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...

  • $\begingroup$ You may want to ask it on MathOverflow (though you might be flamed in response). $\endgroup$ – Yuval Filmus Mar 23 '12 at 4:41
  • $\begingroup$ This is a somewhat complicated matter, and I think it would make a very reasonable MathOverflow question. $\endgroup$ – Alex Kruckman Apr 27 '12 at 3:49

In general, you'll need additional conditions on the structure that $T$ induces on the predicate. You might be interested by stable embeddedness.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.