Ultrafilter

In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") $P$ is a certain subset of $P,$ namely a maximal filter on $P;$ that is, a proper filter on $P$ that cannot be enlarged to a bigger proper filter on $P.$

If $X$ is an arbitrary set, its power set $\wp (X),$ ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on $\wp (X)$ are usually called ultrafilter on the set $X$ .^{[note 1]} An ultrafilter on a set $X$ may be considered as a finitely additive measure on $X$ . In this view, every subset of $X$ is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.^{[citation needed]}

Ultrafilters have many applications in set theory, model theory, topology^[1]^: 186 and combinatorics.^[2]

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 $\,\leq \,$ then

a subset $F\subseteq P$ $F\subseteq P$ is called a filter on $P$ $P$ if
- $F$ is nonempty,
- for every $x,y\in F,$ there exists some element $z\in F$ such that $z\leq x$ and $z\leq y,$ and
- for every $x\in F$ and $y\in P,$ $x\leq y$ implies that $y$ is in $F$ too;
a proper subset $U$ $U$ of $P$ $P$ is called an ultrafilter on $P$ $P$ if
- $U$ is a filter on $P,$ and
- there is no proper filter $F$ on $P$ that properly extends $U$ (that is, such that $U$ is a proper subset of $F$ ).

Types and existence of ultrafilters

Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form $F_{a}=\{x:a\leq x\}$ 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 $\wp (X),$ a principal ultrafilter consists of all subsets of $X$ that contain a given element $x\in X.$ Each ultrafilter on $\wp (X)$ that is also a principal filter is of this form.^[1]^: 187 Therefore, an ultrafilter $U$ on $\wp (X)$ is principal if and only if it contains a finite set.^{[note 2]} If $X$ is infinite, an ultrafilter $U$ on $\wp (X)$ is hence non-principal if and only if it contains the Fréchet filter of cofinite subsets of $X.$ ^{[note 3]}^{[citation needed]} If $X$ is finite, every ultrafilter is principal.^[1]^: 187 If $X$ is infinite then the Fréchet filter is not an ultrafilter on the power set of $X$ but it is an ultrafilter on the finite–cofinite algebra of $X.$

Every filter on 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). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of ZFC; for example, Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.^[3]

Ultrafilter on a 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 $\lnot a$ (the latter being the Boolean complement of $a$ ):

If $P$ is a Boolean algebra and $F$ is a proper filter on $P,$ 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 $a\in F$ or ( $\lnot a$ ) $\in F.$ ^[1]^: 186

A proof that 1. and 2. are equivalent is also given in (Burris, Sankappanavar, 2012, Corollary 3.13, p.133).^[4]

Moreover, ultrafilters on a Boolean algebra can be related to maximal ideals and homomorphisms to the 2-element Boolean algebra {true, false} (also known as 2-valued morphisms) 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 on a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".^{[citation needed]}

Ultrafilter on the power set of a set

Given an arbitrary set $X,$ its power set $\wp (X),$ 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 $\wp (X)$ 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 $\wp (X)$ 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 4]} $A$ or its relative complement $X\setminus A$ is an element of $U.$

Another way of looking at ultrafilters on a power set $\wp (X)$ is as follows: for a given ultrafilter $U$ define a function $m$ on $\wp (X)$ 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 $\wp (X),$ 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 $A\in F$ and $m(A)=0$ if $X\setminus A\in F,$ leaving $m$ undefined elsewhere.^[5]

Applications

Ultrafilters on power sets 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 $D_{a}=\left\{U\in G:a\in U\right\}.$ 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 $\wp (S),$ 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 (for example, Kirman and Sondermann, 1972).^[6] Mihara (1997,^[7] 1999)^[8] shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.

Notes

^ ^a ^b If $X$ happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on $\wp (X)$ or an (ultra)filter just on $X$ is meant; both kinds of (ultra)filters are quite different. Some authors^{[citation needed]} 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 $\wp (X)$ ".
^ To see the "if" direction: If $\left\{x_{1},\ldots ,x_{n}\right\}\in U,$ then $\left\{x_{1}\right\}\in U,{\text{ or }}\ldots {\text{ or }}\left\{s_{n}\right\}\in U,$ by induction on $n,$ using Nr.2 of the above characterization theorem. That is, some $\left\{x_{i}\right\}$ is the principal element of $U.$
^ $U$ is non-principal if and only if it contains no finite set, that is, (by Nr.3 of the above characterization theorem) if and only if it contains every cofinite set, that is, every member of the Fréchet filter.
^ Properties 1 and 3 imply that $A$ and $X\setminus A$ cannot both be elements of $U.$

References

^ ^a ^b ^c ^d Davey, B. A.; Priestley, H. A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
^ Goldbring, Isaac (2021). "Ultrafilter methods in combinatorics". Snapshots of Modern Mathematics from Oberwolfach. Marta Maggioni, Sophia Jahns. doi:10.14760/SNAP-2021-006-EN.
^ Halbeisen, L. J. (2012). Combinatorial Set Theory. Springer Monographs in Mathematics. Springer.
^ Burris, Stanley N.; Sankappanavar, H. P. (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
^ "Notes on Ultrafilters" (PDF).
^ Kirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators". Journal of Economic Theory. 5 (2): 267–277. 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. CiteSeerX 10.1.1.200.520. doi:10.1007/s001990050157. S2CID 15398169. Archived from the original (PDF) on 2011-08-12Reprinted in K. V. Velupillai, S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.{{cite journal}}: CS1 maint: postscript (link)
^ Mihara, H. R. (1999). "Arrow's theorem, countably many agents, and more visible invisible dictators". Journal of Mathematical Economics. 32 (3): 267–277. CiteSeerX 10.1.1.199.1970. doi:10.1016/S0304-4068(98)00061-5.

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