Zhao, Dongsheng and Xu, Luoshan - Uniqueness of directed complete posets based on Scott closed set lattices

lmcs:1530 - Logical Methods in Computer Science, May 16, 2018, Volume 14, Issue 2
Uniqueness of directed complete posets based on Scott closed set lattices

Authors: Zhao, Dongsheng and Xu, Luoshan

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.

Source : oai:arXiv.org:1709.03700
DOI : 10.23638/LMCS-14(2:10)2018
Volume: Volume 14, Issue 2
Published on: May 16, 2018
Submitted on: July 14, 2016
Keywords: Mathematics - General Topology