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. 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 ${ℝ}^n$ ordered by reverse inclusion is a continuous lattice.

Remarks.

• Every algebraic lattice is continuous.

• Every continuous meet semilattice is meet continuous.