Uniqueness of directed complete posets based on Scott closed set lattices

TL;DR

This paper investigates the conditions under which directed complete posets (dcpos) are uniquely determined by their Scott-closed set lattices, introducing new concepts and proving uniqueness for various classes of dcpos.

Contribution

It introduces the notions of down-linear and quasicontinuous elements, and proves $C_{\sigma}$-uniqueness for classes including all quasicontinuous dcpos and specific examples.

Findings

Certain classes of dcpos are $C_{\sigma}$-unique.

$C_{\sigma}$-uniqueness does not imply bounded sobriety.

Introduces new concepts of down-linear and quasicontinuous elements.

Abstract

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.