`Abstract and Applied AnalysisVolume 2014 (2014), Article ID 328153, 11 pageshttp://dx.doi.org/10.1155/2014/328153`
Research Article

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

1Department of Mathematics, Liaocheng University, Liaocheng 252059, China
2College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 5 September 2013; Accepted 7 December 2013; Published 5 January 2014

Copyright © 2014 Lingqiang Li and Qiu Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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

Dedicated to the first author’s father Zonghua Li on the occasion of his 60th birthday

#### 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 , if converges to then so does (the closure of ). In [3, 4], by considering a pair of convergence spaces and , Kent and his coauthors introduced a kind of relative topologicalness (resp., regularity) which was called -topologicalness (resp., -regularity). They discussed -topologicalness (resp., -regularity) both by neighborhood (resp., closure) of filter [5] and generalized F (resp., DF)-diagonal condition. When , -topologicalness (resp., -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 -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 -convergence spaces. Later, the theory of these spaces was extensively discussed under different lattice context [819]. A supercategory of generalized stratified -convergence spaces, called levelwise stratified -convergence spaces in this paper, was researched in [2024]. Indeed, a generalized stratified -convergence space is precisely a left-continuous levelwise stratified -convergence space [22].

Lattice-valued K- and F-diagonal conditions for generalized stratified -convergence spaces were studied in [11, 12, 17, 18] and those for levelwise stratified -convergence spaces were discussed in [18, 23]. Both by lattice-valued DF-diagonal condition and -level closures of stratified -filters, the lattice-valued regularity for generalized stratified -convergence spaces was presented in [13] and that for levelwise stratified -convergence spaces was given in [20, 21]. Later, by -level closures of stratified -filters, -regularity for levelwise generalized stratified -convergence spaces was studied in [24]. Recently, -topologicalness and -regularity for generalized stratified -convergence spaces and that for level stratified -convergence spaces were discussed systemically in [25].

In this paper, for generalized stratified -convergence spaces and levelwise stratified -convergence spaces, we will discuss some lattice-valued weaker regularities, -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 -filters and a new lattice-valued -regularity for stratified generalized -convergence spaces. Also, the relationships between lattice-valued weaker regularities and lattice-valued -regularities are established.

#### 2. Preliminaries

In this paper, if not otherwise specified, is always a complete lattice with a top element and a bottom element , which satisfies the distributive law . A lattice with these conditions is called a complete Heyting algebra or a frame. The operation given by is called the residuation with respect to . A complete Heyting algebra is said to be a complete Boolean algebra if it obeys the law of double negation: , .

For a set , the set of functions from to with the pointwise order becomes a complete lattice. Each element of is called an -set (or a fuzzy subset) of . For any , , and , we denote by , , , and the -sets defined by , , , and . Also, we make no difference between a constant function and its value since no confusion will arise. For a crisp subset , let be the characteristic function; that is if and if . Clearly, the characteristic function of a subset can be regarded as a function from to .

Let be a set. A fuzzy partial order (or an -partial order) on [26] is a function such that (1) for every (reflexivity); (2) implies that for all (antisymmetry); (3) for all (transitivity). The pair is called an -partially ordered set.

Let be a function defined by ; then is an -partial order on . The value is interpreted as the degree that is contained in . In the sequel, we use the symbol to denote for simplicity.

Let be an ordinary function. We define and [27] by for and , and for .

##### 2.1. Stratified -(Ultra)filters

A stratified -filter [27] on a set is a function such that for each and each , (F1) , ; (F2) ; (Fs) . A stratified -filter is called tight if for each [5]. It is proved in [27] that all stratified -filters are tight if and only if is a complete Boolean algebra. It is easily seen that for a stratified -filter on , we have , .

The set of all stratified -filters on is ordered by , . It is shown in [27] that the partially ordered set has maximal elements which are called stratified -ultrafilters. The set of all stratified -ultrafilters on is denoted as . Let . Then is an -ultrafilter if and only if for all we have . A stratified -filter is called a stratified -prime filter if for each . And when is a complete Boolean algebra then and is prime whenever is maximal [27].

For each , it is easily seen that is a filter on . For each , take . Let be a filter on . Then, when is a linearly order frame or is prime ( implies or ), the function , defined by , if and if not so, is a stratified -filter on [22]. Also, when is a linearly order frame or is prime, a stratified -ultrafilter takes values in only [10].

Lemma 1 (Jäger [28] for ). Let be a linearly order frame or let be prime. Then, for each , is an ultrafilter on and .

Proof. At first, we check that is an ultrafilter on . For each , we assume that ; that is, ; then . That means . By the arbitrariness of we get that is an ultrafilter on . At second, we check . Note that takes values in only; thus, it suffices to prove that if ; then . Indeed, let ; then ; that is, and so . Therefore, and it follows that by the maximality of .

The following examples belong to the folklore; we list them here because the notations are needed.

Example 2. (1) For each point in a set , the function , is a stratified -filter on . In general, is not a stratified -ultrafilter. But when is a complete Boolean algebra, then it is so.
(2) Let be a family of stratified -filters on ; then , in particular, , is a stratified -filter on .
(3) Let be a function. If , then the function , where defined by . If , then .

There is a natural fuzzy partial order on inherited from . Precisely, for all , if we let , then is an -partially order. For simplicity, we use the symbol to denote the value below.

##### 2.2. Lattice-Valued Convergence Spaces

Definition 3. A generalized stratified -convergence structure [7] on a set is a function satisfying (LC1) , ; and (LC2) , . The pair is called a generalized stratified -convergence space. If further satisfies the strong axiom (LC2) , then the pair is called a strong stratified -convergence space [8, 15, 16].

A function between two generalized stratified -convergence spaces , is called continuous if for all and all we have . The category SL-GCS has as objects all generalized stratified -convergence spaces and as morphisms the continuous functions. This category is topological over SET [7, 10]. For a given source , the initial structure, on is defined by , , .

Definition 4. A collection , where , is called a levelwise stratified -convergence structure on [20] if it satisfies the following:(LL1) for each ; (LL2) implies ;(LL3) implies whenever .The notation, , means that . The pair is called a levelwise stratified -convergence space.

A function between two levelwise stratified -convergence spaces , is called continuous if for all all , and all we have implies . The category SL-LCS has as objects all levelwise stratified -convergence spaces and as morphisms the continuous functions. This category is topological over SET [20, 21]. For a given source , the initial structure, on is defined by , .

#### 3. Lattice-Valued Weaker Regularities

In this section, we will present the definitions, characterizations, and properties of lattice-valued weaker regularities.

Let be a set; a function is usually called an -filter select function on . We define as , . Let denote the set of all -filter select functions on , and let be the subset consisting of all such that for all .

Let . For all , it can be proved that the function , defined by , , is a stratified -filter, which is called the -diagonal filter of [11, 17]. Then we have the following obvious lemma. It may have appeared in some other places.

Lemma 5. Let or . Then(1), ;(2)for each , ;(3) implies ;(4)for all , then . In particular, if then .

##### 3.1. For Generalized Stratified -Convergence Spaces

Let be a generalized stratified -convergence space. We consider the following axioms.

DLK. For each , we have

. Taking as , in DLK.

Replacing by in DLK (resp., ), we obtain a weaker axiom in symbol (resp., ).

Remark 6. The axiom DLK is the dual axiom of LK which appeared in [11], and the axiom is the dual axiom of which appeared in [17].

Definition 7. Let be a generalized stratified -convergence space. Then is called -regular (resp., -regular, -regular, and -regular) if it satisfies the axiom DLK (resp., , , and ).

Lemma 8 (Li and Jin [25]). Let and . We define as . Then satisfies (F1), (F2), and (Fs); thus, we say that is nearly a stratified -filter. If then .

Lemma 9. Let and . Then and .

Proof. For each , we have that is, . It follows that . From the above lemma we have that is a stratified -filter on .

By the above two lemmas, we get the following characteristic theorem.

Theorem 10. Let be a generalized stratified -convergence space. Then is -regular if and only if, for each resp., , whenever .

Proof. We prove only for -regularity. Assume the given condition is satisfied, let and . By Lemma 9 we have and and so DLK holds; that is, is -regular.
Conversely, let , with . By Lemma 8, . It follows by DLK that
Thus, the requirement is satisfied.

Corollary 11. A generalized stratified -convergence space is -regular (resp., -regular) if and only if for each (resp., ) with for all , we have whenever .

The following theorem considers lattice-valued weaker regularities w.r.t. the initial structures.

Theorem 12. Let be the initial structure relative to the source with each being injective. Then if each is -regular (resp., -regular), then the same is true of .

Proof. We prove only for -regularity. Let . Fix ; define as if and if . Then for each , by it follows that (In particular, if , , then , ).
For each and each , it follows that Hence, , and then, for each , Therefore, . Then, for each , Here, the last inequality holds because each is -regular. Now, we have proved that is -regular.

The following theorem gives the relationship between types of lattice-valued weaker regularities.

Theorem 13. Let be a complete Boolean algebra. Then -regularity -regularity and -regularity -regularity.

Proof. We check only the equivalence -regularity -regularity. The other equivalence is similar. Obviously, -regularity -regularity. Conversely, let be -regular. Note that when is a complete Boolean algebra, then for every stratified -filter there exists a stratified -ultrafilter containing it. Thus, for each , there is some such that for all . Assume that with . Then it is easily seen that and . By Theorem 10, Thus, is -regular.

As a consequence, we obtain that when is a complete Boolean algebra, Theorem 12 holds for -regularity and -regularity.

Obviously, -regularity -regularity and -regularity -regularity. The following example shows that the reverse inclusions do not hold generally.

Example 14. Let and with ordering , and , . Then becomes a complete Boolean algebra. Obviously, and are all stratified -ultrafilters on . Thus, it is easily seen that the function defined by is a generalized stratified -convergence structure on .
(1)   satisfies . Let with . Then , . Thus, for each , we have . Then the axiom , and thus the axiom holds obviously.
(2) does not satisfy . Let be defined by . Then, for each , we have . For each , Taking , then , and , . It follows that It follows that the axiom and thus the axiom DLK does not hold.

##### 3.2. For Levelwise Stratified -Convergence Spaces

Let be a levelwise stratified -convergence space. We consider the following axioms:

DLLK. For each and each with , . Then , , whenever .

Replacing by in DLLK, we obtain a weaker axiom in symbol .

Remark 15. The axiom DLLK is a special case of the regular axiom (R2) in [23] with and .

Definition 16. Let be a levelwise stratified -convergence space. Then is called -regular (resp., -regular) if it satisfies the axiom DLLK (resp., ).

For -regularity (-regularity), we have the following characteristic theorem.

Theorem 17. Let be a levelwise stratified -convergence space. Then is -regular (resp., -regular) if and only if for each and each resp., and each with , , we have that implies whenever .

Proof. We prove only for -regularity. Assume the given condition is satisfied; let satisfy the condition in DLLK and . By Lemma 9 we have and . By the given condition, we have and then . So, the axiom DLLK holds; that is, is -regular. Conversely, Let and with , . Suppose that and . By Lemma 8, , so, . It follows by DLLK that as desired.

The following theorem shows that -regular is an initial property relative to any family of injection functions.

Theorem 18. Let be the initial structure relative to the source with each being injective. If each is -regular, then the same is true of .

Proof. Let and satisfy for all . Fix ; define as if and if . Then for each . Indeed, if , then , and if , then there exists an such that and so . Let . Similar to Theorem 12, we have for all . Because each is continuous, thus . Then since each is -regular. It follows that by the definition of initial structure. We have proved that is -regular.

Theorem 19. Let be a complete Boolean algebra. Then -regularity -regularity.

Proof. The proof is similar to Theorem 13 and thus it is omitted.

As a consequence, we obtain that when is a complete Boolean algebra, then Theorem 18 holds for -regularity.

The last theorem gives the relationship between -regularity for generalized stratified -convergence space and -regularity for levelwise stratified -convergence space.

Let be a generalized stratified -convergence space. It is proved in [22] that the pair , where if and only if , is a levelwise stratified -convergence space.

Theorem 20. Let be a generalized stratified -convergence space. Then is -regular (resp., -regular) if and only if is -regular (resp., -regular).

Proof. We prove only for -regularity. Let be -regular. Take and with , ; then we have . Take with ; then we have ; that is, . By Theorem 10 we obtain . Then ; that is, . It follows by Theorem 17 that is -regular.
Conversely, assume that is -regular. Let us take with and take with . Then if for , we have and . It follows by Theorem 17 that ; that is, . By the arbitrariness of we note that . It follows by Theorem 10 that is -regular.

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

##### 4.1. For Generalized Stratified -Convergence Spaces

Generally, -regularity relates to two different generalized stratified -convergence structures on the same underlying set. Thus, in this section, we add the lowercases as the superscript of and use , to denote different generalized stratified -convergence structures.

At first, we give the notion of closures of stratified -filters and then introduce a new -regularity.

Definition 21. Let be a generalized stratified -convergence space. For each , the -set defined by is called the closure of w.r.t .

Lemma 22. Let be a generalized stratified -convergence space. Then for all and all we get the following:(1);(2) implies ;(3) and the equality holds if is a complete Boolean algebra;(4)if is a complete Boolean algebra, then , , and .

Proof. (1) For each , by we get . So, . Take in (1); we obtain .
(2) It follows from the property (F2) of stratified -filters.
(3) For each we have When is a complete Boolean algebra, then , . So, the “” in the above inequality can be replaced by “”. Thus, .
(5) Let be a complete Boolean algebra. That follows because, for each , there exists an -ultrafilter such that . To prove , it suffices to check that since the reverse inequality holds by (2). Indeed, because each stratified -ultrafilter is prime we have

Theorem 23. Let be a generalized stratified -convergence space. For each , the function defined by
is a stratified -filter, called the closure of .

Proof. (F1) That is obvious. By Lemma 22(1) we have Thus, .

(F2) Firstly, note that whenever . It follows that . Conversely,

(Fs) For all , it follows that by .

It is easily seen that the following lemma holds. We omit the routine proof.

Lemma 24. Let be a generalized stratified -convergence space. Then, for each , .

Definition 25. Let be a pair of generalized stratified -convergence spaces. Then is called -regular if and only if, for each , we have .

Remark 26. When , a generalized stratified -convergence space reduces to a convergence space. It is easily seen that is precisely the filter generated by as a filterbasis [29]. And the -regularity reduces to the corresponding crisp notion in [3].

The following theorem shows that -regularity is preserved under initial constructions.

Theorem 27. Let be pairs of generalized stratified -convergence spaces with each being -regular. If   (resp., ) is the initial structure on relative to the source (resp., ), then is -regular.

Proof. At first, we check below that for each and each we have . Indeed, for each ,
It follow that, for each and each , Thus, for all . It follows by each being -regular that Thus, is -regular.

When , Kent and Richardson [6] studied the relationships between weaker regularities and -regularity. Now we discuss them for the general case.

Definition 28. A generalized (strong) stratified -convergence space is called
(i) a (strong) -Kent convergence space [10] if , , ;
(ii) pretopological [11] if , , , where , defined by , , is called the stratified neighborhood -filter of w.r.t. , and when is a strong stratified -convergence space, then is pretopological if and only if it satisfies for all [17];
(iii) ultrapretopological if it is pretopological and for each , there exists a stratified -ultrafilter such that ;
(iv) topological [11] if there exists a stratified -topology such that , , we have , where is called the interior of w.r.t. [11, 30].

Proposition 29. Let be a strong stratified -Kent convergence space which is -regular relative to every ultrapretopological generalized stratified -convergence structure . Then is -regular.

Proof. Let with , . Let be the ultrapretopological generalized stratified -convergence structure defined by , . From we have . For each with , it follows that for each , , which means . Thus, that is, . Because is a strong -Kent convergence space, then it follows that , and so That is, . It follows by the assumption that is -regular. Thus . By Theorem 17 we know that is -regular.

It is easily seen that when is a complete Boolean algebra, then the above proposition holds for -regularity.

Lemma 30. Let be a topological generalized stratified -convergence space and let be the stratified -topology corresponding to . Then if and only if for all .

Proof. We need only to check the sufficiency. Note that to for each , and if [11, 30]. It follows that, for each ,

Theorem 31. Let be a linearly order frame or let be prime. A topological generalized stratified -convergence space is -regular if and only if it is -regular for every ultrapretopological generalized stratified -convergence structure .

Proof. Note that a topological generalized stratified -convergence space is natural a strong stratified -Kent convergence space [17]. Then the sufficiency follows by Proposition 29. Thus, we prove only the necessity. Let be -regular and let be an arbitrary ultrapretopological generalized stratified -convergence structure with . Then, for each , there exists a such that . Obviously, and then by .
Let be defined by , for all . Then for each . For each , we check below . Here, is the stratified -topology corresponding to . For each , it follows by Lemma 1 that ; that is, Note that . For each , it follows that , which means that there exists an such that and . Thus, . Fix ; we have or .
Case 1. ; that is, . Because is topological, then . From , we get and then ; indeed, since takes values in .
Case 2. ; that is, . We assume that ; it follows by equality (25) that . Because is an ultrafilter on , then and so . As we have known and is ultrapretopological; hence, , then by it follows that . Now, A contradiction! Thus, if , then .
Combining Cases 1 and 2 we get that if then . It follows immediately that .
Next we prove that . By Lemma 30, we need only to check that for all . Indeed,
Then, for each , where the first and the second equalities hold by the pretopologicalness of , the first inequality holds by Lemma 24, the second inequality holds by Lemma 5(4), and the last inequality holds because is -regular. Then it follows that is -regular.

Remark 32. To prove that Theorem 31 holds for -regularity, it seems that must be a complete Boolean algebra. If we further assume that is linearly ordered or is prime then . Thus, we guess that Theorem 31 holds for -regularity only if .

##### 4.2. For Levelwise Stratified -Convergence Spaces

Definition 33 (see [31]). Let be a levelwise stratified -convergence space. For each , the -set defined by is called -level closure of w.r.t. .

It is easily seen that -level closures of -sets have similar properties to closures of -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 -level closures of stratified -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 -level closures of stratified -filters.

Definition 34. Let