[FOM] Strongly Minimal and Minimal Structures.

[Jizhan Hong]

In case anyone is wondering, Prof. John Baldwin has given me the
answers. Let me take this opportunity to thank Dr. Baldwin again.

Also, while I am writing this message, Artem Chernikov sent me a
message to confirm the following proofs too. Thanks Artem Chernikov. I
also want to thank Grant Olney Passmore for his helpful advice.

> 1. Given a first order language L and its extension L', and an
> L'-structure M, if M as a L'-structure is strongly minimal, is it
> still strongly minimal as an L-structure? I believe it's not, but I
> just couldn't find an explicit counter-example.

M is still L-strongly minimal. Since for any L-formula $\phi(x,y)$,
there exist some $m$ and $n$ such that M says that for all y,
$\phi(x,y)$ has at most $m$ elements or $\neg \phi(x,y)$ has at most
$n$ elements. Thus any $N$ L-elementary equivalent to $M$, $N$ is
L-minimal, meaning $M$ is still $L$-strongly minimal.

>
> 2. This is sort of related to the first one: By extending the
> language, can one always get a strongly minimal expansion for any
> minimal structure?
>

By Problem 1, this is possible iff the given structure is strongly
minimal already.

> 3. Is the conjecture that any minimal field is algebraically closed
> solved? (I know that F. Wagner proved it for the positive
> characteristic case)
>

It is still open.
Artem Chernikov said F. Wagner confirmed that it's widely open.

Thanks a lot for the help!
Jizhan Hong