Difference between revisions 115528834 and 115528835 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)

:''y'' ≤ ''sup D'',

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 set in which they did not alre(contracted; show full)

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'').

==Generalizations==

⏎
⏎
*[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.55.903&rep=rep1&type=pdf Synthetic domain theory]
*[http://homepages.inf.ed.ac.uk/als/Research/topological-domain-theory.html Topological domain theory]
*A [[continuity space]] is a generalization of metric spaces and [[poset]]s, that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.

==See also==
*[[Scott domain]]
*[[Scott information system]]
*[[Type theory]]
*[[Category theory]]

== Further reading ==
*{{cite encyclopedia | author = G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott | encyclopedia = Encyclopedia of Mathematics and its Applications | title = Continuous Lattices and Domains | year = 2003 | publisher = Cambridge University Press | volume = 93 | id = ISBN 0-521-80338-1 }}
*{{cite conference | author = [[Samson Abramsky|S. Abramsky]], A. Jung | year = 1994 | title = Domain theory | booktitle = Handbook of Logic in Computer Science | editor = S. Abramsky, D. M. Gabbay, T. S. E. Maibaum, editors, | volume = III | publisher = Oxford University Press | id = ISBN 0-19-853762-X | url = http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf | format = PDF | accessdate = 2007-10-13 }}  
*{{cite book | author = Alex Simpson | title = Mathematical Structures for Semantics | url = http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/l3.ps | accessdate = 2007-10-13 | year = 2001-2002 | chapter = Part III: Topological Spaces from a Computational Perspective }}
*{{cite journal | title = Data types as lattices | author = D. S. Scott | authorlink = Dana Scott | year = 1975 | journal = Proceedings of the International Summer Institute and Logic Colloquium, Kiel'', in ''Lecture Notes in Mathematics | volume = 499 | pages = 579–651 | publisher = Springer-Verlag }}
*{{cite book | author = Carl A. Gunter | title = Semantics of Programming Languages | year = 1992 | publisher = MIT Press }}
*{{cite book | author = B. A. Davey and H. A. Priestley | title = Introduction to Lattices and Order | edition = 2nd | year = 2002 | publisher =  Cambridge University Press | isbn = 0-521-78451-4 }}
*{{cite conference | author = Carl Hewitt and Henry Baker | month = August | year = 1977 | title = Actors and Continuous Functionals | booktitle = Proceedings of IFIP Working Conference on Formal Description of Programming Concepts }}

[[Category{{DEFAULTSORT:Domain tTheory|}}
[[Category:Domain theory]]
[[Category:Fixed points]]

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