Weng Kin Ho ; Jean Goubault-Larrecq ; Achim Jung ; Xiaoyong Xi - The Ho-Zhao Problem

lmcs:1529 - Logical Methods in Computer Science, January 17, 2018, Volume 14, Issue 1 - https://doi.org/10.23638/LMCS-14(1:7)2018
The Ho-Zhao Problem

Authors: Weng Kin Ho ; Jean Goubault-Larrecq ; Achim Jung ; Xiaoyong Xi

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.

Volume: Volume 14, Issue 1
Published on: January 17, 2018
Accepted on: January 17, 2018
Submitted on: July 13, 2016
Keywords: Computer Science - Logic in Computer Science,06B35


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