Uniqueness of directed complete posets based on Scott closed set lattices

In analogy to a result due to Drake and Thron about topological spaces, this paper studies the dcpos (directed complete posets) which are fully determined, among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will be called $C_{\sigma }$-unique). We introduce the notions of down-linear element and quasicontinuous element in dcpos, and use them to prove that dcpos of certain classes, including all quasicontinuous dcpos as well as Johnstone’s and Kou’s examples, are $C_{\sigma }$-unique. As a consequence, $C_{\sigma }$-unique dcpos with their Scott topologies need not be bounded sober.