---
tag: Q2P5D74Q1V
kind: concept
name: Domain theory
msc: [06B35, 68Q55, 03B70]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_theory
same_as:
  wikidata: Q1079166
  wikipedia: Domain theory
---

## Summary

Domain theory studies special partially ordered sets, called domains, whose elements are read as pieces of partial information.
Its basic objects are [[directed-complete partial orders|7TSXP95QN6]], the [[way-below relation|RDM7J5J9HQ]], [[Scott-continuous maps|C6GAHS1Z2P]] and the [[Scott topology|ZR8A03FV67]].
Dana Scott created it in the late 1960s to give a mathematical semantics for programming languages and for the [[lambda calculus|BD756KR8FM]].
Recursive definitions are interpreted as [[least fixed points|DBD64TH52D]] of continuous maps.

## Explanations

### L1 human

A program that has not finished yet has produced only part of its answer.
Domain theory orders such partial answers by "carries less information than", and treats a full answer as the limit of partial ones.
For example, partial functions from $\N$ to $\N$ are ordered by extension.
A recursive definition of a function is then the limit of the finite approximations produced by unfolding the recursion.
The theory sits between order theory, topology and computer science.