The Ho-Zhao problem

Research output: Contribution to journalArticlepeer-review


  • Weng Kin Ho
  • Jean Goubault-Larrecq
  • Achim Jung
  • Xiaoyong Xi

Colleges, School and Institutes

External organisations

  • Nanyang Technological University
  • LSV, ENS Paris-Saclay, CNRS, Universite Paris-Saclay, 94230 Cachan, France
  • Jiangsu Normal University, Shanghai, China


Given a poset P, the set Γ(P) of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory C of Posd (the category of posets and Scott-continuous maps) is said to be Γ-faithful if for any posets P and Q in C, Γ(P) ∼= Γ(Q) implies P ∼= Q. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are Γ-faithful, while Posd is not. Ho & Zhao (2009) asked whether the category DCPO of dcpos is Γ-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 Γ-faithful. This subcategory subsumes all currently known Γ-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.


Original languageEnglish
Article number7
Number of pages19
JournalLogical Methods in Computer Science
Issue number1
Publication statusPublished - 17 Jan 2018


  • Ho-Zhao problem, Scott topology, Scott-closed sets, sobrification, Johnstone's counterexample