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.
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 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.
Formally, a non-empty partially ordered set
is called a Scott domain if the following hold: Since the empty set certainly has some upper bound, we can conclude the existence of a least element
Thus, when a top element (the infimum of the empty set) is adjoined to a Scott domain, one can conclude that: 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.
Scott domains are intended to represent partial algebraic data, ordered by information content.
Obviously such a supremum only exists (i.e., makes sense) provided
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.
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.