On (L,M)-fuzzy convergence spaces

doi:10.1016/j.fss.2013.07.007

Fuzzy Sets and Systems

Volume 238, 1 March 2014, Pages 46-70

https://doi.org/10.1016/j.fss.2013.07.007 Get rights and content

Abstract

This paper presents a definition of $(L, M)$ -fuzzy convergence spaces. It is shown that the category $(L, M)$ -FC of $(L, M)$ -fuzzy convergence spaces, which embeds the category $(L, M)$ -FTop of $(L, M)$ -fuzzy topological spaces as a reflective subcategory, is a Cartesian closed topological category. Further, it is proved that the category of (topological) pretopological $(L, M)$ -fuzzy convergence spaces is isomorphic to the category of (topological) $(L, M)$ -fuzzy quasi-coincident neighborhood spaces. Moreover, the relations among $(L, M)$ -fuzzy convergence spaces, pretopological $(L, M)$ -fuzzy convergence spaces and topological $(L, M)$ -fuzzy convergence spaces are investigated in the categorical sense.

Introduction

Chang [2] first introduced fuzzy set theory to topology. In Changʼs I-topology, each open set was fuzzy, but the topology comprised by those open sets is a crisp subset of I-powerset $I^{X}$ . In a completely different direction, Höhle [7] presented the notion of a fuzzy topology as an I-subset of a powerset $2^{X}$ . Ying [28] studied Höhleʼs topology from a logical point of view and called it fuzzifying topology. Kubiak [14] and Šostak [22] independently extended Höhleʼs topology to M-subset of $L^{X}$ and I-subset of $I^{X}$ , respectively, which is called $(L, M)$ -fuzzy topology.

Convergence theory of filters provides a good tool for interpreting topology. In crisp situation, there are close relations between topological spaces and convergence spaces. With the development of fuzzy set theory, many researchers extended convergence structures to fuzzy setting and discussed the relations between fuzzy convergence structures and fuzzy topologies. In the framework of L-topology, Lowen [18] defined the concept of a prefilter as a subset of $I^{X}$ in order to study the theory of fuzzy topology. Also, Min [19] proposed fuzzy limit structures by means of prefilters. Höhle and Šostak [8] introduced the notion of a (stratified) L-filter as a mapping from $L^{X}$ to L and showed that stratified L-filters played an important role in the development of fuzzy convergence spaces. Using stratified L-filters, Jäger [9] introduced stratified L-fuzzy convergence spaces (which are called L-generalized convergence spaces in [10]) and proved that the category of topological stratified L-generalized convergence spaces and the category of stratified L-topological spaces are isomorphic. There are some following works to study the properties of this kind of fuzzy convergence structures [4], [5], [11], [12], [13], [16], [17], [25]. In the situation of L-fuzzifying topology, Xu [24] introduced fuzzifying convergence structures and proved that topological fuzzifying convergence structures and fuzzifying topologies are equivalent. Later, Yao [26] introduced L-fuzzifying convergence structures using L-filters of ordinary subsets (a mapping from $2^{X}$ to L) and showed that the category of topological L-fuzzifying convergence spaces is isomorphic to that of L-fuzzifying topological spaces when L is a completely distributive lattice. Afterwards, Wu and Fang [23] introduced L-ordered fuzzifying convergence structures and studied its relations with L-fuzzifying convergence structures. In L-fuzzy setting, Güloǧlu et al. [6] introduced the concept of an I-fuzzy convergence structure by means of I-filters and showed that there is a one-to-one correspondence between topological I-fuzzy convergence structures and I-fuzzy topologies. Later, Pang and Fang [20] used L-filters to define L-fuzzy Q-convergence structures and proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to that of L-fuzzy topological spaces. In both of the above convergence spaces, the objects of converging are fuzzy points, which is compatible with the L-fuzzy setting. However, the convergence of an L-filter is not fuzzy, i.e., an L-filter $F$ either converges to a fuzzy point or not. Hence, we claim that neither of the above convergence structures may be the best one in L-fuzzy setting. The aim of this paper is to propose the concept of an $(L, M)$ -fuzzy convergence structure which assigns every $(L, M)$ -fuzzy filter a certain degree of converging to a fuzzy point. We will show that this kind of $(L, M)$ -fuzzy convergence structures not only can characterize $(L, M)$ -fuzzy topology, but also has many nice categorical properties.

This paper is organized as follows. In Section 2, we recall some necessary concepts and notations. In Section 3, we give the definition of $(L, M)$ -fuzzy convergence structures and show that the resulting category $(L, M)$ -FC embeds the category $(L, M)$ -FTop of $(L, M)$ -fuzzy topological spaces as a reflective subcategory. In Section 4, we propose the conceptions of pretopological and topological $(L, M)$ -fuzzy convergence spaces and proved that the resulting categories are isomorphic to the categories of $(L, M)$ -fuzzy quasi-coincident neighborhood spaces and topological $(L, M)$ -fuzzy quasi-coincident neighborhood spaces, respectively. In Section 5, we investigate the relations among $(L, M)$ -fuzzy convergence spaces, pretopological $(L, M)$ -fuzzy convergence spaces and topological $(L, M)$ -fuzzy convergence spaces. In Section 6, we show the category $(L, M)$ -FC of $(L, M)$ -fuzzy convergence spaces is a Cartesian closed topological category.

Section snippets

Preliminaries

Throughout this paper, both L and M denote completely distributive lattices and ′ is an order-reversing involution on L. The smallest element and the largest element in L (M) are denoted by $⊥_{L}$ ( $⊥_{M}$ ) and $⊤_{L}$ ( $⊤_{M}$ ), respectively. For $a, b \in L$ , we say that a is wedge below b in L, in symbols $a ≺ b$ , if for every subset $D \subseteq L$ , $⋁ D \geq b$ implies $d \geq a$ for some $d \in D$ . A complete lattice L is completely distributive if and only if $b = ⋁ {a \in L | a ≺ b}$ for each $b \in L$ . An element a in L is called co-prime if $a \leq b \lor c$ implies $a \leq b$ or $a \leq c$

$(L, M)$ -fuzzy convergence spaces

The aim of this section is to introduce the concept of $(L, M)$ -fuzzy convergence structure and study the relations between $(L, M)$ -fuzzy convergence structures and $(L, M)$ -fuzzy topologies.

Definition 3.1

An $(L, M)$ -fuzzy convergence structure on X is a mapping $c : F_{L M} (X) \to M^{J (L^{X})}$ which satisfies:

(LFC1)
$c (\hat{q} (x_{λ})) (x_{λ}) = ⊤_{M}$ ;
(LFC2)
$F \leq G \Rightarrow c (F) \leq c (G)$ .

The pair

(X, c)

is called an

(L, M)

-fuzzy convergence space.

A continuous mapping between $(L, M)$ -fuzzy convergence spaces $(X, c)$ and $(Y, d)$ is a mapping $f : X \to Y$ such that for all $F \in F_{L M} (X)$ , $x_{λ} \in J (L^{X})$ , $c (F) (x_{λ}$

Pretopological and topological $(L, M)$ -fuzzy convergence spaces

In this section, the concepts of pretopological and topological $(L, M)$ -fuzzy convergence spaces are proposed. The results show that the category of pretopological $(L, M)$ -fuzzy convergence spaces is isomorphic to the category of $(L, M)$ -fuzzy quasi-coincident neighborhood spaces, and the category of topological $(L, M)$ -fuzzy convergence spaces is isomorphic to the category of topological $(L, M)$ -fuzzy quasi-coincident neighborhood spaces.

Definition 4.1

An $(L, M)$ -fuzzy convergence structure c on X is called

Relations among $(L, M)$ -FC, $(L, M)$ -FPC and $(L, M)$ -FTC

In this section, we will discuss the relations among $(L, M)$ -fuzzy convergence spaces, pretopological $(L, M)$ -fuzzy convergence spaces and topological $(L, M)$ -fuzzy convergence spaces. Also, the relations between L-fuzzy (i.e., $(L, L)$ -fuzzy) convergence spaces and L-fuzzy Q-convergence spaces are investigated.

We write $C_{l f p c} (X)$ for the set of all pretopological $(L, M)$ -fuzzy convergence structures on X and still write ≤ for the restriction of the order on $C_{l f c} (X)$ to $C_{l f p c} (X)$ . Then the following theorem

$(L, M)$ -FC is a Cartesian closed topological category

In this section, we assume that $⊥_{L}$ is prime. In order to show that the category $(L, M)$ -FC is Cartesian closed, we first list some definitions and lemmas for $(L, M)$ -fuzzy filters. Since all these results are similar to those for stratified L-filters in [8], [9] and they can be checked easily, we will only give some necessary proofs.

Lemma 6.1

In $(F_{L M} (X), \leq)$ , every nonempty family ${F_{i}}_{i \in I}$ of $(L, M)$ -fuzzy filters has an infimum $⋀_{i \in I} F_{i}$ , which can be calculated as $\forall A \in L^{X}, (\underset{i \in I}{⋀} F_{i}) (A) = \underset{i \in I}{⋀} F_{i} (A) .$

Lemma 6.2

For a nonempty family ${F$

Conclusions

In this paper, we defined a new kind of lattice-valued convergence spaces, called $(L, M)$ -fuzzy convergence spaces. Such spaces have a convergence structure that gives an $(L, M)$ -fuzzy filter a grade of convergence to a fuzzy point. We also introduced pretopological and topological $(L, M)$ -fuzzy convergence spaces and investigated the relations among $(L, M)$ -fuzzy convergence spaces, pretopological $(L, M)$ -fuzzy convergence spaces, topological $(L, M)$ -fuzzy convergence spaces and L-fuzzy Q-convergence

Acknowledgements

The author is thankful to the anonymous reviewers and Prof. J. Gutiérrez García, the Area Editor, for their careful reading and constructive comments.

References (28)

C.L. Chang
Fuzzy topological spaces
J. Math. Anal. Appl.
(1968)
J.M. Fang
Relationships between L-ordered convergence structures and strong L-topologies
Fuzzy Sets Syst.
(2010)
M. Güloğlu et al.
Convergence in I-fuzzy topological spaces
Fuzzy Sets Syst.
(2005)
U. Höhle
Probabilistic metrization of fuzzy uniformities
Fuzzy Sets Syst.
(1982)
G. Jäger
Subcategories of lattice-valued convergence spaces
Fuzzy Sets Syst.
(2005)
G. Jäger
Pretopological and topological lattice-valued convergence spaces
Fuzzy Sets Syst.
(2007)
G. Jäger
Lattice-valued convergence spaces and regularity
Fuzzy Sets Syst.
(2008)
L.Q. Li et al.
On adjunctions between Lim, $S L - Top$ , and $S L - Lim$
Fuzzy Sets Syst.
(2011)
L.Q. Li et al.
On stratified L-convergence spaces: Pretopological axioms and diagonal axioms
Fuzzy Sets Syst.
(2012)
R. Lowen
Convergence in fuzzy topological spaces
Appl. Gen. Topol.
(1979)

K.C. Min

Fuzzy limit spaces

Fuzzy Sets Syst.

(1989)

B. Pang et al.

L-fuzzy Q-convergence structures

Fuzzy Sets Syst.

(2011)

L.S. Xu

Characterizations of fuzzifying topologies by some limit structures

Fuzzy Sets Syst.

(2001)

W. Yao

On many-valued stratified L-fuzzy convergence spaces

Fuzzy Sets Syst.

(2008)

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^☆: The paper is supported by the Graduate Scientific and Technological Innovation Project of Beijing Institute of Technology (2013CX10039) and the National Natural Science Foundation of China (11201437).

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On (L,M)-fuzzy convergence spaces☆

Abstract

Introduction

Section snippets

Preliminaries

(L,M)-fuzzy convergence spaces

Pretopological and topological (L,M)-fuzzy convergence spaces

Relations among (L,M)-FC, (L,M)-FPC and (L,M)-FTC

(L,M)-FC is a Cartesian closed topological category

Conclusions

Acknowledgements

J. Math. Anal. Appl.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Appl. Gen. Topol.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

On $(L, M)$ -fuzzy convergence spaces☆

$(L, M)$ -fuzzy convergence spaces

Pretopological and topological $(L, M)$ -fuzzy convergence spaces

Relations among $(L, M)$ -FC, $(L, M)$ -FPC and $(L, M)$ -FTC

$(L, M)$ -FC is a Cartesian closed topological category