stylistic:tone, linguistic:voice_change
Definition

A subset F of a partially ordered set (P, ≤) is a filter or dual ideal if the following are satisfied:

If, additionally, F ≠ P, then F is said to be a proper filter. Authors in set theory and mathematical logic often require all filters to be proper; this article will eschew that convention. An ultrafilter is a filter contained in no other proper filter.