Domain theory

Remarks

Example: the interval $[0, 1]$ is a continuous domain that is not algebraic; $\set{y : y \ll x} = [0, x)$ for $x > 0$, with supremum $x$, while the only compact element is $0$. Example: the partial functions from $\N$ to $\N$, ordered by extension, form an algebraic domain whose compact elements are the finite partial functions. Non-example: the lattice of closed subsets of $[0, 1]$, ordered by inclusion, is a dcpo, but only $\varnothing$ is way below $[0, 1]$, so it is not a continuous domain. A central tool is the least fixed point: a Scott-continuous map $f$ on a pointed dcpo has the least fixed point $\bigvee_n f^n(\bot)$. Such fixed points give the meaning of recursive programs and of terms of the lambda calculus.