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| The '''ultraproduct''' is a [[mathematics|mathematical]] construction that appears mainly in [[abstract algebra]] and in [[model theory]], a branch of [[mathematical logic]]. An ultraproduct is a quotient of the direct product of a family of [[structure (mathematical logic)|structures]]. All factors need to have the same [[signature (logic)|signature]]. The '''ultrapower''' is the special case of this construction in which all factors are equal.
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| For example, ultrapowers can be used to construct new [[field (mathematics)|field]]s from given ones. The [[hyperreal numbers]], an ultrapower of the [[real numbers]], are a special case of this.
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| Some striking applications of ultraproducts include very elegant proofs of the [[compactness theorem]] and the [[completeness theorem]], [[H. Jerome Keisler|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 of [[non-standard analysis]], which was pioneered (as an application of the compactness theorem) by [[Abraham Robinson]].
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| ==Definition==
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| The general method for getting ultraproducts uses an index set ''I'', a [[structure (mathematical logic)|structure]] ''M''<sub>''i''</sub> for each element ''i'' of ''I'' (all of the same [[signature (logic)|signature]]), and an [[ultrafilter]] ''U'' on ''I''. The usual choice is for ''I'' to be infinite and ''U'' to contain all [[cofinite]] subsets of ''I''. Otherwise the ultrafilter is [[Principal ultrafilter|principal]], and the ultraproduct is isomorphic to one of the factors.
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| Algebraic operations on the [[Cartesian product]]
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| :<math>\prod_{i \in I} M_i </math> | |
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| are defined in the usual way (for example, for a binary function +, (''a'' + ''b'') <sub>''i''</sub> = ''a''<sub>''i''</sub> + ''b''<sub>''i''</sub> ), and an [[equivalence relation]] is defined by ''a'' ~ ''b'' if and only if
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| :<math>\left\{ i \in I: a_i = b_i \right\}\in U,</math>
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| and the '''ultraproduct''' is the [[quotient set]] with respect to ~. The ultraproduct is therefore sometimes denoted by
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| :<math>\prod_{i\in I}M_i / U . </math>
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| One may define a finitely additive [[Measure (mathematics)|measure]] ''m'' on the index set ''I'' by saying ''m''(''A'') = 1 if ''A'' ∈ ''U'' and = 0 otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal [[almost everywhere]] on the index set. The ultraproduct is the set of equivalence classes thus generated.
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| Other [[relation (mathematics)|relation]]s can be extended the same way:
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| :<math>R([a^1],\dots,[a^n]) \iff \left\{ i \in I: R^{M_i}(a^1_i,\dots,a^n_i) \right\}\in U,</math>
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| where [''a''] denotes the equivalence class of ''a'' with respect to ~.
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| In particular, if every ''M''<sub>''i''</sub> is an [[ordered field]], then so is the ultraproduct.
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| An '''ultrapower''' is an ultraproduct for which all the factors ''M''<sub>''i''</sub> are equal:
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| : <math>M^\kappa/U=\prod_{\alpha<\kappa}M/U.\,</math>
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| More generally, the construction above can be carried out whenever ''U'' is a [[filter (mathematics)|filter]] on ''I''; the resulting model <math>\prod_{i\in I}M_i / U</math> is then called a '''reduced product'''.
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| == Examples ==
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| The [[hyperreal numbers]] are the ultraproduct of one copy of the [[real 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 ''ω'' given by ''ω''<sub>''i''</sub> = ''i'' defines an equivalence class representing a hyperreal number that is greater than any real number.
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| Analogously, one can define [[nonstandard integer]]s, [[nonstandard complex numbers]], etc., by taking the ultraproduct of copies of the corresponding structures.
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| As an example of the carrying over of relations into the ultraproduct, consider the sequence ''ψ'' defined by ''ψ''<sub>''i''</sub> = 2''i''. Because ''ψ''<sub>''i''</sub> > ''ω''<sub>''i''</sub> = ''i'' for all ''i'', it follows that the equivalence class of ''ψ''<sub>''i''</sub> = 2''i'' is greater than the equivalence class of ''ω''<sub>''i''</sub> = ''i'', so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let ''χ''<sub>''i''</sub> = ''i'' for ''i'' not equal to 7, but ''χ''<sub>7</sub> = 8. The set of indices on which ''ω'' and ''χ'' agree is a member of any ultrafilter (because ''ω'' and ''χ'' agree almost everywhere), so ''ω'' and ''χ'' belong to the same equivalence class.
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| In the theory of [[large cardinal]]s, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter ''U''. Properties of this ultrafilter ''U'' have a strong influence on (higher order) properties of the ultraproduct; for example, if ''U'' is σ-complete, then the ultraproduct will again be well-founded. (See [[measurable cardinal]] for the prototypical example.)
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| ==Łoś's theorem==
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| Łoś's theorem, also called ''the fundamental theorem of ultraproducts'', is due to [[Jerzy Łoś]] (the surname is pronounced {{IPA-pl|ˈwɔɕ|}}, approximately "wash"). It states that any [[first-order predicate calculus|first-order]] formula is true in the ultraproduct if and only if the set of indices ''i'' such that the formula is true in ''M''<sub>''i''</sub> is a member of ''U''. More precisely:
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| Let σ be a signature, <math> U </math> an ultrafilter over a set <math> I </math>, and for each <math> i \in I </math> let <math> M_{i} </math> be a σ-structure. Let <math> M </math> be the ultraproduct of the <math> M_{i} </math> with respect to <math>U</math>, that is, <math> M = \prod_{ i\in I }M_i/U.</math> Then, for each <math> a^{1}, \ldots, a^{n} \in \prod M_{i} </math>, where <math> a^{k} = (a^{k}_{i})_{i \in I} </math>, and for every σ-formula <math>\phi</math>,
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| :<math> M \models \phi[[a^1], \ldots, [a^n]] \iff \{ i \in I : M_{i} \models \phi[a^1_{i}, \ldots, a^n_{i} ] \} \in U.</math> | |
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| The theorem is proved by induction on the complexity of the formula <math>\phi</math>. The fact that <math>U</math> is an ultrafilter (and not just a filter) is used in the negation clause, and the [[axiom of choice]] is needed at the existential quantifier step. As an application, one obtains the [[transfer principle|transfer theorem]] for [[hyperreal number|hyperreal fields]].
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| ===Examples===
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| Let ''R'' be a unary relation in the structure ''M'', and form the ultrapower of ''M''. Then the set <math>S=\{x \in M|R x\}</math> has an analog ''<sup>*</sup>S'' in the ultrapower, and first-order formulas involving S are also valid for ''<sup>*</sup>S''. For example, let ''M'' be the reals, and let ''Rx'' hold if ''x'' is a rational number. Then in ''M'' we can say that for any pair of rationals ''x'' and ''y'', there exists another number ''z'' such that ''z'' is not rational, and ''x'' < ''z'' < ''y''. Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that ''<sup>*</sup>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.
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| Consider, however, the [[Archimedean property]] of the reals, which states that there is no real number ''x'' such that ''x'' > 1, ''x'' > 1 +1 , ''x'' > 1 + 1 + 1, ... 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 ''ω'' above.
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| ==Ultralimit==
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| :''For the ultraproduct of a sequence of metric spaces, see [[Ultralimit]].''
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| In [[model theory]] and [[set theory]], an '''ultralimit''' or '''limiting ultrapower''' is a [[direct limit]] of a sequence of ultrapowers.
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| Beginning with a structure, ''A''<sub>0</sub>, and an ultrafilter, ''D''<sub>0</sub>, form an ultrapower, ''A''<sub>1</sub>. Then repeat the process to form ''A''<sub>2</sub>, and so forth. For each ''n'' there is a canonical diagonal embedding <math>A_n\to A_{n+1}</math>. At limit stages, such as ''A''<sub>ω</sub>, form the direct limit of earlier stages. One may continue into the transfinite.
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| ==References==
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| * {{ cite book | last=Bell | first=John Lane | coauthors=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 edition | origyear=1969 | publisher=[[Dover Publications]] | isbn=0-486-44979-3 }}
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| * {{ cite book | last=Burris | first =Stanley N. | coauthors=Sankappanavar, H.P. | title=A Course in Universal Algebra | origyear=1981 | year=2000 | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html | edition=Millennium edition}}
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| [[Category:Model theory]]
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| [[Category:Universal algebra]]
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| [[Category:Non-standard analysis]]
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