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Ultrafilter

In , particularly in and , an ultrafilter on a set X is a maximal on the power set \mathcal{P}(X), consisting of a collection \mathcal{U} \subseteq \mathcal{P}(X) such that X \in \mathcal{U}, \emptyset \notin \mathcal{U}, \mathcal{U} is closed under finite intersections (if A, B \in \mathcal{U}, then A \cap B \in \mathcal{U}), upward closed (if A \in \mathcal{U} and A \subseteq B \subseteq X, then B \in \mathcal{U}), and for every A \subseteq X, exactly one of A \in \mathcal{U} or X \setminus A \in \mathcal{U} holds. Ultrafilters are classified into two principal types: principal ultrafilters, which are generated by a single element x \in X as \{A \subseteq X : x \in A\} and thus contain all sets containing x; and nonprincipal (or free) ultrafilters, which contain no finite sets and have empty total intersection \bigcap_{A \in \mathcal{U}} A = \emptyset. Principal ultrafilters exist on any nonempty set and are straightforward to construct, while nonprincipal ultrafilters require the axiom of choice and exist on every infinite set, as guaranteed by the ultrafilter lemma (or ultrafilter theorem), which states that every filter on X extends to an ultrafilter via Zorn's lemma. The existence of $2^{2^{|X|}} ultrafilters on a set X underscores their abundance, particularly on infinite sets like the natural numbers \mathbb{N}, where the space of ultrafilters \beta \mathbb{N} plays a key role in advanced constructions. The concept of ultrafilters originated in early 20th-century work on convergence in and Boolean algebras, with precursors in Frigyes Riesz's 1908 studies on accumulation points and formalization by in 1937 under the influence of the Bourbaki group; Alfred Tarski's 1930 prime ideal provided an equivalent formulation in . Beyond their foundational role in extending filters to maximal consistent families—effectively providing a "two-valued measure" on subsets—ultrafilters enable powerful constructions across mathematics, including ultraproducts in (preserving properties via Łoś's ), the Stone-Čech compactification in for embedding discrete spaces into compact Hausdorff spaces, and proofs of combinatorial s like via partition properties. They also facilitate for hyperreal numbers, algebraic embeddings (e.g., ultraproducts of finite fields yielding the complex numbers), and applications in , , and even for modeling decisive coalitions in voting.

Ultrafilters on partially ordered sets

Definition and basic properties

In a partially ordered set (poset) (P, \leq), a filter is a non-empty subset F \subseteq P such that:
  • It is upward closed: for all x \in F, the principal upset [x) = \{z \in P \mid x \leq z\} \subseteq F.
  • It is upward directed: for all x, y \in F, there exists z \in F such that x \leq z and y \leq z.
An ultrafilter on (P, \leq) is a maximal proper filter, meaning F is a proper filter (i.e., F \neq P) that is not properly contained in any other proper filter on P. Ultrafilters inherit the basic properties of , including properness (they do not contain a bottom element if one exists in the poset) and under finite meets (intersections of finitely many elements in F remain in F, provided the meets exist in the poset). These properties ensure that ultrafilters are nonempty, upward closed subsets that maintain consistency under finite operations while achieving maximality. The maximality of an ultrafilter implies that every proper on the poset can be extended to an ultrafilter under suitable conditions, such as when the poset admits the necessary order structure. Principal ultrafilters, which are generated by a single element a \in P as the upset [a), provide a simple example of this structure when the poset has such minimal generating elements.

Types and existence

Ultrafilters on a partially ordered set (P, \leq) are classified into two main types: principal and non-principal. A principal ultrafilter is one generated by a single element u \in P, defined as the collection \mathcal{U} = \{ a \in P \mid a \geq u \}. This set forms an ultrafilter precisely when it is maximal among proper filters, which occurs if u is such that no proper extension is possible without including elements incomparable or below u in a way that violates filter properties. In contrast, a non-principal ultrafilter lacks a single generating element and cannot be expressed as the upset of any individual u. Such ultrafilters exist on posets and extend filters like the Fréchet filter analog (upsets avoiding finite intersections), capturing "large" subsets without fixed points. An ultrafilter is characterized as a maximal proper in the poset of all proper filters on P, ordered by : it cannot be properly contained in any larger proper filter. Equivalently, for every x \in P \setminus \mathcal{U}, there exists y \in \mathcal{U} such that there is no z \in P with x \leq z and y \leq z. The existence of ultrafilters relies on from . To extend a given filter \mathcal{F} on P, consider the collection \mathcal{L} of all proper filters on P that properly contain \mathcal{F}, partially ordered by inclusion. This \mathcal{L} is nonempty since \mathcal{F} itself is in it (assuming \mathcal{F} is proper). For any \{ \mathcal{G}_\alpha \} in \mathcal{L}, the union \bigcup \mathcal{G}_\alpha is an upper bound: it is upward closed, closed under finite meets (as each \mathcal{G}_\alpha is), and proper (since if it contained a bottom element or violated properness, some \mathcal{G}_\alpha would). By , \mathcal{L} has a maximal element \mathcal{U}, which is an ultrafilter extending \mathcal{F}. This construction applies to any poset where filters are well-defined, ensuring every such filter extends to an ultrafilter. The concept of ultrafilter was introduced by in 1937, initially in the context of filter theory for studying in topological spaces, though its abstract formulation on posets emerged soon after in works on and s.

Ultrafilters on Boolean algebras

Definition and characterization

In a Boolean algebra B, an ultrafilter is a proper U \subseteq B that is maximal with respect to inclusion among all proper filters, or equivalently, a filter such that for every a \in B, exactly one of a or its complement \neg a belongs to U. This closure under complements ensures that U decides every element definitively, assigning it to either the "true" or "false" side without overlap or omission, leveraging the Boolean structure where every element has a unique complement. Ultrafilters on Boolean algebras admit several equivalent characterizations that highlight their structural role. One such is as prime filters: U is prime if for all a, b \in B, a \vee b \in U if and only if a \in U or b \in U. In Boolean algebras, every ultrafilter is prime, and conversely, every prime filter is an ultrafilter, distinguishing them from the general case in distributive lattices where the implication is one-way. Another characterization identifies ultrafilters with Boolean algebra homomorphisms \phi: B \to \{0,1\}, where \{0,1\} is the two-element Boolean algebra with the discrete order, and U = \phi^{-1}(1); this correspondence underscores ultrafilters as points in the Stone space of B, embedding the algebra into its dual representation. Key properties follow from these definitions. The intersection of all ultrafilters on B consists solely of the unit element $1_B, the principal filter generated by the top element, as every non-unit element can be mapped to 0 under some homomorphism. In non-atomic Boolean algebras, where no minimal positive elements exist, all ultrafilters are non-principal, meaning none fixates on a single generator. A representative example arises in the power set \mathcal{P}(S) for an S, ordered by . Here, non-principal ultrafilters extend the cofinite filter—the collection of all subsets with finite complement—and contain no finite sets, ensuring the intersection of all sets in such an ultrafilter is empty. These cofinite ultrafilters illustrate how the is typically invoked to extend the cofinite filter to a maximal one.

Relation to ideals and maximal filters

In a Boolean algebra B, there is a natural duality between ultrafilters and maximal ideals: the complement of an ultrafilter U \subseteq B (defined as \{ b \in B \mid b' \in U \}, where b' denotes the complement of b) forms a , and conversely, the complement of any in B is an ultrafilter. This duality arises because ultrafilters are precisely the maximal proper in B, meaning no larger proper filter properly contains them, while maximal ideals are the maximal proper downward-closed sets closed under finite joins. A key characterization follows from this duality: a filter F \subseteq B is an ultrafilter if and only if its dual ideal (the complement as defined above) is maximal. To see this, suppose F is an ultrafilter; if the dual ideal I were properly contained in a larger proper ideal J, then the complement of J would be a proper filter properly containing F, contradicting maximality of F. Conversely, if I is maximal, then F = B \setminus I is maximal among proper filters by a symmetric argument. Existence of such maximal elements relies on applied to the poset of proper ideals ordered by inclusion, ensuring every chain has an upper bound (their union, which remains proper), yielding a maximal ideal whose complement is an ultrafilter. This correspondence is central to Stone's representation theorem, which embeds B as the algebra of clopen sets in its S(B), the set of all ultrafilters on B equipped with the generated by basis sets U_a = \{ U \in S(B) \mid a \in U \} for a \in B. Here, each ultrafilter corresponds to a point in S(B), and the theorem establishes a isomorphism B \cong \{ U_a \mid a \in B \}, preserving the structure and highlighting ultrafilters as the "points" dual to elements of B. In complete Boolean algebras, ultrafilters exhibit additional preservation properties: they map arbitrary existing suprema to suprema in the , as the via clopen sets in the Stone space ensures that joins (when defined) correspond to unions of basis sets. For instance, if \bigvee S exists in B for S \subseteq B, then \bigvee \{ U_s \mid s \in S \} = U_{\bigvee S}. Regarding countability, in countable algebras (such as the free Boolean algebra on countably many generators), the set of non-principal ultrafilters has $2^{\aleph_0}, reflecting the many "free" choices consistent with the properties.

Ultrafilters on power sets

Construction from sets

An X is defined as a maximal proper on the power set \mathcal{P}(X) ordered by . Specifically, it is a collection \mathcal{U} \subseteq \mathcal{P}(X) such that X \in \mathcal{U}, \mathcal{U} is closed under finite intersections (if A, B \in \mathcal{U}, then A \cap B \in \mathcal{U}), and it is upward closed (if A \in \mathcal{U} and A \subseteq B \subseteq X, then B \in \mathcal{U}). This structure leverages the properties of \mathcal{P}(X), where ultrafilters correspond to homomorphisms to the two-element \{0,1\}. A key property is that for every subset A \subseteq X, exactly one of A or its complement X \setminus A belongs to \mathcal{U}; this ensures the ultrafilter "decides" membership for every of X, making it maximal among proper filters. The \emptyset is never in \mathcal{U}, as that would violate properness. Ultrafilters on X can be constructed by starting with a filter base—a collection of subsets with the —and extending it maximally to a , then to an ultrafilter using applied to the of proper filters ordered by inclusion. This extension is guaranteed by the ultrafilter lemma, which states that every proper on \mathcal{P}(X) is contained in some ultrafilter. On infinite sets, uniform ultrafilters arise in this construction, where all sets in \mathcal{U} have the same , often equal to |X|. A concrete example is the Fréchet filter on the natural numbers \mathbb{N}, consisting of all cofinite subsets (those with finite complement). This is a proper filter that satisfies the and can be extended to an ultrafilter on \mathcal{P}(\mathbb{N}) via the ultrafilter lemma.

Principal and non-principal ultrafilters

A principal ultrafilter on the power set \mathcal{P}(X) of a set X is one generated by a fixed x \in X, defined as U_x = \{ A \subseteq X \mid x \in A \}. This construction always exists for any x \in X and yields an ultrafilter because it satisfies the filter properties: X \in U_x, it is upward closed, closed under finite intersections, and maximal since for any A \subseteq X, either x \in A or x \in X \setminus A. On finite sets, all ultrafilters are principal. Principal ultrafilters are finitely generated by the \{x\} and thus countably generated. In contrast, a non-principal ultrafilter (also called ) on \mathcal{P}(X) for infinite X has no fixed point, meaning there is no x \in X such that \{x\} \in U; equivalently, U contains no finite sets. Such ultrafilters contain all cofinite subsets of X, extending the Fréchet filter of cofinite sets, and thus consist solely of infinite subsets. Their existence on infinite X requires the , via the ultrafilter lemma, which states that every proper filter on \mathcal{P}(X) extends to an ultrafilter using . Non-principal ultrafilters are uniform, meaning every set in U has the same as X. To construct a principal ultrafilter, simply select any x \in X and take U_x as defined above, which is the principal filter generated by the singleton \{x\}. For non-principal ultrafilters, one applies the to extend the Fréchet filter of cofinite sets to a maximal filter; alternatively, consider the product space \{0,1\}^X as a , where ultrafilters correspond to homomorphisms to \{0,1\}, and modulo the ideal of "null" sets (functions zero with respect to a choice-extended measure), yielding a non-principal ultrafilter on \mathcal{P}(X). On the natural numbers \mathbb{N}, non-principal ultrafilters can satisfy additional partition properties; for instance, a selective ultrafilter U on \mathbb{N} ensures that for any partition \{S_n : n \in \mathbb{N}\} of \mathbb{N} into sets not in U, there exists A \in U such that |A \cap S_n| \leq 1 for all n.

Advanced properties and constructions

Ultrafilter lemmas and extensions

The ultrafilter lemma asserts that every filter on a set can be extended to an ultrafilter. This result is a consequence of applied to the of all filters containing a given filter \mathcal{F}, ordered by . To prove it, consider the collection \mathfrak{U} of all filters on a set X that properly contain \mathcal{F}. This poset is nonempty since \mathcal{F} \in \mathfrak{U}. For any chain \mathcal{C} \subseteq \mathfrak{U}, the union \bigcup \mathcal{C} forms a filter finer than \mathcal{F}, serving as an upper bound in \mathfrak{U}. By , \mathfrak{U} has a maximal element \mathcal{U}, which contains \mathcal{F}. To show \mathcal{U} is an ultrafilter, suppose there exists Y \subseteq X such that neither \{Z \cap Y : Z \in \mathcal{U}\} nor \{Z \cap (X \setminus Y) : Z \in \mathcal{U}\} generates a filter. Then finite intersections from these would yield the empty set in \mathcal{U}, contradicting that \mathcal{U} is a filter. Thus, for every Y, exactly one of these generates a filter finer than \mathcal{U}, implying \mathcal{U} is maximal and hence an ultrafilter. Extensions of filters include refinements, where a filter \mathcal{G} refines \mathcal{F} if \mathcal{F} \subseteq \mathcal{G}, as guaranteed by the ultrafilter lemma for maximal refinements. Principalization refers to extending a filter generated by a singleton \{x\} to the principal ultrafilter consisting of all subsets containing x. Ultrapowers provide a key construction using ultrafilters. Given a structure \mathcal{A} and an ultrafilter \mathcal{U} on an index set I, the ultrapower \mathcal{A}^\mathcal{U} has universe consisting of equivalence classes of functions f : I \to A (where A is the universe of \mathcal{A}), with f \sim g if \{i \in I : f(i) = g(i)\} \in \mathcal{U}. Operations and relations are defined pointwise, so + = [i \mapsto f(i) + g(i)], and a formula \phi([f_1], \dots, [f_n]) holds in \mathcal{A}^\mathcal{U} if and only if \{i : \mathcal{A} \models \phi(f_1(i), \dots, f_n(i))\} \in \mathcal{U} (Łoś's theorem). This yields an elementary embedding of \mathcal{A} into \mathcal{A}^\mathcal{U}, useful for nonstandard models. In Boolean-valued models, ultrafilters play a role in specializing models over a complete Boolean algebra B. For a full B-valued model (M, \|\cdot\|), an ultrafilter U on B induces a classical model M/U via the quotient map, where \phi holds in M/U if \|\phi\| \in U. This specialization preserves satisfaction for formulas, linking ultrafilter regularity to saturation properties of the resulting model. Ultrafilters on finite sets are always principal, generated by a singleton, since any filter must decide membership for each element, leading to a fixed point. On infinite sets, non-principal ultrafilters exist assuming the axiom of choice, as extensions of the Fréchet filter of cofinite sets via the ultrafilter lemma.

Stone-Čech compactification

The Stone-Čech compactification of a discrete space X, denoted \beta X, was independently introduced by Eduard Čech and Marshall Stone in 1937 as a universal compactification that embeds X densely into a compact Hausdorff space while preserving continuous extensions. Čech's construction emphasized bicompact (now compact Hausdorff) extensions for completely regular spaces, while Stone's approach leveraged Boolean algebra duality to represent topological spaces via rings of continuous functions. The construction identifies \beta X with the set of all ultrafilters on the power set of X, where each point x \in X corresponds to the principal ultrafilter \mathbf{u}_x = \{A \subseteq X \mid x \in A\}. The topology on \beta X is generated by the basis \{U_A \mid A \subseteq X\}, where U_A = \{\mathbf{p} \in \beta X \mid A \in \mathbf{p}\}, making X homeomorphic to the set of principal ultrafilters and dense in \beta X. The points in \beta X \setminus X correspond precisely to the free (non-principal) ultrafilters on X, which can be viewed as "points at infinity." \beta X is a compact Hausdorff space, and its universal property ensures that every bounded continuous function f: X \to \mathbb{R} extends uniquely to a continuous function \tilde{f}: \beta X \to \mathbb{R}; more generally, for any compact Y, every continuous f: X \to Y extends uniquely to \tilde{f}: \beta X \to Y. This extension property arises from the , where \beta X is the of the [C^*-algebra](/page/C*-algebra) C_b(X) of bounded continuous real-valued functions on X, with the Gelfand transform providing the C_b(X) \cong C(\beta X). A key convergence theorem states that a net (x_\alpha) in X converges to an ultrafilter \mathbf{p} \in \beta X if and only if, for every A \in \mathbf{p}, the net is eventually in A; thus, ultrafilters in \beta X converge precisely when their adherent sets (limit points) coincide under this filter convergence.

Applications

In topology and analysis

In topology, ultrafilters provide a generalization of nets for defining convergence in arbitrary topological spaces. Specifically, an ultrafilter \mathcal{U} on a set X converges to a point x \in X in a topological space (X, \tau) if every open neighborhood U of x belongs to \mathcal{U}. This notion captures the limit points of the space more comprehensively than sequences, as every net in X generates a filter whose adherent ultrafilters determine the limit points of the net. A key result is that a space X is compact if and only if every ultrafilter on X converges to at least one point in X. Furthermore, X is Hausdorff if and only if every convergent ultrafilter has a unique limit. Ultrafilters play a crucial role in proving , which states that the product of any collection of compact topological spaces is compact. One standard proof proceeds by considering an ultrafilter \mathcal{U} on the and showing that the projected ultrafilter on each converges, thereby ensuring the existence of a limit point in the product space via the ultrafilter lemma. This approach highlights the equivalence between for Hausdorff spaces and the ultrafilter lemma in ZF set theory. In , non-principal ultrafilters enable the construction of generalized for bounded sequences, such as Banach limits on \ell^\infty. A Banach limit is a linear functional \phi: \ell^\infty \to \mathbb{R} that extends the usual on convergent sequences, satisfies \phi((x_n)) \geq 0 for non-negative sequences, and is shift-invariant: \phi((x_{n+1})) = \phi((x_n)). Such functionals arise as ultralimits along a free (non-principal) ultrafilter \mathcal{U} on \mathbb{N}, defined by \phi((x_n)) = \lim_{\mathcal{U}} x_n, where the exists for bounded sequences by of the product [m,M]^\mathbb{N} for m \leq x_n \leq M. Free ultrafilters ensure the is non-trivial and shift-invariant. Ultrafilters also underpin , where the ultrapower \mathbb{R}^\mathbb{N}/\mathcal{U} via a non-principal ultrafilter \mathcal{U} on \mathbb{N} yields the hyperreal numbers * \mathbb{R}, a non-Archimedean extension of \mathbb{R}. Elements are equivalence classes [(r_n)]_\mathcal{U} with r_n \in \mathbb{R}, ordered by [(r_n)] < [(s_n)] if \{n : r_n < s_n\} \in \mathcal{U}. Non-principal ultrafilters introduce infinitesimals (e.g., [(1/n)]_\mathcal{U} > 0 but smaller than any positive real) and infinite numbers, violating the : for any positive real r, there exists N \in * \mathbb{N} \setminus \mathbb{N} with N > r. This extension facilitates intuitive proofs in analysis, such as defining the integral of a f: [a,b] \to \mathbb{R} as \int_a^b f(x) \, dx = \mathrm{st} \left( \frac{b-a}{N} \sum_{k=1}^N f(a + k \cdot \frac{b-a}{N}) \right), where N \in * \mathbb{N} \setminus \mathbb{N} is infinite, \mathrm{st} is the standard part map, and the sum approximates the Riemann integral via transfer principle. A notable example in is Solovay's model of ZF + , constructed via forcing from a model with an , where every set of reals is Lebesgue measurable (and has the Baire property). This consistency result implies the failure of the ultrafilter lemma, as the existence of a non-principal ultrafilter on \mathbb{N} would yield a non-Lebesgue measurable set via the associated Banach limit or Vitali-type construction; thus, no such "measurable" ultrafilters (in the sense compatible with universal Lebesgue measurability) exist in the model. In the topology of the Stone-Čech compactification \beta \mathbb{N} \setminus \mathbb{N}, P-points are non-principal ultrafilters \mathcal{U} on \mathbb{N} such that for any countable collection \{A_n\} \subseteq \mathcal{U}, there exists B \in \mathcal{U} with B \subseteq^* A_n (finite difference) for all n. These points are "selective" in the growth of sets and exist under the , playing a role in characterizing selective ultrafilters and applications to partition calculus in topological dynamics.

In logic and model theory

In model theory, ultrafilters play a central role in the construction of , which allow for the creation of new structures that preserve properties from a family of given structures. Given a family of structures \{M_i \mid i \in I\} in a \mathcal{L} and an ultrafilter \mathcal{U} on the I, the \prod_{i \in I} M_i / \mathcal{U} is formed by taking the \prod_{i \in I} M_i and quotienting by the where two elements (a_i)_{i \in I} and (b_i)_{i \in I} are equivalent if \{i \in I \mid a_i = b_i\} \in \mathcal{U}. The operations and relations on the ultraproduct are defined componentwise, respecting the ultrafilter. This construction, introduced by Jerzy Łoś, enables the transfer of logical properties across models. A fundamental result is Łoś's theorem, which states that for any first-order \phi(\bar{x}) and (a_i)_{i \in I} in the product, the satisfies \prod M_i / \mathcal{U} \models \phi[(a_i)/\mathcal{U}] if and only if \{i \in I \mid M_i \models \phi(\bar{a}_i)\} \in \mathcal{U}. This theorem implies that preserve first-order : the of models of a is again a model of that . Consequently, are instrumental in constructing non-standard models; for instance, taking countably many copies of the natural numbers \mathbb{N} and forming the \mathbb{N}^\omega / \mathcal{U} with respect to a non-principal ultrafilter \mathcal{U} on \omega yields a non-standard model of Peano containing infinite "natural numbers" that satisfy the axioms but extend beyond the standard integers. Such models are elementarily equivalent to \mathbb{N} yet exhibit non-standard elements, facilitating the study of properties in a broader context. Ultraproducts also aid in realizing saturated models, which are useful for embedding and homogeneity properties in model theory. Specifically, for a countable structure M, the ultrapower M^\omega / \mathcal{U} with respect to a non-principal ultrafilter \mathcal{U} on \omega is countably saturated, meaning it realizes every consistent type over countable parameter sets. This saturation ensures that the model is "rich" enough to embed smaller models while preserving first-order properties, a key tool in classification and stability theory. Beyond , ultrafilters appear in broader logical contexts, such as set-theoretic forcing, where a generic filter for a forcing poset \mathbb{P} extends to a generic ultrafilter on the complete associated with \mathbb{P}, defining the generic extension V[G] that adds new sets while preserving axioms like ZFC. In , the ultrafilter lemma is equivalent to the for propositional logic and can be used in some proofs of compactness for , though the latter follows from , which is provable in ZF and establishes the equivalence between syntactic provability and semantic validity. A prominent application is Abraham Robinson's development of non-standard analysis, where the hyperreal numbers \mathbb{R}^* are constructed as the ultraproduct \mathbb{R}^\mathbb{N} / \mathcal{U} over a free ultrafilter \mathcal{U} on \mathbb{N}. This field extends the reals with infinitesimals and infinite numbers, allowing rigorous formulations of intuitive infinitesimal arguments in and while preserving properties of the reals.

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