A characterization of the consistent Hoare powerdomains over dcpos

Abstract

It has been shown that for a dcpo P , the Scott closure of Γ c (P ) in Γ(P ) is a consistent Hoare powerdomain of P , where Γ c (P ) is the family of nonempty, consistent and Scott closed subsets of P , and Γ(P ) is the collection of all nonempty Scott closed subsets of P . In this paper, by introducing the notion of a −existing set, we present a direct characterization of the consistent Hoare powerdomain: the set of all −existing Scott closed subsets of a dcpo P is exactly the consistent Hoare powerdomain of P . We also introduce the concept of an F -Scott closed set over each dcpo-∨ ↑ -semilattice. We prove that the Scott closed set lattice of a dcpo P is isomorphic to the family of all F -Scott closed sets of P 's consistent Hoare powerdomain.