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Continuous poset

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

Definitions

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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.[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 .

Properties

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The interpolation property

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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

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For any two elements of a continuous dcpo , the following two conditions are equivalent.[1]: p.61, Proposition I-1.19(i) 

  • 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 .[1]: p.61, Proposition I-1.19(ii) 

For a dcpo , the following conditions are equivalent.[1]: Theorem I-1.10 

  • 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

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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 .[2]: p.56, Theorem 44 

A continuous complete lattice is often called a continuous lattice.

Examples

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Lattices of open sets

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For a topological space , the following conditions are equivalent.

References

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  1. ^ a b c d e Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001.
  2. ^ Grätzer, George (2011). Lattice Theory: Foundation. Basel: Springer. doi:10.1007/978-3-0348-0018-1. ISBN 978-3-0348-0017-4. LCCN 2011921250. MR 2768581. Zbl 1233.06001.
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