通过拟合子尺度的典型扩展

Tomáš Jakl, Anna Laura Suarez
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引用次数: 0

摘要

我们以最近的一个结果为基础,这个结果指出,一个框架 $L$ 的强精确滤波器框架 $\mathsf{SE}(L)$ 与拟合子线程的 coframe$\mathsf{S}_o(L)$ 是反同构的。众所周知,$L$ 的精确滤波器集合$\mathsf{E}(L)$ 是这个框架的子球面。我们考虑了 $\mathsf{SE}(L)$ 的其他几个子集合:分别是完全素数滤波器和斯科特开滤波器的交集的集合 $\mathcal{J}(\mathsf{CP}(L))$ 和 $\mathcal{J}(\mathsf{SO}(L))$ ,以及滤波器框架的正则元素集合 $\mathsf{R}(L)$ 。我们证明所有这些都是 $\mathsf{SE}(L)$ 的子域,因此它们对应于 $\mathsf{S}_o(L)$ 的子域,而这些子域都有精确的描述。通过使用伯克霍夫的极性理论,我们可以证明上述所有结构都具有普适性,而这些普适性是典型扩展的变体。我们还展示了其中一些子集合如何被描述为极性,并给出了三个以滤波器晶格为基础的新的等价子适配性定义。
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Canonical extensions via fitted sublocales
We build on a recent result stating that the frame $\mathsf{SE}(L)$ of strongly exact filters for a frame $L$ is anti-isomorphic to the coframe $\mathsf{S}_o(L)$ of fitted sublocales. The collection $\mathsf{E}(L)$ of exact filters of $L$ is known to be a sublocale of this frame. We consider several other subcollections of $\mathsf{SE}(L)$: the collections $\mathcal{J}(\mathsf{CP}(L))$ and $\mathcal{J}(\mathsf{SO}(L))$ of intersections of completely prime and Scott-open filters, respectively, and the collection $\mathsf{R}(L)$ of regular elements of the frame of filters. We show that all of these are sublocales of $\mathsf{SE}(L)$, and as such they correspond to subcolocales of $\mathsf{S}_o(L)$, which all turn out to have a concise description. By using the theory of polarities of Birkhoff, one can show that all of the structures mentioned above enjoy universal properties which are variations of that of the canonical extension. We also show how some of these subcollections can be described as polarities and give three new equivalent definitions of subfitness in terms of the lattice of filters.
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