Can you characterize property (1) below?
Are there interesting examples?
- Every partial isomorphism $f:M\to N$ between saturated models of $T$ such that $\mathrm{dom}f$ and $\mathrm{range}f$ are algebraically closed (in the respective models) is elementary
Ideally one would want a syntactic characterization. Some sort of elimination of quantifiers? For instance, the following variation of (1) is equivalent to model-completeness which equivalent elimination of "alternations" of quantifiers.
- Every partial isomorphism $f:M\to N$ between saturated models of $T$ such that $\mathrm{dom}f$ and $\mathrm{range}f$ are substructures that model of $T$ is elementary
Example removed
Your condition is equivalent to elimination of quantifiers "down to quantifiers over the algebraic closure".
Let's say an existential formula is explicitly algebraically bounded existential (EABE) if it is quantifier-free, or if it has the form $\exists x\,(\varphi(x,\overline{y})\land \exists^{\leq k} z\, \varphi(z,\overline{y}) \land \theta(x,\overline{y}))$, where $\theta$ is EABE, $\varphi$ is any formula, and $k\in \omega$.
For any tuple $\overline{a}$, let the EABE-type of $\overline{a}$, $\text{tp}^{\text{EABE}}(\overline{a})$ be the subset of $\text{tp}(\overline{a})$ consisting just of Boolean combinations of EABE formulas. Note that the EABE-type of $\overline{a}$ includes the information about which formulas are algebraic: if $\varphi(x,\overline{a})$ is algebraic, then $\text{tp}^{\text{EABE}}(\overline{a})$ includes the formula $\exists x\,(\varphi(x,\overline{a})\land \exists^{\leq k} z\, \varphi(z,\overline{a}) \land \top(x,\overline{a}))$ for some $k$, while if $\varphi(x,\overline{a})$ is not algebraic, $\text{tp}^{\text{EABE}}(\overline{a})$ contains the negation of all these formulas.
I think I've set this up correctly so that $\text{tp}^{\text{EABE}}(\overline{a})$ exactly describes the isomorphism type of $\text{acl}(\overline{a})$, since it knows which formulas are algebraic, and for each finite set of algebraic formulas, it can enumerate the realizations of these formulas and describe the isomorphism type of the substructure generated by these elements. Given that,
Claim: Your condition 1 is equivalent to the condition that every formula is equivalent to a boolean combination of EABE formulas (and you can equivalently drop the saturation condition on $M$ and $N$).
The implication from my condition to yours (without the saturation condition) is easy, since any isomorphism between algebraically closed subsets preserves truth of EABE formulas.
Conversely, given your condition, we have that $\text{tp}(\overline{a})$ is uniquely determined by its restriction to $\text{tp}^{\text{EABE}}(\overline{a})$, and by the usual compactness argument we get quantifier-elimination down to Boolean combinations of EABE formulas.
It's possible that by being a little more careful, you could come up with a nicer representation, especially in a theory which eliminates $\exists^\infty$. For example, are arbitrary Boolean combinations of EABE formulas really necessary? Can you control the form of the witnessing algebraic formulas (besides just saying that they're also equivalent to Boolean combinations of EABE formulas)?
A good example of a theory with these properties is ACFA. In fact, in this theory you can get away with a single existential quantifier, algebraic boundedness given by a polynomial, and no Boolean combinations (see the Completions and QE section of the wiki page).
I don't understand your proposed example in expansions of a stable theory. Why should $\mathcal{U}_1$ and $\mathcal{U}_2$ even be elementarily equivalent?