Consider two structures which are elementary equivalent. If one is minimal/o-minimal, can the same be said about the other?
The pooint is that one cannot quantify over sets (in first-order logic), but perhaps definable sets are "special". As an example, I believe that in some cases of models of theories which admit quantifier elimination, such as Real Closed Fields, the conclusion is true.
Nevertheless, I fail to find an exhaustive approach to the problem.
Thank you in advance for any help.
O-minimality is preserved by elementary equivalence (it is in "Definable sets in ordered structures II" by Knight, Pillay and Steinhorn, thanks to Alex Kruckman for the reference).
Minimality however is not preserved. E.g. every definable set in $(\mathbb N, <)$ is finite or cofinite, I think it has quantifier elimination in the language where you add $0$ and the successor function $s(x) = x+1$. However it has order, so is unstable and therefore not all models are minimal.
This is the reason that there is a notion of "strong minimality" which says that a structure and all structures elementarily equivalent to it are minimal.