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{\Large\bf Teori zincirleri}
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David PIERCE

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\textit{Mimar Sinan G\"uzel Sanatlar \"Universitesi, Matematik B\"ol\"um\"u, Bomonti, \c Si\c sli 34380, \.Istanbul}
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\small{\textit{dpierce@msgsu.edu.tr}}
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{\bf \"Ozet}
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\noindent 
Bir cismin cebirsel kapan\i\c s\i, cisimler zincirinin bir bile\c simidir.  Bu bile\c sim, bir cisimdir, \c c\"unk\"u cisim aksiyomlar\i, $\forall\exists$ bi\c cimindedir, \"orne\u gin $\forall x\;\exists y\;(x=0\lor xy=1)$.  \.Iki-boyutlu vekt\"or uzaylar\i\ aksiyomlar\i, $\forall\exists$ bi\c ciminde olamaz, \c c\"unk\"u bir $2$-boyutlu vekt\"or uzaylar\i\ zincirinin bile\c simi, $1$-boyutlu olabilir \cite{MR2505433}.  Ters olarak, e\u ger bir teorinin modelleri zincirinin her bile\c simi, bu teorinin modeliyse, bu teoriye $\forall\exists$ aksiyomlar verilebilir \cite{MR0103812,MR0089813}.  \"Orne\u gin sadece \c carpma i\c cin bir i\c saret kullanarak bile grup teorinin aksiyomlar\i, $\forall\exists$ bi\c ciminde yaz\i labilir.

Bildi\u gim kadar\i\ ile, \emph{teori} zincirleri ara\c st\i rmalar\i, \"Ozcan Kasal \cite{Kasal} ve (ba\u g\i ms\i z olarak) Alice Medvedev \cite{Medvedev} taraf\i ndan ba\c slat\i lm\i\c st\i r.  $T_0\subseteq T_1\subseteq T_2\subseteq\cdots$, bir teoriler zinciri olsun.  E\u ger her $T_k$ teorisi, modellerine g\"ore tam (\emph{model complete}) veya istikrarl\i\ (\emph{stable}) ise, $\bigcup_{k\in\upomega}T_k$ bile\c simi de ayn\i\ \"ozelli\u gi ta\c s\i r \cite{Medvedev-preprint}.  Ama her $T_k$ teorisi \c cok istikrarl\i\ (\emph{superstable}) iken, zincirin bile\c simi \c cok istikrarl\i\ olmayabilir \cite{Medvedev-preprint}; ve her $T_k$ teorisinin model arkada\c s\i\ (model companion) olurken zincirin bile\c siminin model arkada\c s\i\ olmayabilir \cite{2013arXiv1303.6759K}.  \.Inceleyece\u gimiz \"orneklerde $T_k$, ya $k$-konumlu lineer ba\u g\i ms\i zl\i k i\c saretiyle vekt\"or uzaylar\i\ teorisi \cite{MR2505433}, ya da $k$ tane de\u gi\c smeli t\"urevleme i\c saretiyle cisimler teorisidir \cite{2007arXiv0708.2769P}.  Her durumda, modellerine g\"ore tam ve istikrarl\i\ olan yeni bir teori \c c\i kar \cite{2013arXiv1303.6759K}.


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\noindent \small{{\bf 2010 AMS Konu S\i n\i fland\i r\i lmas\i: 03C10, 03C60, 12H05, 13N15}\\
\noindent \small{{\bf Anahtar Kelimeler:} chain of structures, chain of theories, model complete theory, stable theory.}
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\bibitem{MR0103812}
Chen~Chung Chang, \emph{On unions of chains of models}, Proc. Amer. Math. Soc.
  \textbf{10} (1959), 120--127. \MR{MR0103812 (21 \#2576)}

\bibitem{2013arXiv1303.6759K}
{\"O}.~{Kasal} and D.~{Pierce}, \emph{{Chains of Theories and
  Companionability}}, ArXiv e-prints (2013), {\tt arXiv:1303.6759 [math.LO]}.

\bibitem{Kasal}
{\"O}zcan Kasal, \emph{Model theory of derivation spaces}, Ph.D. thesis, Middle
  East Technical University, Ankara, February 2010.

\bibitem{MR0089813}
Jerzy {\L}o{\'s} and Roman Suszko, \emph{On the extending of models ({IV}):
  {I}nfinite sums of models}, Fund. Math. \textbf{44} (1957), 52--60.
  \MR{MR0089813 (19,724c)}

\bibitem{Medvedev}
Alice Medvedev, \emph{{$\mathbb Q$ACFA}}, Talk given at Recent Developments in
  Model Theory, Ol{\'e}ron, France,
  \url{http://modeltheory2011.univ-lyon1.fr/abstracts.html}, June 2011.

\bibitem{Medvedev-preprint}
\bysame, \emph{{$\mathbb Q$ACFA}}, preprint,
  \url{http://math.berkeley.edu/~alice/grouplessqacfa.pdf}, November 2012.

\bibitem{2007arXiv0708.2769P}
David Pierce, \emph{{Fields with several commuting derivations}}, ArXiv
  e-prints (2007), {\tt arXiv:0708.2769 [math.LO]}, to appear in the Journal of
  Symbolic Logic.

\bibitem{MR2505433}
\bysame, \emph{Model-theory of vector-spaces over unspecified fields}, Arch.
  Math. Logic \textbf{48} (2009), no.~5, 421--436. \MR{MR2505433}

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