论文标题
位置的D-Sublocales和$ t_d $双重性的Coframe
The coframe of D-sublocales of a locale and the $T_D$ duality
论文作者
论文摘要
探索了\ emph {d-sublocale}的概念。这是$ t_d $ spaces双重性的概念类似物。当且仅当相应的本地地图保留被覆盖的质量的属性时，框架$ l $的sublocale $ s $是d-sublocale。结果表明，对于$ l $，那些也是d-sublocales的sublocales的系统形成了密集的sublocale $ \ mathsf {s} _d（l）$的coframe $ \ mathsf {s}}（s}}（l）$的系统。还显示出空间化$ \ mathsf {sp} _d [\ mathsf {s} _d（l）] $ of $ \ mathsf {s} _d _d（l）$精确地由$ t_d $ spatial的$ l $的d-sublocales组成。此外，框架使我们有$ \ Mathsf {s} _d（l）\ cong \ Mathcal {p}（\ Mathsf {pt} _d（l））$ - 也就是说，那些d-sublocales off-sublocales-space-space-spaces-subspace-subspace-以$ t_d $ -spatial $ s的特征为$ -Spatial frameSs，例如$ -Spatial frameSs s s s. \ mathsf {s}（l）的布尔化。
The notion of \emph{D-sublocale} is explored. This is the notion analogue to that of sublocale in the duality of $T_D$spaces. A sublocale $S$ of a frame $L$ is a D-sublocale if and only if the corresponding localic map preserves the property of being a covered prime. It is shown that for a frame $L$ the system of those sublocales which are also D-sublocales form a dense sublocale $\mathsf{S}_D(L)$ of the coframe $\mathsf{S}(L)$ of all its sublocales. It is also shown that the spatialization $\mathsf{sp}_D[\mathsf{S}_D(L)]$ of $\mathsf{S}_D(L)$ consists precisely of those D-sublocales of $L$ which are $T_D$-spatial. Additionally, frames such that we have $\mathsf{S}_D(L)\cong \mathcal{P}(\mathsf{pt}_D(L))$ -- that is, those such that D-sublocales perfectly represent subspaces -- are characterized as those $T_D$-spatial frames such that $\mathsf{S}_D(L)$ is the Booleanization of \mathsf{S}(L).
