Meaning of ultrafilter | Babel Free

Definitions

1. A device that performs ultrafiltration.
2. A proper filter which has a law of dichotomy for complements.
3. A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.

Equivalents

Examples

“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 large as possible without including the empty set⟳. That can be prevented by not allowing any pair of disjoint sets to be both included. If, given a pair of complementary subsets, one of them is prevented from being included, then all subsets of it should be prevented from being included as well, by the second rule⟳. That takes care of all subsets disjoint from the other complementary subset, which should then be included, in order⟳ to make⟳ the filter approach⟳ maximality, i.e., turn⟳ it into an ultrafilter.”

“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 contain⟳ all the elements of the poset.”

CEFR level