格子拓扑中若干问题的研究

Some important properties of L-topological spaces, induced.I(L)-topological spaces, fuzzy real lines and fuzzy metric spaces in.topology on lattices are obtained..Based on the research of remote-neighborhood mappings, a theory about uniformity and metric on completely distributive lattices is introduced. It can directly reflect the characteristics of pointwise topology on lattices, i.e. it can directly reflect the relation between a point (or molecule) and its Q-neighborhood or R-neighborhood.A correspondence between normal subsets of a base and continuous.sub-domains in a continuous domain is established, and an algebraic.characterization of continuous sub-domains which is independent of.embeding-projection pairs is given. This result is important for.appications to develop an information system-based method for solving.effectively continuous domain equations. An base framework for quantitative domain theory based on the concept of stratifications in L-fuzzy theory is proposed.A new category of domain (i.e.the category of alegebraic domains and Scott continuous functions of approximation-order-preserving)and its full subcategory (i.e.the caterory of pointed algebraic domains and Scott.continuous functions of approximation-order-preserving)are constructed, and then the maximal Cartesian closed full subcategories of these categories are determined.

研究把序结构与拓扑结构融为一体的格上拓扑理论与其相关的应用问题。通过对格上度量性质的研究界定出与度量理论相匹配的紧性与仿紧性概念，建立格上拓扑点式度量化理论的完善体系；以Domain理论的抽象基为工具研究格上拓扑理论中的一些特殊范畴，并建立相应的表示理论，Stone型对偶定理等范畴的对应关系与其逻辑解释，推动格上拓扑学的发展。.