The role of matroids in model theory

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712Independence for more general theoriesTo extend this dimension theory beyond uncountably categoricaltheories, we can:1. drop or weaken the condition on independent n-types2. generalise the notion of matroid, or3. allow other closure operators instead of acl.Real closed fields are an example of a pregeometric theory:Algebraic closure is a matroid. But the theory is not uncountablycategorical, and this is an example of 1.We will interpret a part of the machinery of stability theory asdoing 2 in order to get 3.
812Semimodular latticesDefinition (Wilcox)(A, B) is a modular pair if for all C ∈ [A ∧ B, B] we have(A ∨ C) ∧ B = C.A lattice is semimodular (M-symmetric) if being a modular pair isa symmetric relation.• The lattice of closed sets of a matroid is semimodular; theclosures of elements are its atoms.• A semimodular lattice that is generated by its atoms can beinterpreted as a matroid.
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The role of matroids in model theory

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