FI代数上基于模糊滤子的一致拓扑空间

拓扑结构是逻辑代数领域的重要研究内容之一,为了揭示FI代数上的拓扑结构,基于模糊滤子诱导的同余关系在FI代数上构造一致拓扑空间并讨论其拓扑性质,证明了:(i)一致拓扑空间是非连通、局部紧的完全正则空间;(ii)一致拓扑空间是T_0空间当且仅当是T_1当且仅当是T_2空间;(iii)FI代数中蕴涵算子关于一致拓扑是连续的,从而构成拓扑FI代数.同时,获得了一致拓扑空间是紧空间的充分必要条件.最后,讨论了商空间的性质.该研究对从拓扑层面进一步揭示FI代数内部特征具有一定的促进作用.

Uniform topological spaces based on fuzzy filters in FI-algebras

Topological structure is one of important research topics in the field of logic algebra. In order to describe the topological structure of FI-algebras, uniform topological spaces are established and some of their properties are discussed based on the congruences induced by fuzzy filters. The following conclusions are proved:(i) Every uniform topological space is disconnected, locally compact and completely regular. (ii) A uniform topological space is a T₀ space if and only if it is a T₁ space if and only if it is a T₂ space. (iii) The implication operation in an FI-algebra is continuous under the uniform topology, which makes the FI-algebra to be topological. Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact are obtained. Finally, some properties of uniform topology on the quotient spaces are discussed. The results of this paper take a positive role to reveal internal features of FI-algebras on a topological level.