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.
Original language | English |
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Article number | 7 |
Number of pages | 19 |
Journal | Logical Methods in Computer Science |
Volume | 14 |
Issue number | 1 |
DOIs | |
Publication status | Published - 17 Jan 2018 |
Keywords
- Ho-Zhao problem
- Scott topology
- Scott-closed sets
- sobrification
- Johnstone's counterexample