C-minimal theory: Difference between revisions

Revision as of 01:38, 15 February 2022

Revision as of 01:38, 15 February 2022

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# <math>\forall xyzw\, [ C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z)) ],</math>

# <math>\forall xy\, [ x\neq y \rightarrow \exists z\neq y\, C(x;y,z) ].</math>

A '''C-minimal structure''' is a [[structure (mathematical logic)|structure]] ''M'', in a [[signature (logic)|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'';''bcb,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.