否定非对合剩余格上基于正规模糊理想的一致拓扑空间

Uniform topological spaces base on normal fuzzy ideals in negative non-involutive residuated lattices

拓扑结构是逻辑代数研究领域的重要研究内容之一,为了揭示否定非对合剩余格上的拓扑结构,基于正规模糊理想诱导的同余关系在否定非对合剩余格上构造一致拓扑空间并讨论其拓扑性质.证明了:(1)一致拓扑空间是第一可数,零维,非连通,局部紧的完全正则空间;(2)一致拓扑空间是T1空间当且仅当是T2空间;(3)否定非对合剩余格中格运算和伴随运算关于一致拓扑都是连续的,从而构成拓扑否定非对合剩余格.同时,获得了一致拓扑空间是紧空间和离散空间的充分必要条件.最后,讨论了拓扑否定非对合剩余格中代数同构与拓扑同胚间的关系.对从拓扑层面进一步揭示否定非对合剩余格的内部特征具有一定的促进作用.

Topological structure is one of important research contents in the field of logical algebra. In order to describe the topological structure of negative non-involutive residuated lattices, based on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some of their properties are discussed. The following conclusions are proved:(1) every uniform topological space is first-countable, zero-dimensional, disconnected,locally compact and completely regular.(2) a uniform topological space is a T1 space iff it is a T2 space.(3) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous under the uniform topology, which make the negative non-involutive residuated lattice to be topological negative non-involutive residuated lattice.Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices on a topological level.