Let (D,⊑) be a poset (in fact it only needs to be a preorder. Its ideal completion is the order
(Id(D),⊆)
consisting of the set Id(D) of all ideals of D, ordered under inclusion.
This construction is a "standard way" of obtaining an algebraic cpo from any preorder.
The ideal completion has the following properties:
- (D,⊑) can be embedded in (Id(D),⊆), by sending x∈D to its principal ideal ↓x. (This is essentially the Yoneda lemma.)
- (Id(D),⊆) is directed-complete.
- The principal ideals of (D,⊑) are compact elements of (Id(D),⊆).
- (Id(D),⊆) is algebraic.