The Ho-Zhao problem
Research output: Contribution to journal › Article › peer-review
Authors
Colleges, School and Institutes
External organisations
- Nanyang Technological University, Singapore
- LSV, ENS Paris-Saclay, CNRS, Universite Paris-Saclay, 94230 Cachan, France
- Jiangsu Normal University, Shanghai, China
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.
Details
| Original language | English |
|---|---|
| Article number | 7 |
| Number of pages | 19 |
| Journal | Logical Methods in Computer Science |
| Volume | 14 |
| Issue number | 1 |
| Publication status | Published - 17 Jan 2018 |
Keywords
- Ho-Zhao problem, Scott topology, Scott-closed sets, sobrification, Johnstone's counterexample