have ϕ^(λ)=λ(y). For each ℱ∈ℱLs(X),
(11)kLϕℱ(λ)=ℱ(ϕ^(λ))=ℱ(λ(y))=tightλ(y)=[y](λ);thatis,kLϕℱ=[y].
Taking 𝒢=[x]∧[y], then lim𝒢(x)=lim𝒢(y)=0, and limkLϕ𝒢(x)=lim[y](x)=α, limkLϕ𝒢(y)=lim[y](y)=1. It follows that
(12)α=⋀z∈Xlimϕ(z)(z)≰0=[limkLϕ𝒢,lim𝒢].
It follows that the axiom DLK* and thus the axiom DLK does not hold.


## 3.2.  For Levelwise StratifiedL-Convergence Spaces

Let(X,q¯) be a levelwise stratified L-convergence space. We consider the following axioms:DLLK. For each ϕ∈Σ(X) and each α∈L with ∀z∈X, ϕ(z)→qαz. Then ∀ℱ∈ℱLs(X), ∀x∈X, ℱ→qαx whenever kLϕℱ→qαx.ReplacingℱLs(X) by 𝒰Ls(X) in DLLK, we obtain a weaker axiom in symbol DLLK*.Remark 15.
The axiomDLLK is a special case of the regular axiom (R2) in [23] with J=X and ψ=id.Definition 16.
Let(X,q¯) be a levelwise stratified L-convergence space. Then (X,q¯) is called k-regular (resp., k*-regular) if it satisfies the axiom DLLK (resp., DLLK*).Fork-regularity (k*-regularity), we have the following characteristic theorem.Theorem 17.
Let(X,q¯) be a levelwise stratified L-convergence space. Then (X,q¯) is k-regular (resp., k*-regular) if and only if for each ℱ∈ℱLs(X) and each ϕ∈Σ(X)(resp., ϕ∈Σ*(X)) and each α∈L with ∀z∈X, ϕ(z)→qαz, we have that ℱ→qαx implies ℱϕ→qαx whenever ℱϕ∈ℱLs(X).Proof.
We prove only fork-regularity. Assume the given condition is satisfied; let ϕ∈Σ(X) satisfy the condition in DLLK and kLϕℱ→qαx. By Lemma 9 we have (kLϕℱ)ϕ∈ℱLs(X) and (kLϕℱ)ϕ≤ℱ. By the given condition, we have (kLϕℱ)ϕ→qαx and then ℱ→qαx. So, the axiom DLLK holds; that is,  (X,q¯) is k-regular. Conversely, Let ϕ∈Σ(X) and α∈L with ∀z∈X, ϕ(z)→qαz. Suppose that ℱ→qαx and ℱϕ∈ℱLs(X). By Lemma 8, kLϕ(ℱϕ)≥ℱ, so, kLϕ(ℱϕ)→qαx. It follows by DLLK that ℱϕ→qαx as desired.The following theorem shows thatk-regular is an initial property relative to any family of injection functions.Theorem 18.
Let(X,q¯) be the initial structure relative to the source (X→fi(Xi,qi¯))i∈I with each fi:X→Xi being injective. If each (Xi,qi¯) is k-regular, then the same is true of (X,q¯).Proof.
Letϕ∈Σ(X) and α∈L satisfy ϕ(x)→qαx for all x∈X. Fix i∈I; define ϕi∈Σ(Xi) as ϕi(y)=[y] if y∉fi(X) and ϕi(y)=fi⇒(ϕ(fi-1(y))) if y∈fi(X). Then ϕi(y)→qαy for each y∈Xi. Indeed, if y∉fi(X), then ϕi(y)=[y]→qαy, and if y∈fi(X), then there exists an x∈X such that fi(x)=y and so ϕi(y)=fi⇒(ϕ(x))→qαfi(x)=y. Let kLϕℱ→qαx. Similar to Theorem 12, we have fi⇒(kLϕℱ)=kLϕi(fi⇒(ℱ)) for all i∈I. Because each fi is continuous, thus kLϕi(fi⇒(ℱ))=fi⇒(kLϕℱ)→qαifi(x). Then fi⇒(ℱ)→qαifi(x) since each (X,qi¯) is k-regular. It follows that ℱ→qαx by the definition of initial structure. We have proved that (X,q¯) is k-regular.Theorem 19.
LetL be a complete Boolean algebra. Then k-regularity ⇔k*-regularity.Proof.
The proof is similar to Theorem13 and thus it is omitted.As a consequence, we obtain that whenL is a complete Boolean algebra, then Theorem 18 holds for k*-regularity.The last theorem gives the relationship betweenk-regularity for generalized stratified L-convergence space and k-regularity for levelwise stratified L-convergence space.Let(X,lim) be a generalized stratified L-convergence space. It is proved in [22] that the pair (X,qlim¯), where ℱ→(qlim)αx if and only if limℱ(x)≥α, is a levelwise stratified L-convergence space.Theorem 20.
Let(X,lim) be a generalized stratified L-convergence space. Then (X,lim) is k-regular (resp., k*-regular) if and only if (X,qlim¯) is k-regular (resp., k*-regular).Proof.
We prove only fork-regularity. Let (X,lim) be k-regular. Take ϕ∈Σ(X) and α∈L with ∀z∈X, ϕ(z)→(qlim)αz; then we have α≤⋀y∈Xlimϕ(y)(y). Take ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X); then we have ℱ→(qlim)αx; that is,  limℱ(x)≥α. By Theorem 10 we obtain α≤⋀y∈Xlimϕ(y)(y)≤[limℱ,limℱϕ]. Then limℱϕ(x)≥α; that is, ℱϕ→(qlim)αx. It follows by Theorem 17 that (X,qlim¯) is k-regular.
Conversely, assume that(X,qlim¯) is k-regular. Let us take ϕ∈Σ(X) with ⋀y∈Xlimϕ(y)(y)=α and take ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X). Then if limℱ(x)=β for x∈X, we have ϕ(y)→(qlim)α∧βy and ℱ→(qlim)α∧βx. It follows by Theorem 17 that ℱϕ→(qlim)α∧βx; that is,  limℱϕ(x)≥α∧β. By the arbitrariness of x we note that ⋀y∈Xlimϕ(y)(y)=α≤[limℱ,limℱϕ]. It follows by Theorem 10 that (X,lim) is k-regular.


## 4. On the Relationship between Weaker Regularity andp-Regularity

### 4.1. For Generalized StratifiedL-Convergence Spaces

Generally,p-regularity relates to two different generalized stratified L-convergence structures on the same underlying set. Thus, in this section, we add the lowercases p,q as the superscript of lim and use limp, limq to denote different generalized stratified L-convergence structures.At first, we give the notion of closures of stratifiedL-filters and then introduce a new p-regularity.Definition 21.
Let(X,limp) be a generalized stratified L-convergence space. For each λ∈LX, the L-set λ¯p∈LX defined by
(13)∀x∈X,λ¯p(x)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))
is called the closure of λ w.r.t (X,limp).Lemma 22.
Let(X,limp) be a generalized stratified L-convergence space. Then for all λ,μ∈LX and all α∈L we get the following:(1)
λ≤λ¯p;(2)
λ≤μ implies λ¯p≤μ¯p;(3)
(β∧λ)¯p≥β∧λ¯p and the equality holds if L is a complete Boolean algebra;(4)
ifL is a complete Boolean algebra, then ∀x∈X, λ¯p(x)=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)), and (λ∨μ)¯p=λ¯p∨μ¯p.Proof.
(1) For eachx∈X, by limp[x](x)=1 we get λ¯p(x)≥[x](λ)=λ(x). So, λ≤λ¯p. Take λ=1 in (1); we obtain 1¯p=1.
(2) It follows from the property (F2) of stratifiedL-filters.
(3) For eachx∈X we have
(14)(β∧λ)¯p(x)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(β∧λ))=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(β)∧ℱ(λ))≥⋁ℱ∈ℱLs(X)(limpℱ(x)∧β∧ℱ(λ))=β∧⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))=β∧λ¯p(x).
When L is a complete Boolean algebra, then ∀ℱ∈ℱLs(X), ℱ(β)=β. So, the “≥” in the above inequality can be replaced by “=”. Thus, (β∧λ)¯p=β∧λ¯p.
(5) LetL be a complete Boolean algebra. That λ¯p(x)=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)) follows because, for each ℱ∈ℱLs(X), there exists an L-ultrafilter 𝒢 such that ℱ≤𝒢. To prove (λ∨μ)¯p=λ¯p∨μ¯p, it suffices to check that (λ∨μ)¯p≤λ¯p∨μ¯p since the reverse inequality holds by (2). Indeed, because each stratified L-ultrafilter is prime we have
(15)λ¯p(x)∨μ¯p(x)=(⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)))∨(⋁𝒢∈𝒰Ls(X)(limp𝒢(x)∧𝒢(μ)))=⋁ℱ,𝒢∈𝒰Ls(X)((limpℱ(x)∧ℱ(λ))∨(limp𝒢(x)∧𝒢(μ)))≥⋁ℱ∈𝒰Ls(X)((limpℱ(x)∧ℱ(λ))∨(limpℱ(x)∧ℱ(μ)))=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧(ℱ(λ)∨ℱ(μ)))=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ∨μ))=(λ∨μ)¯p(x).Theorem 23.
Let(X,limp) be a generalized stratified L-convergence space. For each ℱ∈ℱLs(X), the function ℱ¯p:LX→L defined by
(16)∀λ∈LX,ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])
is a stratifiedL-filter, called the closure of ℱ.Proof.
(F1) Thatℱ¯p(1)=1 is obvious. By Lemma 22(1) we have
(17)ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])≤⋁μ∈LX(ℱ(μ)∧[μ,λ])≤ℱ(λ).
Thus, ℱ¯p(0)=0.(F2) Firstly, note thatℱ¯p(λ)≤ℱ¯p(μ) whenever λ≤μ. It follows that ℱ¯p(λ∧μ)≤ℱ¯p(λ)∧ℱ¯p(μ). Conversely,
(18)ℱ¯p(λ)∧ℱ¯p(μ)=⋁a∈LX(ℱ(a)∧[a¯p,λ])∧⋁b∈LX(ℱ(b)∧[b¯p,μ])=⋁a,b∈LX(ℱ(a)∧ℱ(b)∧[a¯p,λ]∧[b¯p,μ])≤⋁a,b∈LX(ℱ(a∧b)∧[(a∧b)¯p,λ∧μ])≤⋁c∈LX(ℱ(c)∧[c¯p,λ∧μ])=ℱ¯p(λ∧μ).(Fs) For allβ∈L, it follows that ℱ¯p(β)=⋁μ∈LX(ℱ(μ)∧[μ¯p,β])≥ℱ(1)∧β=β by 1¯p=1.It is easily seen that the following lemma holds. We omit the routine proof.Lemma 24.
Let(X,limp) be a generalized stratified L-convergence space. Then, for each ℱ,𝒢∈ℱLs(X), [ℱ,𝒢]≤[ℱ¯p,𝒢¯p].Definition 25.
Let(X,limp,limq) be a pair of generalized stratified L-convergence spaces. Then (X,limq) is called p-regular if and only if, for each ℱ∈ℱLs(X), we have limqℱ≤limqℱ¯p.Remark 26.
WhenL={0,1}, a generalized stratified L-convergence space reduces to a convergence space. It is easily seen that ℱ¯p is precisely the filter generated by {A¯:A∈𝔽} as a filterbasis [29]. And the p-regularity reduces to the corresponding crisp notion in [3].The following theorem shows thatp-regularity is preserved under initial constructions.Theorem 27.
Let{(Xi,limqi,limpi)}i∈I be pairs of generalized stratified L-convergence spaces with each limqi being pi-regular. If  limq (resp., limp) is the initial structure on X relative to the source (X→fi(Xi,limqi))i∈I (resp., (X→fi(Xi,limpi))i∈I), then (X,limq) is p-regular.Proof.
At first, we check below that for eachi∈I and each λi∈LXi we have (fi←(λi))¯p≤fi←((λi)¯pi). Indeed, for each x∈X,
(19)(fi←(λi))¯p(x)=⋁𝒢∈ℱLs(X)(limp𝒢(x)∧𝒢(fi←(λi)))=⋁𝒢∈ℱLs(X)((⋀j∈Ilimpjfj⇒(𝒢)(fj(x)))∧𝒢(fi←(λi)))≤⋁𝒢∈ℱLs(X)(limpifi⇒(𝒢)(fi(x))∧fi⇒(𝒢)(λi))≤⋁𝒢i∈ℱLs(Xi)(limpi𝒢i(fi(x))∧𝒢i(λi))=fi←((λi)¯pi)(x).
It follow that, for eachℱ∈ℱLs(X) and each λi∈LXi,
(20)fi⇒(ℱ¯p)(λi)=ℱ¯p(fi←(λ))=⋁μ∈LX([μ¯p,fi←(λi)]∧ℱ(μ))≥⋁μi∈LXi([(fi←(μi))¯p,fi←(λi)]∧ℱ(fi←(μi)))≥⋁μi∈LXi([fi←((μi)¯pi),fi←(λi)]∧ℱ(fi←(μi)))≥⋁μi∈LXi([(μi)¯pi,λi]∧fi⇒(ℱ)(μi))=(fi⇒(ℱ))¯pi(λi).
Thus, fi⇒(ℱ¯p)≥(fi⇒(ℱ))¯pi for all i∈I. It follows by each (Xi,limqi) being pi-regular that
(21)limqℱ¯p(x)=⋀i∈Ilimqifi⇒(ℱ¯p)(fi(x))≥⋀i∈Ilimqi(fi⇒(ℱ))¯pi(fi(x))≥⋀i∈Ilimqifi⇒(ℱ)(fi(x))=limqℱ(x).
Thus, (X,limq) is p-regular.WhenL={0,1}, Kent and Richardson [6] studied the relationships between weaker regularities and p-regularity. Now we discuss them for the general case.Definition 28.
A generalized (strong) stratifiedL-convergence space (X,limq) is called
(i) a (strong)L-Kent convergence space [10] if ∀ℱ∈ℱLs(X), ∀x∈X, limqℱ(x)≤limq(ℱ∧[x])(x);
(ii) pretopological [11] if ∀ℱ∈ℱLs(X), ∀x∈X, limqℱ(x)=[𝒰q(x),ℱ], where 𝒰q(x), defined by ∀λ∈LX, 𝒰q(x)(λ)=⋀ℱ∈ℱLs(X)(limqℱ(x)→ℱ(λ)), is called the stratified neighborhood L-filter of x w.r.t. limq, and when (X,limq) is a strong stratified L-convergence space, then (X,limq) is pretopological if and only if it satisfies limq𝒰q(x)(x)=1 for all x∈X [17];
(iii) ultrapretopological if it is pretopological and for eachx∈X, there exists a stratified L-ultrafilter ℱx such that 𝒰q(x)=[x]∧ℱx;
(iv) topological [11] if there exists a stratified L-topology 𝒯 such that ∀λ∈LX, ∀x∈X, we have 𝒰q(x)(λ)=int(λ)(x), where int(λ)=⋁μ∈𝒯(μ∧[μ,λ]) is called the interior of λ w.r.t. 𝒯 [11, 30].Proposition 29.
Let(X,limq) be a strong stratified L-Kent convergence space which is p-regular relative to every ultrapretopological generalized stratified L-convergence structure limp≤limq. Then (X,limq) is k′*-regular.Proof.
Letϕ∈Σ*(X) with ∀y∈X, limqϕ(y)(y)=1. Let limp be the ultrapretopological generalized stratified L-convergence structure defined by ∀y∈X, 𝒰p(y)=ϕ(y)∧[y]. From ϕ(y)≥𝒰p(y) we have limpϕ(y)(y)=1. For each ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X), it follows that for each λ∈LX, λ¯p(y)=⋁ℱ∈ℱLs(X)(limpℱ(y)∧ℱ(λ))≥ϕ(y)(λ), which means λ¯p≥ϕ^(λ). Thus,
(22)ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])≤⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ])=ℱϕ(λ);
that is, ℱ¯p≤ℱϕ. Because (X,limq) is a strong L-Kent convergence space, then it follows that limq𝒰p(y)=limq(ϕ(y)∧[y])(y)≥limqϕ(y)(y)=1, and so
(23)∀𝒢∈ℱLs(X),∀y∈X,limp𝒢(y)=[𝒰p(y),𝒢]=limq𝒰p(y)∧[𝒰p(y),𝒢]≤(LC2′)limq𝒢(y).
That is, limp≤limq. It follows by the assumption that (X,limq) is p-regular. Thus limqℱϕ(x)≥limqℱ¯p(x)≥limqℱ(x). By Theorem 17 we know that (X,limq) is k′*-regular.It is easily seen that whenL is a complete Boolean algebra, then the above proposition holds for k′-regularity.Lemma 30.
Let(X,limq) be a topological generalized stratified L-convergence space and let 𝒯 be the stratified L-topology corresponding to limq. Then ℱ≥𝒰q(x) if and only if ℱ(μ)≥𝒰q(x)(μ) for all μ∈𝒯.Proof.
We need only to check the sufficiency. Note that to for eachμ∈LX, 𝒰q(x)(μ)=int(μ)(x) and 𝒰q(x)(μ)=int(μ)(x)=μ(x) if μ∈𝒯 [11, 30]. It follows that, for each λ∈LX,
(24)ℱ(λ)=⋁μ∈LX(ℱ(μ)∧[μ,λ])≥⋁μ∈𝒯(ℱ(μ)∧[μ,λ])≥⋁μ∈𝒯(𝒰q(x)(μ)∧[μ,λ])=⋁μ∈𝒯(μ(x)∧[μ,λ])=int(λ)(x)=𝒰q(x)(λ).Theorem 31.
LetL be a linearly order frame or let 0∈L be prime. A topological generalized stratified L-convergence space (X,limq) is k′*-regular if and only if it is p-regular for every ultrapretopological generalized stratified L-convergence structure limp≤limq.Proof.
Note that a topological generalized stratifiedL-convergence space is natural a strong stratified L-Kent convergence space [17]. Then the sufficiency follows by Proposition 29. Thus, we prove only the necessity. Let (X,limq) be k′*-regular and let limp be an arbitrary ultrapretopological generalized stratified L-convergence structure with limp≤limq. Then, for each y∈X, there exists a ℋy∈𝒰Ls(X) such that 𝒰p(y)=ℋy∧[y]. Obviously, limpℋy(y)≥limp𝒰p(y)(y)=1 and then limqℋy(y)=1 by limp≤limq.
Letϕ∈Σ*(X) be defined by ϕ(y)=ℋy, for all y∈X. Then limqϕ(y)(y)=1 for each y∈X. For each λ∈𝒯, we check below [λ¯p,ϕ^(λ)]=1. Here, 𝒯 is the stratified L-topology corresponding to limq. For each ϕ(y)∈𝒰Ls(X), it follows by Lemma 1 that ϕ(y)𝔽ϕ(y)=ϕ(y); that is,
(25)ϕ^(λ)(y)=ϕ(y)(λ)={1,ιλ∈𝔽ϕ(y);0,ιλ∉𝔽ϕ(y).
Note that [λ¯p,ϕ^(λ)]=⋀y∈ι(λ¯p)(λ¯p(y)→ϕ(y)(λ)). For each y∈ι(λ¯p), it follows that λ¯p(y)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))>0, which means that there exists an ℱy∈ℱLs(X) such that limpℱy(y)>0 and ℱy(λ)>0. Thus, ℱy(1ιλ)≥ℱy(λ)>0. Fix y∈ι(λ¯p); we have y∈ιλ or y∈X-ιλ.
Case 1. y∈ιλ; that is,  λ(y)>0. Because (X,limq) is topological, then λ(y)=𝒰q(y)(λ)>0. From limqϕ(y)(y)=1, we get ϕ(y)≥𝒰q(y) and then ϕ(y)(λ)>0; indeed, ϕ(y)(λ)=1 since ϕ(y)∈𝒰Ls(X) takes values in {0,1}.
Case 2. y∈X-ιλ; that is,  λ(y)=0. We assume that ϕ(y)(λ)≠1; it follows by equality (25) that ιλ∉𝔽ϕ(y). Because 𝔽ϕ(y) is an ultrafilter on X, then X-ι(λ)∈𝔽ϕ(y) and so ϕ(y)(1X-ιλ)=1. As we have known limpℱy(y)>0 and (X,limp) is ultrapretopological; hence, limpℱy(y)=[𝒰p(y),ℱy]>0, then by 𝒰p(y)(1X-ιλ)=ϕ(y)(1X-ιλ)∧[y](1X-ιλ)=1 it follows that ℱy(1X-ιλ)>0. Now,
(26)0=ℱy(1ιλ∧1X-ιλ)≥ℱy(1ιλ)∧ℱy(1X-ιλ)>0.
A contradiction! Thus, if y∈X-ιλ, then ϕ(y)(λ)=1.
Combining Cases 1 and 2 we get that ify∈ι(λ¯p) then ϕ^(λ)(y)=1. It follows immediately that [λ¯p,ϕ^(λ)]=1.
Next we prove thatkLϕ(𝒰q(x)¯p)≥𝒰q(x). By Lemma 30, we need only to check that kLϕ(𝒰q(x)¯p)(λ)≥𝒰q(x)(λ) for all λ∈𝒯. Indeed,
(27)kLϕ(𝒰q(x)¯p)(λ)=𝒰q(x)¯p(ϕ^(λ))=⋁μ∈LX(𝒰q(x)(μ)∧[μ¯p,ϕ^(λ)])≥𝒰q(x)(λ)∧[λ¯p,ϕ^(λ)]=𝒰q(x)(λ).
Then, for eachℱ∈ℱLs(X),
(28)limqℱ(x)=[𝒰q(x),ℱ]≤[𝒰q(x)¯p,ℱ¯p]≤[kLϕ(𝒰q(x)¯p),kLϕ(ℱ¯p)]≤[𝒰q(x),kLϕ(ℱ¯p)]=limqkLϕ(ℱ¯p)(x)≤limqℱ¯p(x),
where the first and the second equalities hold by the pretopologicalness of (X,limq), the first inequality holds by Lemma 24, the second inequality holds by Lemma 5(4), and the last inequality holds because (X,limq) is k′*-regular. Then it follows that (X,limq) is p-regular.Remark 32.
To prove that Theorem31 holds for k′-regularity, it seems that L must be a complete Boolean algebra. If we further assume that L is linearly ordered or 0∈L is prime then L={0,1}. Thus, we guess that Theorem 31 holds for k′-regularity only if L={0,1}.


### 4.2.  For Levelwise StratifiedL-Convergence Spaces

Definition 33 (see [31]).
Let(X,p¯) be a levelwise stratified L-convergence space. For each λ∈LX, the L-set λ¯pα∈LX defined by
(29)∀x∈X,λ¯pα(x)=⋁ℱ∈cpα(x)ℱ(λ),cpα(x)={ℱ∈ℱLs(X):ℱ⟶pαx}
is called α-level closure of λ w.r.t. (X,p¯).It is easily seen thatα-level closures of L-sets have similar properties to closures of L-sets. We do not list them but use them directly.In [20], Boustique and Richardson modified Jäger’s definition [11] and introduced a notion of α-level closures of stratified L-filters. In [25], we give an equivalent characterization of Boustique and Richardson's definition. This characterization seems more simple and more intuitive. Thus, we use it as the definition of α-level closures of stratified L-filters.Definition 34.
Let(X,p¯) be a levelwise stratified L-convergence space. For each α∈L and each ℱ∈ℱLs(X), it is easily seen that the function ℱ¯pα:LX→L, defined by ∀λ∈LX, ℱ¯pα(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯pα,λ]), is a stratified L-filter; then ℱ¯pα is called the α-level closure of ℱ w.r.t. (X,p¯).Definition 35 (see [24]).
Let(X,p¯,q¯) be a pair of levelwise stratified L-convergence spaces. Then (X,q¯) is called p-regular if, for each α∈L and each ℱ∈ℱLs(X), we have ℱ¯pα→qαx whenever ℱ→qαx.It is proved in [25] that p-regularity is preserved under initial constructions. Now, we look at the relationships between weaker regularities and p-regularity.Definition 36.
A levelwise stratifiedL-convergence space (X,q¯) is called(i)
anL-Kent convergence space if [x]∧ℱ→qαx whenever ℱ→qαx;(ii)
pretopological [23] if ℱ→qαx if and only if ℱ≥𝒰qα(x)=∧{ℱ∣ℱ→qαx};(iii)
ultrapretopological if, for eachx∈X and each α∈L, there exists a stratified L-ultrafilter ℱx such that 𝒰qα(x)=[x]∧ℱx;(iv)
topological [23] if there exists a stratified L-topology 𝒯α for each α∈L such that ∀λ∈LX, ∀x∈X, we have 𝒰qα(x)(λ)=intα(λ)(x), where intα(λ) is the interior of λ w.r.t. 𝒯α.Proposition 37.
Let(X,q¯) be a levelwise stratified L-Kent convergence space which is p-regular relative to every ultrapretopological levelwise stratified L-convergence structure p¯≥q¯. Here for p¯≥q¯, we mean that ℱ→pαx implies ℱ→qαx. Then (X,q¯) is k*-regular.Proof.
Letϕ∈Σ*(X) and α∈L with ∀y∈X, ϕ(y)→qαy. Let p¯ be the ultrapretopological levelwise stratified L-convergence structure defined by ∀α∈L, ∀y∈X, 𝒰pα(y)=ϕ(y)∧[y]. From ϕ(y)≥𝒰pα(y) we have ϕ(y)→pαy. For each ℱ∈ℱLs(X) such that ℱϕ∈ℱLs(X) and ℱ→qαx, it follows that for each λ∈LX, λ¯pα(y)=⋁ℱ∈cpα(y)ℱ(λ)≥ϕ(y)(λ), which means λ¯pα≥ϕ^(λ). Thus,
(30)ℱ¯pα(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯pα,λ])≤⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ])=ℱϕ(λ);
that is, ℱ¯pα≤ℱϕ. Because (X,q¯) is an L-Kent convergence space, then it follows by ϕ(y)→qαy that 𝒰pα(y)=ϕ(y)∧[y]→qαy. Thus, p¯≥q¯; then (X,q¯) is p-regular by the assumption. It follows that ℱ¯pα→qαx and then ℱϕ→qαx by ℱ¯pα≤ℱϕ. By Theorem 17 we know that (X,q¯) is k*-regular.It is easily seen that whenL is a complete Boolean algebra, then the above proposition holds for k-regularity.Lemma 38.
Let(X,q¯) be a topological levelwise stratified L-convergence space and let 𝒯α(α∈L) be the stratified L-topologies corresponding to q¯. Then ℱ≥𝒰qα(x) if and only if ℱ(μ)≥𝒰qα(x)(μ) for all μ∈𝒯α.Proof.
The proof is similar to Lemma30 and thus it is omitted.Theorem 39.
LetL be a linearly order frame or let 0∈L be prime. A topological levelwise stratified L-convergence space (X,q¯) is k*-regular if and only if it is p-regular for every ultrapretopological levelwise stratified L-convergence structure p¯≥q¯.Proof.
The sufficiency follows by Proposition37. We prove only the necessity. Let (X,q¯) be k*-regular and let p¯ be an arbitrary ultrapretopological levelwise stratified L-convergence structure with p¯≥q¯. Fix α∈L; then, for each y∈X, there exists a ℋy∈𝒰Ls(X) such that 𝒰pα(y)=ℋy∧[y]. Obviously, ℋy→pαy and then ℋy→qαy by p¯≥q¯.
Letϕ∈Σ*(X) be defined by ϕ(y)=ℋy, for all y∈X. For each λ∈𝒯α, we check below [λ¯pα,ϕ^(λ)]=1. Here, 𝒯α(α∈L) are the stratified L-topologies corresponding to q¯.
Note that[λ¯pα,ϕ^(λ)]=⋀y∈ι(λ¯pα)(λ¯pα(y)→ϕ(y)(λ)). For each y∈ι(λ¯pα), it follows that λ¯pα(y)=⋁ℱ∈cpα(y)ℱ(λ)>0, which means that there exists an ℱy→pαy such that ℱy(λ)>0. Thus, ℱy(1ιλ)≥ℱy(λ)>0. Fix y∈ι(λ¯pα); then y∈ιλ or y∈X-ιλ.
Case 1. y∈ιλ; that is, λ(y)>0. Because (X,q¯) is topological, thus λ(y)=𝒰qα(y)(λ)=∧{ℱ(λ)∣ℱ→qαy}>0. From ϕ(y)→qαy, we get ϕ(y)(λ)>0; indeed, ϕ(y)(λ)=1 since ϕ(y)∈𝒰Ls(X) takes values in {0,1}.
Case 2. y∈X-ιλ; that is,  λ(y)=0. We assume that ϕ(y)(λ)≠1; it follows by equality (25) that ιλ∉𝔽ϕ(y). Because 𝔽ϕ(y) is an ultrafilter on X, then X-ι(λ)∈𝔽ϕ(y) and so ϕ(y)(1X-ιλ)=1. As we have known ℱy→pαy; hence, ℱy≥𝒰pα(y)=ϕ(y)∧[y]; then ℱy(1X-ιλ)≥ϕ(y)(1X-ιλ)∧1X-ιλ(y)=1. Now,
(31)0=ℱy(1ιλ∧1X-ιλ)≥ℱy(1ιλ)∧ℱy(1X-ιλ)=ℱy(1ιλ)>0.
A contradiction! Thus, ify∈X-ιλ, then ϕ(y)(λ)=1.
Combining of Cases 1 and 2 we get that ify∈ι(λ¯pα) then ϕ^(λ)(y)=1. It follows immediately that [λ¯pα,ϕ^(λ)]=1. Then similar to Lemma 30 we have kLϕ(𝒰qα(x)¯pα)≥𝒰qα(x). Let ℱ→qαx; then ℱ≥𝒰qα(x) by the topologicalness of q¯. Hence, ℱ¯pα≥𝒰qα(x)¯pα and then kLϕ(ℱ¯pα)≥kLϕ(𝒰qα(x)¯pα)≥𝒰qα(x), which means kLϕ(ℱ¯pα)→qαx. Because (X,q¯) is k*-regular, then ℱ¯pα→qαx. It follows that (X,q¯) is p-regular.Remark 40.
Similar to Remark32, we guess that Theorem 39 holds for k-regularity only if L={0,1}.


## 4.1. For Generalized StratifiedL-Convergence Spaces

Generally,p-regularity relates to two different generalized stratified L-convergence structures on the same underlying set. Thus, in this section, we add the lowercases p,q as the superscript of lim and use limp, limq to denote different generalized stratified L-convergence structures.At first, we give the notion of closures of stratifiedL-filters and then introduce a new p-regularity.Definition 21.
Let(X,limp) be a generalized stratified L-convergence space. For each λ∈LX, the L-set λ¯p∈LX defined by
(13)∀x∈X,λ¯p(x)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))
is called the closure of λ w.r.t (X,limp).Lemma 22.
Let(X,limp) be a generalized stratified L-convergence space. Then for all λ,μ∈LX and all α∈L we get the following:(1)
λ≤λ¯p;(2)
λ≤μ implies λ¯p≤μ¯p;(3)
(β∧λ)¯p≥β∧λ¯p and the equality holds if L is a complete Boolean algebra;(4)
ifL is a complete Boolean algebra, then ∀x∈X, λ¯p(x)=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)), and (λ∨μ)¯p=λ¯p∨μ¯p.Proof.
(1) For eachx∈X, by limp[x](x)=1 we get λ¯p(x)≥[x](λ)=λ(x). So, λ≤λ¯p. Take λ=1 in (1); we obtain 1¯p=1.
(2) It follows from the property (F2) of stratifiedL-filters.
(3) For eachx∈X we have
(14)(β∧λ)¯p(x)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(β∧λ))=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(β)∧ℱ(λ))≥⋁ℱ∈ℱLs(X)(limpℱ(x)∧β∧ℱ(λ))=β∧⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))=β∧λ¯p(x).
When L is a complete Boolean algebra, then ∀ℱ∈ℱLs(X), ℱ(β)=β. So, the “≥” in the above inequality can be replaced by “=”. Thus, (β∧λ)¯p=β∧λ¯p.
(5) LetL be a complete Boolean algebra. That λ¯p(x)=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)) follows because, for each ℱ∈ℱLs(X), there exists an L-ultrafilter 𝒢 such that ℱ≤𝒢. To prove (λ∨μ)¯p=λ¯p∨μ¯p, it suffices to check that (λ∨μ)¯p≤λ¯p∨μ¯p since the reverse inequality holds by (2). Indeed, because each stratified L-ultrafilter is prime we have
(15)λ¯p(x)∨μ¯p(x)=(⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ)))∨(⋁𝒢∈𝒰Ls(X)(limp𝒢(x)∧𝒢(μ)))=⋁ℱ,𝒢∈𝒰Ls(X)((limpℱ(x)∧ℱ(λ))∨(limp𝒢(x)∧𝒢(μ)))≥⋁ℱ∈𝒰Ls(X)((limpℱ(x)∧ℱ(λ))∨(limpℱ(x)∧ℱ(μ)))=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧(ℱ(λ)∨ℱ(μ)))=⋁ℱ∈𝒰Ls(X)(limpℱ(x)∧ℱ(λ∨μ))=(λ∨μ)¯p(x).Theorem 23.
Let(X,limp) be a generalized stratified L-convergence space. For each ℱ∈ℱLs(X), the function ℱ¯p:LX→L defined by
(16)∀λ∈LX,ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])
is a stratifiedL-filter, called the closure of ℱ.Proof.
(F1) Thatℱ¯p(1)=1 is obvious. By Lemma 22(1) we have
(17)ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])≤⋁μ∈LX(ℱ(μ)∧[μ,λ])≤ℱ(λ).
Thus, ℱ¯p(0)=0.(F2) Firstly, note thatℱ¯p(λ)≤ℱ¯p(μ) whenever λ≤μ. It follows that ℱ¯p(λ∧μ)≤ℱ¯p(λ)∧ℱ¯p(μ). Conversely,
(18)ℱ¯p(λ)∧ℱ¯p(μ)=⋁a∈LX(ℱ(a)∧[a¯p,λ])∧⋁b∈LX(ℱ(b)∧[b¯p,μ])=⋁a,b∈LX(ℱ(a)∧ℱ(b)∧[a¯p,λ]∧[b¯p,μ])≤⋁a,b∈LX(ℱ(a∧b)∧[(a∧b)¯p,λ∧μ])≤⋁c∈LX(ℱ(c)∧[c¯p,λ∧μ])=ℱ¯p(λ∧μ).(Fs) For allβ∈L, it follows that ℱ¯p(β)=⋁μ∈LX(ℱ(μ)∧[μ¯p,β])≥ℱ(1)∧β=β by 1¯p=1.It is easily seen that the following lemma holds. We omit the routine proof.Lemma 24.
Let(X,limp) be a generalized stratified L-convergence space. Then, for each ℱ,𝒢∈ℱLs(X), [ℱ,𝒢]≤[ℱ¯p,𝒢¯p].Definition 25.
Let(X,limp,limq) be a pair of generalized stratified L-convergence spaces. Then (X,limq) is called p-regular if and only if, for each ℱ∈ℱLs(X), we have limqℱ≤limqℱ¯p.Remark 26.
WhenL={0,1}, a generalized stratified L-convergence space reduces to a convergence space. It is easily seen that ℱ¯p is precisely the filter generated by {A¯:A∈𝔽} as a filterbasis [29]. And the p-regularity reduces to the corresponding crisp notion in [3].The following theorem shows thatp-regularity is preserved under initial constructions.Theorem 27.
Let{(Xi,limqi,limpi)}i∈I be pairs of generalized stratified L-convergence spaces with each limqi being pi-regular. If  limq (resp., limp) is the initial structure on X relative to the source (X→fi(Xi,limqi))i∈I (resp., (X→fi(Xi,limpi))i∈I), then (X,limq) is p-regular.Proof.
At first, we check below that for eachi∈I and each λi∈LXi we have (fi←(λi))¯p≤fi←((λi)¯pi). Indeed, for each x∈X,
(19)(fi←(λi))¯p(x)=⋁𝒢∈ℱLs(X)(limp𝒢(x)∧𝒢(fi←(λi)))=⋁𝒢∈ℱLs(X)((⋀j∈Ilimpjfj⇒(𝒢)(fj(x)))∧𝒢(fi←(λi)))≤⋁𝒢∈ℱLs(X)(limpifi⇒(𝒢)(fi(x))∧fi⇒(𝒢)(λi))≤⋁𝒢i∈ℱLs(Xi)(limpi𝒢i(fi(x))∧𝒢i(λi))=fi←((λi)¯pi)(x).
It follow that, for eachℱ∈ℱLs(X) and each λi∈LXi,
(20)fi⇒(ℱ¯p)(λi)=ℱ¯p(fi←(λ))=⋁μ∈LX([μ¯p,fi←(λi)]∧ℱ(μ))≥⋁μi∈LXi([(fi←(μi))¯p,fi←(λi)]∧ℱ(fi←(μi)))≥⋁μi∈LXi([fi←((μi)¯pi),fi←(λi)]∧ℱ(fi←(μi)))≥⋁μi∈LXi([(μi)¯pi,λi]∧fi⇒(ℱ)(μi))=(fi⇒(ℱ))¯pi(λi).
Thus, fi⇒(ℱ¯p)≥(fi⇒(ℱ))¯pi for all i∈I. It follows by each (Xi,limqi) being pi-regular that
(21)limqℱ¯p(x)=⋀i∈Ilimqifi⇒(ℱ¯p)(fi(x))≥⋀i∈Ilimqi(fi⇒(ℱ))¯pi(fi(x))≥⋀i∈Ilimqifi⇒(ℱ)(fi(x))=limqℱ(x).
Thus, (X,limq) is p-regular.WhenL={0,1}, Kent and Richardson [6] studied the relationships between weaker regularities and p-regularity. Now we discuss them for the general case.Definition 28.
A generalized (strong) stratifiedL-convergence space (X,limq) is called
(i) a (strong)L-Kent convergence space [10] if ∀ℱ∈ℱLs(X), ∀x∈X, limqℱ(x)≤limq(ℱ∧[x])(x);
(ii) pretopological [11] if ∀ℱ∈ℱLs(X), ∀x∈X, limqℱ(x)=[𝒰q(x),ℱ], where 𝒰q(x), defined by ∀λ∈LX, 𝒰q(x)(λ)=⋀ℱ∈ℱLs(X)(limqℱ(x)→ℱ(λ)), is called the stratified neighborhood L-filter of x w.r.t. limq, and when (X,limq) is a strong stratified L-convergence space, then (X,limq) is pretopological if and only if it satisfies limq𝒰q(x)(x)=1 for all x∈X [17];
(iii) ultrapretopological if it is pretopological and for eachx∈X, there exists a stratified L-ultrafilter ℱx such that 𝒰q(x)=[x]∧ℱx;
(iv) topological [11] if there exists a stratified L-topology 𝒯 such that ∀λ∈LX, ∀x∈X, we have 𝒰q(x)(λ)=int(λ)(x), where int(λ)=⋁μ∈𝒯(μ∧[μ,λ]) is called the interior of λ w.r.t. 𝒯 [11, 30].Proposition 29.
Let(X,limq) be a strong stratified L-Kent convergence space which is p-regular relative to every ultrapretopological generalized stratified L-convergence structure limp≤limq. Then (X,limq) is k′*-regular.Proof.
Letϕ∈Σ*(X) with ∀y∈X, limqϕ(y)(y)=1. Let limp be the ultrapretopological generalized stratified L-convergence structure defined by ∀y∈X, 𝒰p(y)=ϕ(y)∧[y]. From ϕ(y)≥𝒰p(y) we have limpϕ(y)(y)=1. For each ℱ∈ℱLs(X) with ℱϕ∈ℱLs(X), it follows that for each λ∈LX, λ¯p(y)=⋁ℱ∈ℱLs(X)(limpℱ(y)∧ℱ(λ))≥ϕ(y)(λ), which means λ¯p≥ϕ^(λ). Thus,
(22)ℱ¯p(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯p,λ])≤⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ])=ℱϕ(λ);
that is, ℱ¯p≤ℱϕ. Because (X,limq) is a strong L-Kent convergence space, then it follows that limq𝒰p(y)=limq(ϕ(y)∧[y])(y)≥limqϕ(y)(y)=1, and so
(23)∀𝒢∈ℱLs(X),∀y∈X,limp𝒢(y)=[𝒰p(y),𝒢]=limq𝒰p(y)∧[𝒰p(y),𝒢]≤(LC2′)limq𝒢(y).
That is, limp≤limq. It follows by the assumption that (X,limq) is p-regular. Thus limqℱϕ(x)≥limqℱ¯p(x)≥limqℱ(x). By Theorem 17 we know that (X,limq) is k′*-regular.It is easily seen that whenL is a complete Boolean algebra, then the above proposition holds for k′-regularity.Lemma 30.
Let(X,limq) be a topological generalized stratified L-convergence space and let 𝒯 be the stratified L-topology corresponding to limq. Then ℱ≥𝒰q(x) if and only if ℱ(μ)≥𝒰q(x)(μ) for all μ∈𝒯.Proof.
We need only to check the sufficiency. Note that to for eachμ∈LX, 𝒰q(x)(μ)=int(μ)(x) and 𝒰q(x)(μ)=int(μ)(x)=μ(x) if μ∈𝒯 [11, 30]. It follows that, for each λ∈LX,
(24)ℱ(λ)=⋁μ∈LX(ℱ(μ)∧[μ,λ])≥⋁μ∈𝒯(ℱ(μ)∧[μ,λ])≥⋁μ∈𝒯(𝒰q(x)(μ)∧[μ,λ])=⋁μ∈𝒯(μ(x)∧[μ,λ])=int(λ)(x)=𝒰q(x)(λ).Theorem 31.
LetL be a linearly order frame or let 0∈L be prime. A topological generalized stratified L-convergence space (X,limq) is k′*-regular if and only if it is p-regular for every ultrapretopological generalized stratified L-convergence structure limp≤limq.Proof.
Note that a topological generalized stratifiedL-convergence space is natural a strong stratified L-Kent convergence space [17]. Then the sufficiency follows by Proposition 29. Thus, we prove only the necessity. Let (X,limq) be k′*-regular and let limp be an arbitrary ultrapretopological generalized stratified L-convergence structure with limp≤limq. Then, for each y∈X, there exists a ℋy∈𝒰Ls(X) such that 𝒰p(y)=ℋy∧[y]. Obviously, limpℋy(y)≥limp𝒰p(y)(y)=1 and then limqℋy(y)=1 by limp≤limq.
Letϕ∈Σ*(X) be defined by ϕ(y)=ℋy, for all y∈X. Then limqϕ(y)(y)=1 for each y∈X. For each λ∈𝒯, we check below [λ¯p,ϕ^(λ)]=1. Here, 𝒯 is the stratified L-topology corresponding to limq. For each ϕ(y)∈𝒰Ls(X), it follows by Lemma 1 that ϕ(y)𝔽ϕ(y)=ϕ(y); that is,
(25)ϕ^(λ)(y)=ϕ(y)(λ)={1,ιλ∈𝔽ϕ(y);0,ιλ∉𝔽ϕ(y).
Note that [λ¯p,ϕ^(λ)]=⋀y∈ι(λ¯p)(λ¯p(y)→ϕ(y)(λ)). For each y∈ι(λ¯p), it follows that λ¯p(y)=⋁ℱ∈ℱLs(X)(limpℱ(x)∧ℱ(λ))>0, which means that there exists an ℱy∈ℱLs(X) such that limpℱy(y)>0 and ℱy(λ)>0. Thus, ℱy(1ιλ)≥ℱy(λ)>0. Fix y∈ι(λ¯p); we have y∈ιλ or y∈X-ιλ.
Case 1. y∈ιλ; that is,  λ(y)>0. Because (X,limq) is topological, then λ(y)=𝒰q(y)(λ)>0. From limqϕ(y)(y)=1, we get ϕ(y)≥𝒰q(y) and then ϕ(y)(λ)>0; indeed, ϕ(y)(λ)=1 since ϕ(y)∈𝒰Ls(X) takes values in {0,1}.
Case 2. y∈X-ιλ; that is,  λ(y)=0. We assume that ϕ(y)(λ)≠1; it follows by equality (25) that ιλ∉𝔽ϕ(y). Because 𝔽ϕ(y) is an ultrafilter on X, then X-ι(λ)∈𝔽ϕ(y) and so ϕ(y)(1X-ιλ)=1. As we have known limpℱy(y)>0 and (X,limp) is ultrapretopological; hence, limpℱy(y)=[𝒰p(y),ℱy]>0, then by 𝒰p(y)(1X-ιλ)=ϕ(y)(1X-ιλ)∧[y](1X-ιλ)=1 it follows that ℱy(1X-ιλ)>0. Now,
(26)0=ℱy(1ιλ∧1X-ιλ)≥ℱy(1ιλ)∧ℱy(1X-ιλ)>0.
A contradiction! Thus, if y∈X-ιλ, then ϕ(y)(λ)=1.
Combining Cases 1 and 2 we get that ify∈ι(λ¯p) then ϕ^(λ)(y)=1. It follows immediately that [λ¯p,ϕ^(λ)]=1.
Next we prove thatkLϕ(𝒰q(x)¯p)≥𝒰q(x). By Lemma 30, we need only to check that kLϕ(𝒰q(x)¯p)(λ)≥𝒰q(x)(λ) for all λ∈𝒯. Indeed,
(27)kLϕ(𝒰q(x)¯p)(λ)=𝒰q(x)¯p(ϕ^(λ))=⋁μ∈LX(𝒰q(x)(μ)∧[μ¯p,ϕ^(λ)])≥𝒰q(x)(λ)∧[λ¯p,ϕ^(λ)]=𝒰q(x)(λ).
Then, for eachℱ∈ℱLs(X),
(28)limqℱ(x)=[𝒰q(x),ℱ]≤[𝒰q(x)¯p,ℱ¯p]≤[kLϕ(𝒰q(x)¯p),kLϕ(ℱ¯p)]≤[𝒰q(x),kLϕ(ℱ¯p)]=limqkLϕ(ℱ¯p)(x)≤limqℱ¯p(x),
where the first and the second equalities hold by the pretopologicalness of (X,limq), the first inequality holds by Lemma 24, the second inequality holds by Lemma 5(4), and the last inequality holds because (X,limq) is k′*-regular. Then it follows that (X,limq) is p-regular.Remark 32.
To prove that Theorem31 holds for k′-regularity, it seems that L must be a complete Boolean algebra. If we further assume that L is linearly ordered or 0∈L is prime then L={0,1}. Thus, we guess that Theorem 31 holds for k′-regularity only if L={0,1}.


## 4.2.  For Levelwise StratifiedL-Convergence Spaces

Definition 33 (see [31]).
Let(X,p¯) be a levelwise stratified L-convergence space. For each λ∈LX, the L-set λ¯pα∈LX defined by
(29)∀x∈X,λ¯pα(x)=⋁ℱ∈cpα(x)ℱ(λ),cpα(x)={ℱ∈ℱLs(X):ℱ⟶pαx}
is called α-level closure of λ w.r.t. (X,p¯).It is easily seen thatα-level closures of L-sets have similar properties to closures of L-sets. We do not list them but use them directly.In [20], Boustique and Richardson modified Jäger’s definition [11] and introduced a notion of α-level closures of stratified L-filters. In [25], we give an equivalent characterization of Boustique and Richardson's definition. This characterization seems more simple and more intuitive. Thus, we use it as the definition of α-level closures of stratified L-filters.Definition 34.
Let(X,p¯) be a levelwise stratified L-convergence space. For each α∈L and each ℱ∈ℱLs(X), it is easily seen that the function ℱ¯pα:LX→L, defined by ∀λ∈LX, ℱ¯pα(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯pα,λ]), is a stratified L-filter; then ℱ¯pα is called the α-level closure of ℱ w.r.t. (X,p¯).Definition 35 (see [24]).
Let(X,p¯,q¯) be a pair of levelwise stratified L-convergence spaces. Then (X,q¯) is called p-regular if, for each α∈L and each ℱ∈ℱLs(X), we have ℱ¯pα→qαx whenever ℱ→qαx.It is proved in [25] that p-regularity is preserved under initial constructions. Now, we look at the relationships between weaker regularities and p-regularity.Definition 36.
A levelwise stratifiedL-convergence space (X,q¯) is called(i)
anL-Kent convergence space if [x]∧ℱ→qαx whenever ℱ→qαx;(ii)
pretopological [23] if ℱ→qαx if and only if ℱ≥𝒰qα(x)=∧{ℱ∣ℱ→qαx};(iii)
ultrapretopological if, for eachx∈X and each α∈L, there exists a stratified L-ultrafilter ℱx such that 𝒰qα(x)=[x]∧ℱx;(iv)
topological [23] if there exists a stratified L-topology 𝒯α for each α∈L such that ∀λ∈LX, ∀x∈X, we have 𝒰qα(x)(λ)=intα(λ)(x), where intα(λ) is the interior of λ w.r.t. 𝒯α.Proposition 37.
Let(X,q¯) be a levelwise stratified L-Kent convergence space which is p-regular relative to every ultrapretopological levelwise stratified L-convergence structure p¯≥q¯. Here for p¯≥q¯, we mean that ℱ→pαx implies ℱ→qαx. Then (X,q¯) is k*-regular.Proof.
Letϕ∈Σ*(X) and α∈L with ∀y∈X, ϕ(y)→qαy. Let p¯ be the ultrapretopological levelwise stratified L-convergence structure defined by ∀α∈L, ∀y∈X, 𝒰pα(y)=ϕ(y)∧[y]. From ϕ(y)≥𝒰pα(y) we have ϕ(y)→pαy. For each ℱ∈ℱLs(X) such that ℱϕ∈ℱLs(X) and ℱ→qαx, it follows that for each λ∈LX, λ¯pα(y)=⋁ℱ∈cpα(y)ℱ(λ)≥ϕ(y)(λ), which means λ¯pα≥ϕ^(λ). Thus,
(30)ℱ¯pα(λ)=⋁μ∈LX(ℱ(μ)∧[μ¯pα,λ])≤⋁μ∈LX(ℱ(μ)∧[ϕ^(μ),λ])=ℱϕ(λ);
that is, ℱ¯pα≤ℱϕ. Because (X,q¯) is an L-Kent convergence space, then it follows by ϕ(y)→qαy that 𝒰pα(y)=ϕ(y)∧[y]→qαy. Thus, p¯≥q¯; then (X,q¯) is p-regular by the assumption. It follows that ℱ¯pα→qαx and then ℱϕ→qαx by ℱ¯pα≤ℱϕ. By Theorem 17 we know that (X,q¯) is k*-regular.It is easily seen that whenL is a complete Boolean algebra, then the above proposition holds for k-regularity.Lemma 38.
Let(X,q¯) be a topological levelwise stratified L-convergence space and let 𝒯α(α∈L) be the stratified L-topologies corresponding to q¯. Then ℱ≥𝒰qα(x) if and only if ℱ(μ)≥𝒰qα(x)(μ) for all μ∈𝒯α.Proof.
The proof is similar to Lemma30 and thus it is omitted.Theorem 39.
LetL be a linearly order frame or let 0∈L be prime. A topological levelwise stratified L-convergence space (X,q¯) is k*-regular if and only if it is p-regular for every ultrapretopological levelwise stratified L-convergence structure p¯≥q¯.Proof.
The sufficiency follows by Proposition37. We prove only the necessity. Let (X,q¯) be k*-regular and let p¯ be an arbitrary ultrapretopological levelwise stratified L-convergence structure with p¯≥q¯. Fix α∈L; then, for each y∈X, there exists a ℋy∈𝒰Ls(X) such that 𝒰pα(y)=ℋy∧[y]. Obviously, ℋy→pαy and then ℋy→qαy by p¯≥q¯.
Letϕ∈Σ*(X) be defined by ϕ(y)=ℋy, for all y∈X. For each λ∈𝒯α, we check below [λ¯pα,ϕ^(λ)]=1. Here, 𝒯α(α∈L) are the stratified L-topologies corresponding to q¯.
Note that[λ¯pα,ϕ^(λ)]=⋀y∈ι(λ¯pα)(λ¯pα(y)→ϕ(y)(λ)). For each y∈ι(λ¯pα), it follows that λ¯pα(y)=⋁ℱ∈cpα(y)ℱ(λ)>0, which means that there exists an ℱy→pαy such that ℱy(λ)>0. Thus, ℱy(1ιλ)≥ℱy(λ)>0. Fix y∈ι(λ¯pα); then y∈ιλ or y∈X-ιλ.
Case 1. y∈ιλ; that is, λ(y)>0. Because (X,q¯) is topological, thus λ(y)=𝒰qα(y)(λ)=∧{ℱ(λ)∣ℱ→qαy}>0. From ϕ(y)→qαy, we get ϕ(y)(λ)>0; indeed, ϕ(y)(λ)=1 since ϕ(y)∈𝒰Ls(X) takes values in {0,1}.
Case 2. y∈X-ιλ; that is,  λ(y)=0. We assume that ϕ(y)(λ)≠1; it follows by equality (25) that ιλ∉𝔽ϕ(y). Because 𝔽ϕ(y) is an ultrafilter on X, then X-ι(λ)∈𝔽ϕ(y) and so ϕ(y)(1X-ιλ)=1. As we have known ℱy→pαy; hence, ℱy≥𝒰pα(y)=ϕ(y)∧[y]; then ℱy(1X-ιλ)≥ϕ(y)(1X-ιλ)∧1X-ιλ(y)=1. Now,
(31)0=ℱy(1ιλ∧1X-ιλ)≥ℱy(1ιλ)∧ℱy(1X-ιλ)=ℱy(1ιλ)>0.
A contradiction! Thus, ify∈X-ιλ, then ϕ(y)(λ)=1.
Combining of Cases 1 and 2 we get that ify∈ι(λ¯pα) then ϕ^(λ)(y)=1. It follows immediately that [λ¯pα,ϕ^(λ)]=1. Then similar to Lemma 30 we have kLϕ(𝒰qα(x)¯pα)≥𝒰qα(x). Let ℱ→qαx; then ℱ≥𝒰qα(x) by the topologicalness of q¯. Hence, ℱ¯pα≥𝒰qα(x)¯pα and then kLϕ(ℱ¯pα)≥kLϕ(𝒰qα(x)¯pα)≥𝒰qα(x), which means kLϕ(ℱ¯pα)→qαx. Because (X,q¯) is k*-regular, then ℱ¯pα→qαx. It follows that (X,q¯) is p-regular.Remark 40.
Similar to Remark32, we guess that Theorem 39 holds for k-regularity only if L={0,1}.


## 5. Conclusions

In this paper, we introduce some weaker regularities for levelwise stratifiedL-convergence spaces and generalized stratified L-convergence spaces and study their characterizations and properties. For generalized stratified L-convergence spaces, we also investigate a notion of closures of stratified L-filters and then define by it a new p-regularity which is different from the p-regularity in [25] defined by the notion of α-level closures of stratified L-filters. At last, we discuss the relationships between weaker regularities and p-regularities. In addition, it seems that the p-regularity (for generalized stratified L-convergence spaces in [25]) has close relationships with k-regularity and k*-regularity. But we fail to establish those relationships for it is difficult to find an appropriate definition for ultrapretopological generalized stratified L-convergence spaces.



---

*Source: 328153-2014-01-12.xml*