Ideal completion (order)

Let (D, \sqsubseteq) be a poset (in fact it only needs to be a preorder. Its ideal completion is the order

  (\text{Id}(D), \subseteq)

consisting of the set \text{Id}(D) of all ideals of D, ordered under inclusion.

This construction is a "standard way" of obtaining an algebraic cpo from any preorder.

Properties

The ideal completion has the following properties:

1. (D, \sqsubseteq) can be embedded in (\text{Id}(D), \subseteq), by sending x \in D to its principal ideal \mathop{\downarrow} x. (This is essentially the Yoneda lemma.)
2. (\text{Id}(D), \subseteq) is directed-complete.
3. The principal ideals of (D, \sqsubseteq) are compact elements of (\text{Id}(D), \subseteq).
4. (\text{Id}(D), \subseteq) is algebraic.

See also

• Ideal
• Powerdomains

References

• Proofs can be found at PlanetMath: ideal completion