L-Interval Valued Fuzzy Filter Space And Its Category L-IVF

The value of a fuzzy set can be taken not only as a number of the interval [0,1], but also as a closed interval of the interval [0,1]. Therefore the interval valued fuzzy set has been defined and studied. Many fuzzy mathematicians have paid much attention to this special fuzzy set, which has been used in the study of fundamental theory and the field of application, such as fuzzy control, fuzzy information analysis and so on. In this paper, the concept of L—interval valued fuzzy set for a complete De Morgan algebra with central elements is first introduced, which is a generalization of interval valued fuzzy set. Then the denfinition of L—interval valued fuzzy filter space is given based on former notions and its topological properties are mainly discussed;The basic mappings between two L—interval valued fuzzy filter spaces are introduced and studied in a very systematic way, such as continuous mapping, open mapping and home-omorphism. Furthermore, a new notion of Hausdoff L—interval valued fuzzy filter space is defined successfully and whether Hausdoffness has products or not is deeply studied. Finallly, the category of L—interval valued fuzzy filter space L—IVF and the category of Hausdoff L—interval valued fuzzy filter space L—IVFHaus are studied carefully. By the study of the existence and concrete construction of equalizers, coequalizers, products, co-prducts of the categories L—IVF and L—IVFHaus, the conclusion that neither of them is complete, but L—IVF is co-complete is proved. At the same time, we get the conclusions that neither of them is a weak topological category and L—IVF has weak topological full subcategories as well as non-weak topological full subcategories. The following are the construction and main contents of this paper: Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and conclusions of lattice theory, fuzzy set and category theory ralated to the interval valued fuzzy set for convience of our following discussion.Chapter Two L—interval valued fuzzy filter space. Firstly, based on the concept of L—interval valued fuzzy set the L— interval valued fuzzy filter space is defined and many ways of generation to a new L—interval valued fuzzy filter space are discussed.Secondly, the basic mappings between two L—interval valued fuzzy filter spaces are introduced and studied in a very systematic way, such as continuous mapping, open mapping and homeomorphism, besides, their characteristic theorms are given. Thirdly, the definition of product L—interval valued fuzzy filter space is given and the product filter of a family of L—interval valued fuzzy filters is charactized. Finally, in the L—interval valued fuzzy filter space, a new notion of Hausdoffness is defined and the L—interval valued fuzzy filter limit of the sequence is constructed. Consequently, we get a series of good conclusions that the L—interval valued fuzzy filter limit of the sequence will be unique if it converges in a L—interval valued fuzzy filter space and Hausdoffness is arbitrarily producted.Chapter Three The category of L—interval valued fuzzy filter space L—TVF. By the study of the existence and concrete construction of equalizers, coequalizers, products, co-prducts of the categories L—TVF and L—IVFHaus, the conclusion that neither of them is complete, but L—IVF is co-complete is proved. At the same time, we get the conclusions that neither of them is weak topological category and L—IVF has weak topological full subcategories as well as non-weak topological full subcategories.