Ideal completion

Remarks

$\operatorname{Idl}(P)$ is a directed-complete partial order: the supremum of a directed family of ideals is its union. The map $x \mapsto {\downarrow} x$ is an order embedding, and the principal ideals ${\downarrow} x$ are exactly the compact elements of $\operatorname{Idl}(P)$, so $\operatorname{Idl}(P)$ is an algebraic dcpo. When $P$ is a lattice, the order ideals are its lattice ideals and $\operatorname{Idl}(P)$ is an algebraic lattice if $P$ has a least element. Example: for $P = \N$, the ideals are the sets $\set{0, \ldots, n}$ and $\N$ itself, so $\operatorname{Idl}(\N) \cong \N \cup \set{\infty}$. Non-example: in $\N$, the set $\set{0, 2}$ is not an ideal, because $1 \le 2$ but $1 \notin \set{0, 2}$.