Filtered in Meet Semilattice

Theorem

Let $\struct {S, \preceq}$ be a meet semilattice.

Let $H$ be a non-empty upper section of $S$.

Then $H$ is filtered if and only if

$\forall x, y \in H: x \wedge y \in H$

Proof

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

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_0:41