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I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language.

Explicitly, if T is an L-theory and LP=L∪{P} (a new predicate). Under which conditions of T we have that the new theory TP obtained by a suitable interpretation of the predicate P becomes simple? or NIP? or stable?

I know, for example, that if T is simple and eliminates the quantifier ∃∞ then TP is simple (Chatzidakis, Pillay). But, what other theorems like this are known? Are there easy examples witnessing the failure of this ``preserving nice properties'' phenomena?

Thank you in advance for the possible answers...

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    $\begingroup$ What do you mean by a "suitable interpretation" of $P$? For any theory $T$, there are predicates that preserve all properties of the theory (e.g., the empty predicate), and there are predicates that destroy all nice properties (e.g., an elementhood predicate satisfying ZFC). (And, of course, for most niceness properties except elimination of imaginaries and the like, adding a predicate can never make the theory nicer than it already was.) $\endgroup$
    – Emil Jeřábek
    Commented Jul 10, 2015 at 9:46

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