Scott Domain

In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete 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 lattices, being different only in possibly lacking a greatest element.

Formally, a non-empty partially ordered set (D, ≤) 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.

Since the empty set certainly has some upper bound, we can conclude the existence of a least element (the supremum of the empty set) from bounded completeness. Also note that, while the term "Scott domain" is widely used with this definition, the term "domain" does not have such a general meaning: it may be used to refer to many structures in domain theory and is usually explained before it is used. Yet, "domain" is the term that Scott himself originally used for these structures. Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications.

It should be remarked that the property of being bounded complete is equivalent to the existence of all non-empty infima. 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:

1. the new top element is compact (since the order was directed complete before) and
2. 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 are closely related to Scott information systems, which constitute a "syntactic" representation of Scott domains.