The [Łos-Tarski preservation theorem](http://en.wikipedia.org/wiki/%C5%81o%C5%9B%E2%80%93Tarski_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 $L$*?
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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](http://dx.doi.org/10.1145/1379759.1379763) ([preprint](http://www.mit.edu/~brossman/hpt-jacm-final.pdf))



Higher-order preservation theorems?
There are a number of preservation results concerning the infinitary logic $L\_{\omega\_{1},\omega}$, as well as its so-called *admissible fragments*. Such results include an analogue of the one you mentioned about existential sentences.


Here are some basic sources in this direction:


H.J. Keisler, **Model theory for infinitary logic.** Logic with countable conjunctions and finite quantifiers. Studies in Logic and the Foundations of Mathematics, Vol. 62. North-Holland Publishing Co., Amsterdam-London, 1971 **[see Chapter 7].**


The following paper also establishes a preservation theorem for "end extensions" (which were called "outer extensions" in the old days).


S. Feferman, *Persistent and invariant formulas for outer extensions*.
**Compositio Math.** 20 1968 29–52 (1968). 


See this [recent paper](http://www.springerlink.com/content/h76g2125172j0071/) for survey of related work to Feferman's result.


