Uniqueness of directed complete posets based on Scott closed set lattices

Dongsheng Zhao, Luoshan Xu

Published 2017-09-12Version 1

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 a dcpo will be called a $C_{\sigma}$-unique dcpo, or $C_{\sigma}$-unique in short). We shall introduce the notions of down-linear element and quasicontinuous element in dcpos, and use them to prove that dcpos of certain class, 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. These results will help to obtain a complete characterization of $C_{\sigma}$-unique dcpos in the future.