In a strongly minimal theory, model-theoretic algebraic closure is a <pregeometry>. Given finite tuples $\bar a,\bar b$ of the same type, the induced correspondence extends to an isomorphism between their algebraic closures. If $c\in\operatorname{acl}(\bar a)$, transport it through this isomorphism. If $c\notin\operatorname{acl}(\bar a)$, its type is the unique generic one over $\bar a$; choose a corresponding element outside $\operatorname{acl}(\bar b)$. Exchange ensures that this choice has the transported type. Hence every finite partial elementary map extends, and every model is aleph-zero-homogeneous.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 144 / 4 / b / Solution

{wiki=Matroid#Closure_operators}

A pregeometry is a closure operator with finite character and exchange. Algebraic closure in a strongly minimal theory is a pregeometry.


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 Incoming links: Pregeometry