基于Scott闭集格的有向完备偏序集的忠实性
Faithfulness of Directed Complete Posets based on Scott Closed Set Lattices
摘要：
Thron证明了拓扑空间$X$具有如下性质：$C(X)$同构于$C(Y)$，蕴含$X$同胚于$Y$当且仅当$X$是sober且$T_D$，其中$C(X)$和$C(Y)$分别表示$X$和$T_0$空间$Y$的闭集格。当我们考虑dcpos（有向完备偏序集）配备了它们的Scott拓扑时，一个类似的问题出现了：哪些dcpos $P$具有这样的性质：对于任何dcpo PDMATHAMT，$C_σ(P)$同构于$C_σ(Q)$意味着$P$同构于PDMATHAMT（这样的dcpo $P$将被称为Scott闭集格忠实，或简称为SCL忠实）？这里，$C_σ(P)$和$C_σ(Q)$分别表示$P$和$Q$的Scott闭集的格。根据$C_σ(P)$对连续（拟连续）dcpo的刻画，可以很容易地推导出每个连续（拟连续）dcpo都是SCL-忠实的。注意，每个连续（拟连续）dcpo的Scott空间是sober的。与Thron的结果相比，人们自然会问是否每个SCL忠实的dcpo都是清醒的（使用Scott拓扑）。本文证明了某些dcpos类是SCL-忠实的，这些类包含某些dcpos类的Scott拓扑不是有界的。这些结果将有助于在未来获得一个完整的描述SCL忠实dcpos。
英文摘要：
By Thron, a topological space $X$ has the property that $C(X)$ isomorphic to $C(Y)$ implies $X$ is homeomorphic to $Y$ iff $X$ is sober and $T_D$, where $C(X)$ and $C(Y)$ denote the lattices of closed sets of $X$ and $T_0$ space $Y$, respectively. When we consider dcpos (directed complete posets) equipped their Scott topologies, a similar question arises: which dcpos $P$ have the property that for any dcpo $Q$, $C_σ(P)$ isomorphic to $C_σ(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_σ(P)$ and $C_σ(Q)$ denote the lattices of Scott closed sets of $P$ and $Q$, respectively. Following a characterization of continuous (quasicontinuous) dcpos in terms of $C_σ(P)$, one easily deduces that every continuous (quasicontinuous) dcpo is SCL-faithful. Note that the Scott space of every continuous (quasicontinuous) dcpo is sober. Compared with Thron's result, one naturally asks whether every SCL-faithful dcpo is sober (with the Scott topology). In this paper we shall prove that some classes of dcpos are SCL-faithful, these classes contain some dcpos whose Scott topologies are not bounded sober. These results will help to obtain a complete characterization of SCL-faithful dcpos in the future.
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