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 Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 2007, Volume 7, Issue 3, Pages 13–44 (Mi vngu265)

Conservative extensions of models with weakly o-minimal theories

B. S. Baizhanov

Institute for Problems of Informatics and Control Sciences, Almaty

Abstract: Let $M\prec N$. It is said that a pair of models $(M,N)$ is conservative pair and $N$ is conservative extension of $M$ if for any finite tuple of elements $\overline{\alpha}$ from $N$, $\mathrm{tp}(\overline{\alpha}|M)$ is definable. We say that elementary extension $N$ of $M$ is $D$-good if any definable $q\in S(M\cup\overline{\alpha})$ ($\overline{\alpha}\in N\setminus M$) is realized in $N$ and $N$ is $CD$-good if any non-isolated one-type $q\in S_1(M\cup\overline{\alpha})$ ($\overline{\alpha}\in N\setminus M$), which is determined (approximated) by definable $\phi$-type, is realized in $N$.
We prove that any model $M$ of any weakly o-minimal theory except one, theory of discrete order with ends, has conservative extension. The central point in our paper is the criterion of the existence of the $CD$-$\omega$-saturated conservative extension of an arbitrary model of weakly o-minimal theory (Theorem 2). As corollary of this proof it follows the existence of $CD$-$\omega$-saturated conservative extension for any model of any weakly o-minimal theory except one and the results on omitting of natural family of definable one-types and all non-definable types (Corollary 5). The existence of conservative and $CD$-$\omega$-saturated conservative extensions for o-minimal theories have been proved accordingly in D. Marker, “Omitting types in o-minimal theories”, The Journal of Symbolic Logic, Vol. 51(1986), P. 63–74., Y. Baisalov, B. Poizat, “Paires de structures o-minimales”, The Journal of Symbolic Logic, Vol. 63(1998), P. 570–578.

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Document Type: Article
UDC: 510.67

Citation: B. S. Baizhanov, “Conservative extensions of models with weakly o-minimal theories”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 7:3 (2007), 13–44

Citation in format AMSBIB
\Bibitem{Bai07} \by B.~S.~Baizhanov \paper Conservative extensions of models with weakly o-minimal theories \jour Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. \yr 2007 \vol 7 \issue 3 \pages 13--44 \mathnet{http://mi.mathnet.ru/vngu265}