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 Template:Nowrap that satisfies the following axioms.

1. $\mathrm{\forall }xyz[C(x;y,z)\to C(x;z,y)],$
2. $\mathrm{\forall }xyz[C(x;y,z)\to \mathrm{\neg }C(y;x,z)],$
3. $\mathrm{\forall }xyzw[C(x;y,z)\to (C(w;y,z)\vee C(x;w,z))],$
4. $\mathrm{\forall }xy[x\ne y\to \mathrm{\exists }z\ne yC(x;y,z)].$

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 Template:Nowrap, 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 Template:Abs_p denote its p-adic absolute value. Then the relation defined by $C(a;b,c)⟺|b-c{|}_p<|a-c{|}_p$ 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

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