The Ho-Zhao Problem

نویسندگان

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

Given a poset P , the set, Γ(P), of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory C of Pos d (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 Pos d 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.

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عنوان ژورنال:
  • Logical Methods in Computer Science

دوره 14  شماره 

صفحات  -

تاریخ انتشار 2018