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!

  • $\begingroup$ Are there relevant properties of acl in $\omega$-categorical theories besides that for $A$ finite acl$(A)$ is finite and its cardinality is bounded by some function of $|A|$? $\endgroup$ Commented May 28, 2015 at 7:07
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    $\begingroup$ Note that even when $\text{acl}$ does give a pregeometry, it doesn't necessarily tell you anything useful. For example, in the random graph, or more generally any Fraïssé limit with disjoint amalgamation, $\text{acl}(A) = A$ for all sets $A$ (this trivially satisfies exchange). The thing about strongly minimal sets that makes the pregeometry so useful is that there is a unique non-algebraic $1$-type over any set, which implies homogeneity: any bijection between bases extends to an automorphism. $\endgroup$ Commented May 28, 2015 at 17:58

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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.

  • $\begingroup$ Thank you for this. This is indeed the sort of thing I was looking for, but maybe less severe (if that makes any sense). Maybe a reference of "Exchange fails + some other property yields result". Does this make any sense? $\endgroup$ Commented May 28, 2015 at 18:39
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    $\begingroup$ I don't think your theory is uncountably categorical. One can vary the sizes of equivalence classes and obtain many models. $\endgroup$ Commented May 28, 2015 at 23:13
  • $\begingroup$ @LevonHaykazyan oh, of course you're right. Well, it's still $\omega$-stable, which is close enough! $\endgroup$ Commented May 28, 2015 at 23:17
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    $\begingroup$ @AlexKruckman: Here is an example of a theory which is totally categorical but exchange fails (very badly). Consider the collection of sets of the form $\{a,b\}, \{a\}$ where $a,b \in \mathbb{Z}$. Our language will have a single unary binary $R(x,y)$ where $R$ holds on a pair of sets iff they have an element in common. It isn't hard to demonstrate that this model is totally categorical nor is it hard to demonstrate that exchange does not hold with acl $\endgroup$ Commented Jul 26, 2015 at 22:19
  • $\begingroup$ Yes, nice example. $\endgroup$ Commented Jul 26, 2015 at 22:22

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