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

1. the set $\operatorname{wb}(a)=\lbrace u\in P\mid u\ll a\rbrace$ is a directed set,
2. $\bigvee \operatorname{wb}(a)$ exists, and
3. $a=\bigvee \operatorname{wb}(a)$ .

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 $u\vee v\le \big(\bigvee F\big)\vee \big(\bigvee G\big)=\bigvee H$ , or $u\vee v\ll a$ , implying that $\operatorname{wb}(a)$ is directed.

Examples.

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

Bibliography

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).