continuous poset
A poset $P$ is said to be continuous^{} if for every $a\in P$

1.
the set $\mathrm{wb}(a)=\{u\in P\mid u\ll a\}$ is a directed set^{},

2.
$\bigvee \mathrm{wb}(a)$ exists, and

3.
$a=\bigvee \mathrm{wb}(a)$.
In the first condition, $\ll $ indicates the way below relation on $P$. It is true that in any poset, if $b:=\bigvee \mathrm{wb}(a)$ exists, then $b\le a$. So for a poset to be continuous, we require that $a\le b$.
A continuous lattice is a complete lattice^{} whose underlying poset is continuous. Note that if $P$ is a complete lattice, condition 1 above is automatically satisfied: suppose $u,v\ll a$ and $D\subseteq P$ with $a\le \bigvee D$, then there are finite subsets $F,G$ of $D$ with $u\le \bigvee F$ and $v\le \bigvee G$. Then $H:=F\cup G\subseteq D$ is finite and $u\vee v\le \left(\bigvee F\right)\vee \left(\bigvee G\right)=\bigvee H$, or $u\vee v\ll a$, implying that $\mathrm{wb}(a)$ is directed.
Examples.
 1.

2.
A chain is continuous iff it is complete.

3.
The lattice of ideals of a ring is continuous.

4.
The set of all lower semicontinuous functions from a fixed compact^{} topological space^{} into the extended real numbers is a continuous lattice.

5.
The set of all closed convex subsets of a compact convex subset of ${\mathbb{R}}^{n}$ ordered by reverse inclusion is a continuous lattice.
Remarks.

•
Every algebraic lattice is continuous.

•
Every continuous meet semilattice is meet continuous.
References
 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  20130322 16:43:18 
Last modified on  20130322 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 