Ultrafilter - Generalization To Partial Orders

Generalization To Partial Orders

In order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. Formally, this states that any filter that properly contains an ultrafilter has to be equal to the whole poset. An important special case of the concept occurs if the considered poset is a Boolean algebra, as in the case of an ultrafilter on a set (defined as a filter of the corresponding powerset). In this case, ultrafilters are characterized by containing, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the latter being the Boolean complement of a).

Ultrafilters on a Boolean algebra can be identified with prime ideals, maximal ideals, and homomorphisms to the 2-element Boolean algebra {true, false}, as follows:

• Maximal ideals of a Boolean algebra are the same as prime ideals.
• 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 of a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".

Let us see another theorem which could be used for the definition of the concept of “ultrafilter”. Let B denote a Boolean algebra and F a proper filter in it. F is an ultrafilter iff:

for all, if, then or

(To avoid confusion: the sign denotes the join operation of the Boolean algebra, and logical connectives are rendered by English circumlocutions.) See details (and proof) in.