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.
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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.
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∀ x y z [ C ( x ; y z ) → C ( x ; z y ) ] , -
∀ x y z [ C ( x ; y z ) → ¬ C ( y ; x z ) ] , -
∀ x y z w [ C ( x ; y z ) → ( C ( w ; y z ) ∨ C ( x ; w z ) ) ] , -
∀ x y [ x ≠ y → ∃ 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 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
For a prime number p and a p-adic number a let |a|p denote its p-adic norm. Then the relation defined by