Editing Boolean prime ideal theorem (section)

==Prime ideal theorems==
An [[Ideal (order theory)|order ideal]] is a (non-empty) [[Directed set|directed]] [[lower set]]. If the considered [[partially ordered set]] (poset) has binary [[Supremum|suprema]] (a.k.a. [[Join and meet|joins]]), as do the posets within this article, then this is equivalently characterized as a non-empty lower set {{mvar|I}} that is closed for binary suprema (that is, <math>x, y \in I</math> implies <math>x \vee y \in I</math>). An ideal {{mvar|I}} is prime if its set-theoretic complement in the poset is a [[Filter (mathematics)|filter]] (that is, <math>x \wedge y \in I</math> implies <math>x \in I</math> or <math>y \in I</math>). Ideals are proper if they are not equal to the whole poset.

Historically, the first statement relating to later prime ideal theorems was in fact referring to filters—subsets that are ideals with respect to the [[Duality (order theory)|dual]] order. The ultrafilter lemma states that every filter on a set is contained within some maximal (proper) filter—an ''ultrafilter''. Recall that filters on sets are proper filters of the Boolean algebra of its [[powerset]]. In this special case, maximal filters (i.e. filters that are not strict subsets of any proper filter) and prime filters (i.e. filters that with each union of subsets ''X'' and ''Y'' contain also ''X'' or ''Y'') coincide. The dual of this statement thus assures that every ideal of a powerset is contained in a prime ideal.

The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form. ''Weak prime ideal theorems'' state that every ''non-trivial'' algebra of a certain class has at least one prime ideal. In contrast, ''strong prime ideal theorems'' require that every ideal that is disjoint from a given filter can be extended to a prime ideal that is still disjoint from that filter. In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras (BPI) is valid.

Another variation of similar theorems is obtained by replacing each occurrence of ''prime ideal'' by ''maximal ideal''. The corresponding maximal ideal theorems (MIT) are often—though not always—stronger than their PIT equivalents.