Filtered in Meet Semilattice with Finite Infima

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Theorem

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

Let $H$ be a non-empty lower subset of $S$.

Then $H$ is filtered if and only if

for every non-empty finite subset $A$ of $H$, $\inf A \in H$


Proof

This follows by mutatis mutandis of the proof of Directed in Join Semilattice with Finite Suprema.

$\blacksquare$


Sources