Difference between revisions 115528848 and 115528849 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) orders can be cast into various [[category theory|categories]] of dcpos, using functions which are monotone, Scott-continuous, or even more specialized as morphisms. Finally, note that the term ''domain'' itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.

== Important results ==

A poset ''D'' is a dcpo if and only if each chain in ''D'' has a supremum. 
However, directed sets are strictly more powerful than chains.

If ''f'' is a continuous function on a poset ''D'' then it has a least fixed point, given as the least upper bound of all finite iterations of ''f'' on the least element 0: ''V''<sub>''n'' in '''N'''</sub> ''f'' <sup>''n''</sup>(0).  This is the [[Kleene fixed-point theorem]].

==Generalizations==
(contracted; show full)[[Category:Domain theory]]
[[Category:Fixed points (mathematics)]]

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