C-minimal theory: Difference between revisions

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A ''C''-relation is a ternary relation ''C''(''x'';''yz'') that satisfies the following axioms.

# <math>\forall xyz\, [ C(x;yz)\rightarrow C(x;zy) ],</math>,

# <math>\forall xyz\, [ C(x;yz)\rightarrow\neg C(y;xz) ],</math>,

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

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

A '''C-minimal structure''' is a [[structure (mathematical logic)|structure]] ''M'', in a [[signature (mathematical 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'';''bc''), where ''b'' and ''c'' are elements of ''M''.