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; y, z) that satisfies the following axioms.

  1. ∀ x y z [ C ( x ; y , z ) → C ( x ; z , y ) ] , {\displaystyle \forall xyz\,[C(x;y,z)\rightarrow C(x;z,y)],}
  2. ∀ x y z [ C ( x ; y , z ) → ¬ C ( y ; x , z ) ] , {\displaystyle \forall xyz\,[C(x;y,z)\rightarrow \neg C(y;x,z)],}
  3. ∀ x y z w [ C ( x ; y , z ) → ( C ( w ; y , z ) ∨ C ( x ; w , z ) ) ] , {\displaystyle \forall xyzw\,[C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z))],}
  4. ∀ x y [ x ≠ y → ∃ z ≠ y C ( x ; y , z ) ] . {\displaystyle \forall xy\,[x\neq y\rightarrow \exists z\neq 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 C(x; b, c), 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 absolute value. Then the relation defined by C ( a ; b , c ) ⟺ | b − c | p < | a − c | p {\displaystyle C(a;b,c)\iff |b-c|_{p}<|a-c|_{p}} is a C-relation, and the theory of Qp with addition and this relation is C-minimal. The theory of Qp as a field, however, is not C-minimal.

References

  • Macpherson, Dugald; Steinhorn, Charles (1996), "On variants of o-minimality", Annals of Pure and Applied Logic, 79 (2): 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 (2): 113–162, doi:10.1016/0168-0072(94)90064-7
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