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.

Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1607.03576

• 03B70 - Logic in computer science
• 03E70 - Nonclassical and second-order set theories
• 68Q42 - Grammars and rewriting systems