Properties Of M-L-closure Systems And Fuzzy Number

Closure systems (i.e. topped∩-strcutures) is a structure involved in many areas of mathematics and computer science. Considering the importance of multi-agent/multi-source systems in information science, paper [18] introduces the notion of M-closure systems (a generalization of closure systems). In this paper, definitions of M-L-closure systems, continuous mapping, open mapping and closed mapping between M-L-closure systems are given, and properties on those mappings are discussed. It is proved that M-L-CS, the category of M-L-closure systems and continuous mappings between them, is a topological construct. As an application, notions of product M-L-closure spaces, sum M-L-closure spaces and quotient M-L-closure spaces are defined.Fuzzy number is a special kind of fuzzy set, which has been applied in many areas, such as fuzzy control and fuzzy information analysis. This article defines the metrics pi (i=1,2,3) in fuzzy number set (?), compares the relationship between them, and discussed completeness of ((?),ρ1) and ((?), ρ3), arc connectivity and local arc connectivity of ((?)b,ρi)(i=1,2,3)((?)b is bounded (?)).The key points and the main contents of this paper are as follows:Chapter one are preliminares, which mainly introduces the elementary knowl-edge related to fuzzy sets, category and fuzzy number.Chapter two are special mappings and categorical properties of M-L-closure systems. It firstly defines the M-L-closure systems, continuous mapping, open map-ping and closed mapping of M-L-closure systems. Furthermore, some properties on those mappings are discussed. Finally, it is proved that M-L-CS is a topological construct. Based on the category theory, the notions of product M-L-closure spaces, sum M-L-closure spaces and quotient M-L-closure spaces are defined.Chapter three are some properties of three metrics on fuzzy number (?), which fristly defines the metrics ρi (i=1,2,3) in fuzzy number set (?). Secondly,we compare the relationship between metrics ρi(i=1,2,3). Thirdly, we discussed completeness of ((?),ρi) and((?).ρ3), arc connectivity and local arc connectivity of ((?),ρi)(i=1,2,3)((?)b is bounded(?)).