ultrafilters

Noun

ultrafilter (plural ultrafilters)

1. A device that performs ultrafiltration.
2. (whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements.
   • A proper filter of a set has these three properties: (1) it does not contain the empty set, (2) if it contains a subset then it contains all supersets of that subset, and (3) if it contains a pair of subsets then it also contains their intersection. To make it into an ultrafilter it must be made as...
3. A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.
   • An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must...

See all 4 meanings of ultrafilter