C-minimal theory: Difference between revisions

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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 fieldsfield]]s with a (Krull) [[valuation (algebra)|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 {{nowrap|''C''(''x''; ''y'', ''z'')}} that satisfies the following axioms.

# <math>\forall xyz\, [ C(x;y,z)\rightarrow C(x;z,y) ],</math>

# <math>\forall xyz\, [ C(x;y,z)\rightarrow\neg C(y;x,z) ],</math>

# <math>\forall xyzw\, [ C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z)) ],</math>

# <math>\forall xy\, [ x\neq y \rightarrow \exists z\neq y\, C(x;y,z) ].</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 {{nowrap|''C''(''x''; ''b'', ''c'')}}, 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 {{abs|''a''|}}_''p'' denote its ''p''-adic absolute value. Then the relation defined by <math>C(a; b, c) \iff |b-c|_p < |a-c|_p</math> is a ''C''-relation, and the theory of '''Q'''_''p'' with addition and this relation is C-minimal. The theory of '''Q'''_''p'' as a field, however, is not C-minimal.

== References ==

* {{Citationcitation | 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| doi-access=free }}

* {{Citationcitation | 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| doi-access=free }}

{{Mathematical logic}}