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In [[model theory]], a branch of [[mathematical logic]], a '''C-minimal theory''' is a theory that is "minimal" with respect to a [[Triadic relation|ternary relation]] ''C'' with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example. | |||
This notion was defined in analogy to the [[o-minimal theory|o-minimal theories]], which are "minimal" (in the same sense) with respect to a linear order. | |||
==Definition== | |||
A ''C''-relation is a ternary relation ''C''(''x'';''yz'') that satisfies the following axioms. | |||
# <math>\forall xyz\, [ C(x;yz)\rightarrow C(x;zy) ],</math> | |||
# <math>\forall xyz\, [ C(x;yz)\rightarrow\neg C(y;xz) ],</math> | |||
# <math>\forall xyzw\, [ C(x;yz)\rightarrow (C(w;yz)\vee C(x;wz)) ],</math> | |||
# <math>\forall xy\, [ x\neq y \rightarrow \exists z\neq y\, C(x;yz) ].</math> | |||
A '''C-minimal structure''' is a [[structure (mathematical logic)|structure]] ''M'', in a [[signature (logic)|signature]] containing the symbol ''C'', such that ''C'' satisfies the above axioms and every set of elements of ''M'' that is definable with parameters in ''M'' is a Boolean combination of instances of ''C'', i.e. of formulas of the form ''C''(''x'';''bc''), where ''b'' and ''c'' are elements of ''M''. | |||
A theory is called '''C-minimal''' if all of its models are C-minimal. A structure is called '''strongly C-minimal''' if its theory is C-minimal. One can construct C-minimal structures which are not strongly C-minimal. | |||
==Example== | |||
For a [[prime number]] ''p'' and a [[p-adic number|''p''-adic number]] ''a'' let |''a''|<sub>''p''</sub> denote its ''p''-adic norm. Then the relation defined by <math>C(a;bc) \iff |b-c|_p < |a-c|_p</math> is a ''C''-relation, and the theory of '''Q'''<sub>''p''</sub> is with addition and this relation is C-minimal. The theory of '''Q'''<sub>''p''</sub> as a field, however, is not C-minimal. | |||
==References== | |||
* {{Citation | last1=Macpherson | first1=Dugald | author1-link=Dugald Macpherson | last2=Steinhorn | first2=Charles | title=On variants of o-minimality | doi=10.1016/0168-0072(95)00037-2 | year=1996 | journal=Annals of Pure and Applied Logic | volume=79 | pages=165–209 | issue=2}} | |||
* {{Citation | last1=Haskell | first1=Deirdre | last2=Macpherson | first2=Dugald | author2-link=Dugald Macpherson | title=Cell decompositions of C-minimal structures | year=1994 | journal=Annals of Pure and Applied Logic | volume=66 | pages=113–162 | doi=10.1016/0168-0072(94)90064-7 | issue=2}} | |||
[[Category:Model theory]] | |||
Revision as of 09:45, 25 January 2014
In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation C with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example.
This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order.
Definition
A C-relation is a ternary relation C(x;yz) that satisfies the following axioms.
![{\displaystyle \forall xyz\,[C(x;yz)\rightarrow C(x;zy)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afebee6a325f010c7f74c09740762c28f11d4b58)
![{\displaystyle \forall xyz\,[C(x;yz)\rightarrow \neg C(y;xz)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1723a04a55cd8bcd6afccc00cef5d54dfef8cf96)
![{\displaystyle \forall xyzw\,[C(x;yz)\rightarrow (C(w;yz)\vee C(x;wz))],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0624bbcce59871d55b827beda403a3957d1427)
![{\displaystyle \forall xy\,[x\neq y\rightarrow \exists z\neq y\,C(x;yz)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d61d4a11db08f52744cf2ba9a9802aef8b3df27)
A C-minimal structure is a structure M, in a signature containing the symbol C, such that C satisfies the above axioms and every set of elements of M that is definable with parameters in M is a Boolean combination of instances of C, i.e. of formulas of the form C(x;bc), where b and c are elements of M.
A theory is called C-minimal if all of its models are C-minimal. A structure is called strongly C-minimal if its theory is C-minimal. One can construct C-minimal structures which are not strongly C-minimal.
Example
For a prime number p and a p-adic number a let |a|p denote its p-adic norm. Then the relation defined by is a C-relation, and the theory of Qp is with addition and this relation is C-minimal. The theory of Qp as a field, however, is not C-minimal.
References
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