Editing Boolean algebra (structure) (section)

== Ideals and filters ==
{{Main|Ideal (order theory)|Filter (mathematics)}}
An ''ideal'' of the Boolean algebra {{mvar|A}} is a nonempty subset {{mvar|I}} such that for all {{mvar|x}}, {{mvar|y}} in {{mvar|I}} we have {{math|{{var|x}} ∨ {{var|y}}}} in {{mvar|I}} and for all {{mvar|a}} in {{mvar|A}} we have {{math|{{var|a}} ∧ {{var|x}}}} in {{mvar|I}}. This notion of ideal coincides with the notion of [[ring ideal]] in the Boolean ring {{mvar|A}}. An ideal {{mvar|I}} of {{mvar|A}} is called ''prime'' if {{math|{{var|I}} ≠ {{var|A}}}} and if {{math|{{var|a}} ∧ {{var|b}}}} in {{mvar|I}} always implies {{mvar|a}} in {{mvar|I}} or {{mvar|b}} in {{mvar|I}}. Furthermore, for every {{math|{{var|a}} ∈ {{var|A}}}} we have that {{math|{{var|a}} ∧ &minus;{{var|a}} {{=}} 0 ∈ {{var|I}}}}, and then if {{mvar|I}} is prime we have {{math|{{var|a}} ∈ {{var|I}}}} or {{math|&minus;{{var|a}} ∈ {{var|I}}}} for every {{math|{{var|a}} ∈ {{var|A}}}}. An ideal {{mvar|I}} of {{mvar|A}} is called ''maximal'' if {{math|{{var|I}} ≠ {{var|A}}}} and if the only ideal properly containing {{mvar|I}} is {{mvar|A}} itself. For an ideal {{mvar|I}}, if {{math|{{var|a}} ∉ {{var|I}}}} and {{math|&minus;{{var|a}} ∉ {{var|I}}}}, then {{math|{{var|I}} ∪ {{mset|{{var|a}}}}}} or {{math|{{var|I}} ∪ {{mset|&minus;{{var|a}}}}}} is contained in another proper ideal {{mvar|J}}. Hence, such an {{mvar|I}} is not maximal, and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of [[prime ideal]] and [[maximal ideal]] in the Boolean ring {{mvar|A}}.

The dual of an ''ideal'' is a ''filter''. A ''filter'' of the Boolean algebra {{mvar|A}} is a nonempty subset {{mvar|p}} such that for all {{mvar|x}}, {{mvar|y}} in {{mvar|p}} we have {{math|{{var|x}} ∧ {{var|y}}}} in {{mvar|p}} and for all {{mvar|a}} in {{mvar|A}} we have {{math|{{var|a}} ∨ {{var|x}}}} in {{mvar|p}}. The dual of a ''maximal'' (or ''prime'') ''ideal'' in a Boolean algebra is ''[[ultrafilter]]''. Ultrafilters can alternatively be described as [[2-valued morphism]]s from {{mvar|A}} to the two-element Boolean algebra. The statement ''every filter in a Boolean algebra can be extended to an ultrafilter'' is called the ''[[Boolean prime ideal theorem#The ultrafilter lemma|ultrafilter lemma]]'' and cannot be proven in [[Zermelo–Fraenkel set theory]] (ZF), if [[Zermelo–Fraenkel set theory|ZF]] is [[consistent]]. Within ZF, the ultrafilter lemma is strictly weaker than the [[axiom of choice]].
The ultrafilter lemma has many equivalent formulations: ''every Boolean algebra has an ultrafilter'', ''every ideal in a Boolean algebra can be extended to a prime ideal'', etc.