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The Ho-Zhao problem
Citation
Ho, W. K., Goubault-Larrecq, J., Jung, A., & Xi, X. (2018). The Ho-Zhao problem. Logical Methods in Computer Science, 14(1), 1-19. https://lmcs.episciences.org/4218
Abstract
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.
Date Issued
2018
Publisher
Logical Methods in Computer Science
Journal
Logical Methods in Computer Science