Jinming Fang

Relationships between L-ordered convergence structures and strong L-topologies

By making use of the intrinsic fuzzy inclusion orders on the fuzzy power set, the relationships between L-ordered convergence spaces and strong L-topological spaces are researched, where L is a commutative unital quantale. It is shown that the category of strong L-topological spaces can be embedded in the category of L-ordered convergence spaces as a reflective subcategory. As a result, we observe that there is a Galois correspondence between the category of L-ordered convergence spaces and the category of strong L-topological spaces. Further, it is proved that the class of all strong L-topological L-ordered convergence spaces precisely is the class of all strong L-topological spaces, and the class of spaces with non-idempotent L-ordered interior operators is characterized as a subclass of the class of L-ordered convergence spaces.

I-FTOP is isomorphic to I-FQN and I-AITOP

In this paper, we give two characterizations of I-FTOP, the category of I-fuzzy topological spaces and I-fuzzy continuous mappings. First, we construct the category of I-fuzzy quasi-coincident neighborhood spaces and its continuous mappings, denoted I-FQN, and then show that I-FTOP is isomorphic to I-FQN. Second, we construct the category of antichain I-topological spaces and its continuous mappings, denoted I-AITOP, and show that I-FTOP is isomorphic to I-AITOP.

On separation axioms in I-fuzzy topological spaces

In this paper, we introduce the notions of some low-level separation axioms and investigate some of their properties and the relations between them in the general framework of I-fuzzy topological spaces.

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.

Extension of Shi's quasi-uniformities in a Kubiak–Šostak sense

Abstract The aim of this paper is to study the extension of Shi Fu-Guis quasi-uniformities in a Kubiak–Sostak sense. The relationship between this extension of Shi Fu-Guis quasi-uniform molecular lattices (Shis QUML) and the corresponding Kubiak–Sostak extension of Wang Guo-Juns topological molecular lattices (Wangs TML) is discussed. QUML and TML denote the categories of Shis QUML and Wangs TML, respectively. CD is the category of completely distributive lattices with complete lattice morphisms as morphisms. We prove that FQUML—the category of fuzzy quasi-uniform molecular lattices (the extension of Shis QUML)—is a topological category over CD op and FTML—the category of fuzzy topological molecular lattices (the extension of Wangs TML)—can be embedded in FQUML. We also prove that FQUML is isomorphic to QUML c ( M ) when M is a completely distributive lattice with multiplicative property, where QUML c ( M ) is the co-tower extension of QUML. Finally, we study the Kubiak–Sostak extension of Huttons quasi-uniformities.

One-to-one correspondence between fuzzifying topologies and fuzzy preorders

This paper is devoted to the discussion of the relationship between fuzzifying topologies and fuzzy relations. A saturated fuzzifying topology can be generated by a reflexive fuzzy relation; and conversely, a fuzzy preorder can be generated by a fuzzifying topology. In fact, there exists a one-to-one correspondence between the set of all saturated fuzzifying topologies and that of all fuzzy preorders.

Categories isomorphic to the Kubiak-Šostak extension of TML

Abstract In this paper, we study Wang Guo-Juns topological molecular lattices (Wangs TML) in a Kubiak–Sostak sense and construct a category—the Kubiak–Sostak extension of TML, denoted by FTML. Then we prove that FTML is topological over CD op and TML is bireflective full subcategory of FTML. Finally, we introduce several important categories to characterize FTML.

Sums of L-fuzzy topological spaces

It follows from L-FTOP being topological over SET that there is L-fuzzy topological sum in L-FTOP. In this paper, we use the final topologies constructed in Section 3 of Rodabaugh [Powerset operator foundations of variable-basis fuzzy topology, in: U. Hohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, the Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston/Dordrecht/London, 1999, pp. 91-116] to characterize L-fuzzy topological sum internally and establish connections between L-fuzzy topological sum and its factor spaces. We create a functor @w from L-FYS (the category of L-fuzzifying topological space) to L-FTOP and show that @w has a right-adjoint, hence there exists an adjunction from L-FYS to L-FTOP. Moreover @w preserves L-fuzzy topological sums that already exists in both L-FYS and L-FTOP. Finally, we examine certain additivity property of L-fuzzy topological spaces. These results imply that the topological sum in L-FTOP, a necessary consequence of L-FTOP being a topological construct, is now better understood.

Lattice-valued semiuniform convergence spaces

In this paper, two kinds of lattice-valued semiuniform convergence spaces are proposed, namely stratified L-semiuniform convergence spaces and stratified L-ordered semiuniform convergence spaces respectively. It is shown that (i) the category of stratified L-semiuniform convergence spaces is topological; (ii) the category of stratified L-ordered semiuniform convergence spaces is a bireflective full subcategory of the category of stratified L-semiuniform convergence spaces, and hence it is topological; (iii) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are Cartesian-closed; (iv) the category of stratified L-semiuniform convergence spaces is extensional; (v) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are closed under the formation of products of quotient mappings. In case that L is the two-point chain, both coincide with the category of semiuniform convergence spaces in the classical case.

L-fuzzy closure systems

This paper studies the generalization of fuzzy closure operators and fuzzy closure systems, introduced by Belohlavek in 2001, and introduces the concepts of strong L-fuzzy closure systems and strong L-fuzzy closure operators. It is shown that a strong L-fuzzy closure system is precisely the fuzzy system in opposition to the crisp system, and a strong L-fuzzy closure operator is a suitable closure operator that has a close relation to a strong L-fuzzy closure system. It is also shown that there is a Galois correspondence between the category of (strong) L-fuzzy closure system spaces and that of (strong) L-fuzzy closure spaces.