li Yue

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.

On fuzzy pseudo-metric spaces

This paper attempts to endow the fuzzy pseudo-metric in the sense of George and Veeramani with many-valued topological structures. It is shown that George and Veeramanis fuzzy pseudo-metric can be characterized by a family of compatible ordinary pseudo-metrics called pseudo-metric chain in this paper. Then, two kinds of many-valued topological structures-fuzzifying topology and fuzzifying uniformity-are generated by this fuzzy pseudo-metric, and their properties are also investigated.

L-FUZZY UNIFORM SPACES

The aim of this paper is to study L-fuzzy uniformizable spaces. A new kind of topological fuzzy remote neighborhood system is defined and used for investigating the relationship between L-fuzzy co-topology and L-fuzzy (quasi-)uniformity. It is showed that this fuzzy remote neighborhood system is different from that in [23] when U is an Lfuzzy quasi-uniformity and they will be coincident when U is an L-fuzzy uniformity. It is also showed that each L-fuzzy co-topological space is L-fuzzy quasi-uniformizable.

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.

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.

Lattice-valued induced fuzzy topological spaces

The aim of this paper is to study induced LM-fuzzy topological spaces. Firstly, we study the stratifications of LM-fuzzy topologies and show that SLM-FTOP-the category of stratified LM-fuzzy topological spaces-is a coreflective full subcategory of LM-FTOP. Secondly, we study Kubiak-Lowen functors @w and @i in LM-fuzzy topological spaces. We prove that M-FYS is isomorphic to ILM-FTOP-the category of induced LM-fuzzy topological spaces-and ILM-FTOP is a reflective and coreflective full subcategory of SLM-FTOP. Finally, we study weakly induced modification of LM-fuzzy topologies and show that WILM-FTOP-the category of weakly induced LM-fuzzy topological spaces-is a reflective and coreflective full subcategory of LM-FTOP.

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.

On (L,M)-fuzzy quasi-uniform spaces

An (L,M)-fuzzy topology is a graded extension of topological spaces handling M-valued families of L-fuzzy subsets of a referential, where L and M are completely distributive lattices. When M reduces to the set 2={0,1}, a (2,M)-fuzzy topology is called a fuzzifying topology after Ying. Sostak introduced the notion (L,M)-fuzzy uniform spaces. The aim of this paper is to study the relationship between (2,M)-fuzzy quasi-uniform spaces and (L,M)-fuzzy quasi-uniform spaces as well as the relationship between (2,M)-fuzzy quasi-uniform spaces and pointwise (L,M)-fuzzy quasi-uniform spaces-the extension of Shis L-quasi-uniform space in a Kubiak-Sostak sense. It is shown that the category of (2,M)-fuzzy quasi-uniform spaces can be embedded in the category of stratified (L,M)-fuzzy quasi-uniform spaces as a both reflective and coreflective full subcategory; and the former category can also be embedded in the category of pointwise (L,M)-fuzzy quasi-uniform spaces.

diagonal conditions and Continuous extension theorem

Abstract In this paper, we obtain a generalization of Kowalsky diagonal condition and that of Fischer diagonal condition respectively, namely Kowalsky ⊤-diagonal condition and Fischer ⊤-diagonal condition. We show that our Fischer ⊤-diagonal condition assures a complete-MV-algebra-valued convergence space, proposed in this paper, is strong L -topological, and Kowalsky ⊤-diagonal condition assures a principle (or pretopological) complete-MV-algebra-valued convergence space is strong L -topological also. As applications, we give a “dual form” of our Fischer ⊤-diagonal condition and obtain a concept of regular ⊤-convergence space. In addition, we present an extension theorem for continuous maps from a dense subspace to a regular ⊤-convergence space to show that our ⊤-diagonal conditions works indeed.

Urysohn closedness on completely distributive lattices

The purpose of this paper is to investigate the Urysohn closedness of co-topology on lattices which are not required to have an order-reversing involution. The concept of Urysohn closedness on lattices is introduced and characterizations of Urysohn closedness are given in terms of ideals and nets. These characterizations are obtained mainly through the introduction of a kind of convergence for ideals and nets that we call L-Urysohn convergence.