\begin{definition}[Definition:Meet-Continuous Lattice]
Let $\left({S, \preceq}\right)$ be a meet semilattice.
Then $\left({S, \preceq}\right)$ is '''meet-continuous''' {{iff}}
:$\left({S, \preceq}\right)$ is up-complete and
::(MC): for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$
\end{definition}