DCPO中二元链的反射包络：正则性、极大Γ-忠实性及内部反射公式
The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula
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中文总结 AI 辅助
本文在范畴论框架下，研究DCPO中二元链的反射包络R₂的正则性、极大Γ-忠实性，给出2-反射的内部构造及相关判据，丰富了dcpo范畴的反射子范畴理论。
中文摘要 AI 辅助
设\boldsymbol{\text{DCPO}}为dcpo与Scott连续映射构成的范畴，\boldsymbol{R_2}为二元链\boldsymbol{2}的反射包络。我们证明\boldsymbol{R_2}是\boldsymbol{\text{DCPO}}中最小的非离散正则反射满子范畴，它对极限和Skula闭子dcpo封闭，且包含所有 sober dcpo。我们还证明\boldsymbol{R_2}是\boldsymbol{\text{DCPO}}中严格包含弱支配dcpos的极大Γ-忠实满子范畴。最后，我们通过超限迭代给出\boldsymbol{2}-反射的内部具体构造，并给出\boldsymbol{\text{DCPO}}的反射满子范畴的判据，其类似于\boldsymbol{T_0}拓扑空间的Keimel-Lawson条件。
英文摘要
Let \(\DCPO\) be the category of dcpos and Scott-continuous maps, and let \(\Rtwo\) be the reflective hull of the two-element chain \(2\). We prove that \(\Rtwo\) is the least non-discrete proper reflective full subcategory of \(\DCPO\). It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that \(\Rtwo\) is the maximal \(Γ\)-faithful full subcategory of \(\DCPO\) that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of \(2\)-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of \(\DCPO\), analogous to the Keimel--Lawson conditions for \(T_0\) topological spaces.
