  It has been shown that for a dcpo P, the Scott closure of \Gamma_c(P) in
\Gamma(P) is a consistent Hoare powerdomain of P, where \Gamma_c(P) is the
family of nonempty, consistent and Scott closed subsets of P, and \Gamma(P) is
the collection of all nonempty Scott closed subsets of P. In this paper, by
introducing the notion of a \vee-existing set, we present a direct
characterization of the consistent Hoare powerdomain: the set of all
\vee-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-\vee-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.
