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In the [[mathematics|mathematical]] fields of [[order theory|order]] and [[domain theory]], a '''Scott domain''' is an [[algebraic poset|algebraic]], [[bounded complete]] [[complete partial order|cpo]]. It has been named in honour of [[Dana S. Scott]], who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to [[algebraic lattice]]s, being different only in possibly lacking a [[greatest element]]. They are also closely related to [[Scott information system]]s, which constitute a "syntactic" representation of Scott domains.
 
While the term "Scott domain" is widely used with the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications.
 
==Definition==
Formally, a non-empty [[partially ordered set]] (''D'', ≤) is called a ''Scott domain'' if the following hold:
 
* ''D'' is [[complete partial order|directed complete]], i.e. all [[directed set|directed subsets]] of ''D'' have a [[supremum]].
* ''D'' is [[bounded complete]], i.e. all subsets of ''D'' that have some [[upper bound]] have a supremum.
* ''D'' is [[algebraic poset|algebraic]], i.e. every element of ''D'' can be obtained as the supremum of a directed set of [[compact element]]s of ''D''.
 
==Properties==
Since the empty set certainly has some upper bound, we can conclude the existence of a [[least element]] <math>\bot</math> (the supremum of the empty set) from bounded completeness.  
 
The property of being bounded complete is equivalent to the existence of [[infimum|infima]] of all non-empty subsets of ''D''. It is well known that the existence of all infima implies the existence of all suprema and thus makes a partially ordered set into a [[complete lattice]]. Thus, when a top element (the infimum of the empty set) is adjoined to a Scott domain, one can conclude that:
# the new top element is compact (since the order was directed complete before) and
# the resulting poset will be an [[algebraic lattice]] (i.e. a complete lattice that is algebraic).
Consequently, Scott domains are in a sense "almost" algebraic lattices.
 
Scott domains become [[topological space]]s by introducing the [[Scott continuity|Scott topology]].
 
==Explanation==
Scott domains are intended to represent ''partial algebraic data'', ordered by information content. An element <math>x \in D</math> is a piece of data that might not be fully defined. The statement <math>x \sqsubseteq y</math> means "<math>y</math> contains all the information that <math>x</math> does".
 
With this interpretation we can see that the [[supremum]] <math>\sqcup X</math> of a subset <math>X \subseteq D</math> is the element that contains all the information that ''any'' element of <math>X</math> contains, but ''no more''. Obviously such a supremum only exists (i.e., makes sense) provided <math>X</math> does not contain inconsistent information; hence the domain is directed and bounded complete, but not ''all'' suprema necessarily exist. The algebraicity axiom essentially ensures that all elements get all their information from (non-strictly) lower down in the ordering; in particular, the jump from compact or "finite" to non-compact or "infinite" elements does not covertly introduce any extra information that cannot be reached at some finite stage. The bottom element is the supremum of the empty set, i.e. the element containing no information at all; its existence is implied by bounded completeness, since, vacuously, the empty set has an upper bound in any non-empty poset.
 
On the other hand, the infimum <math>\sqcap X</math> is the element that contains all the information that is shared by ''all'' elements of <math>X</math>, and ''no less''; if <math>X</math> contains inconsistent information, then its elements have no information in common and so its infimum is <math>\bot</math>. In this way all infima exist, but not all infima are necessarily interesting.
 
This definition in terms of partial data allows an algebra to be defined as the limit of a sequence of increasingly more defined partial algebras — in other words a fixed point of an operator that adds progressively more information to the algebra. For more information, see [[Domain theory]].
 
==Examples==
* Every finite poset is directed complete and algebraic. Thus any bounded complete finite poset trivially is a Scott domain.
 
* The [[natural numbers]] with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in this direction, see the article on [[algebraic lattice]]s.
 
* Consider the set of all finite and infinite words over the alphabet {0,1}, ordered by the [[prefix order]] on words. Thus, a word ''w'' is smaller than some word ''v'' if ''w'' is a prefix of ''v'', i.e. if there is some (finite or infinite) word ''v' '' such that ''w'' ''v' '' = ''v''. For example 101 ≤ 10110. The empty word is the bottom element of this ordering and every directed set (which is always a [[total order|chain]]) is easily seen to have a supremum. Likewise, one immediately verifies bounded completeness. However, the resulting poset is certainly missing a top having many maximal elements instead (like 111... or 000...). It is also algebraic, since every finite word happens to be compact and we certainly can approximate infinite words by chains of finite ones. Thus this is a Scott domain which is not an algebraic lattice.
 
* For a negative example, consider the [[real number]]s in the unit interval [0,1], ordered by their natural order. This bounded complete cpo is not algebraic. In fact its only compact element is 0.
 
==Literature==
''See the literature given for [[domain theory]].''
 
{{DEFAULTSORT:Scott Domain}}
[[Category:Domain theory]]
[[Category:Order theory]]

Latest revision as of 19:38, 7 January 2015

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