p-Topologicalness and p-regularity for lattice-valued convergence spaces

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ScienceDirect Fuzzy Sets and Systems 238 (2014) 26–45 www.elsevier.com/locate/fss

p-Topologicalness and p-regularity for lattice-valued convergence spaces ✩ Lingqiang Li, Qiu Jin ∗ Department of Mathematics, Liaocheng University, Liaocheng 252059, PR China Received 27 July 2012; received in revised form 23 August 2013; accepted 25 August 2013 Available online 2 September 2013

Abstract In this paper, p-topologicalness and p-regularity for Jäger’s stratified L-generalized convergence spaces and those for Boustique et al.’s stratified L-convergence spaces are studied through two different methods. One method generalizes the well-known Fischer’s (dual) diagonal condition, the other method uses the (neighborhood) closure of stratified L-filter. In addition, the p-topologicalness and p-regularity w.r.t. initial constructions and final constructions are also discussed. © 2013 Elsevier B.V. All rights reserved. Keywords: Lattice-valued convergence space; p-Topological; p-Regular; Diagonal axiom; Closure (neighborhood) L-filter

1. Introduction Stratified lattice-valued convergence spaces were first defined in [12] and then developed in a series of papers [13–15] for the case that the lattice is a complete Heyting algebra (or a frame). Later, the theory of these spaces was generalized to the lattice context of complete residuated lattices [28]. In [25], the lattice situation was further extended to enriched cl-premonoids. These lattice-valued convergence spaces are usually called stratified L-generalized convergence spaces. Recently, a subcategory of stratified L-generalized convergence space, called stratified strong L-generalized convergence spaces were also discussed frequently [6,21–23]. When L is a frame, Flores et al. introduced a kind of lattice-valued convergence spaces [8] which are called levelwise stratified L-convergence spaces in this paper. They also proposed a so-called left-continuity condition and proved that levelwise stratified L-convergence spaces that satisfy the left-continuity condition are precisely stratified L-generalized convergence spaces. In [3], Boustique and Richardson investigated a weaker lattice-valued convergence space by removing one convergence axiom in Flores et al.’s definition. Fischer’s diagonal condition (F-diagonal condition for short) plays an important role in the theory of convergence spaces. The condition F is necessary and sufficient for a convergence structure to be a topology. Furthermore, a dual version of F (DF-diagonal condition for short) is necessary and sufficient for a convergence space to be regular [7,20]. Topologicalness can also be characterized by the requirement that for each filter F, if F converges to x then so does ✩

This work is supported by NSFC (11371130) and A Project of Hunan Province Science and Technology Program (2012RS4029) and the Ke Yan Foundation of Liaocheng University (318011310) and the Natural Science Foundation of Shandong Province (ZR2013AQ011). * Corresponding author. E-mail addresses: [email protected] (L. Li), [email protected] (Q. Jin). 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.08.012

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the filter U (F) (the neighborhood of F) [10,20]. Regularity can also be characterized by the requirement that for each filter F, if F converges to x then so does the filter F (the closure of F) [1,5]. In [19,27], by considering a pair of convergence spaces (X, p) and (X, q), Kent and his co-authors 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 [10] and generalized F (resp., DF)-diagonal condition. When p = q, p-topologicalness (resp., p-regularity) is precisely topologicalness (resp., regularity). The topologicalness for stratified L-generalized convergence spaces was studied by lattice-valued F-diagonal conditions in [15,23,25] and by neighborhood of stratified L-filter in [17]. In [15], the regularity for stratified L-generalized convergence spaces was discussed by lattice-valued DF-diagonal condition when L being a frame and by closure of stratified L-filter when L being a complete Boolean algebra. In [9], when L is a frame, Flores and Richardson studied topologicalness and regularity for levelwise stratified L-convergence spaces by lattice-valued F-diagonal condition and its’ duality. Later, Boustique and Richardson [3] characterized regularity for levelwise stratified L-convergence space by closure of stratified L-filter. In [24], Losert et al. discussed p-regularity for levelwise stratified L-convergence spaces by closure of stratified L-filter. But they did not consider p-regularity by a suitable lattice-valued DF-diagonal condition. To our knowledge, there is no work on p-regularity for stratified L-generalized convergence spaces. In this paper, the main aim is to study p-regularity and its’ dual notion p-topologicalness for stratified L-generalized convergence spaces and that for levelwise stratified L-convergence spaces. The contents are arranged as follows. Section 2 recalls some basic notions as preliminary. Section 3 presents p-pretopological axioms and p-topological axioms. The relationships between p-pretopological axioms and p-topological axioms are discussed in detail. The p-topologicalness is studied both by lattice-valued F-diagonal axioms and neighborhood of stratified L-filter. The p-topologicalness w.r.t. initial constructions and final constructions is studied. Section 4 focuses on p-regular axioms. Firstly, a simple representation for closure of stratified L-filter is presented. Then p-regularity is discussed by a suitable lattice-valued DF-diagonal condition and closure of stratified L-filter simultaneously. Lastly, the p-regularity w.r.t. initial constructions and final constructions is studied. 2. Preliminaries 2.1. L-Sets and L-orders 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∈I βi ) = 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 set X, the set LX of functions from X to L with the pointwise order becomes a complete lattice. Each element X X of LX is called denote by α  ∧ λ, α → λ,   an L-set (or a fuzzy subset) of X. For any λ ∈ L , K ⊆ L and α ∈ L, we K and K the L-sets defined by (α ∧ λ)(x) = α ∧ λ(x), (α → λ)(x) = α → λ(x), ( K )(x) = μ∈K μ(x)  and ( K )(x) = μ∈K μ(x). We make no difference between a constant function and its value since no confusion will arise. Definition 2.1. (See [2].) Let X be a set. A fuzzy partial order (or, an L-partial order) on X 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.

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 Let R : LX × LX −→ L be a function defined by R(λ, μ) = x∈X (λ(x) → μ(x)), then R is an L-partial order on LX . The value R(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 R(λ, μ) for simplicity. f → : LX −→ LY and f ← : LY −→ LX [11] by f → (A)(y) =  Let f : X −→ Y beXa function. We define ← (B) = B ◦ f for B ∈ LY . A(x) for A ∈ L and y ∈ Y , and f f (x)=y The following lemma is useful to the subsequent section. Lemma 2.2. (See [29].) Let f : X −→ Y be a function. For λ, μ, ν ∈ LX , {λi }i∈I , {μi }i∈I ⊆ LX , ω ∈ LY , α ∈ L, we have   (1) λ  μ implies [λ,  ν]  [μ, ν]; (2) [λ, i∈I μi ] = i∈I [λ, μi ]; (3) [ i∈I λi , μ] = i∈I [λi , μ]; (4) [α ∧ λ, μ] = α → [λ, μ]; (5) λ ∧ [λ, μ]  μ; (6) [λ, μ]  [f → (λ), f → (μ)]; (7) [f → (λ), ω] = [λ, f ← (ω)]. 2.2. Stratified L-(ultra)filters A stratified L-filter [11] 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 ∧ λ)  α ∧ F(λ). In the presence of (F1) and (F2), the stratification condition (Fs) is equivalent to (Fs ) F(α)  α. A stratified L-filter F is called tight if F(α) = α for each α ∈ L [10]. It is proved in [11] that all stratified L-filters are tight iff L is a complete Boolean algebra. The set FLs (X) of all stratified L-filters on X is ordered by F  G ⇔ ∀λ ∈ LX , F(λ)  G(λ). It is shown in [11] that the partially ordered set (FLs (X), ) has maximal elements which are called stratified L-ultrafilters. The set of all stratified L-ultrafilter on X is denoted as ULs (X). 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 . X There is a natural fuzzy partial order on FLs (X) inherited from L(L ) . Precisely, if we let   s   X FL (X) (F, G) = LL (F, G) = F(λ) → G(λ) λ∈LX

for all F, G ∈ FLs (X), 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. Lemma 2.3. (See [11].) (1) A stratified L-filter F is an L-ultrafilter iff for all λ ∈ LX we have F(λ) = F(λ → 0) → 0. s (2) Let L be a complete Boolean algebra and F ∈ FL (X). Then (i) F = F G ∈U s (X) G, (ii) F is prime whenever L F is maximal. The following examples belong to the folklore, we list them here because the notations are needed. Example 2.4. (1) For each point x in a set X, it is obvious that the function [x] : LX −→ L, [x](λ) = λ(x) is a stratified L-filter on X, called the principal L-filter generated by 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 ∈J Fj is also a stratified L-filter on X. Let F0 denote the meet of all stratified L-filters on X, i.e., the smallest stratified L-filter on X. (3) Let f : X −→ Y be a function. If F is a stratified L-filter on X, then the function f ⇒ (F) : LY −→ L defined by λ → F(λ ◦ f ) is a stratified L-filter on Y , called the image of F under f . If F is a stratified L-ultrafilter, then f ⇒ (F) is also a stratified L-ultrafilter.  If f is a surjective function and G is a stratified L-filter on Y , then the function f ⇐ (G) : LX −→ L defined by λ → {G(μ): f ← (μ)  λ} is a stratified L-filter on Y , called the inverse image of G under f . The following lemma can be proved easily.

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Lemma 2.5. Let f : X −→ Y be a function and F, G ∈ FLs (X). Then (1) For all λ, μ ∈ LX , F(λ) ∧ [λ, μ]  F(μ); (2) [F, G]  [f ⇒ (F), f ⇒ (G)]. 2.3. Lattice-valued convergence spaces Definition 2.6. A stratified L-generalized convergence structure [12] on a set X is a function limq : FLs (X) −→ LX satisfying (LC1) limq [x](x) = 1 for every x ∈ X; and (LC2) ∀F , G ∈ FLs (X), F  G ⇒ limq F  limq G. The pair (X, limq ) is called a stratified L-generalized convergence space. If limq further satisfies the strong axiom (LC2 ) ∀x ∈ X, ∀F , G ∈ FLs (X), [F, G] ∧ limq F(x)  limq G(x), then the pair (X, limq ) is called a stratified strong L-generalized convergence space [6,21,22].

A function f : X −→ X between two stratified L-generalized convergence spaces (X, limq ), (X , limq ) is called

continuous if for all F ∈ FLs (X) and all x ∈ X we have limq F(x)  limq f ⇒ (F)(f (x)). The category SL-GCS has as objects all stratified L-generalized convergence spaces and as morphisms the continuous functions. This category is topological over SET [12]. In SL-GCS initial and final structures can be easily described [13]. fi

For a given −→ (Xi , limqi ))i∈I , the initial structure, limq on X is defined by ∀F ∈ FLs (X), ∀x ∈ X,  sourceq(X q ⇒ i lim F(x) = i∈I lim fi (F)(fi (x)). fi

For a given sink ((Xi , limqi ) −→ X)i∈I , the final structure, limq on X is defined by ⎧ F  [x]; ⎨ 1,  qi limq F(x) = lim Gi (xi ), F  [x]. ⎩ s ⇒ i∈I,xi ∈Xi ,Gi ∈FL (Xi ),fi (xi )=x,fi (Gi )F

Remark 2.7. Let limq be final structure defined above and x ∈ X. If x ∈  limq F(x) = limqi Gi (xi )

i∈I fi (Xi ),

then it is easy to show that

i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F

no mater F  [x] or F  [x]. Thus, when X =  limq F(x) =

the final structure limq can be simplified as  limqi Gi (xi ) F ∈ FLs (X), x ∈ X .

i∈I fi (Xi ),

i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F

Definition 2.8. A collection q = (qα )α∈L , where qα : FLs (X) −→ P(X), is called a levelwise stratified L-convergence structure on X [3] if it satisfies: qα (LL1) [x] → x for each x ∈ X, qα qα (LL2) G  F → x implies G → x, qα

qβ

(LL3) F → x implies F → x whenever β  α. qα The notation, F → x, means that x ∈ qα (F). The pair (X, q) is called a levelwise stratified L-convergence space. A function f : X −→ X between two levelwise stratified L-convergence spaces (X, q), (X , q ) is called continuqα

qα

ous if for all F ∈ FLs (X) and all x ∈ X and all α ∈ L we have F → x implies f ⇒ (F) → 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 [3,4]. The initial and final structures in SL-LCS can be defined similarly to that in SL-GCS. qαi fi qα For a given source (X −→(Xi , q i ))i∈I , the initial structure, q on X is defined by F → x ⇔ ∀i ∈ I, fi⇒ (F) → fi (x) (F ∈ FLs (X), x ∈ X, α ∈ L).

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L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45 fi

For a given sink ((Xi , qi ) −→ X), the final structure, q = (qα )α∈L on X is defined as 

F  [x], x∈ / i∈I fi (Xi ); qα F →x⇔ qαi F  fi⇒ (Gi ), ∃i ∈ I, xi ∈ Xi , Gi ∈ FLs (Xi ) s.t. f (xi ) = x, Gi → xi .

Thus, when X = i∈I fi (Xi ), the final structure q can be simplified as qα

qαi

F → x ⇔ ∃i ∈ I, xi ∈ Xi , Gi ∈ FLs (Xi ) s.t. f (xi ) = x, Gi → xi , fi⇒ (Gi )  F. 3. p-Topological axioms In this section, we shall generalize the pretopological axioms and topological axioms for stratified L-generalized convergence spaces and that for levelwise stratified L-convergence spaces to the p-pretopological and p-topological cases. 3.1. p-Pretopological axioms for stratified L-generalized convergence spaces In the following, we use (X, limp , limq ) to denote a pair of stratified L-generalized convergence spaces (X, limp ) and (X, limq ). p s Let (X, lim  ) be a stratified L-generalized convergence space. Then a function Up : X −→ FL (X) defined by Up (x)(λ) = F ∈F s (X) (limp F(x) → F(λ)) is called the stratified L-neighborhood function of limp [12]. Similarly, L for each α ∈ L, a function Upα : X −→ FLs (X) defined by    Upα (x) = lpα (x), lpα (x) = F ∈ FLs (X): limp F(x)  α is called the α-level stratified L-neighborhood function of limp [14]. Definition 3.1. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then limq is called p-pretopological if the following axiom is satisfied. p-(Lp): ∀F ∈ FLs (X), ∀x ∈ X, limq F(x)  [Up (x), F]. It is easily seen that when limp = limq , the p-pretopologicalness is precisely the pretopologicalness in [14]. Proposition 3.2. Let (X, limq ) be a stratified L-generalized convergence space. Then limq is pretopological iff it is p-pretopological for each limp  limq . Proof. Let (X, limq ) be pretopological and limp  limq . Then by limp  limq we have Up (x)  Uq (x) and by pretopologicalness of limq we get limq F(x)  [Uq (x), F]  [Up (x), F]. Thus limq is p-pretopological. The converse implication is obvious. 2 When limp = limq , p-pretopological axiom was proved to split into some weaker axioms [14,16,23]. Now, we extend them to the general case. p-(Lu): ∀x ∈ X, limq Up (x)(x) = 1. p-(Lpw1): ∀α ∈ L, ∀x ∈ X, [α ∧ Up (x)]  Uqα (x), where      α ∧ Up (x) = F ∈ FLs (X): α  Up (x), F . Intuitively, [α ∧ Up (x)] is the least stratified L-filter  larger than thefunction λ → α ∧ Up (x)(λ). p-(Lpw2): ∀{Fj }j ∈J ⊆ FLs (X), ∀x ∈ X, limq ( j ∈J Fj )(x)  j ∈J limp Fj (x). When limp = limq , we omit the prefix “p” in symbols p-(Lp), p-(Lu), p-(Lpw1) and p-(Lpw2). This simplification is also used for the subsequent p-topological axioms and p-regular axioms. Under this case, the “” in the above axioms can be replaced with “=”. It is proved that: (1) (Lpw1) + (Lpw2) ⇔ (Lp) [14], (2) (Lp) ⇒ (Lu) and (Lp) ⇔ (Lu) if limp = limq is a stratified strong L-generalized convergence structure [23]. For general limp and limq , we have the following results.

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Theorem 3.3. (1) p-(Lp) ⇒ p-(Lpw1), p-(Lpw2) and if limp satisfies (Lpw1), then p-(Lpw2) ⇒ p-(Lp). (2) p-(Lp) ⇒ p-(Lu) and p-(Lu) ⇒ p-(Lp) if limq is a stratified strong L-generalized convergence structure. Proof. (1) p-(Lp) ⇒ p-(Lpw1). 

 α ∧ Up (x) =



F



p -(Lp)



α[Up (x),F ]

αlimq

F = Uqα (x).

F (x)

p-(Lp) ⇒ p-(Lpw2).       p -(Lp)   p q Up (x), Fi  lim Fi (x)  Up (x), Fi = lim Fi (x). i∈I

i∈I

i∈I

i∈I

Let limp  satisfy (Lpw1). For each F ∈ FLs (X) and each x ∈ X, let [Up (x), F] = α. Then by (Lpw1) we have F  Upα (x) = {G: limp G(x)  α} and so limq F(x)  limq Upα (x)

p -(Lpw2)





limp G(x)  α,

limp G (x)α

i.e., the axiom p-(Lp) is satisfied. (2) Take F = Up (x) in p-(Lp) we get p-(Lu). Let limq be a stratified strong L-generalized convergence structure, then by p-(Lu) and (LC2 ) we have     Up (x), F = limq Up (x)(x) ∧ Up (x), F  limq F(x), i.e., the axiom p-(Lp) holds.

2

The following example shows that p-(Lpw1) + p-(Lpw2)  p-(Lp) generally. Example 3.4. Let L be the linearly ordered frame ({0, α, 1}, ∧, 1) with 0 < α < 1. Assume X = {x, y}. For each F ∈ FLs (X) and z ∈ X, let  1, F  [z]; limp F(z) = 0, others. In [14], it is proved that (X, limp ) is a stratified L-generalized convergence space and for each z ∈ X, Up (z) = [z]. For x, y ∈ X, it is easily seen that the functions Fx , Fy : LX −→ L defined by ⎧ ⎧ ⎨ 1, λ(x)  α, λ(y) = 1; ⎨ 1, λ(y)  α, λ(x) = 1; ∀λ ∈ LX , Fx (λ) = 0, λ(x) = 0; Fy (λ) = 0, λ(y) = 0; ⎩ ⎩ α, otherwise; α, otherwise; are all stratified L-filters on X. For each F ∈ FLs (X) and each z ∈ X, let  1, F  [z] or F  Fz ; q lim F(z) = 0, others, Then (X, limq ) is a stratified L-generalized convergence space. And for each z ∈ X, Uq1 (z) = Uqα (z) = Uq (z) = [z] ∧ Fz , where ⎧ ⎨ 1, λ = 1;  [z] ∧ Fz (λ) = 0, λ(z) = 0; ⎩ α, otherwise; and [z] ∧ Fz = [z], Fz . (1) (X, limp , limq ) satisfies p-(Lpw1). Indeed,     0 ∧ Up (z)  F0 = Uq0 (z), 1 ∧ Up (z) = Up (z) = [z]  [z] ∧ Fz = Uq1 (z).

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In addition, note that (α ∧ [z])(λ) =



0, λ(z)=0; α, otherwise.

Compared α ∧ [z] with [z] ∧ Fz , we find that [z] ∧ Fz is the least stratified L-filter larger than α ∧ [z], i.e., [α ∧ [z]] = [z] ∧ Fz . Thus     α ∧ Up (z) = α ∧ [z] = [z] ∧ Fz = Uqα (z). (2) (X, limp , limq ) satisfies p-(Lpw2).   limq ( j ∈J Fj )(z)  j ∈J limp Fj (z). For all {Fj }j ∈J ⊆ FLs (X) and all z ∈ X, we need to check the inequality   (z) = 0, then the desired inequality holds obviously. If j ∈J limp Fj (z) = 1, then ∀j ∈ J we have If j ∈J limp Fj  q  Fj  [z] and so j ∈J Fj  [z]. Thus lim ( j ∈J Fj )(z) = 1 and the desired inequality holds. (3) (X, limp , limq ) does not satisfy p-(Lp). Take F = [z] ∧ Fz , we have limq F(z) = 0 < α = [Up (z), F] = [[z], [z] ∧ Fz ]. In general, we have no p-(Lp) ⇒ (Lpw1) of limp . Indeed, consider limp in the above example and limq defined by ∀F ∈ FLs (X), ∀z ∈ X, limq F(z) ≡ 1. It is easily seen that the axiom p-(Lp) is satisfied but [α ∧ Up (z)] = [α ∧ [z]] = [z] ∧ Fz < [z] = Upα (z) as observed in the above example. 3.2. p-Pretopological axioms for levelwise stratified L-convergence spaces In the following, we use (X, p, q) to denote a pair of levelwise stratified L-convergence spaces (X, p) and (X, q). Let (X, p) be a levelwise stratified L-convergence space. For each α ∈ L, a function Upα : X −→ FLs (X), defined  pα by Upα (x) = lpα (x), lpα (x) = {F ∈ FLs (X): F → x}, is called the α-level stratified L-neighborhood function of p. It is easy to check that the following axioms are equivalent: qα p-(LLp): ∀F ∈ FLs (X), ∀x ∈ X, F  Upα (x) ⇒ F → x, qα

p-(LLu): ∀x ∈ X, ∀α ∈ L, Upα (x) → x,

 pα qα p-(LLpw2): ∀{Fj }j ∈J ⊆ FLs (X), ∀x ∈ X, ∀j ∈ J, Fj → x ⇒ j ∈J Fj → x. When p = q, we omit the prefix “p” in symbols p-(LLp), p-(LLu) and p-(LLpw2). This simplification is also used for the subsequent p-topological axioms and p-regular axioms. It is observed easily that the “⇒” in the above axioms can be replaced with “⇔” when p = q [9,15,17,25]. Definition 3.5. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. Then q is called p-pretopological if either of the above three axioms is satisfied. Similar to Proposition 3.2, we get the following proposition. Proposition 3.6. Let (X, q) be a stratified L-generalized convergence space. Then q is pretopological iff it is p-pretopological for each p  q, i.e., ∀α ∈ L, pα ⊆ qα . 3.3. p-Topological axioms for stratified L-generalized convergence spaces In the literatures [14,15,17,23,25], lattice-valued Fischer’s diagonal conditions and lattice-valued Gähler’s neighborhood conditions were used to study topologicalness for stratified L-generalized convergence spaces. In this subsection, by generalizing these conditions, we shall study p-topologicalness for stratified L-generalized convergence spaces. At first, we fix some notations. Let J, X be any set and φ : J −→ FLs (X) be any function. Then a function ˆ φˆ : LX → LJ is defined as ∀λ ∈ LX , ∀j ∈ J, φ(λ)(j ) = φ(j )(λ). For all F ∈ FLs (J ), it is proved that the function X X ˆ is a stratified L-filter, which is called the L-diagonal filkL φF : L −→ L defined by ∀λ ∈ L , kL φF(λ) = F(φ(λ)) p ter of F under φ [14,23]. Let (X, lim ) be a stratified L-generalized convergence space. Take φ = Up : X −→ FLs (X) (resp., φ = Upα : X −→ FLs (X), α ∈ L), then for each F ∈ FLs (X), the stratified L-filter Up (F) := kL Up F (resp., Upα (F) := kL Upα F , α ∈ L) is called (resp., α-level) stratified L-neighborhood filter of F w.r.t. limp [17]. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. We consider the following axiom

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p-(Lf): Let J , X be any sets, ψ : J −→ X and φ : J −→ FLs (X) be functions. limq ψ ⇒ (F)(x) ∧

∀F ∈ FLs (J ), ∀x ∈ X,



 limp φ(j ) ψ(j )  limq kL φF(x).

j ∈J

When limp = limq , p-(Lf) is precisely the lattice-valued Fischer’s diagonal axiom (Lf) in [15]. Restricting J = X and ψ = id in p-(Lf), we obtain a weak axiom p-(Lk), which is the generalization of lattice-valued Kowalski’s diagonal axiom (Lk) in [14]. It is proved in [15] that (Lf) ⇔ (Lk) + (Lpw2). Now, we examine that whether the equivalence p-(Lf) ⇔ p-(Lk) + p-(Lpw2) holds. Proposition 3.7. (1) p-(Lf) ⇒ p-(Lpw2) + p-(Lk), and (2) p-(Lk) ⇒ p-(Lf) if limp satisfies (Lpw2). Proof. (1) Obviously, p-(Lf) ⇒ p-(Lk). Now, we check p-(Lf) ⇒ p-(Lpw2). Let {Fj }j ∈J ⊆ FLs (X)  and x ∈ X. Take ψ(j ) ≡ x, φ(j ) = Fj and F = F0 in p-(Lf), then it is easily seen that ψ ⇒ (F0 ) = [x], kL φF0 = j ∈J Fj and 

limp Fj (x) = limq [x](x) ∧

j ∈J



limp Fj (x)  limq kL φF0 (x) = limq

j ∈J



 Fj (x).

j ∈J

 : X −→ (2) Let J, X be any sets, ψ : J −→ X and φ : J −→ FLs (X) be functions. Then we define a function φ s (x) = {φ(j ): j ∈ J, ψ(j ) = x} if there exists j ∈ J such that ψ(j ) = x and φ (x) = [x] if not so. Let FL (X) as φ  ◦ ψ(j )  φ(j ) and ψ ⇒ (F). Indeed, for each j ∈ J , we have φ F ∈ FLs (J ). Then we check below that kL φF  kL φ  X ˆ  ◦ ψ(j )(λ)  φ(j )(λ), i.e., φ (λ) ◦ ψ  φ(λ). Thus so for each λ ∈ L , φ    ˆ (λ) = F φ (λ) ◦ ψ  F φ(λ) ψ ⇒ (F)(λ) = ψ ⇒ (F) φ kL φ = kL φF(λ). Additionally, 

p

lim φ (x)(x) =

x∈X



lim

p

x∈X

=



 

    φ(j ) (x) = limp φ(j ) ψ(j ) x∈X ψ(j )=x

ψ(j )=x

 lim φ(j ) ψ(j ) , p

j ∈J

where the second equality uses (Lpw2). Then p-(Lf) follows by limq ψ ⇒ (F)(x) ∧



  (x)(x) limp φ(j ) ψ(j ) = limq ψ ⇒ (F)(x) ∧ limp φ

j ∈J

x∈X p -(Lk)

ψ ⇒ (F)(x)  limq kL φF(x).  limq kL φ

2

Corollary 3.8. If limp satisfies (Lpw2) then p-(Lf) ⇔ p-(Lk) + p-(Lpw2). The following example shows that p-(Lf) maybe not imply (Lpw2) of limp . Thus it seems that the additional condition (Lpw2) in the above corollary cannot be deleted. Example 3.9. Let L, X and Fz (z ∈ X) be defined as in Example 3.4. For each F ∈ FLs (X) and each z ∈ X, let limq F(z) ≡ 1 and limp F(z) =



1, F >F0 ; 0, others.

Then (X, limp , limq ) is a pair of stratified L-generalized convergence  spaces. Obviously, the axiomp-(Lf) is satisfied. But limp does not fulfill the axiom (Lpw2) since F =F0 limp F(z) = 1 > 0 = limp F0 (z) = limp ( F =F0 F)(z). 2

34

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. We generalize Gähler’s neighborhood condition as follows. p-(Lg): ∀α ∈ L, ∀F ∈ FLs (X), α  [limq F, limq Upα (F)]. When limp = limq , p-(Lg) is denoted as (Lg). Jäger proved that (Lf) ⇔ (Lg). The following theorem generalizes this equivalence. Theorem 3.10. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then p-(Lf) ⇔ p-(Lg). Proof. p-(Lf) ⇒ p-(Lg). For each α∈ L, let J = {(G, y) ∈ FLs (X) × X | G ∈ lpα (y)}, ψ : J −→ X, (G, y) → y, φ : J −→ FLs (X), (G, y) → G. Then j ∈J limp φ(j )(ψ(j ))  α. Because limp [y](y) = 1  α we have that ψ is a surjective function. Thus for each F ∈ FLs (X), K = ψ ⇐ (F) ∈ FLs (J ) and ψ ⇒ (K) = F . For each λ, μ ∈ LX , it is α  ˆ easily seen that ψ ← (μ)  φ(λ) ⇔μU p (λ). Hence     ˆ ˆ kL φK(λ) = ψ ⇐ (F) φ(λ) = F(μ)  ψ ← (μ)  φ(λ)     α α α  = F(μ)  μ  U p (λ) = F Up (λ) = Up (F)(λ). Thus kL φK = Upα (F). By p-(Lf) we have      limq F, limq Upα (F) = limq ψ ⇒ (K), limq kL φK  limp φ(j ) ψ(j )  α,



j ∈J

i.e., p-(Lg) is satisfied.  p-(Lg) ⇒ p-(Lf). Let J, X, φ, ψ be defined as in p-(Lf) and j ∈J limp φ(j )(ψ(j )) = α. Then for each j ∈ J , s α  ˆ λ ∈ LX we have Upα (ψ(j ))(λ)  φ(j )(λ), i.e., U p (λ) ◦ ψ  φ(λ). For each F ∈ FL (J ), we have     α  α ˆ Upα ψ ⇒ (F) (λ) = ψ ⇒ (F) U p (λ) = F Up (λ) ◦ ψ  F φ(λ) = kL φF(λ). Thus Upα (ψ ⇒ (F))  kL φF . By p-(Lg) we have      α  limq ψ ⇒ (F), limq Upα ψ ⇒ (F)  limq ψ ⇒ (F), limq kL φF , i.e., p-(Lf) is satisfied.

2

When L = {0, 1}, it is easily seen that both p-(Lf) and p-(Lg) characterize p-topologicalness for convergence spaces. Thus we give the following definition. Definition 3.11. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then limq is called p-topological if the axiom p-(Lf) or p-(Lg) is satisfied. The following proposition shows that p-topologicalness is preserved under initial constructions. The proof is similar to Lemma 4.8 in [15], thus it is omitted. Proposition 3.12. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces with each limqi fi

being pi -topological. If limq (resp., limp ) is the initial structure on X relative to the source (X −→ (Xi , limqi ))i∈I fi

(resp., (X −→ (Xi , limpi ))i∈I ), then (X, limq ) is p-topological. Definition 3.13. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified L-generalized convergence spaces. X α  α → Then f is said to be an interior function if f → (U q (λ))  Up (f (λ)) for all λ ∈ L and all α ∈ L. Lemma 3.14. Let f : (X, limq ) −→ (Y, limp ) be an interior function. Then f ⇒ (Uqα (F))  Upα (f ⇒ (F)) for all F ∈ FLs (X) and all α ∈ L.

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

35

← α  α → ← α Proof. For each λ ∈ LY , by f is an interior function we have f → (U q (f (λ)))  Up (f (f (λ)))  Up (λ) and ← ← α α   then Uq (f (λ))  f (Up (λ)). Thus

    ←  ← α α   U f ⇒ Uqα (F) (λ) = Uqα (F) f ← (λ) = F U q f (λ)  F f p (λ)  ⇒  α α  = f ⇒ (F) U p (λ) = Up f (F) (λ). By the arbitrariness of λ, we get f ⇒ (Uqα (F))  Upα (f ⇒ (F)).

2

Proposition 3.15. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces with each limqi fi

being pi -topological. Let limq (resp., limp ) be the final structure on X w.r.t. the sink ((Xi , limqi ) −→ X)i∈I (resp.,

fi ((Xi , limpi ) −→ X)i∈I ). If X = i∈I fi (Xi ) and each fi : (Xi , limpi ) −→ (X, limp ) is an interior function, then (X, limq ) is p-topological. Proof. Let F ∈ FLs (X), α ∈ L and x ∈ X. Then by Remark 2.7 we have 

α ∧ limq F(x) =



α ∧ limqi Gi (xi )

i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F





i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F

limqi Upαi (Gi )(xi )





i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,Upα (fi⇒ (Gi ))Upα (F )

limqi Upαi (Gi )(xi )



=

i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Upα (Gi ))Upα (fi⇒ (Gi ))Upα (F ) i





limqi Upαi (Gi )(xi )

limqi Hi (xi )

i∈I,xi ∈Xi ,Hi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Hi )Upα (F )

= limq Upα (F)(x), where the first inequality holds for pi -topologicalness of limqi , the second equality follows from the above lemma. By the arbitrariness of x, we have α  [limq F, limq Upα (F)]. It follows that limq is p-topological. 2 The above proposition shows that p-topologicalness is preserved under final constructions with some additional conditions. In [27], when L = {0, 1}, Wilde and Kent have proved that these additional conditions cannot be deleted generally. In [15], Jäger found that the axiom (Lf) does not ensure a stratified L-generalized convergence space to be topological. Thus the authors proposed a strong lattice-valued Fischer’s diagonal axiom (Lfs) and proved that (Lfs) ensures precisely a stratified L-generalized convergence space to be topological [23]. In addition, Jäger [17] also introduced a lattice-valued Gähler’s neighborhood condition and proved this condition ensures precisely a stratified strong L-generalized convergence space to be topological. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. p-(Lfs) For any set J , any functions ψ : J −→ X and φ : J −→ FLs (X), and any F ∈ FLs (J ), x ∈ X, if limp φ(j )(ψ(j )) = 1 and φ(j )  [ψ(j )] for every j ∈ J then limq ψ(F)(x) = [Up (x), kL φF]. In [23], it is proved that (Lfs) ⇔ (Lp) + (Lk ): Restricting φ as limp φ(x)(x) = 1 in p-(Lk). When limp = limq , it is easily seen that p-(Lp) ⇔ p-(Lps): ∀F ∈ FLs (x), ∀x ∈ X, limq F(x) = [Up (x), F]. Now, we consider whether the equivalence p-(Lfs) ⇔ p-(Lk ) + p-(Lps) holds. Proposition 3.16. (1) p-(Lfs) ⇒ p-(Lps), (2) p-(Lfs) ⇒ p-(Lk ) and p-(Lk ) + p-(Lps) ⇒ p-(Lfs) if limp satisfies (Lpw2).

36

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

Proof. (1) p-(Lfs) ⇒ p-(Lps). Take J = X, ψ = idX and φ(x) = [x]. Then limp φ(j )(ψ(j )) = 1, φ(j )  [ψ(j )] for every j ∈ J and ψ ⇒ (F) = F , kL φF = F for every F ∈ FLs (X). Thus by p-(Lfs) we have limq F(x) = [Up (x), F]. (2) Let limp satisfy (Lpw2). p-(Lfs) ⇒ p-(Lf ): Restricting φ as limp φ(j )(ψ(j )) = 1 in p-(Lf).  : J −→ F s (X) as ∀j ∈ J , φ (j ) = φ(j ) ∧ [ψ(j )], then φ   φ. Let J, X, ψ, φ be defined as in p-(Lf ). Take φ L p p Since lim satisfies (Lpw2) we have lim φ (j )(ψ(j ))  limp φ(j )(ψ(j )) ∧ limp [ψ(j )](ψ(j )) = 1. Thus for each F ∈ FLs (J ) and each x ∈ X we get by p-(Lfs) and p-(Lps)     F  Up (x), kL φF = limq kL φF(x). limq ψ(F)(x) = Up (x), kL φ Obviously, p-(Lf ) ⇒ p-(Lk ). p-(Lk ) + p-(Lps) ⇒ p-(Lfs). Let J, X, ψ, φ satisfy the condition of p-(Lfs). By p-(Lps), it suffices to check that limq ψ ⇒ (F)(x) = limq kL φF(x) for all F ∈ FLs (J ) and all x ∈ X. ˆ  λ ◦ ψ and thus kL φF  ψ ⇒ (F). Therefore, By ∀j ∈ J , φ(j )  [ψ(j )] we have ∀λ ∈ LX , φ(λ)  q ⇒ q  : X −→ F s (X) as φ (x) = {φ(j ): lim ψ (F)(x)  lim kL φF(x). One the other hand, we define a function φ L  j ∈ J, ψ(j ) = x} if there exists j ∈ J such that 3.7, it  ψ(j ) = x and φ (x) = [x] if not so. Similar to Proposition ψ ⇒ (F) and x∈X limp φ (x)(x) = 1 (using (Lpw2)) and thus by p-(Lk ) we have can be proved that kL φF  kL φ ψ ⇒ (F)(x)  limq kL φF(x). 2 limq ψ ⇒ (F)(x)  limq kL φ Corollary 3.17. If limp satisfies (Lpw2) then p-(Lfs) ⇔ p-(Lk ) + p-(Lps). The following example shows that p-(Lfs) maybe not imply (Lpw2) of limp . Thus it seems that the additional condition (Lpw2) in the preceding corollary cannot be deleted. Example 3.18. Let (X, limp , limq ) be defined as in Example 3.9. We have known that limp does not fulfill the axiom (Lpw2). But (X, limp , limq ) satisfies the axiom p-(Lfs). Indeed, let J, X, ψ, φ satisfy the condition of p-(Lfs). By Up (x) = F0 (see Example 5.6, [14]) we have   ∀F ∈ FLs (J ), ∀z ∈ X, limq ψ ⇒ (F)(z) = 1 = [F0 , kL φF] = Up (z), kL φF , i.e., the axiom p-(Lfs) is satisfied. It is proved in [23] that (Lfs) ⇔ (Lf) + (Lp). Let limp satisfy (Lpw2). From Proposition 3.7 and Proposition 3.16, we obtain that p-(Lf) + p-(Lps) ⇒ p-(Lfs) and p-(Lfs) ⇒ p-(Lps). But we fail to prove p-(Lfs) ⇒ p-(Lf). Therefore, we do not know that whether the equivalence p-(Lf) + p-(Lps) ⇔ p-(Lfs) holds. For stratified strong L-generalized convergence spaces, it is proved in [17] that (Lfs) ⇔ (Lu) + (Lk ) ⇔ (Lgs): ∀F ∈ FLs (X), limp F  limp Up (F). Because (Lu) + (LC2 ) ⇔ (Lps) and Up (F)  F , it is easily seen that (Lgs) can be rewritten as limp F(x) = limp Up (F)(x) = [Up (x), Up (F)]. The following theorem generalizes the above equivalence. Theorem 3.19. Let (X, limp , limq ) be a pair of stratified strong L-generalized convergence space and limp with (Lu). Then p-(Lfs) ⇔ p-(Lgs):   ∀F ∈ FLs (X), ∀x ∈ X, limq F(x) = limq Up (F)(x) = Up (x), Up (F) . Proof. p-(Lfs) ⇒ p-(Lgs). By p-(Lfs) ⇒ p-(Lps) we have limq Up (F)(x) = [Up (x), Up (F)]. Take J = X, ψ = idX and φ = Up . Then ψ ⇒ (F) = F , kL φF = Up (F) for every F ∈ FLs (X), and φ(j )  [ψ(j )] for every j ∈ J . By (Lu) of limp we have limp φ(j )(ψ(j )) = 1 for every j ∈ J . Thus by p-(Lfs) we have limq F(x) = [Up (x), Up (F)]. p-(Lgs) ⇒ p-(Lfs). Firstly, note that if we take F = [x] in p-(Lgs), then we get p-(Lu). Let J, X, φ, ψ be defined as in p-(Lfs). For each F ∈ FLs (J ), by ∀j ∈ J , φ(j )  [ψ(j )] we get kL φF  ψ ⇒ (F). From ∀j ∈ J , limp φ(j )(ψ(j )) = 1 we get φ(j )  Up (ψ(j )), then it follows that Up (ψ ⇒ (F))  kL φF . Thus      limq ψ ⇒ (F)(x) = Up (x), Up ψ ⇒ (F)  Up (x), kL φF    LC2

q ⇒  lim Up (x)(x) ∧ Up (x), ψ (F)  limq ψ ⇒ (F)(x), i.e., p-(Lfs) holds.

2

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

37

Remark 3.20. Because p-(Lfs) does not imply (Lu) of limp (see Example 3.18), thus it seems that the axiom (Lu) in the preceding theorem cannot be deleted generally. It is not an easy thing to discuss whether the axiom p-(Lfs) and the axioms p-(Lgs) are preserved under initial (final) constructions. But we can prove that the axiom (Lfs) is preserved under initial constructions. Lemma 3.21. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces with each limqi being fi

pi -pretopological. If limq (resp., limp ) is the initial structure on X relative to the source (X −→ (Xi , limqi ))i∈I (resp., fi

(X −→ (Xi , limpi ))i∈I ), then (X, limq ) is p-pretopological.  q qi ⇒ s Proof. Let F ∈ F i∈I lim fi (F)(fi (x)) L (X) and x ∈ X. Then by the given condition we have lim F(x) = p pi ⇒ p pi and lim F(x) = i∈I lim fi (F)(fi (x)). For all i ∈ I , because fi : (X, lim ) −→ (X, lim ) is continuous so Upi (fi (x))  fi⇒ (Up (x)). From limqi is pi -pretopological we get          limqi fi⇒ (F) fi (x)  Upi fi (x) , fi⇒ (F)  fi⇒ Up (x) , fi⇒ (F)  Up (x), F .  By the arbitrariness of i we have limq F(x) = i∈I limqi fi⇒ (F)(fi (x))  [Up (x), F]. Thus limq is p-pretopological. 2 When limp = limq , the inequality [Up (x), F]  limp F(x) = limq F(x) is natural true. Thus by the above lemma we obtain that (Lps) ⇔ (Lp) is preserved under initial constructions. In [23], we have known that (Lfs) ⇔ (Lps) + (Lf). By Lemma 4.8 in [15] we know that (Lf) is preserved under initial constructions. Therefore, (Lfs) is preserved under initial constructions. 3.4. p-Topological axioms for levelwise stratified L-convergence spaces In this subsection, a lattice-valued Fischer’s diagonal condition and a lattice-valued Gähler’s neighborhood condition are used to study p-topologicalness for levelwise stratified L-convergence spaces. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. We consider the following axiom pα p-(LLf): Let J, X be any sets, α ∈ L, ψ : J −→ X, and φ : J −→ FLs (X) such that φ(j ) → ψ(j ), for each j ∈ J . qα

qα

Then for each F ∈ FLs (J ) and each x ∈ X, ψ ⇒ (F) → x implies kL φF → x. When limp = limq , p-(LLf) is precisely the lattice-valued Fischer’s diagonal axiom in [8]. Restricting J = X and ψ = id in p-(LLf), we obtain a weaker axiom p-(LLk). Similar to Proposition 3.7, we have the following proposition. Proposition 3.22. (1) p-(LLf) ⇒ p-(LLpw2) + p-(LLk), and (2) p-(LLk) ⇒ p-(LLf) if p satisfies (LLpw2). Corollary 3.23. If p satisfies (LLpw2) then p-(LLf) ⇔ p-(LLk) + p-(LLpw2). Remark 3.24. Let (X, limq ) be a stratified L-generalized convergence space. Then it is easily seen that the pair q

q

(q lim )α

(X, q lim ), where F → x iff limq F(x)  α, is a levelwise stratified L-convergence space. Thus Example 3.18 shows that p-(LLf) does not imply (LLpw2) of p generally. Therefore, we guess that the additional condition (LLpw2) in the above corollary cannot be deleted. Let (X, p) be a levelwise stratified L-convergence space. Then for each F ∈ FLs (X) and each α ∈ L, the stratified L-filter Upα (F) := kL Upα F is called α-level stratified L-neighborhood filter of F w.r.t. (X, p). Theorem 3.25. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. Then p-(LLf) ⇔ p-(LLg): qα qα ∀α ∈ L, ∀F ∈ FLs (X), ∀x ∈ X, F → x ⇒ Upα (F) → x.

38

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

Proof. p-(LLf) ⇒ p-(LLg). For each α ∈ L, let J = {(G, y) ∈ FLs (X) × X | G ∈ lpα (y)}, ψ : J −→ X, (G, y) → y, pα

pα

φ : J −→ FLs (X), (G, y) → G. Then ∀j ∈ J , φ(j ) → ψ(j ). Because [y] → y we have that ψ is a surjective function. Thus for each F ∈ FLs (X), K = ψ ⇐ (F) ∈ FLs (J ) and ψ ⇒ (K) = F . Similar to Theorem 3.10, we have kL φK = qα

qα

Upα (F). Let F = ψ ⇒ (K) → x. Then by p-(LLf) we have kL φK = Upα (F) → x. That is, the axiom p-(LLg) holds. qα

p-(LLg) ⇒ p-(LLf). Let J, X, φ, ψ and α ∈ L satisfy the condition of p-(LLf). If ψ ⇒ (F) → x then it follows by qα qα p-(LLg) that Upα (ψ ⇒ (F)) → x. Similar to Theorem 3.10, we have Upα (ψ ⇒ (F))  kL φF . Thus kL φF → x. That is, p-(LLf) is satisfied. 2 Definition 3.26. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. Then q is called p-topological if the axiom p-(LLf) or p-(LLg) is satisfied. One can prove that p-topologicalness is preserved under initial constructions and finial constructions with similar conditions in Proposition 3.15. At last, we give the relationships between p-topological axiom for stratified L-generalized convergence spaces and that for levelwise stratified L-convergence spaces. Proposition 3.27. Let (X, limp , limq ) be a pair of stratified L-generalized spaces. Then (X, limp , limq ) satisfies p q p-(Lg) iff (X, p lim , q lim ) satisfies p-(LLg). q

Proof. Let (X, limp , limq ) satisfy p-(Lg). If F (q

limq

)α

p

(q lim )α

→

x, then limq F(x)  α. It follows that α  limq Upα (F)(x),

q

i.e., Upαlimp (F) → x. Thus (X, p lim , q lim ) satisfies p-(LLg). p

q

Conversely, let (X, p lim , q lim ) satisfy p-(LLg). Take x ∈ X and F ∈ FLs (X), let limq F(x) = β. For each α ∈ L, q

we have limq F(x)  α ∧β, i.e., F

(q lim )α∧β

→

α∧β lim Up (F)(x)  α ∧ β. Note that p q q

i.e., (X, lim , lim ) satisfies p-(Lg).

2

q

p

q

α∧β

x. It follows from p-(LLg) of (X, p lim , q lim ) that Uplimp (F)

α∧β Up (F)  Upα (F), and so,

q

lim

Upα (F)(x)  α ∧ β

(q lim )α∧β

→

x,

= α ∧ lim F(x). Thus q

4. p-Regularity axioms In this section, we shall study the p-regularity axiom for stratified L-generalized convergence spaces and that for levelwise stratified L-convergence spaces. Generally, two approaches are used to discuss lattice-valued p-regularity. One is by the dual axiom of lattice-valued Fischer’s diagonal axiom and the other is by closures of stratified L-filters. 4.1. The α-level closure of a stratified L-filter Definition 4.1. (See [18].) Let (X, limp ) be a stratified L-generalized convergence space or (X, p)  be a levelwise stratified L-convergence space. For each λ ∈ LX , the L-set λαp ∈ LX defined by ∀x ∈ X, λαp (x) = F ∈lpα (x) F(λ) is called the α-level closure of λ. The following lemma collects some basic properties of α-level closure of L-sets. Lemma 4.2. Let (X, limp ) be a stratified L-generalized convergence space or (X, p) be a levelwise stratified L-convergence space. Then for all λ, μ ∈ LX and all α, β ∈ L we get β (1) λ  λαp ; (2) λ  μ implies λαp  μαp ; (3) If α  β, then λαp  λp . (4) (β ∧ λ)αp  β ∧ λαp and the equality holds if L is a complete Boolean algebra;  (5) For each x ∈ X, let ulpα (x) = lpα (x) ∩ ULs (X). If L is a complete Boolean algebra, then λαp (x) = F ∈ulpα (x) F(λ) and (λ ∨ μ)αp = λαp ∨ μαp .

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

39

p

Proof. (1) For each x ∈ X, by [x] ∈ lα (x) we get λαp (x)  [x](λ) = λ(x). So, λ  λαp . Take λ = 1 in (1), we obtain 1αp = 1. (2) It follows from the property (F2) of stratified L-filters. β (3) It follows from that α  β implies lpα (x) ⊇ lp (x) for each x ∈ X. (4) For each x ∈ X we have    F(β) ∧ F(λ) (β ∧ λ)αp (x) = F(β ∧ λ) = F ∈lpα (x)



F ∈lpα (x)

 β ∧ F(λ) = β ∧ F(λ) = β ∧ λαp (x).

 

F ∈lpα (x)

F ∈lpα (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)αp = β ∧ λαp .  (5) That λαp (x) = F ∈ulpα (x) F(λ) follows from Lemma 2.3. To prove (λ ∨ μ)αp = λαp ∨ μαp , it suffices to check that (λ ∨ μ)αp  λαp ∨ μαp since the reverse inequality holds by (2). Indeed, also by Lemma 2.3 we get that each stratified L-ultrafilter is prime. Therefore       λαp (x) ∨ μαp (x) = F(λ) ∨ G(μ) F ∈ulpα (x)

=



F ,G ∈ulpα (x)

=





G ∈ulpα (x)

F(λ) ∨ G(μ) 





F(λ) ∨ F(μ)

F ∈ulpα (x)

F(λ ∨ μ) = (λ ∨ μ)αp (x).

2

F ∈ulpα (x)

Let L be a complete Boolean algebra. For each α ∈ L, F ∈ FLs (X), Jäger proved that the function F αp : LX −→ L defined by   F αp (λ) = F(μ): μ ∈ LX , ∀x ∈ X, ∀F ∈ lpα (x), F(μ)  λ(x) ,  is a stratified L-filter [16]. Later, he [18] proved that F αp (λ) = μ∈LX ,[μαp ,λ]=1 F(μ). In the above proof, the tightness of the L-filters is used. Thus, as Jäger pointed out, the lattice context should be restricted to be complete Boolean algebra. To generalize the lattice-context, Boustique and Richardson [3] modified Jäger’s definition and proved that their definition coincided with Jäger’s definition when L is a complete Boolean algebra. Definition 4.3. (See [3].) Let (X, limp ) be a stratified L-generalized convergence space or (X, p) be a levelwise stratified L-convergence space. For each F ∈ FLs (X), α ∈ L, the function F αp : LX −→ L defined by   F αp (λ) = F(μ) ∧ β: μ ∈ LX , β ∈ L, μ ∧ β  λ, ∀G ∈ lpα (x), G(μ) ∧ β  λ(x) is a stratified L-filter on X, called the α-level closure of F . The following theorem gives a more simple and intuitive representation for the α-level closure of stratified L-filter. Theorem 4.4. Let (X, limp ) be a stratified L-generalized convergence spaceor (X, p) be a levelwise stratified X α α L-convergence space. For each α ∈ L, F ∈ FLs (X)  and each λ ∈ L , F p (λ) = μ∈LX (F(μ) ∧ [μp , λ]). Thus when α L is a complete Boolean algebra, then Fp (λ) = μ∈LX ,[μαp ,λ]=1 F(μ). Proof. It is easily seen that the condition μ ∧ β  λ is equivalent to β  [μ, λ], and the condition ∀G ∈ lpα (x), G(μ) ∧ β  λ(x) is equivalent to β  [μαp , λ]. Because [μαp , λ]  [μ, λ], then the condition μ ∧ β  λ, in the definition of F αp , can be deleted. It follows that

40

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

F αp (λ) =



      F(μ) ∧ μαp , λ . F(μ) ∧ β: μ ∈ LX , β ∈ L, β  μαp , λ =

2

μ∈LX

Remark 4.5. When L = {0, 1}, a stratified L-generalized convergence space or a levelwise stratified L-convergence space reduces to a convergence space. Then it is easily seen that F 1p is precisely the filter generated by {A: A ∈ F} as a filterbasis [26]. We close this subsection with a useful lemma. Lemma 4.6. Let (X, p) be a levelwise stratified L-convergence space or (X, limp ) be a stratified L-generalized convergence space. Then for each F, G ∈ FLs (X) and each α ∈ L, [F, G]  [F αp , G αp ]. Proof. It holds by the following inequality.   α α F p, Gp = F αp (λ) → G αp (λ) λ∈LX

   

=

F(μ) ∧

λ∈LX



 →

  

 

G(ν) ∧



ν αp , λ

ν∈LX

F(μ) ∧

λ∈LX μ∈LX



μαp , λ

μ∈LX

  

=





μαp , λ



→

 

G(ν) ∧



ν αp , λ









ν∈LX

     F(μ) ∧ μαp , λ → G(μ) ∧ μαp , λ

λ∈LX μ∈LX

 

F(μ) → G(μ) = [F, G].



2

μ∈LX

4.2. p-Regularity axiom for stratified L-generalized convergence spaces Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. We consider the following axiom. p-(DLf): Let J, X be any sets, ψ : J −→ X and φ : J −→ FLs (X) be functions.   ∀F ∈ FLs (J ), ∀x ∈ X, limq kL φF(x) ∧ limp φ(j ) ψ(j )  limq ψ ⇒ (F)(x). j ∈J

When lim = lim , the axiom p-(DLf) is precisely the axiom (LR) in [16]. Next, we give a characterization of p-(DLf) by α-level closure of stratified L-filter. p

q

Lemma 4.7. Let J, X, φ, ψ satisfy the condition of p-(DLf) and α ∈ L. If α  ˆ [φ(μ), ψ ← (λ)] for all λ, μ ∈ LX . Proof. If α 





j ∈J

limp φ(j )(ψ(j )) then φ(j ) ∈ lpα (ψ(j )) for all j ∈ J and         α   μp , λ = G(μ) → λ(x)  G(μ) → λ ψ(j ) j ∈J

x∈X G ∈lpα (x) φ(j )∈lpα (ψ(j ))







j ∈J

G ∈lpα (ψ(j ))

   ˆ φ(j )(μ) → λ ψ(j ) = φ(μ)(j ) → ψ ← (λ)(j )

j ∈J

 ˆ = φ(μ), ψ ← (λ) .

j ∈J

2

limp φ(j )(ψ(j )), then [μαp , λ] 

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

41

Lemma 4.8. Let J, X, φ, ψ satisfy the condition of p-(DLf) and F ∈ FLs (X). We define F φ : LJ −→ L as F φ (λ) =  φ

φ φ ˆ μ∈LX (F(μ) ∧ [φ(μ), λ]). Then F satisfies (F2), (Fs ) and F (1) = 1 and kL φF  F . Proof. It is obvious that F φ (1) = 1. (F2) Firstly, note that F φ (λ)  F φ (μ) whenever λ  μ. It follows F φ (λ ∧ μ)  F φ (λ) ∧ F φ (μ). Conversely,        ˆ ˆ F(a) ∧ φ(a), λ ∧ F(b) ∧ φ(b), μ F φ (λ) ∧ F φ (μ) = a∈LX

=

 

   ˆ ˆ F(a) ∧ F(b) ∧ φ(a), λ ∧ φ(b), μ

a,b∈LX



b∈LX

 



  ˆ ∧ b), λ ∧ μ F(a ∧ b) ∧ φ(a

a,b∈LX





  ˆ F(c) ∧ φ(c), λ ∧ μ = F φ (λ ∧ μ).

c∈LX

 ˆ ˆ (Fs ) For all β ∈ L, it follows that F φ (β) = μ∈LX (F(μ) ∧ [φ(μ), β])  F(1) ∧ β = β by φ(1) = 1. φ X That kL φF  F follows because for each λ ∈ L ,      ˆ ˆ ˆ F(μ) ∧ φ(μ), kL φF φ (λ) = F φ φ(λ) = φ(λ)  F(λ). 2 μ∈LX

Theorem 4.9. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then p-(DLf) ⇔ p-(LRC): ∀α ∈ L, ∀F ∈ FLs (X), α  [limq F, limq F αp ]. Proof. p-(DLf) ⇒ p-(LRC). For each α ∈ L, let J = {(G, y) ∈ FLs (X) × X: G ∈ lpα (y)}, ψ : J −→ X, (G, y) →  y, φ : J −→ FLs (X), (G, y) → G. Then j ∈J limp φ(j )(ψ(j ))  α. Let F ∈ FLs (X), we prove F φ ∈ FLs (J ). By Lemma 4.8, we check only F φ (0) = 0. Indeed,          ˆ F(μ) ∧ φ(μ), G(μ) → 0 F φ (0) = F(μ) ∧ 0 = μ∈LX

(G ,y)∈J

μ∈LX

        = F(μ) ∧ (βμ → 0) , G(μ) → 0 = F(μ) ∧ 

(G ,y)∈J

μ∈LX

μ∈LX

  where (G ,y)∈J G(μ) = βμ . Then from ∀y ∈ X, ([y], y) ∈ J we have y∈X μ(y)  (G ,y)∈J G(μ) = βμ , which means F(μ)  F(βμ ), and so     F(μ) ∧ (βμ → 0)  F(βμ ) ∧ (βμ → 0)  F(0) = 0. F φ (0) = μ∈LX

μ∈LX

For all λ, μ ∈ LX ,      α   μp , λ = G(μ) → λ(y) = y∈X G ∈lpα (y)

=



 φ(j )(μ) → λ ψ(j ) =

j ∈J

 ˆ = φ(μ), ψ (λ) . 

It follows that

←

 

y∈X G ∈lpα (y)



G(μ) → λ(y)

ˆ φ(μ)(j ) → ψ ← (λ)(j )

j ∈J

42

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

      ˆ F(μ) ∧ φ(μ), ψ ← (λ) ψ ⇒ F φ (λ) = F φ ψ ← (λ) = =

 



μ∈LX

F(μ) ∧ μαp , λ



= F αp (λ).

μ∈LX

Thus ψ ⇒ (F φ ) = F αp . By p-(DLf) and kL φF φ  F (Lemma 4.8), we have  limq F(x) → limq F αp (x)  limq kL φF φ (x) → limq ψ ⇒ F φ (x)    limp φ(j ) ψ(j )  α. j ∈J

By the arbitrariness of x, we get [limq F, limq F αp ]  α.  p-(LRC) ⇒ p-(DLf). Let J , X, φ, ψ be defined as in p-(DLf). Then assume that α = j ∈J limp φ(j )(ψ(j )). For each F ∈ FLs (J ), it follows by p-(LRC) that [limq kL φF, limq (kL φF)αp ]  α. For all λ ∈ LX , by Lemma 4.7, we have         α  ˆ kL φF(μ) ∧ μαp , λ = F φ(μ) (kL φF)αp (λ) = ∧ μp , λ μ∈LX

μ∈LX

      ˆ ˆ  F φ(μ) ∧ φ(μ), ψ ← (λ)  F ψ ← (λ) = ψ ⇒ (F)(λ). μ∈LX

Thus (kL φF)αp  ψ ⇒ (F), then p-(DLf) follows by  q    lim kL φF, limq ψ ⇒ (F)  limq kL φF, limq (kL φF)αp  α.

2

Definition 4.10. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then limq is called p-regular if the axiom p-(DLf) or p-(LRC) is satisfied. In [16], Jäger proved that when (Lpw2) is satisfied then regularity has a nice characterization. The next proposition gives a similar result for p-regularity. Proposition 4.11. Let (X, limp , limq ) satisfy p-(Lps). Then limq is p-regular iff for each α ∈ L and each x ∈ X, α  limq Up (x)αp = [Up (x), Up (x)αp ]. Proof. It suffices to prove that α  [Up (x), Up (x)αp ] implies α ∧ limq F(x)  limq F αp (x) for all F ∈ FLs (X). Indeed, by Lemma 4.6      p -(Lps)  limq F αp (x) = Up (x), F αp  Up (x), Up (x)αp ∧ Up (x)αp , F αp  p-(Lps)  = α ∧ limq F(x). 2  α ∧ Up (x), F The following theorem shows that p-regularity is preserved under initial constructions. The proof is similar to Lemma 5.2 in [16], thus it is omitted. Proposition 4.12. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces with each limqi fi

being pi -regular. If limq (resp., limp ) is the initial structure on X relative to the source (X −→ (Xi , limqi ))i∈I (resp., fi

(X −→ (Xi , limpi ))i∈I ), then (X, limq ) is p-regular. Definition 4.13. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified L-generalized convergence spaces or f : (X, q) −→ (Y, p) be a function between levelwise stratified L-convergence spaces. Then f is said to be a closure function if f → (λαq )  f → (λ)αp for all λ ∈ LX and all α ∈ L.

L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45

43

Lemma 4.14. Let f : (X, limq ) −→ (Y, limp ) or f : (X, q) −→ (Y, p) be a closure function. Then f ⇒ (F αq )  f ⇒ (F)αp for all F ∈ FLs (X) and all α ∈ L. Proof. We prove only for stratified L-generalized convergence spaces. For each F ∈ FLs (X), that f ⇒ (F αq )  f ⇒ (F)αp follows by ∀λ ∈ LY        f ⇒ (F)(μ) ∧ μαp , λ = F f ← (μ) ∧ μαp , λ f ⇒ (F)αp (λ) = μ∈LY

μ∈LY

        F f ← f → (ν) ∧ f → (ν)αp , λ  F(ν) ∧ f → (ν)αp , λ ν∈LX







F(ν) ∧ f

→



ν αq



,λ



=

ν∈LX   = F αq f ← (λ) = f ⇒ F αq (λ),



ν∈LX

  F(ν) ∧ ν αq , f ← (λ)

ν∈LX

where the third inequality holds for f being a closure function, and the third equality follows from Lemma 2.2 (7).

2

Proposition 4.15. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces with each limqi fi

being pi -regular. Let limq (resp., limp ) be the final structure on X w.r.t. the sink ((Xi , limqi ) −→ X)i∈I (resp.,

fi ((Xi , limpi ) −→ X)i∈I ). If X = i∈I fi (Xi ) and each fi : (Xi , limpi ) −→ (X, limp ) is a closure function, then (X, limq ) is p-regular. Proof. Let F ∈ FLs (X), α ∈ L and x ∈ X. Then by Remark 2.7 we have   α ∧ limqi Gi (xi ) α ∧ limq F(x) = i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F





i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )F

limqi Gi αpi (xi )





limqi Gi αpi (xi )

i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi )αp F αp



=

limqi Gi αpi (xi )

p i∈I,xi ∈Xi ,Gi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Gi αp )fi⇒ (Gi )αp F α i





limqi Hi (xi )

i∈I,xi ∈Xi ,Hi ∈FLs (Xi ),fi (xi )=x,fi⇒ (Hi )F αp

= limq F αp (x), where the first inequality holds for pi -regularity of (Xi , limqi ), the second equality follows from the above lemma. By the arbitrariness of x, we have α  [limq F, limq F αp ]. It follows that (X, limq ) is p-regular. 2 The above proposition shows that p-regularity is preserved under final constructions with some additional conditions. 4.3. p-Regularity for levelwise stratified L-convergence spaces Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. We consider the following axioms: pα p-(DLLf): Let J, X be any sets, α ∈ L, ψ : J −→ X, and φ : J −→ FLs (X) such that φ(j ) → ψ(j ), for each qα

qα

j ∈ J . Then for each F ∈ FLs (J ) and each x ∈ X, kL φF → x implies ψ ⇒ (F) → x.

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L. Li, Q. Jin / Fuzzy Sets and Systems 238 (2014) 26–45 qα

qα

p-(LLRC): ∀α ∈ L, ∀F ∈ FLs (X), F → x implies F αp → x. When p = q, the axiom p-(DLLf) is precisely the axiom (R2) in [9]. The axiom p-(LLRC) has appeared in [24]. Next, we shall prove that p-(DLLf) ⇔ p-(LLRC). The following two lemmas are needed. The proofs of them are similar to Lemmas 4.7 and 4.8 thus they are omitted. ˆ Lemma 4.16. Let J , X, φ, ψ satisfy the condition of p-(DLLf) and α ∈ L. Then [μαp , λ]  [φ(μ), ψ ← (λ)] for all λ, μ ∈ LX . Lemma 4.17. Let J, X, φ, ψ satisfy the condition of p-(DLLf) and F ∈ FLs (X). We define F φ : LJ −→ L as  ˆ F φ (λ) = μ∈LX (F(μ) ∧ [φ(μ), λ]). Then F φ satisfies (F2), (Fs ) and F φ (1) = 1 and kL φF φ  F . Theorem 4.18. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. Then p-(DLLf) ⇔ p-(LLRC). qα

Proof. p-(LLRC) ⇒ p-(DLLf). Let J , X, φ, ψ be defined as in p-(DLLf). Assume that F ∈ FLs (J ) and kL φF → x. qα

qα

Then (kL φF)αp → x. By Lemma 4.16, we can prove (kL φF)αp  ψ ⇒ (F). Then ψ ⇒ (F) → x, as desired. p-(DLLf) ⇒ p-(LLRC). For each α ∈ L, let J = {(G, y) ∈ FLs (X) × X | G ∈ lpα (y)}, ψ : J −→ X, (G, y) → y, qα

φ : J −→ FLs (X), (G, y) → G. Then ∀j ∈ J , φ(j ) → ψ(j ). Similar to Theorem 4.11, it can be proved that qα

F φ ∈ FLs (J ) and ψ ⇒ (F φ ) = F αp for each F ∈ FLs (X). By p-(DLLf) and kL φF φ  F , we have F → x implies qα

F αp → x.

2

Definition 4.19. Let (X, p, q) be a pair of levelwise stratified L-convergence spaces. Then q is called p-regular if the axiom p-(DLLf) or p-(LLRC) is satisfied. It can be proved that p-regularity is preserved under initial constructions and final constructions with similar restrictions as Proposition 4.15. The following proposition gives the relationship between p-regularity for stratified L-convergence space and p-regularity for levelwise stratified L-convergence space. The proof is similar to Proposition 3.27, thus it is omitted. Proposition 4.20. Let (X, limq , limp ) be a pair of stratified L-generalized convergence spaces. Then limq is p-regular q p iff q lim is p lim -regular. 5. Conclusions This paper focuses on p-topological axioms and p-regular axioms for stratified L-generalized convergence spaces and those for levelwise stratified L-convergence spaces. The p-topologicalness is discussed both by lattice-valued diagonal axioms and by α-level neighborhood of stratified L-filter. The p-regularity is discussed both by lattice-valued diagonal axiom and by α-level closure of stratified L-filter. The p-topologicalness and p-regularity w.r.t. initial constructions and final constructions are also discussed. As a preliminary, some p-pretopological axioms are presented and compared, and a more simple and intuitive representation for the α-level closure of stratified L-filter is given. If we replace stratified L-filters FLs (X) with stratified L-ultrafilters ULs (X), we can define weaker p-topologicalness and weaker p-regularity similarly. It can be proved that weaker p-topologicalness (resp., weaker p-regularity) is equivalent to p-topologicalness (resp., p-regularity) whenever L is a complete Boolean algebra. Acknowledgements The authors thank the reviewers and the area editor for their valuable comments and suggestions. Dedicated to the first author’s mother Guifang Tian on the occasion of her 60th birthday. References [1] H.J. Biesterfeld, Regular convergence spaces, Indag. Math. 28 (1966) 605–607.

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[2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

45

R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, New York, 2002. H. Boustique, G. Richardson, A note on regularity, Fuzzy Sets Syst. 162 (2011) 64–66. H. Boustique, G. Richardson, Regularity: Lattice-valued Cauchy spaces, Fuzzy Sets Syst. 190 (2012) 94–104. C.H. Cook, H.R. Fischer, Regular convergence spaces, Math. Ann. 174 (1967) 1–7. J.M. Fang, Stratified L-ordered convergence structures, Fuzzy Sets Syst. 161 (2010) 2130–2149. H.R. Fischer, Limesräume, Math. Ann. 137 (1959) 269–303. P.V. Flores, R.N. Mohapatra, G. Richardson, Lattice-valued spaces: Fuzzy convergence, Fuzzy Sets Syst. 157 (2006) 2706–2714. P.V. Flores, G. Richardson, Lattice-valued convergence: Diagonal axioms, Fuzzy Sets Syst. 159 (2008) 2520–2528. W. Gähler, Monadic topology – a new concept of generalized topology, in: Recent Developments of General Topology, in: Math. Res., vol. 67, Akademie-Verlag, Berlin, 1992, pp. 136–149. ˘ U. Höhle, A. Sostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, in: Handb. Fuzzy Sets Ser., vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 123–273. G. Jäger, A category of L-fuzzy convergence spaces, Quaest. Math. 24 (2001) 501–517. G. Jäger, Subcategories of lattice-valued convergence spaces, Fuzzy Sets Syst. 156 (2005) 1–24. G. Jäger, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets Syst. 158 (2007) 424–435. G. Jäger, Fischer’s diagonal condition for lattice-valued convergence spaces, Quaest. Math. 31 (2008) 11–25. G. Jäger, Lattice-valued convergence spaces and regularity, Fuzzy Sets Syst. 159 (2008) 2488–2502. G. Jäger, Gähler’s neighbourhood condition for lattice-valued convergence spaces, Fuzzy Sets Syst. 204 (2012) 27–39. G. Jäger, Diagonal conditions for lattice-valued uniform convergence spaces, Fuzzy Sets Syst. 210 (2013) 39–53. D.C. Kent, G.D. Richardson, p-regular convergence spaces, Math. Nachr. 149 (1990) 215–222. D.C. Kent, G.D. Richardson, Convergence spaces and diagonal conditions, Topol. Appl. 70 (1996) 167–174. L. Li, Many-valued convergence, many-valued topology, and many-valued order structure, PhD thesis, Sichuan University, 2008 (in Chinese). L. Li, Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets Syst. 182 (2011) 66–78. L. Li, Q. Jin, On stratified L-convergence spaces: Pretopological axioms and diagonal axioms, Fuzzy Sets Syst. 204 (2012) 40–52. B. Losert, H. Boustique, G. Richardson, Modifications: Lattice-valued structures, Fuzzy Sets Syst. 210 (2013) 54–62. D. Orpen, G. Jäger, Lattice-valued convergence spaces: extending the lattices context, Fuzzy Sets Syst. 190 (2012) 1–20. G. Preuss, Foundations of Topology, Kluwer Academic Publishers, London, 2002. S.A. Wilde, D.C. Kent, p-Topological and p-regular: dual notions in convergence theory, Int. J. Math. Math. Sci. 22 (1999) 1–12. W. Yao, On many-valued stratified L-fuzzy convergence spaces, Fuzzy Sets Syst. 159 (2008) 2503–2519. D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets Syst. 158 (2007) 349–366.