# continuous poset

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

1. 1.

the set $\operatorname{wb}(a)=\{u\in P\mid u\ll a\}$ is a directed set,

2. 2.

$\bigvee\operatorname{wb}(a)$ exists, and

3. 3.

$a=\bigvee\operatorname{wb}(a)$.

In the first condition, $\ll$ indicates the way below relation on $P$. It is true that in any poset, if $b:=\bigvee\operatorname{wb}(a)$ exists, then $b\leq a$. So for a poset to be continuous, we require that $a\leq b$.

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 $a\leq\bigvee D$, then there are finite subsets $F,G$ of $D$ with $u\leq\bigvee F$ and $v\leq\bigvee G$. Then $H:=F\cup G\subseteq D$ is finite and $u\vee v\leq\big{(}\bigvee F\big{)}\vee\big{(}\bigvee G\big{)}=\bigvee H$, or $u\vee v\ll a$, implying that $\operatorname{wb}(a)$ is directed.

Examples.

1. 1.

Any finite poset is continuous, and so is any finite lattice (since it is complete).

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 compact topological space into the extended real numbers is a continuous lattice.

5. 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.

## 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 ContinuousPoset 2013-03-22 16:43:18 2013-03-22 16:43:18 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 06B35 continuous lattice