In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words, it is a collection of subsets of X {\displaystyle X} that satisfies the definition of a filter on X {\displaystyle X} and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X {\displaystyle X} that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X {\displaystyle X} can also be characterized as a filter on X {\displaystyle X} with the property that for every subset A {\displaystyle A} of X {\displaystyle X} either A {\displaystyle A} or its complement X ∖ A {\displaystyle X\setminus A} belongs to the ultrafilter. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set P ( X ) {\displaystyle {\mathcal {P}}(X)} and the partial order is subset inclusion ⊆ . {\displaystyle \,\subseteq .} This article deals specifically with ultrafilters on a set and does not cover the more general notion. There are two types of ultrafilter on a set. A principal ultrafilter on X {\displaystyle X} is the collection of all subsets of X {\displaystyle X} that contain a fixed element x ∈ X {\displaystyle x\in X} . The ultrafilters that are not principal are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exist models of ZF where every ultrafilter on a set is principal. Ultrafilters have many applications in set theory, model theory, and topology. Usually, only free ultrafilters lead to non-trivial constructions. For example, an ultraproduct modulo a principal ultrafilter is always isomorphic to one of the factors, while an ultraproduct modulo a free ultrafilter usually has a more complex structure.
Definitions
Given an arbitrary set X , {\displaystyle X,} an ultrafilter on X {\displaystyle X} is a non-empty family U {\displaystyle U} of subsets of X {\displaystyle X} such that:
Proper or non-degenerate: The empty set is not an element of U . {\displaystyle U.}
Upward closed in X {\displaystyle X} : If A ∈ U {\displaystyle A\in U} and if B ⊆ X {\displaystyle B\subseteq X} is any superset of A {\displaystyle A} (that is, if A ⊆ B ⊆ X {\displaystyle A\subseteq B\subseteq X} ) then B ∈ U . {\displaystyle B\in U.}
π−system: If A {\displaystyle A} and B {\displaystyle B} are elements of U {\displaystyle U} then so is their intersection A ∩ B . {\displaystyle A\cap B.}
If A ⊆ X {\displaystyle A\subseteq X} then either A {\displaystyle A} or its complement X ∖ A {\displaystyle X\setminus A} is an element of U . {\displaystyle U.}
Properties (1), (2), and (3) are the defining properties of a filter on X . {\displaystyle X.} Some authors do not include non-degeneracy (which is property (1) above) in their definition of "filter". However, the definition of "ultrafilter" (and also of "prefilter" and "filter subbase") always includes non-degeneracy as a defining condition. This article requires that all filters be proper although a filter might be described as "proper" for emphasis. A filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter subbase is a non-empty family of sets that is contained in some (proper) filter. The smallest (relative to ⊆ {\displaystyle \subseteq } ) filter containing a given filter subbase is said to be generated by the filter subbase. The upward closure in X {\displaystyle X} of a family of sets P {\displaystyle P} is the set
P ↑ X := { S : A ⊆ S ⊆ X for some A ∈ P } . {\displaystyle P^{\uparrow X}:=\{S:A\subseteq S\subseteq X{\text{ for some }}A\in P\}.}
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