C-minimal theory

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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 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[edit]

A C-relation is a ternary relation C(x;yz) that satisfies the following axioms.

  1. \forall xyz\, [ C(x;yz)\rightarrow C(x;zy) ],
  2. \forall xyz\, [ C(x;yz)\rightarrow\neg C(y;xz) ],
  3. \forall xyzw\, [ C(x;yz)\rightarrow (C(w;yz)\vee C(x;wz)) ],
  4. \forall xy\, [ x\neq y \rightarrow \exists z\neq 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[edit]

For a prime number p and a p-adic number a let |a|p denote its p-adic norm. Then the relation defined by C(a;bc) \iff |b-c|_p < |a-c|_p 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.

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