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

@article{paperbot405,
  title = {Uniqueness of directed complete posets based on Scott closed set lattices},
  author = {Dongsheng Zhao and Luoshan Xu},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 14, Issue 2},
  year = {2018},
  doi = {10.23638/lmcs-14(2:10)2018}
}