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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 $L_P=L\cup \{P\}$ (a new predicate). Under which conditions of $T$ we have that the new theory $T_P$ 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 $\exists^\infty$ then $T_P$ 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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You may want to ask it on MathOverflow (though you might be flamed in response). –  Yuval Filmus Mar 23 '12 at 4:41
    
This is a somewhat complicated matter, and I think it would make a very reasonable MathOverflow question. –  Alex Kruckman Apr 27 '12 at 3:49
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