Bin Pang

Subcategories of the category of L-convex spaces ☆

Abstract In this paper, several types of L-convex spaces are introduced, including stratified L-convex spaces, convex-generated L-convex spaces, weakly induced L-convex spaces and induced L-convex spaces. Their relations are discussed category-theoretically. Firstly, it is shown that there is a Galois correspondence between the category SL-CS of stratified L-convex spaces (resp. the category WIL-CS of weakly induced L-convex spaces) and the category L-CS of L-convex spaces. In particular, SL-CS and WIL-CS are both coreflective subcategories of L-CS. Secondly, it is proved that there is a Galois correspondence between the category CS of convex spaces and the category SL-CS (resp. WIL-CS). Specially, CS can be embedded into SL-CS and WIL-CS as a coreflective subcategory. Finally, it is shown that the category CGL-CS of convex-generated L-convex spaces, the category IL-CS of induced L-convex spaces and CS are isomorphic.

L-FUZZY Q-CONVERGENCE STRUCTURES

Abstract In this paper, we construct a topological category of pretopological L-fuzzy Q-convergence spaces, which contains the category of topological L-fuzzy Q-convergence spaces as a bireflective full subcategory. Considering the connections with L-fuzzy topology, it is proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to the category of topological L-fuzzy quasi-coincident neighborhood spaces, and the latter is isomorphic to the category of L-fuzzy topological spaces. Moreover, we find that our pretopological L-fuzzy Q-convergence spaces can be characterized as a kind of L-fuzzy quasi-coincident neighborhood spaces, which is called strong L-fuzzy quasi-coincident neighborhood space.

On (L,M)-fuzzy convergence spaces

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.

Enriched (L, M)-fuzzy convergence spaces

This paper presents a definition of enriched (L, M)-fuzzy convergence spaces. It is shown that the resulting category E(L, M)-FC is a Cartesian closed topological category, which can embed the category E(L, M)-FTop of enriched (L, M)- fuzzy topological spaces as a reflective subcategory. Also, it is proved that the category of topological enriched (L, M)-fuzzy convergence spaces is isomorphic to E(L, M)-FTop and the category of pretopological enriched (L, M)-fuzzy convergence spaces is isomorphic to the category of enriched (L, M)-fuzzy quasi-coincident neighborhood spaces.

Stratified L -ordered filter spaces

Abstract In this paper, stratified L-ordered filter spaces and stratified L-ordered Cauchy spaces are introduced. It is shown that the category of stratified L-ordered filter spaces is a strongly Cartesian closed topological category and the category of stratified L-ordered Cauchy spaces is a Cartesian closed topological category. Also, the relationships among stratified L-ordered filter spaces, stratified L-ordered Cauchy spaces, stratified L-filter spaces and stratified L-Cauchy spaces are investigated.

A new definition of order relation for the introduction of algebraic fuzzy closure operators

Abstract In this paper, a new approach to order relation between fuzzy sets is provided, which is called well inclusion order between fuzzy sets. Based on this new order relation, the concept of algebraic fuzzy closure operators is introduced. It is shown that there is a categorical isomorphism between algebraic fuzzy closure operators and fuzzy convex structures. Also, the relationship between fuzzy closure systems and fuzzy convex structures is investigated. It is proved that the category of fuzzy convex spaces is a bicoreflective subcategory of the category of fuzzy closure system spaces.

Degrees of compactness in (L,M)-fuzzy convergence spaces and its applications

Based on completely distributive lattices L and M, we define the degrees of compactness of (L,M)-fuzzy convergence spaces, (L,M)-fuzzy topological spaces, (L,M)-fuzzy pseudo-quasi-metric spaces and pointwise (L,M)-fuzzy quasi-uniform spaces. It is shown that (1) the Tychonoff Theorem with respect to the compactness degrees holds in (L,M)-fuzzy convergence spaces and (L,M)-fuzzy topological spaces; (2) the compactness degrees of an (L,M)-fuzzy pseudo-quasi-metric space and a pointwise (L,M)-fuzzy quasi-uniform space are equal to the compactness degrees of their induced (L,M)-fuzzy topological spaces, respectively; (3) an (L,M)-fuzzy pseudo-quasi-metric space can induce a pointwise (L,M)-fuzzy quasi-uniform space and their compactness degrees are equal.

Strong inclusion orders between L-subsets and its applications in L-convex spaces

Abstract In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.

The category of stratified L-filter spaces

Abstract In this paper, the concepts of stratified L-filter space, complete stratified L-filter space and symmetric stratified L-Kent convergence space are introduced. It is shown that (1) the category of stratified L-filter spaces with Cauchy continuous maps is a strong topological universe; (2) the category of complete stratified L-filter spaces, as a bicoreflective subcategory of the category of stratified L-filter spaces, is isomorphic to the category of symmetric stratified L-Kent convergence spaces; (3) the category of complete stratified L-filter spaces, as an isomorphism-closed full subcategory of the category of stratified L-filter spaces, is strongly Cartesian closed.

Several types of enriched (L,M)-fuzzy convergence spaces

Abstract In this paper, several types of enriched ( L , M ) -fuzzy convergence spaces are introduced, including enriched ( L , M ) -fuzzy Kent convergence spaces, enriched ( L , M ) -fuzzy limit spaces, pretopological enriched ( L , M ) -fuzzy convergence spaces, topological enriched ( L , M ) -fuzzy convergence spaces and enriched ( L , M ) -fuzzy Choquet convergence spaces. These concepts generalize the concepts of Kent convergence spaces, of limit spaces, of pretopological convergence spaces, of topological convergence spaces and of Choquet convergence spaces in general topology to the setting of ( L , M ) -fuzzy topology. Also, their categorical properties and their mutual categorical relations are investigated.