Faithfulness of Directed Complete Posets based on Scott Closed Set Lattices

TL;DR

This paper investigates which directed complete posets (dcpos) can be uniquely identified by their lattices of Scott closed sets, extending classical results from topology to domain theory and revealing classes of SCL-faithful dcpos.

Contribution

The paper characterizes classes of SCL-faithful dcpos, including some with non-bounded sober Scott topologies, advancing understanding of their structural properties.

Findings

Some classes of dcpos are SCL-faithful.

Continuous and quasicontinuous dcpos are SCL-faithful.

Not all SCL-faithful dcpos have bounded sober Scott topologies.

Abstract

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 $Q$, $C_\sigma(P)$ isomorphic to $C_\sigma(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_{\sigma}(P)$ and $C_{\sigma}(Q)$ denote the lattices of Scott closed sets of $P$ and $Q$, respectively. Following a characterization of continuous (quasicontinuous) dcpos in terms of $C_{\sigma}(P)$, 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.