C-minimal theory

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.
# $forall\; xyz,\; [\; C(x;yz)\; ightarrow\; C(x;zy)\; ]\; ,$
# $forall\; xyz,\; [\; C(x;yz)\; ightarrow\; eg\; C(y;xz)\; ]\; ,$
# $forall\; xyzw,\; [\; C(x;yz)\; ightarrow\; (C(w;yz)vee\; C(x;wz))\; ]\; ,$
# $forall\; xy,\; [\; x\; eq\; y\; ightarrow\; exists\; z\; eq\; y,\; C(x;yz)\; ]\; .$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 Q_"p" is with addition and this relation is C-minimal. The theory of Q_"p" as a field, however, is not C-minimal.

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