AI 中文总结
本文引入d-谱空间范畴，证明其为经典谱空间的双拓扑扩展，二者存在范畴嵌入关系，且d-谱空间与d-格的谱对应，还具备patch空间、de Groot对偶等相关性质。
AI 中文摘要
我们引入并研究d-谱空间的范畴，这是Stone和Hochster提出的经典谱空间的双拓扑类似物。d-谱空间是紧致、d- sober的双拓扑空间，满足两个开集格均为coherent frame，其中d-sobriety是Jung和Moshier提出的双拓扑意义下的sobriety概念。我们证明谱空间的范畴可作为同时是反射和余反射的满子范畴嵌入到d-谱空间的范畴中。此外，我们证明d-谱空间恰好是d-格的谱。该结果的关键在于与d-谱空间关联的紧开集构成的d-格，以及针对d-格的谱构造。我们还证明d-谱空间的patch空间是d-布尔的，且d-谱空间的de Groot对偶仍是d-谱的，这与谱空间对应的经典性质一致。我们的结果表明，d-谱空间构成了谱空间框架的自然且性质良好的双拓扑扩展。
英文摘要
We introduce and study the category of \emph{d-spectral spaces}, a bitopological analogue of the classical spectral spaces of Stone and Hochster. A d-spectral space is a compact, d-sober bitopological space such that both open set lattices are coherent frames, where d-sobriety is the bitopological notion of sobriety due to Jung and Moshier. We show that the category of spectral spaces embeds into the category of d-spectral spaces as a simultaneously reflective and coreflective full subcategory. Moreover, we prove that d-spectral spaces are precisely the spectra of d-lattices. Key to this result is the d-lattice of compact open sets associated to a d-spectral space and the spectrum construction for d-lattices. We also show that the patch space of a d-spectral space is d-Boolean and that the de Groot dual of a d-spectral space is again d-spectral, mirroring the corresponding classical properties of spectral spaces. Our results demonstrate that d-spectral spaces form a natural and well-behaved bitopological extension of the spectral space framework.
Comments23 pages
