Continuous poset

In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.

Definitions

Let be two elements of a preordered set. Then we say that approximates, or that is way-below, if the following two equivalent conditions are satisfied.

For any directed set such that, there is a such that.
For any ideal such that,.

If approximates, we write. The approximation relation is a transitive relation that is weaker than the original order, also antisymmetric if is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if satisfies the ascending chain condition.
For any, let
Then is an upper set, and a lower set. If is an upper-semilattice, is a directed set, and therefore an ideal.
A preordered set is called a continuous preordered set if for any, the subset is directed and.

Properties

The interpolation property

For any two elements of a continuous preordered set, if and only if for any directed set such that, there is a such that. From this follows the interpolation property of the continuous preordered set : for any such that there is a such that.

Continuous dcpos

For any two elements of a continuous directed-complete [partially ordered set|dcpo], the following two conditions are equivalent.

and.
For any directed set such that, there is a such that and.

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any such that and, there is a such that and.
For a dcpo, the following conditions are equivalent.

is continuous.
The supremum map from the partially ordered set of ideals of to has a left adjoint.

In this case, the actual left adjoint is

Continuous complete lattices

For any two elements of a complete lattice, if and only if for any subset such that, there is a finite subset such that.
Let be a complete lattice. Then the following conditions are equivalent.

is continuous.
The supremum map from the complete lattice of ideals of to preserves arbitrary infima.
For any family of directed sets of,.
is isomorphic to the image of a Scott-continuous idempotent map on the direct power of arbitrarily many two-point lattices.

A continuous complete lattice is often called a continuous lattice.

Examples

Lattices of open sets

For a topological space, the following conditions are equivalent.

The complete Heyting algebra of open sets of is a continuous complete Heyting algebra.
The sobrification of is a compact space">compact set">compact space
is an exponentiable object in the category of topological spaces. That is, the functor has a right adjoint.