The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages.
Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology.
The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus.
Using again just the syntactic transformations available in this formalism, one can obtain so-called fixed-point combinators (the best-known of which is the Y combinator); these, by definition, have the property that f(Y(f)) = Y(f) for all functions f. To formulate such a denotational semantics, one might first try to construct a model for the lambda calculus, in which a genuine (total) function is associated with each lambda term.
Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result.
The important step to finding a model for the lambda calculus is to consider only those functions (on such a partially ordered set) that are guaranteed to have least fixed points.
Computation then is modeled by applying monotone functions repeatedly on elements of the domain in order to refine a result.
Domains provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below.
The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory.
A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the order theory glossary.
Hence we can view directed subsets as consistent specifications, i.e. as sets of partial results in which no two elements are contradictory.
Naturally, one has a special interest in those domains of computations in which all consistent specifications converge, i.e. in orders in which all directed sets have a least upper bound.
From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a least element.
For example, in the natural subset-inclusion ordering on some powerset, any infinite element (i.e. set) is much more "informative" than any of its finite subsets.
A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation.
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 already occur.
Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of finite elements.
From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones.
All of these classes of orders can be cast into various categories of dcpos, using functions that are monotone, Scott-continuous, or even more specialized as morphisms.
Finally, note that the term domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.
If f is a continuous function on a domain D then it has a least fixed point, given as the least upper bound of all finite iterations of f on the least element ⊥: This is the Kleene fixed-point theorem.