定向完备偏序集的拓扑对偶
首发时间:2021-04-15
摘要:本文以Scott拓扑为桥梁,借助定向完备偏序集的理想,定义了定向完备偏序集的对偶空间。研究了对偶空间的拓扑性质,并由此引入了$L$-空间的概念。建立了定向完备偏序集的拓扑对偶,证明了定向完备偏序集带有相应映射构成的范畴与$L$-空间带有$F$-连续映射构成的范畴对偶等价。最后,将上述思想方法应用到完备格上,建立了完备格的拓扑对偶,证明了完备格带有完备格同态构成的范畴对偶等价于$CL$-空间与特定映射构成的范畴。
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A Topological Duality for Directed Complete Posets
Abstract:In this paper, the dual space for directed complete posets is defined by endowing the ideal posets with Scott topology. The new concept of $L$-space is introduced by studying the topological properties of the dual space. With this concept, a topological duality for dire\\-cted complete posets is developed, and a dual equivalence between the category of directed complete posets with corresponding mappings and the category of $L$-space with $F$-continuous maps is showed. Finally, following by this idea, a topological duality for complete lattices is established and categorical equivalence between the category of complete lattices with compl\\-ete lattices homomorphisms and the category of $CL$-space with corresponding maps is pro\\-vided.
Keywords: Dcpo Complete lattice Stone duality
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