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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