The Ho-Zhao Problem

Abstract

Given a poset $P$, the set, $\Gamma (P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of ${\mathbf{Pos}}_d$ (the category of posets and Scott-continuous maps) is said to be $\Gamma$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $\Gamma (P)\cong \Gamma (Q)$ implies $P\cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $\Gamma$-faithful, while ${\mathbf{Pos}}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $\Gamma$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $\Gamma$-faithful. This subcategory subsumes all currently known $\Gamma$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

19 pages, 4 figures