完全分配格上的拓扑结构

Topological Structures on Completely Distributive Lattices

本文在完全分配格上建立了T-结构理论(它是拟邻近结构理论在完全分配格上的推广),讨论了拟一致结构、T-结构和余拓扑的相互诱导问题。用范畴的观点讨论了这三种结构的关系,证明了T-分子格范畴同构于全有界拟一致分子格范畴;拓扑分子格范畴同构于交完备T-分子格范畴;一个完全分配格上全体T-结构在集合包含序之下构成完备格。本文的结果完善了完全分配格上的拓扑结构框架,推广了分明拓扑学和不分明拓扑学的相应理论。

In this paper, the theory of T-structures on completely distributive lattices (which is the generalization of the theory of quasi-proximities) is established. From the categorical point of view, the author studies the interrelations among quasi-uniformities, T-structures and co-topologies. The author proves that the category of T-molecular lattices is isomorphic with the category of totally bounded quasi-uniformities, and the category of topological molecular lattices is isomorphic with the category of co-perfect T-molecular lattics. The order of T-structures is discussed and it is shown that all T-structures on a completely distributive lattices forma a complete lattice. Finally, the inverse images of Tstructures under generalized order homomorphisms are discussed. The results of this paper complete the framework of the topological structure on completely distributive lattices, and generalize the corresponding theory in general and fuzzy in topology.