Filter [mde-QP0H]

April 14, 2025
Matt Earnshaw

A filter in a poset \(P\) is a downwards directed, upwards closed subset of \(P\).

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Ultrafilter [mde-INQP]

April 14, 2025
Matt Earnshaw

an ultrafilter in a poset \(P\) is a filter which is maximal (with respect to inclusion) amongst all proper filters (those not equal to \(P\)).

an ultrafilter in a set \(X\) is an ultrafilter in \(\mathcal {P}(X)\).