Fu-Gui Shi

L-fuzzy numbers and their properties

In this paper, the notions of L-fuzzy convex sets and L-fuzzy numbers are introduced where L is a completely distributive lattice. The notions of [0,1]-fuzzy convex sets and [0,1]-fuzzy numbers are generalized. Furthermore their properties and characterizations are presented in terms of cut sets of L-fuzzy sets.

A new notion of fuzzy compactness in L -topological spaces

In this paper, a new notion of compactness is introduced in L-topological spaces by means of βa-open cover and Qa-open cover, which is called S*-compactness. Ultra-compactness implies S*-compactness. S*-compactness implies fuzzy compactness, but fuzzy compactness need not imply S*-compactness. If L=[0,1], then strong compactness implies S*-compactness, but S*-compactness need not imply strong compactness. The intersection of an S*-compact L-set and a closed L-set is S*-compact. The continuous image of an S*-compact L-set is S*-compact. A weakly induced L-space (X, I) is S*-compact if and only if(X, [I]) is compact. The Tychonoff Theorem for S*-compactness is true. The L-fuzzy unit interval is S*-compact. Moreover S*-compactness can also be characterized by nets.

A new approach to the fuzzification of matroids

In this paper, the concept of closed fuzzy pre-matroids is generalized to L-fuzzy set theory when L is a complete lattice. It is also called an L-fuzzifying matroid. In the definition of L-fuzzifying matroids, each subset can be regarded as an independent set to some degree. When L is completely distributive, an L-fuzzifying matroid can be characterized by means of its L-fuzzifying rank function. An L-fuzzifying matroid and its L-fuzzifying rank function are one-to-one corresponding.

L-fuzzy interiors and L-fuzzy closures

In this paper, we introduce new definitions of L-fuzzy neighborhood systems, L-fuzzy interior operators and L-fuzzy closure operators. Three characterizations of the category L-FTOP of L-fuzzy topological spaces and their L-fuzzy continuous mappings are presented by means of the category L-FNS of L-fuzzy neighborhood spaces and their continuous mappings, the category L-FIS of L-fuzzy interior spaces and their L-fuzzy continuous mappings, and the category L-FCS of L-fuzzy closure spaces and their L-fuzzy continuous mappings.

A new definition of fuzzy compactness

In this paper, a new definition of fuzzy compactness is presented in L-topological spaces when L is a complete DeMorgan algebra. This definition does not rely on the structure of the basis lattice L and no distributivity in L is required. The intersection of a fuzzy compact L-set and a closed L-set is fuzzy compact. The continuous image of a fuzzy compact L-set is fuzzy compact. If the set of all prime elements of L is order generating, then the Alexander Subbase Theorem is true. When L is a completely distributive DeMorgan algebra, it is equivalent to Lowens fuzzy compactness, and in this case, its many characterizations are presented by means of open L-sets and closed L-sets.

Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed-complete posets

This paper pursues an investigation on quantitative domains via fuzzy sets initiated by the first author. This time we define an L-topology, called the fuzzy Scott topology, on fuzzy dcpos and investigate its properties. Scott convergence of stratified L-filters is also defined and studied. We show that a fuzzy dcpo (X,e) is continuous if and only if for any stratified L-filter on X, Scott convergence coincides with the convergence with respect to the fuzzy Scott topology. At last, we show that the category of fuzzy dcpos with fuzzy Scott continuous maps is Cartesian-closed.

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.

On L-fuzzy topological spaces☆

In this paper, we study L-fuzzy topological spaces, where L represents a completely distributive lattice. We shall investigate the level decomposition of L-fuzzy topologies on X and the corresponding L-fuzzy continuous maps. In addition, we shall establish the representation theorems of L-fuzzy topologies on X.

O-convergence of fuzzy nets and its applications

Abstract In this paper, an O-convergence theory of fuzzy nets is built in L -topological spaces by means of neighborhoods of fuzzy points. It has many nice properties. It can be used to characterize the closed set, open set, T 2 separation axiom and fuzzy compactness.

A New Approach to the Fuzzification of Convex Structures

A new approach to the fuzzification of convex structures is introduced. It is also called an -fuzzifying convex structure. In the definition of -fuzzifying convex structure, each subset can be regarded as a convex set to some degree. An -fuzzifying convex structure can be characterized by means of its -fuzzifying closure operator. An -fuzzifying convex structure and its -fuzzifying closure operator are one-to-one corresponding. The concepts of -fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained in -fuzzifying convex structure.