Category:Meet-Continuous Lattices

This category contains results about Meet-Continuous Lattices.

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 $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$