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\newtheorem{exercise}{Al\i\c st\i rma}

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\begin{document}
\title{Modeller kuram\i\ al\i\c st\i rmalar\i}
\author{David Pierce}
\date{\today}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U}
\maketitle

\begin{exercise}
  Derste sadece i\c slem simgesi olmayan imzalar\i n terimlerini tan\i mlad\i k.
  \c Simdi her imzan\i n terimlerini tan\i mlayaca\u g\i z.
  Tan\i ma g\"ore her de\u gi\c sken bir \textbf{terimdir.}
  Ayr\i ca, rasgele bir imzada,
  \begin{compactenum}[(i)]
  \item
    her de\u gi\c smez bir terimdir, ve
  \item
    her $n$ sayma say\i s\i\ i\c cin,
    her $n$-konumlu $F$ i\c slem simgesi i\c cin,
    t\"um $t_0$, \dots, $t_{n-1}$ terimleri i\c cin
    \begin{equation*}
      Ft_0\cdots t_{n-1}
    \end{equation*}
    ifadesi de bir terimdir.
    E\u ger $n=2$ ise
    \begin{equation*}
    (t_0\mathbin Ft_1)  
    \end{equation*}
    ifadesi de kullan\i labilir.
  \end{compactenum}
  Hi\c c de\u gi\c skenin g\"oz\"ukmedi\u gi terim, \textbf{sabittir.}
Bir $\sig$ imzas\i\ i\c cin $\str A$, imzas\i\ $\sig$ olan bir yap\i\ olsun.
\"Ozyinelemeyle
$\sig$'nin her sabit $t$ teriminin $\str A$'daki
$t^{\str A}$ yorumunu tan\i mlayaca\u g\i z.
\begin{compactenum}[I.]
  \item
  E\u ger $t$ bir de\u gi\c smez ise,
  o zaman $t^{\str A}$ zaten tan\i mlanm\i\c st\i r.
\item
  Ayr\i ca $t$, yukar\i daki $Ft_0\cdots t_{n-1}$ terimiyse
  \begin{equation*}
    t^{\str A}=F^{\str A}(t_0{}^{\str A},\dots,t_{n-1}{}^{\str A}).
  \end{equation*}
\end{compactenum}
\.Imzas\i n\i n hi\c c y\"uklemi olmayan bir yap\i, bir \textbf{cebirdir.}
(\"Orne\u gin gruplar, cisimler, ve vekt\"or uzaylar\i, cebirdir.)
  \c Simdi bir $\str A$ cebirinin imzas\i\ $\sig$ olsun;
$\str B$'nin imzas\i\ ayn\i\ olsun;
  ve $h\colon A\to B$ olsun.
  $\sig$'n\i n her $d$ de\u gi\c smezi i\c cin
  \begin{equation*}
    h(d^{\str A})=d^{\str B}
  \end{equation*}
  olsun, ve her $n$ i\c cin,
  $\sig$'nin her $n$-konumlu $F$ i\c slem simgesi i\c cin,
  $A^n$ kuvvetinin her $\vec a$ eleman\i\ i\c cin
  \begin{equation*}
    h(F^{\str A}(\vec a))=F^{\str B}(h(a_0),\dots,h(a_{n-1}))
  \end{equation*}
  olsun.
  T\"umevar\i mla $\sig$'nin her sabit $t$ terimi i\c cin
  \begin{equation*}
    h(t^{\str A})=t^{\str B}
  \end{equation*}
  g\"osterin.
  (E\u ger ayr\i ca $h$ bir e\c slemeyse, o zaman bir \textbf{izomorfizimdir,}
  ve $\str A\cong\str B$.)
\end{exercise}

\begin{exercise}
Verilen k\"umelerin verilen yap\i larda tan\i mlanabildi\u gini g\"osterin.
\begin{enumerate}[a)]
\item
$\{\text{\c cift say\i lar}\}$, $(\Z,+)$'da
\item
$\{1\}$, $(\Z,\times)$'da
\item
$\{\text{asal say\i lar}\}$, $(\N,\times,1)$'de
\item
  $\{\text{asal say\i lar}\}$, $(\Z,\times,1)$'de
\end{enumerate}
\end{exercise}

\begin{exercise}
  $G$, $(\Z,+,0,-)$ abelyan grubu olsun, ve $H$,
  $G$'nin $\{x+x\colon x\in\Z\}$ altgrubu olsun.
  (Bu durumda $G/H$ b\"ol\"um\"u $\Z_2$.)
  $G$'de
  \begin{equation*}
  \{(x,y)\colon x+H=y+H\}  
  \end{equation*}
    ba\u g\i nt\i s\i n\i\ tan\i mlayan bir form\"ul bulun.
\end{exercise}

\begin{exercise}
Abelyan gruplar\i n $\{+,0,-\}$ imzas\i\ $\sig$ olsun.
Bu imzada \"oyle $\sigma$ ve $\tau$ c\"umlelerini bulun ki
imzas\i\ $\sig$ olan her $\str A$ yap\i s\i\ i\c cin
\begin{gather*}
	\str A\models\sigma\iff\str A\cong\Z_2,\\
	\str A\models\tau\iff\str A\cong\Z_2\oplus\Z_2.
\end{gather*}
\end{exercise}

\begin{exercise}
$\Th{\Z,+}\neq\Th{\Z\oplus\Z,+}$ g\"osterin.
\end{exercise}

\end{document}
