Explicit Descriptions of Power Domains
Let D be a domain. The lower power domain can be defined by
- P = {closure | Ø∈A⊆D} where
-
-
-
- closure = {d∈D | ∃X⊆D, X directed, d = X, and ∀x∈X ∃a∈A x≤a}.
-
-
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.
Read more about this topic: History Of Power Domains
Famous quotes containing the words explicit, descriptions, power and/or domains:
“I think “taste” is a social concept and not an artistic one. I’m willing to show good taste, if I can, in somebody else’s living room, but our reading life is too short for a writer to be in any way polite. Since his words enter into another’s brain in silence and intimacy, he should be as honest and explicit as we are with ourselves.”
—John Updike (b. 1932)
“Our Lamaze instructor . . . assured our class . . . that our cervix muscles would become “naturally numb” as they swelled and stretched, and deep breathing would turn the final explosions of pain into “manageable discomfort.” This descriptions turned out to be as accurate as, say a steward advising passengers aboard the Titanic to prepare for a brisk but bracing swim.”
—Mary Kay Blakely (20th century)
“For some men the power to destroy life becomes the equivalent to the female power to create life.”
—Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 4 (1991)
“I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.”
—Henry David Thoreau (1817–1862)