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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"> | |
| <li>the set {(''x'',''y'') ∈ ''M''<sup>2</sup> : ''x'' < ''y''} is in ''S''<sub>2</sub> | |
| <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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Your responsibility insurances already covers you for your appropriate related to
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Appropriate charges irrespective of the final outcome. In that respect, you already
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Towards the coverage.
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Base. The Best is a wonderful library for supplementary information about the purpose of it. Which means they"ll not cost you until they win you problems
In a insurance claim o-r suit against someone who caused you actual
Damage. Their contingency fees are determined as a commission on any
Therefore there"s every incentive on your plaintiff lawyer to protect, money won
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Assistance is readily available to you from plaintiff solicitors eager to work with
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of choices. In these circumstances, client-lawyer union is crucial: you
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attorney face-to-face in-a pre-paid appropriate plan, and the majority of the attorneys
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(Word count: 417)
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