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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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To many ladies, BMI calculations might appear a quite complicated plus complex process that has no hope of understanding. However, which is completely untrue because BMI calculation for ladies is not certain scientific secret or magic. If you have ever utilized an online BMI calculator for women online, you'd understand by now which if would require you to insert your height and fat before providing a number as a final output.
If the bmi chart shows your BMI level to be 'tween 25 plus 29.9, you're reckoned to be over weight. This will be enjoyed because a notice, requiring we stop the insalubrious stuff and receive active with a reasonable fat-burning exercise procedure.
37. Making time for yourself: Running = sanity. Alone or with friends it has fantastic healing results that last all day bmi chart men. I find doing it early each morning is ideal as I recognize I'll get my run in and "life stuff" during the day will likely not get inside the technique.
The above stated research is not truly the only 1 which have studied Playboy playmates with the assumption that the ladies chosen for this title reflect the societal norm for beauty. For instance Singh had earlier noted that, whereas changes in body structure had been found amidst Playboy Playmates, the total Waist-to-Hip ratio had only improved slightly from .68 to .71. What is the importance of the? A healthy WHR range for a female is considered between .67 plus .80, where a high ratio is associated with decreased fertility (Rempala & Garvey, 2007). The ratio additionally increases during menopause and pregnancy, both, naturally, associated with decreased fertility.
It is fairly uncommon for children to be obese due to health issues. The opposite is true. Children have wellness issues because they are obese. Unless diagnosed by a doctor, health problems ought not to be chosen by parents as an excuse. Similarly with genetics. Although genetics can play a piece, it is very only a especially small part. Fat parents frequently have fat children, yet this is not normally genetic. How about the fact that children mimic the bad eating habits and activity practices of their parents? How frequently have you enjoyed parents plus kids living off Big Macs and different fatty efficiency foods.
The waist-to-hip ratio is a beneficial measure for determining health bmi chart women risk due to the site of fat storage. It is calculated by dividing the ratio of ab girth by cool measuring.
18. Type of Diet: Adhering to a well-balanced, low-fat, wholegrain diet that is higher in carbs has constantly been the number one route for me. I love a good smoothie (see post "Smoothie Operator --quick nutritional training meal") while training. Here's an interesting post w/ superior strategies on eating from Cool Running called "The Runner's Diet".
Ladies, here are several criteria my neighbors agree with when it comes to the most perfect size. Aside from being one which places the weight in range, the perfect size and body should help you to a) stow away a own carry-on luggage, b) chase down a cab and c) dance away inside a favorite dress... all without a caliper care inside the globe.