In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.
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.
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. [1]: p.52, Examples I-1.3, (4)
For any , let
Then is an upper set, and a lower set. If is an upper-semilattice, is a directed set (that is, implies ), and therefore an ideal.
A preordered set is called a continuous preordered set if for any , the subset is directed and .
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 .
For any two elements of a continuous dcpo , the following two conditions are equivalent. [1]: p.61, Proposition I-1.19(i)
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 . [1]: p.61, Proposition I-1.19(ii)
For a dcpo , the following conditions are equivalent. [1]: Theorem I-1.10
In this case, the actual left adjoint is
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.
A continuous complete lattice is often called a continuous lattice.
For a topological space , the following conditions are equivalent.
In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.
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.
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. [1]: p.52, Examples I-1.3, (4)
For any , let
Then is an upper set, and a lower set. If is an upper-semilattice, is a directed set (that is, implies ), and therefore an ideal.
A preordered set is called a continuous preordered set if for any , the subset is directed and .
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 .
For any two elements of a continuous dcpo , the following two conditions are equivalent. [1]: p.61, Proposition I-1.19(i)
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 . [1]: p.61, Proposition I-1.19(ii)
For a dcpo , the following conditions are equivalent. [1]: Theorem I-1.10
In this case, the actual left adjoint is
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.
A continuous complete lattice is often called a continuous lattice.
For a topological space , the following conditions are equivalent.