Generalizations of pregeometries

Question

Combinatorial geometries and pregeometries are important in classifying strongly minimal (as well as O-minimal) theories. More formally, a model of a strongly minimal (or an O-minimal) theory with the algebraic closure operator always forms a pregeometry. This is because in models of these types of theories, the exchange property will always hold. However, I'm curious if anyone has studied models of $\aleph_0-$ categorical theories where the exchange property fails (For the model equipped with the $acl$ operator). Has there been any work on the classification of these theories? Any references would be nice!

Answer 1

Were you hoping that the failure of exchange would give some sort of non-structure result? This is not necessarily an answer to your question, but here's an example showing that this can't happen (or this hypothesis alone is not enough). It's an $\aleph_0$-categorical theory in which $\text{acl}$ does not satisfy exchange, but which is $\omega$-stable (i.e. Nearly as nice as can be).

Let $L = \{E,P\}$. Let $T$ say that $E$ is an equivalence relation with infinitely many infinite classes, and $P$ is a unary predicate which picks out exactly one element from each class. If $a$ is any element not in $P$ and $b$ is the unique element in $P$ which is equivalent to $a$, then $b\in\text{acl}(a)\setminus \text{acl}(\varnothing)$, but $a\notin \text{acl}(b)$, so $\text{acl}$ doesn't satisfy exchange.