Difference between revisions 115528833 and 115528834 on dewiki

'''Domain theory''' is a   branch of [[mathematics]] that studies special kinds of [[partially ordered set]]s (posets) commonly called '''domains'''. Consequently, domain theory can be considered as a branch of [[order theory]]. The field has major applications in [[computer science]], where it is used to specify [[denotational semantics]], especially for [[functional programming|functional programming languages]]. Domain theory formalizes the intu(contracted; show full)
there is some element ''d'' in ''D'' such that 

:''x'' ≤ ''d''. 

Then one also says that ''x'' ''approximates'' ''y'' and writes 

:''x'' 
<<≪ ''y''. 

This does imply that 

:''x'' ≤ ''y'', 

since the singleton set {''y''} is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact: the chain) of finite sets 

:{0}, {0, 1}, {0, 1, 2}, ... 

Since the supremum of this chain is the set of all natural numbers '''N''', this shows that no infinite set is way below '''N'''.

However, being way below some element is a ''relative'' notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these '''finite''' elements ''x'' is that they are way below themselves, i.e. 

:''x'' <<≪ ''x''. 

An element with this property is also called '''[[compact element|compact]]'''. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in [[set theory]] and [[topology]]. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed se(contracted; show full)[[Category:Domain theory|Domain theory]]
[[Category:Fixed points]]

[[fa:نظریه حوزه‌ها]]
[[fr:Théorie des domaines]]
[[ko:도메인 이론]]
[[ja:領域理論]]
[[zh:域理论]]