Example of structure where the exchange principle for model-theoretic algebraic closure does not hold?

Question

I need an example of a structure such that the exchange principle for model-theoretic algebraic closure does not hold, which is, it is not true that whenever $a \in alc(b, A) $ and $a \notin alc(A) $ we have $b \in alc(a, A)$, where $A$ is a set of parameters. I know that it holds in minimal and o-minimal structures, so I have to think about something that does not belong in these classes, but I still have no idea.

Answer 1

Set $A=\emptyset$ for simplicity. What we're looking for is a structure in which some infinite set $B$ of elements all "single out" some element $a$ in a definable way; the idea then is that $a\in acl(b)$ (indeed $dcl(b)$) for each $b\in B$, but since $B$ is infinite we can't obviously "reverse" the process.

There are many ways to do this. I'm particularly a fan of the following structure $\mathcal{S}$:

• The underlying set of $\mathcal{S}$ is $\mathbb{N}^{\mathbb{N}}$, the set of infinite sequences of natural numbers.

• The signature of $\mathcal{S}$ is the "backshift" operation, $$p:(a_0,a_1,a_2,a_3,...)\mapsto (a_1,a_2,a_3,...).$$

Note that we crucially do not have any "lateral" ordering: there's no sense in which $(1,1,1,1,...)$ is "to the left of" $(2,2,2,2,...)$, and so forth.

It's easy to check (think about automorphisms) that we always have $$acl(\alpha)=dcl(\alpha)=\{p^n(\alpha): n\in\mathbb{N}\}.$$ In particular, we have $$\alpha\in acl(\beta)\mbox{ and }\beta\in acl(\alpha)\quad\iff\quad \alpha=\beta.$$ So exchange fails everywhere.

(Note that we can tweak this example to get a structure where exchange holds for $acl$ but fails for $dcl$: just work with $2^\mathbb{N}$ instead of $\mathbb{N}^\mathbb{N}$.)