Study On A New Kind Of Fuzzifying Convrgence Structures

Based on a complete Heyting algebra, a new kind of lattice-valued fuzzify-ing convergence spaces is introduced, called L-ordered fuzzifying convergence spaces, and some problems concerning with it are studied. Being different from the former fuzzifying convergence spaces, the new L-ordered fuzzifying conver-gence spaces are compatible with the inclusion order on L-filters. It is showed that the category of L-ordered fuzzifying convergence spaces still has the desired structural property of Cartesian-closedness and the two subcategories of it can be characterized as the category of L-fuzzifying neighborhood spaces and the cate-gory of L-fuzzifying topological spaces. In addition, the category of L-ordered fuzzifying convergence spaces is a reflective full subcategory of that of fuzzifying convergence spaces, furthermore, the category of L-fuzzifying topological spaces can be embedded in it.This paper is organized as follows:The first section is exordiums and preliminaries. The author introduces the background about convergence structures, the idea about the paper and the main work. At the end of the section. some preliminaries are given.The second section is about the investigation of the new L—ordered fuzzify-ing convergence spaces. Firstly, based on the idea that the convergence structure should be compatible with the inclusion order of L—fiters. the author introduce the new L—ordered fuzzifying convergence spaces, whose rationality is supported by an example. Secondly, some properties about the category of L—ordered fuzzifying convergence spaces are discussed, in which it is worthy to say that the category of L—ordered fuzzifying convergence spaces which is "smaller" than that of fuzzifying convergence spaces still has the desired structural property of Cartesian-closedness which supports that it is meaningful to introduce the L—ordered fuzzifying convergence spaces. The third section is the discussion about the subcategories of the category of L—ordered fuzzifying convergence spaces. In this section, two new categories are proposed, which are called the category of principal L—ordered fuzzifying conver-gence spaces and the category of topological L—ordered fuzzifying convergence spaces. It is proved that they are isomorphie to the category of L—fuzzifying neighborhood spaces and the category of L—fuzzifying topological spaces respec-tively.The fourth section is the research about the relations between the category of L—ordered fuzzifying convergence spaces and other categories. Firstly, it is shown that the L—ordered fuzzifying convergence spaces can reduce to fuzzify-ing convergence spaces naturally, and the differ between them is shown by an example. Secondly, it is proposed that the category of L—ordered fuzzifying con-vergence spaces is a reflective full subcategory of that of fuzzifying convergence spaces, hence it is topological. At the end, it is proved that the category of L—fuzzifying topological spaces can be embedded in that of L—ordered fuzzify-ing convergence spaces.