The Łos-Tarski preservation theorem states that a set of formulas $F$ of first-order language $L$ is preserved under substructures for models of theory $T$ in $L$ precisely when $F$ is equivalent modulo $T$ to a set of universal formulas of $L$. The often-seen corollary is that a formula of $L$ is preserved by embeddings between models of $T$ precisely when it is equivalent modulo $T$ to an existential formula of $L$.
Lyndon's preservation theorem has a similar flavour, relating positive formulas and surjective homomorphisms. Moreover, first-order formulas preserved under homomorphisms are precisely those that are equivalent to an existential positive first-order formula with the same quantifier-rank.
These kinds of preservation results use first-order theories. However, some higher-order logics, such as the monadic fragment of second-order logic, are sufficiently restricted that they retain many of the properties of first-order logic. This motivates my question:
Are any preservation results known for logics $L$ and theories $T$ beyond first-order? In other words, are there results such as: a formula of logic $L$ is preserved under embeddings between models of theory $T$ in $L$ precisely when it is equivalent modulo $T$ to an existential formula of $L$?
Pointers or even key phrases would be appreciated. I'm struggling to find relevant references, which may indicate that I have overlooked a trivial reason why such theorems cannot exist: if this is the case then a hint would be appreciated.
References:
- Wilfrid Hodges, Model Theory, Cambridge University Press, 1993.
- Benjamin Rossman, Homomorphism preservation theorems, Journal of the ACM 55(3), 2008. doi:10.1145/1379759.1379763 (preprint)
