Ultrafilter

This article is about the mathematical concept. For the physical device, see ultrafiltration.

The powerset lattice of the set {1,2,3,4}, with the upper set ↑{1,4} colored yellow. It is a principal filter, but not an ultrafilter, as it can be extended to the larger nontrivial filter ↑{1}, by including also the light green elements. Since ↑{1} cannot be extended any further, it is an ultrafilter.

In the mathematical field of set theory, an ultrafilter on a given partially ordered set (poset) P is a maximal filter on P, that is, a filter on P that cannot be enlarged. Filters and ultrafilters are special subsets of P. If P happens to be a Boolean algebra, each ultrafilter is also a prime filter, and vice versa.^[1]^:186

If X is an arbitrary set, its power set ℘(X), ordered by set inclusion, is always a Boolean algebra, and (ultra)filters on ℘(X) are usually called "(ultra)filters on X".^{[note 1]}Ultrafilters have many applications in set theory, model theory, and topology.^[1]^:186 An ultrafilter on a set X may be considered as a finitely additive measure. In this view, every subset of X is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0).

Ultrafilters on partial orders

In order theory, an ultrafilter is a subset of a partially ordered set that is maximal among all proper filters. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.

Formally, if P is a set, partially ordered by (≤), then

a subset F of P is called a filter on P if
- F is nonempty,
- for every x, y in F, there is some element z in F such that z ≤ x and z ≤ y, and
- for every x in F and y in P, x ≤ y implies that y is in F, too;
a subset U of P is called a ultrafilter on P if
- U is a filter on P, and
- there is no filter F on P such that U ⊊ F ⊊ P.

Special case: Boolean algebra

An important special case of the concept occurs if the considered poset is a Boolean algebra. 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):

If P is a Boolean algebra and F ⊊ P is a proper filter, then the following statements are equivalent:

F is an ultrafilter on P,
F is a prime filter on P,
for each a in P, either a is in F or (¬a) is in F.^[1]^:186

A proof of 1.⇔2. is also given in (Burris, Sankappanavar, 2012, Cor.3.13, p.133)^[2]

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

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".

Special case: ultrafilter on the powerset of a set

Given an arbitrary set X, its power set ℘(X), ordered by set inclusion, is always a Boolean algebra; hence the results of the above section #Special case: Boolean algebra apply. An (ultra)filter on ℘(X) is often called just an "(ultra)filter on X".^{[note 1]} The above formal definitions can be particularized to the powerset case as follows:

Given an arbitrary set X, an ultrafilter on ℘(X) is a set U consisting of subsets of X such that:

The empty set is not an element of U.
If A and B are subsets of X, the set A is a subset of B, and A is an element of U, then B is also an element of U.
If A and B are elements of U, then so is the intersection of A and B.
If A is a subset of X, then either^{[note 2]} A or its relative complement X \ A is an element of U.

A characterization is given by the following theorem. A filter U on ℘(X) is an ultrafilter if any of the following conditions are true:

There is no filter F strictly finer than U, that is, U ⊆ F implies U = F.
If a union A∪B is in U, then A is in U or B is.
For each subset A of X, either^{[note 2]} A is in U or (X \ A) is.

Another way of looking at ultrafilters on a power set ℘(X) is to define a function m on ℘(X) by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Such a function is called a 2-valued morphism. Then m is finitely additive, and hence a content on ℘(X), and every property of elements of X is either true almost everywhere or false almost everywhere. However, m is usually not countably additive, and hence does not define a measure in the usual sense.

For a filter F that is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.

Completeness

The completeness of an ultrafilter U on a powerset is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition implies that the completeness of any powerset ultrafilter is at least $\aleph _{0}$ . An ultrafilter whose completeness is greater than $\aleph _{0}$ —that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or σ-complete.

The completeness of a countably complete nonprincipal ultrafilter on a powerset is always a measurable cardinal.

Types and existence of ultrafilters

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form F_a = {x | a ≤ x} for some (but not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.

For ultrafilters on a powerset ℘(S), a principal ultrafilter consists of all subsets of S that contain a given element s of S. Each ultrafilter on ℘(S) that is also a principal filter is of this form.^[1]^:187 Therefore, an ultrafilter U on ℘(S) is principal if and only if it contains a finite set.^{[note 3]} If S is infinite, an ultrafilter U on ℘(S) is hence non-principal if and only if it contains the Fréchet filter of cofinite subsets of S.^{[note 4]} If S is finite, each ultrafilter is principal.^[1]^:187

One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn's Lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.

Applications

Ultrafilters on powersets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in Stone's representation theorem.

The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element a of P, let D_a = {U ∈ G | a ∈ U}. This is most useful when P is again a Boolean algebra, since in this situation the set of all D_a is a base for a compact Hausdorff topology on G. Especially, when considering the ultrafilters on a powerset ℘(S), the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality |S|.

The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. For example, in constructing hyperreal numbers as an ultraproduct of the real numbers, the domain of discourse is extended from real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead the functions and relations are defined "pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If U is nonprincipal, then the extension thereby obtained is nontrivial.

In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This construction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces.

Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.

In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (e.g., Kirman and Sondermann, 1972^[3]). Mihara (1997,^[4] 1999^[5]) shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.

Ordering on ultrafilters

The Rudin–Keisler ordering is a preorder on the class of powerset ultrafilters defined as follows: if U is an ultrafilter on ℘(X), and V an ultrafilter on ℘(Y), then V ≤_RK U if there exists a function f: X → Y such that

C ∈ V ⇔ f^-1[C] ∈ U

for every subset C of Y.

Ultrafilters U and V are called Rudin–Keisler equivalent, denoted U ≡_RK V, if there exist sets A ∈ U and B ∈ V, and a bijection f: A → B that satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified by fixing A = X, B = Y.)

It is known that ≡_RK is the kernel of ≤_RK, i.e., that U ≡_RK V if and only if U ≤_RK V and V ≤_RK U.

Ultrafilters on ℘(ω)

There are several special properties that an ultrafilter on ℘(ω) may possess, which prove useful in various areas of set theory and topology.

A non-principal ultrafilter U is called a P-point (or weakly selective) if for every partition { C_n | n<ω } of ω such that ∀n<ω: C_n ∉ U, there exists some A ∈ U such that A ∩ C_n is a finite set for each n.
A non-principal ultrafilter U is called Ramsey (or selective) if for every partition { C_n | n<ω } of ω such that ∀n<ω: C_n ∉ U, there exists some A ∈ U such that A ∩ C_n is a singleton set for each n.

It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters.^[6] In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.^[7] Therefore, the existence of these types of ultrafilters is independent of ZFC.

P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of [ω]² there exists an element of the ultrafilter that has a homogeneous color.

An ultrafilter on ℘(ω) is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters.

Notes

1 2 If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on ℘(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter" of a partial ordered set" vs. "on an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of ℘(X)".
1 2 Properties 1 and 3 imply that A and X \ A cannot both be elements of U.
↑ To see the "if" direction: If {s₁,...,s_n} ∈ U, then {s₁} ∈ U, or ..., or {s_n} ∈ U by induction on n, using Nr.2 of the above characterization theorem. That is, some {s_i} is the principal element of U.
↑ U is non-principal iff it contains no finite set, i.e. (by Nr.3 of the above characterization theorem) iff it contains every cofinite set, i.e. every member of the Fréchet filter.

References

1 2 3 4 5 B.A. Davey and H.A. Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
↑ Stanley N. Burris and H.P. Sankappanavar (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
↑ Kirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators". Journal of Economic Theory. 5: 267. doi:10.1016/0022-0531(72)90106-8.
↑ Mihara, H. R. (1997). "Arrow's Theorem and Turing computability" (PDF). Economic Theory. 10 (2): 257–276. doi:10.1007/s001990050157Reprinted in K. V. Velupillai , S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.
↑ Mihara, H. R. (1999). "Arrow's theorem, countably many agents, and more visible invisible dictators". Journal of Mathematical Economics. 32: 267–277. doi:10.1016/S0304-4068(98)00061-5.
↑ Rudin, Walter (1956), "Homogeneity problems in the theory of Čech compactifications", Duke Mathematical Journal, 23 (3): 409–419, doi:10.1215/S0012-7094-56-02337-7
↑ Wimmers, Edward (March 1982), "The Shelah P-point independence theorem", Israel Journal of Mathematics, Hebrew University Magnes Press, 43 (1): 28–48, doi:10.1007/BF02761683