摘要：Given a poset $P$, the set, $\Gamma(P)$, of all Scott closed sets ordered byinclusion forms a complete lattice. A subcategory $\mathbf{C}$ of$\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said tobe $\Gamma$-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 allcontinuous dcpos and the category of bounded complete dcpos are$\Gamma$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) askedwhether the category $\mathbf{DCPO}$ of dcpos is $\Gamma$-faithful. In thispaper, 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 constructthe desired counterexample which relies heavily on Johnstone's famous dcpowhich is not sober in its Scott topology.
关键词：06B35;Computer Science - Logic in Computer Science
