Editing Ultrafilter (section)

==Ultrafilter on a Boolean algebra==

An important special case of the concept occurs if the considered poset is a [[Boolean algebra (structure)|Boolean algebra]].  In this case, ultrafilters are characterized by containing, for each element <math>x</math> of the Boolean algebra, exactly one of the elements <math>x</math> and <math>\lnot x</math> (the latter being the [[Boolean algebra#Nonmonotone laws|Boolean complement]] of <math>x</math>):

If <math display="inline">P</math> is a Boolean algebra and <math>F</math> is a proper filter on <math>P,</math> then the following statements are equivalent:
# <math>F</math> is an ultrafilter on <math>P,</math>
# <math>F</math> is a [[prime filter]] on <math>P,</math>
# for each <math>x \in P,</math> either <math>x \in F</math> or (<math>\lnot x</math>) <math>\in F.</math><ref name="Davey.Priestley.1990"/>{{rp|186}}
A proof that 1. and 2. are equivalent is also given in (Burris, Sankappanavar, 2012, Corollary 3.13, p.133).<ref name="Burris.Sankappanavar.2012">{{cite book|url= http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf|isbn=978-0-9880552-0-9|first1= Stanley N.|last1= Burris|first2= H. P.|last2= Sankappanavar|title=A Course in Universal Algebra|year=2012 |publisher=S. Burris and H.P. Sankappanavar }}</ref>

Moreover, ultrafilters on a Boolean algebra can be related to [[Boolean algebra (structure)#Ideals and filters|maximal ideals]] and [[Boolean algebra (structure)#Homomorphisms and isomorphisms|homomorphisms]] to the 2-element Boolean algebra {true, false} (also known as [[2-valued morphism]]s) as follows:
* Given a homomorphism of a Boolean algebra onto {true, false}, the [[inverse image]] of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.
* Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
* Given an ultrafilter on a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".{{cn|date=July 2016}}