Are there exist any aleph-one categorical theories which are not strongly minimal?

Question

Is every aleph-one categorical theory strongly minimal?Are there exist any aleph-one categorical theories which are not strongly minimal?

Answer 1

Consider the theory of two equal-cardinality infinite sets:

• The language has two unary predicates $U$ and $V$ and a binary relation $E$.

• The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.

This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!

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Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.

Answer 2

Many examples of $\aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $\varphi(x)$ (possibly with parameters satisfying an isolated type over $\emptyset$, so that a realization exists in every model) such that in any model $M\models T$, $X = \varphi(M)$ is a strongly minimal set and $M = \text{acl}(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.

The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.

For some more "natural mathematical examples", consider the following:

• Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $\text{Th}(G)$ is almost strongly minimal. So for example the theory of $\text{SL}_2(\mathbb{C})$ is $\aleph_1$-categorical and almost strongly minimal but not strongly minimal.

• Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $I\subseteq P\times L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $\aleph_1$-categorical and almost strongly minimal but not strongly minimal.

Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(\mathbb{Z}/4\mathbb{Z})^\omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $\mathbb{F}_2$-vector space), but the rest of the structure is not algebraic over it.