Explicit Descriptions of Power Domains
Let D be a domain. The lower power domain can be defined by
- P = {closure | Ø∈A⊆D} where
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- closure = {d∈D | ∃X⊆D, X directed, d = X, and ∀x∈X ∃a∈A x≤a}.
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In other words, P is the collection of downward-closed subsets of D that are also closed under existing least upper bounds of directed sets in D. Note that while the ordering on P is given by the subset relation, least upper bounds do not in general coincide with unions.
It is important to check which properties of domains are preserved by the power domain constructions. For example, the Hoare powerdomain of an ω-complete domain is again ω-complete.
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