The role of matroids in model theory
The role of matroids in model theory
The role of matroids in model theory
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612Strong m<strong>in</strong>imalityDef<strong>in</strong>itionA complete first-order <strong>theory</strong> is called strongly m<strong>in</strong>imal if for all<strong>model</strong>s• algebraic closure is a matroid, and• for all n, all <strong>in</strong>dependent n-tuples have the same type.<strong>The</strong>oremLet T be a countable complete first-order <strong>theory</strong>.If T is strongly m<strong>in</strong>imal, then T is uncountably categorical and thecard<strong>in</strong>ality <strong>of</strong> an uncountable <strong>model</strong> equals its dimension.More generally, T is uncountably categorical if and only if therestriction <strong>of</strong> T to a certa<strong>in</strong> formula is strongly m<strong>in</strong>imal and every<strong>model</strong> <strong>of</strong> T is prime over this strongly m<strong>in</strong>imal part.
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