Theultraproduct is amathematical construction that appears mainly inabstract algebra andmathematical logic, in particular inmodel theory andset theory. An ultraproduct is thequotient set of thedirect product of a family ofstructures. All factors need to have the samesignature. Theultrapower is the special case of this construction in which all factors are equal.

For example, ultrapowers can be used to construct newfields from given ones. Thehyperreal numbers, an ultrapower of thereal numbers, are a special case of this.

Some striking applications of ultraproducts include very elegant proofs of thecompactness theorem and thecompleteness theorem,Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area ofnonstandard analysis, which was pioneered (as an application of the compactness theorem) byAbraham Robinson.

The general method for getting ultraproducts uses an index set${\displaystyle I,}$ astructure${\displaystyle M_i}$ (assumed to be non-empty in this article) for each element${\displaystyle i\in I}$ (all of the samesignature), and anultrafilter${\displaystyle \mathcal{U}}$ on${\displaystyle I.}$

For any two elements${\displaystyle a_{\bullet }={\left(a_i\right)}_{i\in I}}$ and${\displaystyle b_{\bullet }={\left(b_i\right)}_{i\in I}}$ of theCartesian product$\prod _{i\in I}{M}_i,$ declare them to be${\displaystyle \mathcal{U}}$-equivalent, written${\displaystyle a_{\bullet }\sim b_{\bullet }}$ or${\displaystyle a_{\bullet }{=}_{\mathcal{U}}{b}_{\bullet },}$ if and only if the set of indices${\displaystyle \left\{i\in I:a_i=b_i\right\}}$ on which they agree is an element of${\displaystyle \mathcal{U};}$ in symbols,$${\displaystyle a_{\bullet }\sim b_{\bullet }⟺\left\{i\in I:a_i=b_i\right\}\in \mathcal{U},}$$which compares components only relative to the ultrafilter${\displaystyle \mathcal{U}.}$ Thisbinary relation${\displaystyle \sim }$ is anequivalence relation^{[proof 1]} on the Cartesian product${\displaystyle \prod _{i\in I}{M}_i.}$

Theultraproduct of${\displaystyle M_{\bullet }={\left(M_i\right)}_{i\in I}}$ modulo${\displaystyle \mathcal{U}}$ is thequotient set of${\displaystyle \prod _{i\in I}{M}_i}$ with respect to${\displaystyle \sim }$ and is therefore sometimes denoted by${\displaystyle \prod _{i\in I}{M}_i/\mathcal{U}}$ or${\displaystyle {\prod }_{\mathcal{U}}{M}_{\bullet }.}$

Explicitly, if the${\displaystyle \mathcal{U}}$-equivalence class of an element${\displaystyle a\in \prod _{i\in I}{M}_i}$ is denoted by$${\displaystyle a_{\mathcal{U}}:=\{x\in \prod _{i\in I}{M}_i:x\sim a\}}$$then the ultraproduct is the set of all${\displaystyle \mathcal{U}}$-equivalence classes$${\displaystyle {\prod }_{\mathcal{U}}{M}_{\bullet }=\prod _{i\in I}{M}_i/\mathcal{U}:=\left\{a_{\mathcal{U}}:a\in \prod _{i\in I}{M}_i\right\}.}$$

Although${\displaystyle \mathcal{U}}$ was assumed to be an ultrafilter, the construction above can be carried out more generally whenever${\displaystyle \mathcal{U}}$ is merely afilter on${\displaystyle I,}$ in which case the resulting quotient set${\displaystyle \prod _{i\in I}{M}_i/\mathcal{U}}$ is called areduced product.

When${\displaystyle \mathcal{U}}$ is aprincipal ultrafilter (which happens if and only if${\displaystyle \mathcal{U}}$ contains itskernel${\displaystyle \cap \mathcal{U}}$) then the ultraproduct is isomorphic to one of the factors. And so usually,${\displaystyle \mathcal{U}}$ is not aprincipal ultrafilter, which happens if and only if${\displaystyle \mathcal{U}}$ is free (meaning${\displaystyle \cap \mathcal{U}=\varnothing }$), or equivalently, if everycofinite subset of${\displaystyle I}$ is an element of${\displaystyle \mathcal{U}.}$ Since every ultrafilter on a finite set is principal, the index set${\displaystyle I}$ is consequently also usually infinite.

The ultraproduct acts as a filter product space where elements are equal if they are equal only at the filtered components (non-filtered components are ignored under the equivalence). One may define a finitely additivemeasure${\displaystyle m}$ on the index set${\displaystyle I}$ by saying${\displaystyle m(A)=1}$ if${\displaystyle A\in \mathcal{U}}$ and${\displaystyle m(A)=0}$ otherwise. Then two members of the Cartesian product are equivalent precisely if they are equalalmost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.

Finitaryoperations on the Cartesian product${\displaystyle \prod _{i\in I}{M}_i}$ are defined pointwise (for example, if${\displaystyle +}$ is a binary function then${\displaystyle a_i+b_i=(a+b{)}_i}$). Otherrelations can be extended the same way:$${\displaystyle R\left(a_{\mathcal{U}}^1,\dots ,a_{\mathcal{U}}^n\right)⟺\left\{i\in I:R^{M_i}\left(a_i^1,\dots ,a_i^n\right)\right\}\in \mathcal{U},}$$where${\displaystyle a_{\mathcal{U}}}$ denotes the${\displaystyle \mathcal{U}}$-equivalence class of${\displaystyle a}$ with respect to${\displaystyle \sim .}$ In particular, if every${\displaystyle M_i}$ is anordered field then so is the ultraproduct.

Anultrapower is an ultraproduct for which all the factors${\displaystyle M_i}$ are equal. Explicitly, theultrapower of a set${\displaystyle M}$ modulo${\displaystyle \mathcal{U}}$ is the ultraproduct${\displaystyle \prod _{i\in I}{M}_i/\mathcal{U}={\prod }_{\mathcal{U}}{M}_{\bullet }}$ of the indexed family${\displaystyle M_{\bullet }:={\left(M_i\right)}_{i\in I}}$ defined by${\displaystyle M_i:=M}$ for every index${\displaystyle i\in I.}$ The ultrapower may be denoted by${\displaystyle {\prod }_{\mathcal{U}}M}$ or (since${\displaystyle \prod _{i\in I}M}$ is often denoted by${\displaystyle M^I}$) by$${\displaystyle M^I/\mathcal{U}:=\prod _{i\in I}M/\mathcal{U}}$$

For every${\displaystyle m\in M,}$ let${\displaystyle (m{)}_{i\in I}}$ denote the constant map${\displaystyle I\to M}$ that is identically equal to${\displaystyle m.}$ This constant map/tuple is an element of the Cartesian product${\displaystyle M^I=\prod _{i\in I}M}$ and so the assignment${\displaystyle m\mapsto (m{)}_{i\in I}}$ defines a map${\displaystyle M\to \prod _{i\in I}M.}$ Thenatural embedding of${\displaystyle M}$ into${\displaystyle {\prod }_{\mathcal{U}}M}$ is the map${\displaystyle M\to {\prod }_{\mathcal{U}}M}$ that sends an element${\displaystyle m\in M}$ to the${\displaystyle \mathcal{U}}$-equivalence class of the constant tuple${\displaystyle (m{)}_{i\in I}.}$

Thehyperreal numbers are the ultraproduct of one copy of thereal numbers for every natural number, with regard to an ultrafilter over the natural numbers containing all cofinite sets. Their order is the extension of the order of the real numbers. For example, the sequence${\displaystyle \omega }$ given by${\displaystyle {\omega }_i=i}$ defines an equivalence class representing a hyperreal number that is greater than any real number.

Analogously, one can definenonstandard integers,nonstandard complex numbers, etc., by taking the ultraproduct of copies of the corresponding structures.

As an example of the carrying over of relations into the ultraproduct, consider the sequence${\displaystyle \psi }$ defined by${\displaystyle {\psi }_i=2i.}$ Because${\displaystyle {\psi }_i>{\omega }_i=i}$ for all${\displaystyle i,}$ it follows that the equivalence class of${\displaystyle {\psi }_i=2i}$ is greater than the equivalence class of${\displaystyle {\omega }_i=i,}$ so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let${\displaystyle {\chi }_i=i}$ for${\displaystyle i}$ not equal to${\displaystyle 7,}$ but${\displaystyle {\chi }_7=8.}$ The set of indices on which${\displaystyle \omega }$ and${\displaystyle \chi }$ agree is a member of any ultrafilter (because${\displaystyle \omega }$ and${\displaystyle \chi }$ agree almost everywhere), so${\displaystyle \omega }$ and${\displaystyle \chi }$ belong to the same equivalence class.

In the theory oflarge cardinals, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter${\displaystyle \mathcal{U}.}$ Properties of this ultrafilter${\displaystyle \mathcal{U}}$ have a strong influence on (higher order) properties of the ultraproduct; for example, if${\displaystyle \mathcal{U}}$ is${\displaystyle \sigma }$-complete, then the ultraproduct will again be well-founded. (Seemeasurable cardinal for the prototypical example.)

Łoś's theorem, also calledthe fundamental theorem of ultraproducts, is due toJerzy Łoś (the surname is pronounced[ˈwɔɕ], approximately "wash", or[ˈɫɔɕ]). It states that anyfirst-order formula is true in the ultraproduct if and only if the set of indices${\displaystyle i}$ such that the formula is true in${\displaystyle M_i}$ is a member of${\displaystyle \mathcal{U}.}$ More precisely:

Let${\displaystyle \sigma }$ be a signature,${\displaystyle \mathcal{U}}$ an ultrafilter over a set${\displaystyle I,}$ and for each${\displaystyle i\in I}$ let${\displaystyle M_i}$ be a${\displaystyle \sigma }$-structure. Let${\displaystyle {\prod }_{\mathcal{U}}{M}_{\bullet }}$ or${\displaystyle \prod _{i\in I}{M}_i/\mathcal{U}}$ be the ultraproduct of the${\displaystyle M_i}$ with respect to${\displaystyle \mathcal{U}.}$ Then, for each${\displaystyle a^1,\dots ,a^n\in \prod _{i\in I}{M}_i,}$ where${\displaystyle a^k={\left(a_i^k\right)}_{i\in I},}$ and for every${\displaystyle \sigma }$-formula${\displaystyle \varphi ,}$$${\displaystyle {\prod }_{\mathcal{U}}{M}_{\bullet }\models \varphi \left[a_{\mathcal{U}}^1,\dots ,a_{\mathcal{U}}^n\right]⟺\{i\in I:M_i\models \varphi [a_i^1,\dots ,a_i^n]\}\in \mathcal{U}.}$$

The theorem is proved by induction on the complexity of the formula${\displaystyle \varphi .}$ The fact that${\displaystyle \mathcal{U}}$ is an ultrafilter (and not just a filter) is used in the negation clause, and theaxiom of choice is needed at the existential quantifier step. As an application, one obtains thetransfer theorem forhyperreal fields.

Let${\displaystyle R}$ be a unary relation in the structure${\displaystyle M,}$ and form the ultrapower of${\displaystyle M.}$ Then the set${\displaystyle S=\{x\in M:Rx\}}$ has an analog${\displaystyle {}^{\ast }S}$ in the ultrapower, and first-order formulas involving${\displaystyle S}$ are also valid for${\displaystyle {}^{\ast }S.}$ For example, let${\displaystyle M}$ be the reals, and let${\displaystyle Rx}$ hold if${\displaystyle x}$ is a rational number. Then in${\displaystyle M}$ we can say that for any pair of rationals${\displaystyle x}$ and${\displaystyle y,}$ there exists another number${\displaystyle z}$ such that${\displaystyle z}$ is not rational, and Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that${\displaystyle {}^{\ast }S}$ has the same property. That is, we can define a notion of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals.

Consider, however, theArchimedean property of the reals, which states that there is no real number${\displaystyle x}$ such that${\displaystyle x>1,x>1+1,x>1+1+1,\dots }$ for every inequality in the infinite list. Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic. In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number${\displaystyle \omega }$ above.

Inmodel theory andset theory, thedirect limit of a sequence of ultrapowers is often considered. Inmodel theory, this construction can be referred to as anultralimit orlimiting ultrapower.

Beginning with a structure,${\displaystyle A_0}$ and an ultrafilter,${\displaystyle {\mathcal{D}}_0,}$ form an ultrapower,${\displaystyle A_1.}$ Then repeat the process to form${\displaystyle A_2,}$ and so forth. For each${\displaystyle n}$ there is a canonical diagonal embedding${\displaystyle A_n\to A_{n+1}.}$ At limit stages, such as${\displaystyle A_{\omega },}$ form the direct limit of earlier stages. One may continue into the transfinite.

Theultrafilter monad is thecodensity monad of the inclusion of thecategory of finite sets into thecategory of all sets.^[1]

Similarly, theultraproduct monad is the codensity monad of the inclusion of the category${\displaystyle \mathbf{F}\mathbf{i}\mathbf{n}\mathbf{F}\mathbf{a}\mathbf{m}}$ offinitely-indexedfamilies of sets into the category${\displaystyle \mathbf{F}\mathbf{a}\mathbf{m}}$ of allindexed families of sets. So in this sense, ultraproducts are categorically inevitable.^[1] Explicitly, an object of${\displaystyle \mathbf{F}\mathbf{a}\mathbf{m}}$ consists of a non-emptyindex set${\displaystyle I}$ and anindexed family${\displaystyle {\left(M_i\right)}_{i\in I}}$ of sets. A morphism${\displaystyle {\left(N_i\right)}_{j\in J}\to {\left(M_i\right)}_{i\in I}}$ between two objects consists of a function${\displaystyle \varphi :I\to J}$ between the index sets and a${\displaystyle J}$-indexed family${\displaystyle {\left({\varphi }_j\right)}_{j\in J}}$ of function${\displaystyle {\varphi }_j:M_{\varphi (j)}\to N_j.}$ The category${\displaystyle \mathbf{F}\mathbf{i}\mathbf{n}\mathbf{F}\mathbf{a}\mathbf{m}}$ is a full subcategory of this category of${\displaystyle \mathbf{F}\mathbf{a}\mathbf{m}}$ consisting of all objects${\displaystyle {\left(M_i\right)}_{i\in I}}$ whose index set${\displaystyle I}$ is finite. The codensity monad of the inclusion map${\displaystyle \mathbf{F}\mathbf{i}\mathbf{n}\mathbf{F}\mathbf{a}\mathbf{m}\hookrightarrow \mathbf{F}\mathbf{a}\mathbf{m}}$ is then, in essence, given by$${\displaystyle {\left(M_i\right)}_{i\in I}\mapsto {\left(\prod _{i\in I}{M}_i/\mathcal{U}\right)}_{\mathcal{U}\in U(I)}.}$$

• Compactness theorem – Theorem in mathematical logic
• Extender (set theory)
• Löwenheim–Skolem theorem – Existence and cardinality of models of logical theories
• Transfer principle – Concept in model theory
• Ultrafilter – Maximal proper filter

Proofs

• Bell, John Lane; Slomson, Alan B. (2006) [1969].Models and Ultraproducts: An Introduction (reprint of 1974 ed.).Dover Publications.ISBN 0-486-44979-3.
• Burris, Stanley N.; Sankappanavar, H.P. (2000) [1981].A Course in Universal Algebra (Millennium ed.).

• Mathematical logic
• Model theory
• Nonstandard analysis
• Theorems in the foundations of mathematics
• Universal algebra

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