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The question I want to ask is close to but not exactly what stated in the title:

Fix a language $L$, it is known that a statement $\sigma$ is universal in the language if whenever $M$ satisfies $\sigma $ and $N$ is a substructure of $M$ then $N$ also satisfy $\sigma$. It is also known that a statement $\sigma$ is existential if whenever $M$ satisfies $\sigma $ and $N$ is an extension of $M$ then $N$ also satisfy $\sigma$.

I can not find generalization of these criteria for formulas with more quantifiers. I wonder why this is the case?

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up vote 7 down vote accepted

There is one more well-known equivalence for $\forall \exists$ sentences.

Theorem (Chang-Los-Suszko). A theory $T$ is preserved under taking unions of increasing chains of structures if and only if $T$ is equivalent to a set of $\forall \exists$ sentences.

For a proof, see Keisler, "Fundamentals of model theory", Handbook of Mathematical Logic, p. 63.

I found a related paper, which is older and doesn't quite answer your question but may be of interest. R. C. Lyndon, "Properties preserved under algebraic constructions", Bull. Amer. Math. Soc. 65 n. 5 (1959), 287-299, Project Euclid

According to that paper, and MathSciNet, a general solution to your question should be contained in H. J. Keisler, "Theory of models with generalized atomic formulas", J. Symbolic Logic v. 25 (1960) 1-26, MathSciNet, JStor

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You mean Keisler's sandwich theorem? Chang-Keisler, "Model Theory", theorem 5.2.8: A theory has a $\Pi_{2n}$-set of axioms if the class of its models is closed under $n$-sandwiches.

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The result about sandwiches can also be found in the exercises of the model theory chapter in Shoenfield's "Mathematical Logic". – Andreas Blass Jan 9 '12 at 1:09

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