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@YCor - I hope you saw Angelo's comment. (At the end of the question I mentioned Eric Wofsey' example of a non-surjective epimorphic endomorphism of a noetherian ring.)
@LSpice - Thanks for your comment. The main partial result is that (as indicated in the post) the properties are equivalent for posets. I had previously asked if (P2) implies (P1). Apparently nobody knows. So I thought that answers to closely related questions might shed some light on the question asked before, and that, asking many such related questions was a way of maximizing the probability of getting some of them answered. I wish I had better ideas about how to tackle the main question.
@FrançoisG.Dorais - What do you exactly mean by "if $A'$ is any isomorphic structure to $A$, then the isomorphism of $A'$ and $A$ lifts uniquely to an isomorphism of $V(A')$ and $V(A)$"? [I'm asking this because it seems to me that $A'$ equipotent to $A$ does not imply $V_1(A')$ equipotent to $V_1(A)$ --- take $A=\{\varnothing\},A'=\{\{\varnothing\}\}$.]
Very nice! Can't you even start with two complete ordered fields and show that if the Birch-Swinnerton-Dyer Conjecture holds over one, it holds over the other?
