Ultrafilter

An ultrafilter on a set is a maximal proper filter of subsets: it contains exactly one side of every two-part partition. It can be principal, concentrated at one element, or nonprincipal.

Filter (mathematics): A nonempty family of subsets closed under finite intersections and supersets, and not containing the empty set. An ultrafilter is precisely a maximal proper filter.
Maximal element: An element of a partially ordered set is maximal if no strictly larger element exists. Maximality among proper filters yields the decisive partition-selection property.
Ultraproduct: A construction that combines structures by identifying sequences equal on a set belonging to a chosen ultrafilter. The ultrafilter determines which coordinates count as holding almost everywhere.
Filter base: A family of sets whose finite intersections generate a filter. A base generates a filter but need not itself be maximal or satisfy every partition choice.
Proper filter: A filter that does not contain the empty set. Properness prevents an ultrafilter from selecting both sides of a partition.
Principal ultrafilter: The ultrafilter consisting of all subsets containing a fixed element. This is the simplest case: every choice is determined by one point.
Łoś's theorem: A theorem stating that a first-order formula holds in an ultraproduct exactly when it holds on an ultrafilter-large set of factors. It explains how logical truth passes through ultraproducts.
Prime ideal: An ideal in a ring whose quotient ring is an integral domain. In a Boolean algebra, ultrafilters correspond to complements of maximal ideals, a closely related dual description.
Boolean algebra: An algebraic structure with operations corresponding to conjunction, disjunction, and complement. Subsets form a Boolean algebra, where ultrafilters choose one of each element-complement pair.
Nonprincipal ultrafilter: An ultrafilter containing no finite set, when defined on an infinite set. It selects sides without being concentrated at any individual point.

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