Editing Ultrafilter (section)

{{short description|Maximal proper filter}}
{{About|the mathematical concept in [[order theory]]|ultrafilters on [[Set (mathematics)|sets]] specifically|Ultrafilter (set theory)|the physical device|ultrafiltration}}
[[File:Filter vs ultrafilter_210div.svg|thumb|[[Hasse diagram]] of the [[divisor]]s of 210, ordered by the relation ''is divisor of'', with the [[upper set]] ↑14 colored dark green. It is a {{em|principal filter}}, but not an {{em|ultrafilter}}, as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements. Since ↑2 cannot be extended any further, it is an ultrafilter.]]
In the [[Mathematics|mathematical]] field of [[order theory]], an '''ultrafilter''' on a given [[partially ordered set]] (or "poset") <math display="inline">P</math> is a certain subset of <math>P,</math> namely a [[Maximal element|maximal]] [[Filter (mathematics)|filter]] on <math>P;</math> that is, a [[proper filter]] on <math display="inline">P</math> that cannot be enlarged to a bigger proper filter on <math>P.</math>

If <math>X</math> is an arbitrary set, its [[power set]] <math>{\mathcal P}(X),</math> ordered by [[set inclusion]], is always a [[Boolean algebra (structure)|Boolean algebra]] and hence a poset, and ultrafilters on <math>{\mathcal P}(X)</math> are usually called {{em|[[Ultrafilter (set theory)|ultrafilters on the set]]}} <math>X</math>.<ref name="notation warning" group="note">If <math>X</math> happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on <math>{\mathcal P}(X)</math> or an (ultra)filter just on <math>X</math> is meant; both kinds of (ultra)filters are quite different. Some authors{{cn|date=July 2016}} use "(ultra)filter ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on <math>X</math>" to abbreviate "(ultra)filter of <math>{\mathcal P}(X)</math>".<!---guessed from a sentence in section "Types and existence of ultrafilters" which is now commented-out---></ref> An ultrafilter on a set <math>X</math> may be considered as a [[finitely additive]] 0-1-valued [[measure (mathematics)|measure]] on <math>{\mathcal P}(X)</math>. In this view, every subset of <math>X</math> is either considered "[[Almost everywhere|almost everything]]" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.{{r|Kruckman.2012|at=§4}}

Ultrafilters have many applications in set theory, [[model theory]], [[topology]]<ref name="Davey.Priestley.1990">{{cite book|first1= B. A.|last1= Davey|first2= H. A.|last2= Priestley|title= Introduction to Lattices and Order|title-link= Introduction to Lattices and Order|publisher= Cambridge University Press|year= 1990|series= Cambridge Mathematical Textbooks}}</ref>{{rp|186}} and combinatorics.<ref>{{Cite journal|last=Goldbring|first=Isaac|date=2021|others=Marta Maggioni, Sophia Jahns|title=Ultrafilter methods in combinatorics|url=http://publications.mfo.de/handle/mfo/3870|journal=Snapshots of Modern Mathematics from Oberwolfach |language=en|doi=10.14760/SNAP-2021-006-EN}}</ref>