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| {{lowercase|title=o-minimal theory}}
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| In [[mathematical logic]], and more specifically in [[model theory]], an infinite [[structure (mathematical logic)|structure]] (''M'',<,...) which is [[Total order|totally ordered]] by < is called an '''o-minimal structure''' if and only if every [[definable set|definable]] subset ''X'' ⊂ ''M'' (with parameters taken from ''M'') is a finite [[union (set theory)|union]] of [[interval (mathematics)|interval]]s and points.
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| O-minimality can be regarded as a weak form of [[quantifier elimination]]. A structure ''M'' is o-minimal if and only if every formula with one free variable and parameters in ''M'' is equivalent to a quantifier-free formula involving only the ordering, also with parameters in ''M''. This is analogous to the [[strongly minimal theory|minimal]] structures, which are exactly the analogous property down to equality.
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| A [[Theory (mathematical logic)|theory]] ''T'' is an '''o-minimal theory''' if every [[Model theory|model]] of ''T'' is o-minimal. It is known that the complete theory ''T'' of an o-minimal structure is an o-minimal theory.<ref>Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988).</ref> This result is remarkable because the [[complete theory]] of a minimal structure need not be a [[strongly minimal theory]], that is, there may be an elementarily equivalent structure which is not minimal.
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| ==Set-theoretic definition==
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| O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set ''M'' in a set-theoretic manner, as a sequence ''S'' = (''S''<sub>''n''</sub>), ''n'' = 0,1,2,... such that
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| # ''S''<sub>''n''</sub> is a [[Boolean algebra (structure)|boolean algebra]] of subsets of ''M''<sup>''n''</sup>
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| # if ''A'' ∈ ''S''<sub>''n''</sub> then ''M'' × ''A'' and ''A'' ×''M'' are in ''S''<sub>''n''+1</sub>
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| # the set {(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) ∈ ''M''<sup>''n''</sup> : ''x''<sub>1</sub> = ''x''<sub>''n''</sub>} is in ''S''<sub>''n''</sub>
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| # if ''A'' ∈ ''S''<sub>''n''+1</sub> and ''π'' : ''M''<sup>''n''+1</sup> → ''M''<sup>''n''</sup> is the projection map on the first ''n'' coordinates, then ''π''(''A'') ∈ ''S''<sub>''n''</sub>.
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| If ''M'' has a dense linear order without endpoints on it, say <, then a structure ''S'' on ''M'' is called o-minimal if it satisfies the extra axioms
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| <ol start="5">
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| <li>the set {(''x'',''y'') ∈ ''M''<sup>2</sup> : ''x'' < ''y''} is in ''S''<sub>2</sub>
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| <li>the sets in ''S''<sub>1</sub> are precisely the finite unions of intervals and points.
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| </ol>
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| The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.
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| ==Model theoretic definition==
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| O-minimal structures originated in model theory and so have a simpler — but equivalent — definition using the language of model theory.<ref>Marker (2002) p.81</ref> Specifically if ''L'' is a language including a binary relation <, and (''M'',<,...) is an ''L''-structure where < is interpreted to satisfy the axioms of a dense linear order,<ref>The condition that the interpretation of < be dense is not strictly necessary, but it is known that discrete orders lead to essentially trivial o-minimal structures, see, for example, {{MR|0899083}} and {{MR|0943306}}.</ref> then (''M'',<,...) is called an o-minimal structure if for any definable set ''X'' ⊆ ''M'' there are finitely many intervals ''I''<sub>1</sub>,...,''I''<sub>''r''</sub> with endpoints in ''M'' ∪ {±∞} and a finite set ''X''<sub>0</sub> such that
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| :<math>X=X_0\cup I_1\cup\ldots\cup I_r.</math>
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| ==Examples==
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| Examples of o-minimal theories are:
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| * The complete theory of dense linear orders in the language with just the ordering.
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| * RCF, the [[theory]] of [[real closed field]]s.<ref>Marker (2002) p.99</ref>
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| * The complete theory of the [[real number|real field]] with restricted [[analytic function]]s added (i.e. analytic functions on a neighborhood of [0,1]<sup>''n''</sup>, restricted to [0,1]<sup>''n''</sup>; note that the unrestricted sine function has infinitely many roots, and so cannot be definable in an o-minimal structure.)
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| * The complete theory of the real field with a symbol for the [[exponential function]] by [[Wilkie's theorem]]. More generally, the complete theory of the real numbers with [[Pfaffian function]]s added.
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| * The last two examples can be combined: given any o-minimal expansion of the real field (such as the real field with restricted analytic functions), one can define its Pfaffian closure, which is again an o-minimal structure.<ref>Patrick Speisseger, ''Pfaffian sets and o-minimality,'' in: Lecture notes on o-minimal structures and real analytic geometry, C. Miller, J.-P. Rolin, and P. Speissegger (eds.), Fields Institute Communications vol. 62, 2012, pp. 179–218. {{doi|10.1007/978-1-4614-4042-0_5}}</ref> (The Pfaffian closure of a structure is, in particular, closed under Pfaffian chains where arbitrary definable functions are used in place of polynomials.)
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| In the case of RCF, the definable sets are the [[semialgebraic set]]s. Thus the study of o-minimal structures and theories generalises [[real algebraic geometry]]. A major line of current research is based on discovering expansions of the real ordered field that are o-minimal. Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures. There is a cell decomposition theorem,<ref>Marker (2002) p.103</ref> [[Hassler Whitney|Whitney]] and [[Jean-Louis Verdier|Verdier]] [[Stratification (mathematics)|stratification]] theorems and a good notion of dimension and Euler characteristic.
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| ==See also==
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| * [[Semialgebraic set]]
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| * [[Real algebraic geometry]]
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| * [[Strongly minimal theory]]
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| * [[Weakly o-minimal structure]]
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| * [[C-minimal theory]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite book | first=Lou | last=van den Dries | title=Tame Topology and o-minimal Structures | year=1998 | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | volume=248 | location=Cambridge | zbl=0953.03045 | isbn=0-521-59838-9 }}
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| * {{cite journal | journal=[[Bulletin of the American Mathematical Society]] | last=Marker | first=David | volume=37 | year=2000 | pages=351–357 | title=Review of "Tame Topology and o-minimal Structures" | url=http://www.ams.org/bull/2000-37-03/S0273-0979-00-00866-1/S0273-0979-00-00866-1.pdf | doi=10.1090/S0273-0979-00-00866-1 | issue=3 }}
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| * {{cite book | last=Marker | first=David | title=Model theory: An introduction | series=Graduate Texts in Mathematics | volume=217 | location=New York, NY | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-98760-6 | zbl=1003.03034 }}
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| * {{cite journal |last=Pillay |first=Anand |coauthors=Steinhorn, Charles |year=1986 |title=Definable Sets in Ordered Structures I |journal=[[Transactions of the American Mathematical Society]] |volume=295 | issue=2 | pages=565–592 |url=http://www.ams.org/journals/tran/1986-295-02/S0002-9947-1986-0833697-X/S0002-9947-1986-0833697-X.pdf |doi=10.2307/2000052 |jstor=2000052 | zbl=0662.03023 }}
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| * {{cite journal |author=Knight, Julia | authorlink = Julia F. Knight|coauthors=Pillay, Anand; Steinhorn, Charles |year=1986 |title=Definable Sets in Ordered Structures II |journal=[[Transactions of the American Mathematical Society]] |volume=295 |pages=593–605 |doi=10.2307/2000053 |jstor=2000053 |issue=2 | zbl=0662.03024 }}
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| * {{cite journal |last=Pillay |first=Anand |coauthors=Steinhorn, Charles |year=1988 |title=Definable Sets in Ordered Structures III | journal=[[Transactions of the American Mathematical Society]] | volume=309 | pages=469–476 | doi=10.2307/2000920 | jstor=2000920 | issue=2 | zbl=0707.03024 }}
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| * {{cite journal | last=Wilkie | first=A.J. | authorlink=Alex Wilkie | title=Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function | journal=[[Journal of the American Mathematical Society]] | year=1996 | volume=9 | url=https://www.ams.org/jams/1996-9-04/S0894-0347-96-00216-0/S0894-0347-96-00216-0.pdf | doi=10.1090/S0894-0347-96-00216-0 | pages=1051 | issue=4 }}
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| * {{cite journal | last=Denef | first=J. | coauthors=van den Dries, L. | title=''p''-adic and real subanalytic sets | journal=[[Annals of Mathematics]] | volume=54 | year=1989 | jstor=1971463 | pages=79–138 | issue=1}}
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| ==External links==
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| * [http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/ ''Model Theory preprint server'']
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| * [http://www.maths.manchester.ac.uk/raag/ ''Real Algebraic and Analytic Geometry Preprint Server'']
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| [[Category:Model theory]]
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| [[Category:Topology]]
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| [[Category:Real algebraic geometry]]
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As soon as you have manufactured your option for your best grill you'll will need to invest a small time picking out a couple of accessories that will make your outside cooking enjoyable and secure. Moreover, it is constructed by generating use of a forge mechanism which creates a unbiased knife. This is far more expensive as compared to block knife. It is also critical to contemplate the materials that are composed of the Japanese chef knives. Majority of the chefs in the East favor this form of knife Greatest Plastic Cutting Board blade. Japanese chef knives also come in several knife handles. The AccuSharp is a good example of the ideal compact knife sharpeners.
It really is a tiny heavy for me but not to heavy it's a man`s knife, This is a knife that seems to just keep sharp with really little sharpeningfantastic Replica of Hand crafted Bob Kramer design chef knife but a third of the price but this is the exact same steel he uses and its licensed below his name. This knife is low maintenance holds a very good edge, uncomplicated to use, and a pleasure to use all day at all cooking tasks. This chef knife appears to hold a razor like edge for ages.
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