In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete and directed-complete partial order (dcpo). They are 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 lattices, being different only in possibly lacking a greatest element. They are also closely related to Scott information systems, 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. Originally, Dana Scott demanded a complete lattice, and the Russian mathematician Yuri Yershov constructed the isomorphic structure of a directed-complete partial order (dcpo). But this was not recognized until after scientific communications improved after the fall of the Iron Curtain. In honour of their work, a number of mathematical papers now dub this fundamental construction a "Scott–Ershov" domain.
Definition Formally, a non-empty partially ordered set ( D , ≤ ) {\displaystyle (D,\leq )} is called a Scott domain if the following hold:
D is directed-complete, i.e. all 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, i.e. every element of D can be obtained as the supremum of a directed set of compact elements of D.
Properties Since the empty set certainly has some upper bound, we can conclude the existence of a least element ⊥ {\displaystyle \bot } (the supremum of the empty set) from bounded completeness. The property of being bounded-complete is equivalent to the existence of 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. However, removing the top element from a complete lattice does not always produce a Scott domain. (Consider the complete lattice P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} . The finite subsets of N {\displaystyle \mathbb {N} } form a directed set, but have no upper bound in P ( N ) ∖ { N } {\displaystyle {\mathcal {P}}(\mathbb {N} )\setminus \{\mathbb {N} \}} .) Scott domains become topological spaces by introducing the Scott topology.
Explanation Scott domains are intended to represent partial algebraic data, ordered by information content. An element x ∈ D {\displaystyle x\in D} is a piece of data that might not be fully defined. The statement x ≤ y {\displaystyle x\leq y} means " y {\displaystyle y} contains all the information that x {\displaystyle x} does". The bottom element is the element containing no information at all. Compact elements are the elements representing a finite amount of information. With this interpretation we can see that the supremum ⋁ X {\displaystyle \bigvee X} of a subset X ⊆ D {\displaystyle X\subseteq D} is the element that contains all the information that any element of X {\displaystyle X} contains, but no more. Obviously such a supremum only exists (i.e., makes sense) provided X {\displaystyle X} 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. On the other hand, the infimum ⋀ X {\displaystyle \bigwedge X} is the element that contains all the information that is shared by all elements of X {\displaystyle X} , and no less. If X {\displaystyle X} contains no consistent information, then its elements have no information in common and so its infimum is ⊥ {\displaystyle \bot } . In this way all non-empty 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.
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