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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1012 WeightDef<strong>in</strong>itionIn a semimodular lattice, the weight <strong>of</strong> a lattice element A is themaximal n such that there is an <strong>in</strong>dependent sequenceB 0 , B 1 , . . . , B n−1 with A ̸⊥ B 0 , . . . , A ̸⊥ B n−1 .Fact<strong>The</strong> weight 1 elements <strong>of</strong> a semimodular lattice form a matroidunder the follow<strong>in</strong>g closure operator:cl A = {b | ∃A 0 ⊆ A: A 0 <strong>in</strong>dependent and A 0 ̸⊥ b}.If there are ‘enough’ weight 1 elements, and if there is somecontrol over the <strong>in</strong>dependent sets <strong>of</strong> weight 1 elements, then thematroid <strong>of</strong> weight 1 elements helps to understand the <strong>model</strong>s <strong>of</strong> a<strong>theory</strong>. This is the case for superstable theories.
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