论文标题
从属代数的麦克尼尔完成
MacNeille completions of subordination algebras
论文作者
论文摘要
$ \ Mathsf {S5} $ - 从属代数是De Vries代数的自然概括。最近证明，$ \ mathsf {subs5^s} $ of $ \ mathsf {s5} $ - 从属代数和它们之间的兼容从属关系等同于紧凑型Hausdorff Space和封闭关系的类别。 We generalize MacNeille completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries代数。我们还表明，与每个$ \ Mathsf {S5} $ - 从属代数相关的函子圆形理想的框架建立了$ \ Mathsf {subs5^s} $与紧凑的常规框架和预先框架的类别之间的双重等价性。我们的结果是无选择的，并为紧凑的Hausdorff空间提供了对石材样二元性的进一步见解，它们之间具有各种形态。特别是，我们展示了它们如何限制在$ \ mathsf {subs5^s} $的宽子类别中，对应于紧凑的Hausdorff Space之间的连续关系和连续功能。
$\mathsf{S5}$-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each $\mathsf{S5}$-subordination algebra the frame of its round ideals establishes a dual equivalence between $\mathsf{SubS5^S}$ and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces.
