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.
# \forall xyz\, C(x;y,z)\rightarrow C(x;z,y)
# \forall xyz\, C(x;y,z)\rightarrow\neg C(y;x,z)
# \forall xyzw\, C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z))
# \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 $C(a; b, c) \iff , b-c, _p < , a-c, _p$ 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.

References

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{{Mathematical logic

Model theory