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continuous poset
A poset $P$ is said to be continuous if for every $a\in P$
1. the set $\operatorname{wb}(a)=\{u\in P\mid u\ll a\}$ is a directed set,
2. $\bigvee\operatorname{wb}(a)$ exists, and
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. 2. 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).
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