continuous poset

A poset P is said to be continuousMathworldPlanetmathPlanetmath if for every aP

  1. 1.

    the set wb(a)={uPua} is a directed setMathworldPlanetmath,

  2. 2.

    wb(a) exists, and

  3. 3.


In the first condition, indicates the way below relation on P. It is true that in any poset, if b:=wb(a) exists, then ba. So for a poset to be continuous, we require that ab.

A continuous lattice is a complete latticeMathworldPlanetmath whose underlying poset is continuous. Note that if P is a complete lattice, condition 1 above is automatically satisfied: suppose u,va and DP with aD, then there are finite subsets F,G of D with uF and vG. Then H:=FGD is finite and uv(F)(G)=H, or uva, implying that wb(a) is directed.


  1. 1.

    Any finite poset is continuous, and so is any finite latticeMathworldPlanetmath (since it is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath).

  2. 2.

    A chain is continuous iff it is complete.

  3. 3.

    The lattice of ideals of a ring is continuous.

  4. 4.

    The set of all lower semicontinuous functions from a fixed compactPlanetmathPlanetmath topological spaceMathworldPlanetmath into the extended real numbers is a continuous lattice.

  5. 5.

    The set of all closed convex subsets of a compact convex subset of n ordered by reverse inclusion is a continuous lattice.



  • 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title continuous poset
Canonical name ContinuousPoset
Date of creation 2013-03-22 16:43:18
Last modified on 2013-03-22 16:43:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06B35
Defines continuous lattice