# Parameters and strongly minimal sets

Suppose $T$ is a countable complete theory, with monster model $\mathbb{C}$. A definable set $D := \phi(\mathbb{C}, \overline a)$ is strongly minimal if given any other formula $\psi(x, \overline b)$ with $\overline b\in \mathbb{C}$, then $D\cap \psi(\mathbb{C}, \overline b)$ is either finite or cofinite.

One of the important facts about strongly minimal sets is that the closure operator $A\rightarrow acl(A)\cap D$ turns $D$ into a pregeometry. My question is about proving the exchange principle: if $A\subseteq D$; $b, c\in D$; $b\in acl(A\cup \{c\})\setminus acl(A)$, then $c\in acl(A\cup \{b\})$.

Most proofs I find start as follows. Suppose not, and find $b, c$ as above but with $c\not\in acl(A\cup \{b\})$. Now there is a step where we include $A$ and $\overline a$ as constants in the language. Including $A$ is fine, but including $\overline a$ makes me uneasy. It could be the case that $c\in acl(A\cup\{b\} \cup \{\overline a\})$, which would undermine the rest of the proof.

If anyone knows of a good proof of exchange in this setting, I would much appreciate it.

You're right to be worried about this point in the proof. In fact, what you're trying to prove is false: you have the wrong definition of the closure operator!

In a strongly minimal set $$D$$ defined by a formula with parameters $$\varphi(x,\overline{a})$$, the closure operator should be $$X \mapsto \text{acl}_{\overline{a}}(X) = \text{acl}(X\cup\overline{a})\cap D$$.

Here's a counterexample to exchange for your definition of closure, showing that including the parameters is necessary:

Let $$L = \{E,f\}$$, where $$E$$ is a binary relation and $$f$$ is a unary function. Let $$T$$ be the theory asserting that $$E$$ is an equivalence relation with infinitely many infinite classes, and $$f$$ chooses a representative for each $$E$$-class. That is, $$\forall x\, (x\, E\, f(x))$$, and $$\forall x\, \forall y\, (x\, E\, y) \rightarrow (f(x) = f(y))$$.

$$T$$ is a complete theory with quantifier-elimination. Let $$\mathbb{C}$$ be its monster model. For any $$a\in \mathbb{C}$$, the $$E$$-class of $$a$$, defined by $$x\, E\, a$$, is strongly minimal. Call it $$D_a$$. Note that $$\text{acl}(\emptyset) = \emptyset$$, since every element of $$\mathbb{C}$$ has infinitely many conjugates under automorphisms of $$\mathbb{C}$$ (permuting the equivalence classes).

Choose any $$b\in D_a$$ such that $$f(b)\neq b$$, and let $$c = f(b)$$. Now $$c\in (\text{acl}(b)\cap D_a)\setminus (\text{acl}(\emptyset)\cap D_a)$$, but $$b\notin \text{acl}(c)\cap D_a$$, since $$b$$ has infinitely many conjugates under automorphisms of $$\mathbb{C}$$ fixing $$c$$ (take any permutation of $$D_a\setminus \{c\}$$).

What went wrong, of course, is that $$c$$ should really be in the closure of $$\emptyset$$. Indeed, following the correct definition of closure, $$c\in \text{acl}(\emptyset \cup \{a\}) \cap D_a$$. It's the unique element of $$\mathbb{C}$$ satisfying the formula $$(x\, E\, a) \land (f(x) = x)$$.

It's easier to think about / work with strongly minimal sets defined over $$\emptyset$$, so we often add the parameters $$\overline{a}$$ to the language at the start of the discussion. Note that if there are constants for $$\overline{a}$$ in the language, then $$\text{acl}(X\cup\overline{a})\cap D = \text{acl}(X)\cap D$$.

The popular textbooks by Marker and Hodges are both unfortunately sloppy on this point. Hodges only proves exchange over base sets $$A$$ containing the parameters $$\overline{a}$$, but he doesn't explicitly write down the definition of closure in strongly minimal sets defined with parameters. The book by Tent & Ziegler, as usual, gets it right.

• Thanks! One follow up question. If $D$ is strongly minimal and there are several different formulas with possibly different parameters defining $D$, is there a way to make a canonical choice?
– Andy
Commented Nov 23, 2015 at 5:36
• Yes, if you're willing to work in $M^\text{eq}$. In $M^\text{eq}$, definable sets have canonical parameters, up to interdefinability. Given $\varphi(x,y)$, the formula $E_\varphi(y,z)\colon \forall x\, \varphi(x,y) \leftrightarrow \varphi(x,z)$ is a definable equivalence relation, where $b$ and $c$ are equivalent just in case $\varphi(x,b)$ and $\varphi(x,c)$ define the same set. Now the equivalence class $[a]$, as an element of $M^\text{eq}$ has the property that an automorphism of $M$ fixes the set $D = \varphi(M,a)$ if and only if it fixes $[a]$. Commented Nov 23, 2015 at 5:47
• A consequence is that if $\psi(x,b)$ defines the same set $D$, then any automorphism of $M$ that fixes $b$ fixes $D$, and hence fixes $[a]$. It follows that $[a]\in \text{dcl}^{\text{eq}}(b)$ (I should note that I'm calling the monster model $M$). So $[a]$ is a minimal for $D$ in a sense: it's definable from any other parameter (even using a different formula). Now $[b]$ is also a canonical parameter (using a different formula $\psi$), but by the argument above $[a]$ and $[b]$ are interdefinable. So canonical parameters are minimal for $\text{dcl}$ and unique up to interdefinability. Commented Nov 23, 2015 at 5:51
• The upshot for strongly minimal sets is that if $[a]$ is the canonical parameter for $D$ and $\overline{b}$ is some other parameter for $D$, then $\text{acl}_{[a]}(X) \subseteq \text{acl}_{\overline{b}}(X)$, since $X[a]\subseteq \text{acl}(X\overline{b})$. So $\text{acl}_{[a]}$ is the minimal closure relation of the form $\text{acl}_A$ such that $D$ is defined over $A$ (i.e. such that we're sure to get a pregeometry). Commented Nov 23, 2015 at 5:54
• If you're not comfortable working with $M^{\text{eq}}$, another way of describing this canonical closure operator on $D$ is: $b\in \text{cl}(X)$ if and only if $b$ has only finitely many images under automorphisms of $M$ fixing $D$ set-wise and fixing $X$ point-wise. Commented Nov 23, 2015 at 18:02