C-minimal theory

In model theory, a branch of mathematical logic, a C-minimal theory is a theory which is "minimal" with respect to a ternary relation C with certain properties. The theories of the p-adic number fields Q_p 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.

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. $\forall xy[x\neq y\rightarrow \exists z\neq yC(x;yz)]$.

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.

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 $C(a;bc)\iff |b-c|_{p}<|a-c|_{p}$ is a C-relation, and the theory of Q_p is C-minimal with respect to this relation.

References

• Macpherson, Dugald; Steinhorn, Charles (1996), "On variants of o-minimality", Annals of Pure and Applied Logic, 79: 165–209, doi:10.1016/0168-0072(95)00037-2
• Haskell, Deirdre; Macpherson, Dugald (1994), "Cell decompositions of C-minimal structures", Annals of Pure and Applied Logic, 66: 113–162