C-minimal theory explained

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 that satisfies the following axioms.

\forall xyz [ C(x;y,z) → C(x;z,y) ], \forall xyz [ C(x;y,z) → \neg C(y;x,z) ], \forall xyzw [ C(x;y,z) → (C(w;y,z)\vee C(x;w,z)) ], \forall xy [ x ≠ y → \exists z ≠ y C(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, 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 _p denote its p-adic absolute value. Then the relation defined by

C(a; b, c) \iff |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