Definition:Meet-Continuous Lattice

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Let $\left({S, \preceq}\right)$ be a meet semilattice.

Then $\left({S, \preceq}\right)$ is meet-continuous if and only if

$\left({S, \preceq}\right)$ is up-complete and
(MC): for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$

Also known as

Some sources prefer not to hyphenate: meet continuous lattice.