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\begin{document}
\title{Modeller Kuram\i\ (TASLAK)}
\author{David Pierce}
\date{23 Mart 2017}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\mbox{}\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\uppertitleback{\tableofcontents}

\maketitle

\addsec{\"Ons\"oz}

Bildi\u gim ve kulland\i\u g\i m modeller kuram\i\ kitaplar\i\
ilk yay\i m tarihlerine g\"ore
a\c sa\u g\i da
s\i ralanm\i\c st\i r.
\begin{compactdesc}
\item[1956] Robinson \emph{Complete Theories}
\cite{MR0472504}
\item[1963] Robinson 
\emph{Introduction to Model Theory 
and to the Metamathematics of Algebra} \cite{MR0153570}
\item[1965] Robinson \emph{Non-standard Analysis}
\cite{MR1373196}
\item[1967] Shoenfield \emph{Mathematical Logic}
\cite{MR1809685}
\item[1969]  Bell \&\ Slomson 
\emph{Models and Ultraproducts: An Introduction}
\cite{MR0269486}
\item[1973]  Chang \&\ Keisler \emph{Model Theory}
\cite{Chang--Keisler}
\item[1985]  Poizat \emph{Cours de th\'eorie des mod\`eles}
\cite{MR2001a:03072,MR817208}
\item[1993]  Hodges \emph{Model Theory} \cite{MR94e:03002}
\item[1995]  Rothmaler \emph{Introduction to Model Theory}
\cite{MR1800596}
\item[2002] Marker \emph{Model Theory: An introduction}
\cite{MR1924282}
\item[2003] Marcja \&\ Toffalori 
\emph{A Guide to Classical and Modern Model Theory}
\cite{MR1997808} 
\item[2012] Tent \&\ Ziegler \emph{A Course in Model Theory}
\cite{MR2908005}
\end{compactdesc}
T\"urk\c ce ifadelerde yard\i m etti\u gi i\c cin 
Ay\c se Berkman'a te\c sekk\"ur ederim.  
Ali Nesin'in \emph{Analiz IV} kitab\i n\i\ da 
\cite{Nesin-Analiz-IV} 
matematiksel T\"urk\c ce \"orne\u gi olarak kulland\i m.  
Baz\i\ terimler, Gr\"unberg ile Onart \cite{MTS} 
ve Demirta\c s \cite{Demirtas} 
taraf\i ndan yaz\i lm\i\c s kitaplardan al\i nm\i\c st\i r.


\section{Do\u gal say\i lar}

\c Co\u gunlukla s\i ral\i\ bir $n$-li $(a_1,\dots,a_n)$ olarak yaz\i l\i r,
ama bu metinde $(a_0,\dots,a_{n-1})$ tercih ediliyor.

Tan\i ma g\"ore $\upomega$, 
\"oyle $\xi$ k\"umelerinin en k\"u\c c\"u\u g\"ud\"ur ki
\begin{compactenum}[1)]
\item
$\emptyset\in\xi$, ve
\item
her $\alpha$ k\"umesi i\c cin $\alpha\in\xi$ ise $\alpha\cup\{\alpha\}\in\xi$.
\end{compactenum}
O zaman $\upomega$, \textbf{do\u gal say\i lar k\"umesidir.}
\"Ozel olarak
\begin{align*}
0&=\emptyset,&1&=\{0\},&2&=\{0,1\},&3&=\{0,1,2\},
\end{align*}
ve genelde $n\in\upomega$ ise
\begin{equation*}
n=\{0,\dots,n-1\}.
\end{equation*}

Her $A$ k\"umesi i\c cin $A^n$ kuvveti,
elemanlar\i\ $n$'den $A$'ya giden g\"ondermeler olan k\"umedir.
Bu \c sekilde $A^n$ kuvvetinin ayn\i\ eleman\i,
\begin{align*}
	&\vec a,&&(a_0,\dots,a_{n-1}),&&(a_i\colon i<n),&&i\mapsto a_i
\end{align*}
bi\c cimlerinde yaz\i labilir.
\"Ozel olarak
\begin{equation*}
A^0=\{\emptyset\}=\{0\}=1.
\end{equation*}
%Herhangi $B$ k\"umesi i\c cin $A^B$ kuvveti,
%elemanlar\i\ $B$'den $A$'ya giden g\"ondermeler olan k\"ume olarak
%anla\c s\i labilir.
$\upomega\setminus\{0\}$ k\"umesi, $\N$ \textbf{sayma say\i lar\i\ k\"umesidir.}

\section{\.Imzalar ve yap\i lar}

\subsection{\.Imzalar}

$\R$ ger\c cel say\i lar s\i ralanm\i\c s cisminin \emph{imzas\i,}
\begin{equation*}
\{+,0,-,\times,1,<\}
\end{equation*}
k\"umesidir.
Burada
\begin{compactitem}
\item
 $0$ ve $1$, \emph{de\u gi\c smezdir;}
\item
$+$ ve $\times$, \emph{$2$-konumlu i\c slem simgeleridir;}
\item
$-$, $1$-konumlu i\c slem simgesidir;
\item
$<$, $2$-konumlu \emph{y\"uklemdir.}
\end{compactitem}
Genelde bir \textbf{imzan\i n} her eleman\i,
\begin{compactenum}[1)]
\item
ya \textbf{de\u gi\c smezdir,}
\item
ya da bir $n$ sayma say\i s\i\ i\c cin
\begin{compactenum}
\item
ya $n$-konumlu bir \textbf{i\c slem simgesidir,}
\item
ya da $n$-konumlu bir \textbf{y\"uklemdir.}
\end{compactenum}
\end{compactenum}
Bu metinde $\sig$ her zaman bir imza olacak.

\subsection{Yap\i lar}

$\R$ ger\c cel say\i lar s\i ralanm\i\c s cismi bir \emph{yap\i d\i r.}
\.Imzas\i\ $\sig$ olan bir \textbf{yap\i,}
\"oyle bir $(A,S\mapsto S^{\str A})$ s\i ral\i\ ikilisidir ki
\begin{compactenum}[1)]
\item
$A$ bir k\"umedir, ve
\item
$S\mapsto S^{\str A}$, tan\i m k\"umesi $\sig$ olan
bir g\"ondermedir, ve
\begin{compactenum}
\item
$\sig$'nin her $d$ de\u gi\c smezi i\c cin
\begin{equation*}
d^{\str A}\in A;
\end{equation*}
\item
$\sig$'nin her $n$-konumlu $F$ i\c slem simgesi i\c cin
\begin{equation*}
F^{\str A}\colon A^n\to A,
\end{equation*}
yani $F^{\str A}$, $A$'da $n$-konumlu bir \textbf{i\c slemdir;}
\item
$\sig$'nin her $n$-konumlu $R$ y\"uklemi i\c cin
\begin{equation*}
R^{\str A}\included A^n,
\end{equation*}
yani $R^{\str A}$, $A$'da $n$-konumlu bir \textbf{ba\u g\i nt\i d\i r.}
\end{compactenum}
\end{compactenum}
$(A,S\mapsto S^{\str A})$ yap\i s\i,
sadece $\str A$ olarak yaz\i labilir.
Bu yap\i n\i n \textbf{evreni,} $A$'d\i r.
$\sig$'nin her $S$ eleman\i\ i\c cin $S^{\str A}$,
$S$'nin $\str A$'daki \textbf{yorumudur.}
\.Imzas\i\ $\sig$ olan yap\i lar
\begin{equation*}
\Str{\sig}
\end{equation*}
s\i n\i f\i n\i\ olu\c sturur.

E\u ger $\sig=\{S_0,S_1,\dots\}$ ise $\str A$,
$(A,S_0{}^{\str A},S_1{}^{\str A},\dots)$ olarak yaz\i labilir.
E\u ger farkl\i\ bir yap\i\ $\str A$'n\i n evrenini kullanmazsa
$\str A$, $(A,S_0,S_1,\dots)$ olarak yaz\i labilir.
Normalde bir $\str B$ yap\i s\i n\i n evreni $B$, ve saire
(\sayfaya{German} bak\i n);
ama tutarl\i\ olmak zor veya imk\^ans\i zd\i r.

\begin{example}
Ger\c cel say\i lar k\"umesi $R$ ise,
o zaman ger\c cel say\i lar s\i ralanm\i\c s cismi
$(R,+^{\str R},0^{\str R},-^{\str R},\times^{\str R},1^{\str R},<^{\str R})$;
ama normalde $\R$ ifadesi
hem k\"ume hem de s\i ralanm\i\c s cisim i\c cin kullan\i l\i r.
\end{example}

\subsection{\"Ornekler}

\begin{asparaenum}
  \item
  A\c sa\u g\i daki yap\i\ \"ornekleri matematikte s\i k s\i k kullan\i l\i yor.
  \begin{compactenum}
\item
$(\C,+,0,-,\times,1)$ karma\c s\i k say\i lar cismi.
\item
$p$ asal olmak \"uzere $p$-elemanl\i\ $(\F_p,+,0,-,\times,1)$ cismi.
\item
$(\Q,+,0,-,\times,1,<)$ kesirli say\i lar s\i ralanm\i\c s cismi.
\item
Bir $(G,\times,\gid,{}\inv)$ grubu.
\item
$(\Q,+,0,-)$ abelyan grubu.
\item
$(\Z,+,0,-)$ tamsay\i lar abelyan grubu.
\item
$(\Z,+,0,-,\times,1)$ tamsay\i lar de\u gi\c smeli halkas\i.
\item
$(\Q,<)$ kesirli say\i lar do\u grusal s\i ras\i.
\item
$(\upomega,<)$ do\u gal say\i lar do\u grusal s\i ras\i.
\item
$(\N,{}\divides{})$ par\c cal\i\ s\i ras\i.
  \end{compactenum}
	\item
Bir	$(K,+,0,-,\times,1)$ cismi verilirse
\begin{align*}
\Mat^{2\times 2}(K)&=\left\{\begin{bmatrix}x^0_0&x^0_1\\x^1_0&x^1_1\end{bmatrix}\colon x^i_j\in K\right\},&
I&=\begin{bmatrix}1&0\\0&1\end{bmatrix}
\end{align*}
olmak \"uzere $(\Mat^{2\times 2}(K),+,0,-,\times,I)$
de\u gi\c smeli olmayan matrisler halkas\i\ elde edilebilir.
\item
  Bir $\Omega$ k\"umesinin altk\"umeleri
  \begin{equation*}
    \pow{\Omega}
  \end{equation*}
kuvvet k\"umesini olu\c sturur, 
ve bu k\"umeden\mygloss[sym]{$\pow{\Omega}$} 
\begin{compactenum}
\item
$(\pow{\Omega},\included)$ par\c cal\i\ s\i ras\i,
\item
$(\pow{\Omega},\cup,\emptyset,\cap,\Omega,{}')$
\textbf{Boole cebiri} 
\end{compactenum}
elde edilir. 
\item
Herhangi k\"ume, imzas\i\ bo\c s olan bir yap\i d\i r.
\item
E\u ger $E$, bir $A$ k\"umesinde bir denklik ba\u g\i nt\i s\i\ ise,
o zaman $(A,E)$ s\i ral\i\ ikilisi bir yap\i d\i r.
Bu durumda $b\in A$ ise $b$'nin
$\{x\in A\colon x\mathrel Eb\}$ denklik s\i n\i f\i\
$[b]$ olarak yaz\i labilir.  O zaman tan\i ma g\"ore
\begin{equation*}
A/E=\{[x]\colon x\in A\}.
\end{equation*}
\item
$n\in\N$ ise $n$ mod\"ul\"une g\"ore kalanda\c sl\i k,
$\Z$'de bir denklik ba\u g\i nt\i s\i d\i r.
Bu durumda $\Zmod n=\{[x]\colon x\in\Z\}$ ise
\begin{equation*}
(\Zmod n,+,0,-,\times)
\end{equation*}
de\u gi\c smeli halkas\i\ elde edilir.
\item
E\u ger $K$ bir cisim ise, 
o zaman $K$ \"uzerinde vekt\"or uzaylar\i n\i n imzas\i
\begin{equation*}
\{+,\bm0,-\}\cup\{F_a\colon a\in K\}
\end{equation*}
olarak al\i nabilir.
\c Simdi $\str V$,
$K$ \"uzerinde bir vekt\"or uzay\i\ olsun.
E\u ger $\bm u\in V$ ise, o zaman tan\i ma g\"ore
\begin{equation*}
F_a{}^{\str V}(\bm u)=a\cdot\bm u.
\end{equation*}
Her $n$ sayma say\i s\i\ i\c cin
$\str V$'nin imzas\i na yeni $n$-konumlu bir $\parallel_n$ y\"uklemi eklenebilir,
ve bu simgenin $\str V$'deki yorumu,
$n$-konumlu do\u grusal ba\u g\i ml\i l\i k olabilir.
Bu \c sekilde $(\bm u_0,\dots,\bm u_{n-1})\in{\parallel_n}^{\str V}$
ancak ve ancak $K$'nin, biri $0$ olmayan baz\i\ $a_i$ elemanlar\i\ i\c cin
\begin{equation*}
F_{a_0}{}^{\str V}(\bm u_0)+\dots+F_{a_{n-1}}{}^{\str V}(\bm u_{n-1})=\bm 0.
\end{equation*}
O zaman $(V,\parallel_1,\parallel_2,\parallel_3,\dots)$ yap\i s\i\ incelenebilir.
\item
Tekrar $K$ bir cisim olsun, ve $n\in\N$ olsun.
O zaman $K^{n+1}$ kuvveti, bir $V$ i\c c \c carp\i m uzay\i\ olarak anla\c s\i labilir.
$V$'nin her $\bm a$ eleman\i\ 
$\begin{bmatrix}a^0\\\vdots\\a^n\end{bmatrix}$ s\"utun vekt\"or\"u olarak al\i ns\i n.
E\u ger $V^n$ kuvvetinin $(\bm a_j\colon j<n)$ eleman\i\ verilirse,
$i\leq n$ olmak \"uzere
\begin{equation*}
A_i=
\begin{bmatrix}
         a^0_0 &\cdots&         a^0_{n-1} \\
\vdots         &      &\vdots             \\
\widehat{a^i_0}&\cdots&\widehat{a^i_{n-1}}\\
\vdots         &      &\vdots             \\
         a^n_0 &\cdots&         a^n_{n-1}
\end{bmatrix}
\end{equation*}
olsun.
Yani
$n\times(n-1)$'lik $(\bm a_0\;\cdots\;\bm a_{n-1})$ matrisinin $i$'ninci sat\i r\i\ silinirse
kare $A_i$ matrisi elde edilsin.
\c Simdi
\begin{equation*}
X(\bm a_j\colon j<n)=
\begin{bmatrix}
\det(A_0)\\\vdots\\(-1)^j\det(A_j)\\\vdots\\(-1)^{n-1}\det(A_n)
\end{bmatrix}
\end{equation*}
olsun.
Bu \c sekilde $V$'nin her $\bm u$ eleman\i\ i\c cin
\begin{equation*}
\bm u\cdot X(\bm a_j\colon j<n)=\det\begin{bmatrix}\bm u&\bm a_0&\cdots&\bm a_{n-1}\end{bmatrix}.
\end{equation*}
Burada $X$, $V$'de $n$-konumlu i\c slemdir.
E\u ger $n=2$ ise $X(\bm a,\bm b)$,
normal $\bm a\times\bm b$ \c capraz \c carp\i m\i d\i r \cite{MR0209411}.
\end{asparaenum}

\section{\.Ifadeler}

\subsection{Terimler}

Her $n$ do\u gal say\i s\i\ i\c cin
\begin{equation*}
x_n
\end{equation*}
simgesi
\textbf{de\u gi\c sken} olarak anla\c s\i ls\i n.
Bunlar\i n yerine $x$, $y$, $z$ falan kullan\i labilir.
\c Simdi, her imzada,
\emph{\"ozyinelemeyle,}
\textbf{terimler} tan\i mlar:
\begin{compactenum}[1)]
\item
Her de\u gi\c sken, bir terimdir.
\item
her de\u gi\c smez bir terimdir;
\item
her $m$ sayma say\i s\i\ i\c cin,
her $m$-konumlu $F$ i\c slem simgesi i\c cin,
t\"um $t_0$, \dots, $t_{m-1}$ terimleri i\c cin,
\begin{equation*}
Ft_0\cdots t_{n-1}
\end{equation*}
ifadesi de terimdir.
\end{compactenum}
Normalde $n=2$ durumunda $Ft_0t_1$ ifadesinin yerine
$(t_0\mathbin Ft_1)$ kullan\i l\i r.

Terimlerin tan\i m\i\ \"ozyinelemeli oldu\u gundan
t\"umevar\i ml\i\ kan\i tlar m\"umk\"und\"ur:
Bir imzada, bir $A$ k\"umesi
\begin{compactenum}[1)]
\item
  her de\u gi\c sken i\c cerirse,
\item
  her de\u gi\c smez i\c cerirse,
\item
  her $m$ sayma say\i s\i\ i\c cin,
  her $m$-konumlu $F$ i\c slem simgesi i\c cin,
  $A$'n\i n zaten $t_0$, \dots, $t_{m-1}$ terimlerini i\c cerdi\u gi
  $Ft_0\cdots t_{m-1}$ terimini i\c cerirse,
\end{compactenum}
o zaman $A$, her terimi i\c cerir.

Hi\c cbir de\u gi\c skenin g\"oz\"ukmedi\u gi terim \textbf{sabittir.}
Sabit terimlerin tan\i m\i\ da \"ozyinelemeli bi\c cimde konulabilir:

\begin{theorem}\label{thm:fixed}
  Bir imzada, bir $A$ k\"umesi
\begin{compactenum}[1)]
\item
her de\u gi\c smez i\c cersin;
\item
her $m$ sayma say\i s\i\ i\c cin,
her $m$-konumlu $F$ i\c slem simgesi i\c cin,
  $A$'n\i n zaten $t_0$, \dots, $t_{m-1}$ terimlerini i\c cerdi\u gi
$Ft_0\cdots t_{m-1}$ terimi i\c cersin.
\end{compactenum}
O zaman $A$, her sabit terim i\c cerir.
\end{theorem}

\begin{proof}
  \.Imzan\i n sabit olmayan terimleriyle $A$'n\i n elemanlar\i,
  $B$ k\"umesini olu\c stursun.  K\i saca
  \begin{equation*}
    B=A\cup\{\text{sabit olmayan terimler}\}.
  \end{equation*}
  \begin{asparaenum}
  \item
    Her de\u gi\c sken, sabit olmad\i\u g\i ndan $B$'dedir.
  \item
    Her de\u gi\c smez, sabit oldu\u gundan $A$'dad\i r,
    dolay\i s\i yla $B$'dedir.
  \item
    $m\in\N$ ve $F$, $m$-konumlu bir i\c slem simgesi olsun,
    ve $t_0$, \dots, $t_{m-1}$ terimleri $B$'de olsun.
    Bu terimlerin her biri ya sabit de\u gildir ya da $A$'dad\i r.
    Biri sabit de\u gilse $Ft_0\cdots t_{m-1}$ terimi de sabit de\u gildir,
    dolay\i s\i yla bu terim $B$'dedir.
    E\u ger  $t_0$, \dots, $t_{m-1}$ terimlerinin her biri $A$'da ise,
    o zaman varsay\i ma g\"ore $Ft_0\cdots t_{m-1}$ terimi de $A$'dad\i r,
    dolay\i s\i yla bu terim $B$'dedir.
  \end{asparaenum}
  T\"umevar\i mdan $B$ her terim i\c cerir.
  \"Ozel olarak $A$, her sabit terim i\c cerir.
\end{proof}

E\u ger $t$, $\sig$'nin sabit bir terimiyse,
ve $\str A\in\Str{\sig}$ ise,
$t$'nin $\str A$'daki $t^{\str A}$ \emph{yorumunu} tan\i mlamak isteriz.
E\u ger $t$ bir de\u gi\c smez ise, o zaman $t^{\str A}$ zaten tan\i mland\i.
Genel tan\i m \"ozyinelemeli olacak,
ama yapabilmemiz i\c cin bir teorem gerekir.

\begin{theorem}\label{thm:ur}
  E\u ger $t_0$, \dots, $t_{m-1}$, $u_0$, \dots, $u_{m-1}$ terim ise,
  ve $Ft_0\cdots t_{m-1}$ ve $Fu_0\cdots u_{m-1}$ terimleri birbiriyle ayn\i\ ise,
  o zaman $m$'nin her $k$ eleman\i\ i\c cin $t_k$ ve $u_k$ ayn\i d\i r.
\end{theorem}

\begin{proof}
  T\"umevar\i mla her terim i\c cin
  hem sonuna yeni simgeler ekleyerek,
  hem de sonundan simgeler kald\i rarak
  yeni bir terim elde etmeyece\u giz.
  Bunu g\"ostermek yeter\dots
\end{proof}

\c Simdi $t^{\str A}$ yorumuna \"ozyinelemeli bir tan\i m verilebilir:
\begin{compactenum}
\item
  $t$ de\u gi\c smez ise dedi\u gimiz gibi $t^{\str A}$ zaten tan\i mland\i.
\item
  E\u ger $t$, $Ft_0\cdots t_{m-1}$ bi\c cimindeyse
  ve $t_k{}^{\str A}$ yorumlar\i\ tan\i mlan\i rsa
\begin{equation*}
(Ft_0\cdots t_{n-1})^{\str A}=F^{\str A}(t_0{}^{\str A},\dots,t_{n-1}{}^{\str A}).
\end{equation*}
\end{compactenum}
Her durumda $t^{\str A}\in A$.
(Bunun kan\i t\i\ t\"umevar\i mla verilebilir.)

\begin{example}
Her durumda $t_k{}^{\F_p}=1$ ise
\begin{equation*}
(\cdots(t_0+t_1)+\cdots+t_{p-1})^{\F_p}=0.
\end{equation*}
\end{example}

\subsection{Form\"uller}

De\u gi\c skenler ve de\u gi\c smezler,
``b\"ol\"unemez'' terimlerdir,
ama normalde bu \c sekilde konu\c smuyoruz.
Yine de \textbf{b\"ol\"unemez form\"ullerin} iki \c ce\c siti vard\i r:
\begin{compactenum}
  \item
E\u ger $t$ ve $u$ terim ise,
o zaman
\begin{equation*}
t=u
\end{equation*}
denklemi
b\"ol\"unemez bir form\"uld\"ur.
\item
E\u ger $R$, $m$-konumlu y\"uklem ise,
ve $t_0$, \dots, $t_{m-1}$ terim ise,
o zaman
\begin{equation*}
Rt_0\cdots t_{m-1}
\end{equation*}
ifadesi b\"ol\"unemez bir form\"uld\"ur.
\end{compactenum}
Bu tan\i m, \"ozyinelemeli de\u gildir.
Ama \textbf{form\"ullerin} tan\i m\i,
\"ozyinelemelidir:
\begin{compactenum}[1)]
\item
(Tabii ki) her b\"ol\"unemez form\"ul, bir form\"uld\"ur.
\item
  $\phi$ ve $\psi$ form\"ul ise
  \begin{equation*}
  (\phi\land\psi)  
  \end{equation*}
  de form\"uld\"ur.
\item
  $\phi$ form\"ul ise
  \begin{equation*}
   \lnot\phi 
  \end{equation*}
  de form\"uld\"ur.
\item
  $\phi$ form\"ul ve $x$ de\u gi\c sken ise
  \begin{equation*}
   \Exists x\phi 
  \end{equation*}
  de form\"uld\"ur.
\end{compactenum}
Form\"uller i\c cin \Teorem{thm:ur} gibi bir teorem do\u grudur,
dolay\i s\i yla form\"uller k\"umesinde a\c sa\u g\i daki gibi
\"ozyinelemeli tan\i mlar yap\i labilir:

\begin{compactenum}[1)]
\item
  $\phi$ b\"ol\"unemez ise
  \begin{equation*}
    \fv{\phi},
  \end{equation*}
  $\phi$'de g\"oz\"uken de\u gi\c skenlerin olu\c sturdu\u gu k\"umedir.
\item
  $\fv{\phi\land\psi}=\fv{\phi}\cup\fv{\psi}$.
\item
$\fv{\lnot\phi}=\fv{\phi}$.
\item
  $\fv{\Exists x\phi}=\fv{\phi}\setminus\{x\}$.
\end{compactenum}
$\fv{\phi}$ k\"umesinin elemanlar\i,
$\phi$'nin \textbf{serbest de\u gi\c skenleridir.}

\subsection{Do\u gruluk}

Hi\c c serbest de\u gi\c skeni olmayan bir form\"ul, bir \textbf{c\"umledir.}
$\sig$ imzas\i n\i n her c\"umlesi, 
imzas\i\ $\sig$ olan her yap\i nda
ya \emph{yanl\i\c st\i r} ya da \emph{do\u grudur.}
Bu ko\c sullar\i n tan\i mlanmas\i\ i\c cin
daha fazla kavramlar gerekiyor.

Her $\alpha$ b\"ol\"unemez form\"ul\"unde,
bir $x$ de\u gi\c skeninin her ge\c ci\c sinin yerine bir $t$ terimi konulursa
\begin{equation*}
(\alpha^x_t)
\end{equation*}
form\"ul elde edilir.  Ayr\i ca
$y$ $x$'ten farkl\i\ bir de\u gi\c sken olmak \"uzere
\begin{gather*}
	((\phi^x_t)\land(\psi^x_t))\text{ form\"ul\"ud\"ur }((\phi\land\psi)^x_t),\\
	\lnot(\phi^x_t)\text{ form\"ul\"ud\"ur }(\lnot\phi^x_t),\\
	\Exists x\phi\text{ form\"ul\"ud\"ur }(\Exists x\phi^x_t),\\
	\Exists y(\phi^x_t)\text{ form\"ul\"ud\"ur }(\Exists y\phi^x_t).
\end{gather*}

$\signature J$, $\sig$ imzas\i n\i\ kapsayan bir imza olsun,
ve $\str A\in\Str{\sig}$, $\str B\in\Str{\signature J}$ olsun.
E\u ger $\str A$ ve $\str B$'nin evrenleri ayn\i\ (yani $A=B$) ise,
ve $\sig$'nin her $S$ eleman\i
\begin{equation*}
S^{\str A}=S^{\str B}
\end{equation*}
ise, o zaman $\str B$, $\str A$'n\i n $\signature J$'ye bir \textbf{a\c c\i l\i m\i d\i r.}

E\u ger $\str A\in\Str{\sig}$ ve $X\included A$ ise,
o zaman $X$'in her $a$ eleman\i, yeni bir de\u gi\c smez olarak anla\c s\i labilir.
Bu de\u gi\c smezler $\sig$'ye eklenirse
\begin{equation*}
\sig(X)
\end{equation*}
imzas\i\ elde edilsin.
O zaman $\str A$'n\i n $\sig(X)$'e $\str A_X$ a\c c\i l\i m\i\ vard\i r,
ve burada, $X$'in her $a$ eleman\i n\i n yorumu kendisidir:
\begin{equation*}
a^{\str A_X}=a.
\end{equation*}
E\u ger bir $\sigma$ c\"umlesi $\str A$'da \textbf{do\u gru} ise
\begin{equation*}
\str A\models\sigma
\end{equation*}
ifadesini yazar\i z.  Tan\i m\i\ \c simdi verebiliriz.
\begin{compactenum}[1)]
\item
$t^{\str A}=u^{\str A}$ ise
$\str A\models t=u$.
\item
$(t_0{}^{\str A},\dots,t_{n-1}{}^{\str A})\in R^{\str A}$ ise
$\str A\models Rt_0\cdots t_{n-1}$.
\item
$\str A\models\sigma$ ve $\str A\models\tau$ ise $\str A\models(\sigma\land\tau)$.
\item
$\str A\models\sigma$ de\u gilse $\str A\models\lnot\sigma$.
\item
$A$'n\i n bir $a$ eleman\i\ i\c cin $\str A_{\{a\}}\models\phi^x_a$ ise $\str A\models\Exists x\phi$.
\end{compactenum}
E\u ger $\sigma$ $\str A$'da do\u gru de\u gilse \textbf{yanl\i\c st\i r.}

Baz\i\ k\i saltmalardan faydalanabiliriz:
\begin{gather*}
t\neq u\text{ demek }\lnot\;t=u,\\
	(\phi\lor\psi)\text{ demek }\lnot(\lnot\phi\land\lnot\psi),\\
	(\phi\lto\psi)\text{ demek }(\lnot\phi\lor\psi),\\
	(\phi\liff\psi)\text{ demek }((\phi\lto\psi)\land(\psi\lto\phi)),\\	
	\Forall x\phi\text{ demek }\lnot\Exists x\lnot\phi.
\end{gather*}
Baz\i\ ayra\c clar atlanabilir.  \"Orne\u gin
\begin{gather*}
	(\phi\land\psi\land\theta)\text{ demek }(\phi\land(\psi\land\theta)),\\
	(\phi\lto\psi\lto\theta)\text{ demek }(\phi\lto(\psi\lto\theta)),\\
	(\phi\land\psi\lto\theta)\text{ demek }((\phi\land\psi)\lto\theta).	
\end{gather*}

\begin{example}\mbox{}
  \begin{asparaenum}
    \item
$\Forall x\Exists y(x\neq0\lto xy=1)$ c\"umlesi
  her cisimde do\u grudur ama $(\Z,+,0,-,\times,1)$
  halkas\i nda yanl\i\c st\i r.
\item
  $\Forall x\Exists y(y^2=x\lor y^2=-x)$ c\"umlesi
  $(\R,+,0,-,\times,1)$ cisminde do\u grudur
  ama $(\Q,+,0,-,\times,1)$ cisminde yanl\i\c st\i r.
\item
  $\Forall x\Forall y\Exists z(x<y\lto x<z\land z<y)$ c\"umlesi
  $(\Q,<)$ s\i ras\i nda do\u grudur ama $(\Z,<)$ s\i ras\i nda yanl\i\c st\i r.
\item
  $\Forall x\Exists y\Forall zxyz=z$ c\"umlesi
  $(\Q,\times)$ yap\i s\i nda do\u grudur ama
  $(\Z,\times)$ yap\i s\i nda yanl\i\c st\i r.
  \end{asparaenum}
  
\end{example}

\section{Tan\i mlanabilirlik}

\subsection{\.I\c slemler}

$\sig$ imzas\i ndan gelen, de\u gi\c skenleri
$\{x_i\colon i<n\}$ k\"umesinden gelen terimler
\begin{equation*}
\Tm^n(\sig)
\end{equation*}
k\"umesini olu\c stursun.
B\"oyle terimlere $n$-konumlu denebilir,
fakat bu durumda her $n$-konumlu terim $(n+1)$-konumlu dad\i r:
\begin{equation*}
\Tm^0(\sig)\included\Tm^1(\sig)\included\Tm^2(\sig)\included\cdots
\end{equation*}
E\u ger $t\in\Tm^n(\sig)$, $\str A\in\Str{\sig}$, ve $\vec a\in A^n$ ise,
$i<n$ olmak \"uzere
her $x_i$ de\u gi\c skeninin $t$'deki her ge\c ci\c sinin yerine
$a_i$ konuldu\u gu terim
\begin{equation*}
t(\vec a)
\end{equation*}
olarak yaz\i ls\i n.
\"Ozyinelemeli bir tan\i m verilebilir:
\begin{compactenum}
\item
  $i<n$ ise  $x_i(\vec a)$, $a_i$ olur.
\item
  $d$ de\u gi\c smezse $d(\vec a)$, $d$ olur.
\item
  $m$'nin her $k$ eleman\i\ $t_k(\vec a)$, $u_k$ ise
  ve $F$, $m$-konumlu y\"uklem ise
  $Ft_0\cdots t_{m-1}(\vec a)$, $Fu_0\cdots u_{m-1}$ olur.
\end{compactenum}
\c Simdi
\begin{equation*}
  t^{\str A},
	\text{ $A$ k\"umesinin $n$-konumlu }
  \vec a\mapsto t(\vec a)^{\str A}
	\text{ i\c slemi}
\end{equation*}
olsun.
K\i saca
\begin{equation}\label{eqn:tAa=taA}
  t^{\str A}(\vec a)=t(\vec a)^{\str A}.
\end{equation}
\"Ozel olarak
\begin{gather}
  x_i{}^{\str A}(\vec a)=a_i,\notag\\
  d^{\str A}(\vec a)=d^{\str A},\notag\\
  (Ft_0\cdots t_{m-1})^{\str A}(\vec a)
  =F^{\str A}(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a)).\label{eqn:Ft}
\end{gather}
Genelde $X\included A$ ise 
$\{t^{\str A}\colon t\in\Tm^n(\sig(X))\}$ k\"umesinin elemanlar\i,
$\str A$'n\i n $X$ \"uzerinde $n$-konumlu 
\textbf{tan\i mlanabilir} i\c slemleridir.

\begin{example}
\begin{compactenum}[1)]
\item
bir $(G,\times,\gid,{}\inv)$ grubunda
$a\in G$ olmak \"uzere 
\begin{equation*}
x\mapsto a\inv xa
\end{equation*}
i\c slemi tan\i mlanabilir;
\item
bir $(K,+,0,-,\times,1)$ cisminde
$a_i\in K$ olmak \"uzere
\begin{equation*}
x\mapsto\sum_{i=0}^na_ix^i
\end{equation*}
polinom i\c slemi tan\i mlanabilir.
\end{compactenum}
\end{example}

\subsection{Ba\u g\i nt\i lar}

$\sig$ imzas\i ndan gelen, serbest de\u gi\c skenleri
$\{x_i\colon i<n\}$ k\"umesinden gelen form\"uller
\begin{equation*}
\Fm^n(\sig)
\end{equation*}
k\"umesini olu\c stursun.
B\"oyle form\"ullere $n$-konumlu denebilir,
fakat bu durumda her $n$-konumlu form\"ul $(n+1)$-konumlu dad\i r:
\begin{equation*}
\Fm^0(\sig)\included\Fm^1(\sig)\included\Fm^2(\sig)\included\cdots
\end{equation*}
E\u ger $\phi\in\Fm^n(\sig)$, $\str A\in\Str{\sig}$, ve $\vec a\in A^n$ ise
\begin{equation*}
  ((\cdots(\phi^{x_0}_{a_0})\cdots){}^{x_{n-1}}_{a_{n-1}})
\end{equation*}
ifadesinin yerine
\begin{equation*}
\phi(\vec a)
\end{equation*}
yazal\i m.
O zaman
\begin{equation*}
\phi^{\str A}=\{\vec a\in A^n\colon\str A\models\phi(\vec a)\}
\end{equation*}
olsun.
Bu \c sekilde
\begin{equation}\label{eqn:phi^A}
  \vec a\in\phi^{\str A}\iff\str A\models\phi(\vec a).
\end{equation}
\"Ozel olarak $t\in\Tm^n(\sig)$ ise
\begin{equation*}
    (t=x_n)^{\str A}=\{(\vec a,b)\in A^{n+1}\colon t^{\str A}(\vec a)=b^{\str A}\}.
\end{equation*}
E\u ger $t_i\in\Tm^n(\sig)$ ise
\begin{equation}\label{eqn:t0=t1}
		(t_0=t_1)^{\str A}=\{\vec a\in A^n\colon t_0{}^{\str A}(\vec a)=t_1{}^{\str A}(\vec a)\},
\end{equation}
ve $R$ $m$-konumlu y\"uklem ise
\begin{multline}\label{eqn:Rt}
	(Rt_0\cdots t_{m-1})^{\str A}\\
	=\{\vec a\in A^n\colon(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a))\in R^{\str A}\}.
\end{multline}
Ayr\i ca
\begin{gather}\label{eqn:x0=x0}
	(x_0=x_0\land\dots\land x_{n-1}=x_{n-1})^{\str A}=A^n,\\\label{eqn:not-phi}
\lnot\phi^{\str A}=A^n\setminus\phi^{\str A},
\end{gather}
ve $\psi\in\Fm^n(\sig)$ ise
\begin{gather*}
(\phi\land\psi)^{\str A}=\phi^{\str A}\cap\psi^{\str A},\\
(\phi\lor\psi)^{\str A}=\phi^{\str A}\cup\psi^{\str A}.	
\end{gather*}
\c Simdi $\pi$, $A^{n+1}$ kuvvetinden $A^n$ kuvvetine giden
\begin{equation*}
(a_0,\dots,a_n)\mapsto(a_0,\dots,a_{n-1})
\end{equation*}
g\"ondermesi olsun.
$X\included A^{n+1}$ ise $\pi[X]$ (yani $\{\pi(\vec a)\colon\vec a\in X\}$),
$X$'in \textbf{izd\"u\c s\"um\"ud\"ur.}
E\u ger $\theta\in\Fm^{n+1}(\sig)$ ise
$\Exists{x_n}\theta^{\str A}$, $\theta^{\str A}$'n\i n izd\"u\c s\"um\"ud\"ur:
\begin{equation*}
\Exists{x_n}\theta^{\str A}=\pi\left[\theta^{\str A}\right].
\end{equation*}


\c Simdi $X\included A$ olmak \"uzere
\begin{equation*}
\Dfn^n_X(\str A)=\{\phi^{\str A}\colon\phi\in\Fm^n(\sig(X))\}
\end{equation*}
olsun.  Bu k\"umenin elemanlar\i,
$\str A$'n\i n $X$ \"uzerinde $n$-konumlu \textbf{tan\i mlanabilir} ba\u g\i nt\i lar\i d\i r.
K\i saca $X$ \"uzerinde tan\i mlanabilir bir ba\u g\i nt\i,
$X$-tan\i mlanabilirdir.

$\Dfn^n_X(\str A)$, $\pow{A^n}$ Boole cebirinin altcebiridir,
yani $\cup$, $\cap$, ve ${}'$ i\c slemleri alt\i nda kapal\i d\i r,
ve bo\c s k\"umeyi
ve $A^n$ kuvvetinin t\"um\"um\"u i\c erir.

Normalde $(x_0,x_1,x_2)$ de\u gi\c sken listesinin yerine
$(x,y,z)$ kullan\i l\i r.

\begin{example}\mbox{}
\begin{asparaenum}
\item
$(G,\times,\gid,{}\inv)$ bir grup ise
$\{(a,a\inv)\colon a\in G\}$ ba\u g\i nt\i s\i,
$(G,\times,\gid)$ yap\i s\i nda
\begin{equation*}
xy=\gid
\end{equation*}
form\"ul\"u taraf\i ndan tan\i mlan\i r.
\"Ozel olarak bu form\"ul ve
\begin{equation*}
y=x\inv
\end{equation*}
form\"ul\"u,
ayn\i\ ba\u g\i nt\i s\i n\i\ tan\i mlar.
Ayr\i ca $(G,\times)$ yap\i s\i nda $\{\gid\}$ k\"umesi,
\begin{equation*}
\Forall y(xy=y\land yx=y)
\end{equation*}
taraf\i ndan tan\i mlan\i r.  Asl\i nda
\begin{align*}
&\Forall y xy=y,&	
&\Exists y xy=y
\end{align*}
form\"ullerinden her biri kullan\i labilir.
Sonu\c c olarak $(G,\times)$ yap\i s\i nda
$\{(a,a\inv)\colon a\in G\}$ ba\u g\i nt\i s\i,
\begin{equation*}
\Exists zxyz=xz
\end{equation*}
form\"ul\"u taraf\i ndan tan\i mlan\i r.
Bu nedenle $(G,\times)$ yap\i s\i na da
grup denebilir.
\item
Benzer \c sekilde $(K,+,0,-,\times,1)$ bir cisim ise
$(K,+,\times)$ yap\i s\i nda $\{(a,-a)\colon a\in K\}$,
$\{0\}$, ve $\{1\}$ ba\u g\i nt\i lar\i\ $\emptyset$-tan\i mla\-nabilir.
\item
$(\R,+,\times)$ cisminde $\{(x,y)\colon x\leq y\}$ s\i ralamas\i,
\begin{equation*}
\Exists zx+zz=y
\end{equation*}
taraf\i ndan tan\i mlan\i r.
\item
Bir grubun \textbf{merkezi}
$\Forall yxy=yx$
taraf\i ndan tan\i mlan\i r.
\end{asparaenum}
\end{example}

\c Simdi b\"ol\"unemez form\"ul taraf\i ndan 
tan\i mlanan ba\u g\i nt\i lara \textbf{temel} densin.
O zaman $\bigcup_{n\in\upomega}\Dfn^n_A(\str A)$ birle\c siminin her eleman\i,
kesi\c simler, t\"umleyenler, ve izd\"u\c s\"umler al\i narak
temel tan\i mlanabilir ba\u g\i nt\i lardan elde edilibilir.

Bir form\"ulde $\exists x$ ifadesi (ve $\lnot\Exists x\lnot$ ifadesinin
$\forall x$ k\i saltmas\i) bir \textbf{niceleyicidir.}
\c Simdi
\begin{equation*}
\Fm^n_0(\sig)=\{\phi\in\Fm^n(\sig)\colon\phi\text{ niceleyicisiz}\}
\end{equation*}
olsun.
Bu k\"umeye \"ozyinelemeli bir tan\i m verilebilir,
yani \Teorem{thm:fixed} gibi bir teorem vard\i r:

\begin{theorem}\label{thm:qf}
  Bir $A$ k\"umesi i\c cin,
\begin{compactenum}
\item
  \begin{compactenum}
    \item
      $\{t,u\}\included\Tm^n(\sig)$ ise
      $t=u\in A$ olsun;
    \item
      $R$,
      $\sig$'nin $m$-konumlu bir y\"uklem ise,
      ve $\{t_0,\dots,t_{m-1}\}\included\Tm^n(\sig)$ ise,
      o zaman
      $Rt_0\cdots t_{m-1}\in A$ olsun;
  \end{compactenum}
\item
  $\phi\in A$ ise $\lnot\phi\in A$ olsun;
\item
  $\{\phi,\psi\}\included A$ ise $(\phi\land\psi)\in A$ olsun.
\end{compactenum}
O zaman $\Fm^n_0\included A$.
\end{theorem}






O zaman
\begin{equation*}
\{\phi^{\str A}\colon\phi\in\Fm^n_0(\sig(X))\}
\end{equation*}
k\"umesi, $\pow{A^n}$ Boole cebirinin, 
t\"um temel tan\i mlanabilir ba\u g\i nt\i lar\i\ i\c ceren
altcebirlerinin en k\"u\c c\"u\u g\"ud\"ur.

$\Dfn^1_A(\str A)$ k\"umesinin elemanlar\i,
$\str A$'n\i n tan\i mlanabilir k\"umeleridir.
$A$'n\i n her sonlu $\{a_0,\dots,a_{n-1}\}$ altk\"umesi
\begin{equation*}
x=a_0\lor\dots\lor x=a_{n-1}
\end{equation*}
form\"ul\"u taraf\i ndan tan\i mlan\i r.
O zaman t\"umleyeni sonlu olan k\"umeler de tan\i mlanabilir.

\section{G\"ondermeler}

Bu b\"ol\"umde bir $\sig$ imzas\i nda
$\str A$ ve $\str B$, imzas\i\ $\sig$ olan yap\i\ olacaklar,
ve $h\colon A\to B$.
E\u ger $\vec a\in A^n$ ise
\begin{equation}\label{eqn:h-not}
h(\vec a)=(h(a_0),\dots,h(a_{n-1}))
\end{equation}
anla\c s\i labilir.
O zaman $X\included A^n$ ise
\begin{equation*}
h[X]=\{h(\vec a)\colon\vec a\in X\}.
\end{equation*}

\subsection{Homomorfizimler}

E\u ger $\sig$'nin her
\begin{compactenum}[1)]
\item
$d$ de\u gi\c smezi i\c cin
\begin{equation*}
h(d^{\str A})=d^{\str B},
\end{equation*}
\item
$F$ i\c slem simgesi i\c cin
\begin{equation}\label{eqn:hF}
h\circ F^{\str A}=F^{\str B}\circ h,
\end{equation}
\item
$R$ y\"uklemi i\c cin
\begin{equation}\label{eqn:hRincRB}
h[R^{\str A}]\included R^{\str B}
\end{equation}
\end{compactenum}
ise, o zaman $h$, $\str A$'dan $\str B$'ye giden bir \textbf{homomorfizim}
veya \textbf{benzer yap\i\ d\"on\"u\c s\"um\"ud\"ur.}
Bu durumda
\begin{equation*}
h\colon\str A\to\str B
\end{equation*}
ifadesini yazal\i m.

\begin{example}\mbox{}
\begin{asparaenum}
\item
$x\mapsto[x]\colon(\Z,+,\times)\to(\Zmod n,+,\times)$.
\item
$x\mapsto x\colon(\N,{}\divides{})\to(\N,\leq)$.
\item
$(x,y)\mapsto mx+ny\colon(\Zmod m\oplus\Zmod n,+)\to(\Zmod{mn},+)$.
\end{asparaenum}
\end{example}

\begin{theorem}\label{thm:hom-tm}
$h\colon\str A\to\str B$ ve
$t$, $\sig$'nin bir terimi ise
\begin{equation}\label{eqn:htA=tBh}
h\circ t^{\str A}=t^{\str B}\circ h.
\end{equation}
\end{theorem}

\begin{proof}
T\"umevar\i m kullanaca\u g\i z.
\begin{compactenum}
\item
$h(x_i{}^{\str A}(\vec a))
=h(a_i)
=x_i{}^{\str B}(h(\vec a))$.
\item
$d$ de\u gi\c smeziyse
$h(d^{\str A}(\vec a))
=h(d^{\str A})
=d^{\str B}
=d^{\str B}(h(\vec a))$.
\item
Bir $m$ i\c cin $F$, $m$-konumlu bir y\"uklem ise,
ve $t$'nin $t_i$ oldu\u gu durumda \eqref{eqn:htA=tBh} iddias\i\ do\u gru ise
\begin{multline*}
h((Ft_0\cdots t_{m-1})^{\str A}(\vec a))\\
\begin{aligned}
&=h(F^{\str A}(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a))&&\text{[\eqref{eqn:Ft}]}\\
&=F^{\str B}(h(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a)))&&\text{[\eqref{eqn:hF}]}\\
&=F^{\str B}(h(t_0{}^{\str A}(\vec a)),\dots,h(t_{m-1}{}^{\str A}(\vec a)))&&\text{[\eqref{eqn:h-not}]}\\
&=F^{\str B}(t_0{}^{\str B}(h(\vec a)),\dots,t_{m-1}{}^{\str B}(h(\vec a)))&&\text{[hipotez]}\\
&=(Ft_0\cdots t_{m-1})^{\str B}(h(\vec a)).&&\text{[\eqref{eqn:Ft}]}
\end{aligned}
\end{multline*}
\end{compactenum}
T\"umevar\i mla her $t$ i\c cin iddia do\u grudur.
\end{proof}

\begin{theorem}\label{thm:hom}
$h\colon\str A\to\str B$
ve $\phi$, $\sig$'in b\"ol\"unemez bir form\"ul\"u ise
\begin{equation*}
h[\phi^{\str A}]\included\phi^{\str B}.
\end{equation*}
\end{theorem}

\begin{proof}
$\phi$'nin iki durumu vard\i r.
\begin{align*}
	\vec a\in(t_0=t_1)^{\str A}
&\implies	t_0(\vec a)^{\str A}=t_1(\vec a)^{\str A},&&\text{[\eqref{eqn:t0=t1}]}\\
&\implies	h(t_0(\vec a)^{\str A})=h(t_1(\vec a)^{\str A})&&\\
&\implies	h(t_0{}^{\str A}(\vec a))=h(t_1{}^{\str A}(\vec a))&&\text{[\eqref{eqn:tAa=taA}]}\\
&\implies	t_0{}^{\str B}(h(\vec a))=t_1{}^{\str B}(h(\vec a))&&\text{[\Teorem{thm:hom-tm}]}\\	
&\implies	t_0(h(\vec a))^{\str B}=t_1(h(\vec a))^{\str B}&&\text{[\eqref{eqn:tAa=taA}]}\\
&\implies h(\vec a)\in(t_0=t_1)^{\str B},&&\text{[\eqref{eqn:t0=t1}]}
\end{align*}
\begin{multline*}
\vec a\in(Rt_0\cdots t_{m-1})^{\str A}\\
\begin{aligned}
&\implies	(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a))\in R^{\str A}&&\text{[\eqref{eqn:Rt}]}\\
&\implies	h(t_0{}^{\str A}(\vec a),\dots,t_{m-1}{}^{\str A}(\vec a))\in R^{\str B}&&\text{[\eqref{eqn:hRincRB}]}\\
&\implies	(h(t_0{}^{\str A}(\vec a)),\dots,h(t_{m-1}{}^{\str A}(\vec a)))\in R^{\str B}&&\text{[\eqref{eqn:h-not}]}\\
&\implies	(t_0{}^{\str B}(h(\vec a)),\dots,t_{m-1}{}^{\str B}(h(\vec a)))\in R^{\str B}&&\text{[\Teorem{thm:hom-tm}]}\\
&\implies h(\vec a)\in(Rt_0\cdots t_{m-1})^{\str B}.&&\text{[\eqref{eqn:Rt}]}
\end{aligned}
\end{multline*}
Her durumda $\vec a\in\phi^{\str A}\implies h(\vec a)\in\phi^{\str B}$.
\end{proof}

\subsection{G\"ommeler}

$h\colon\str A\to\str B$ olsun.
E\u ger
\begin{compactenum}[1)]
\item
$h$ birebir ve
\item
$\sig$'nin her $R$ y\"uklemi i\c cin $h[R^{\str A}]=R^{\str B}\cap h[A]$
\end{compactenum}
ise, o zaman $h$,
$\str A$'nin $\str B$'ye bir \textbf{g\"ommesidir,}
ve bu durumda
\begin{equation*}
h\colon\str A\embedded\str B
\end{equation*}
ifadesini yazabiliriz.

\begin{example}\mbox{}
\begin{asparaenum}
\item
$\Zmod n=\{[0],\dots,[n-1]\}$,
ama $i\in n$ olmak \"uzere $h([i])=i$ ise $h$,
$(\Zmod n,+)$ grubunun $(\Z,+)$ grubuna bir g\"ommesi de\u gildir
\c c\"unk\"u $[1]+[n-1]=[0]$ ama $1+n-1=n$,
ve $h([0])\neq n$.
\item
$[x]\mapsto[mx]\colon(\Zmod n,+)\embedded(\Zmod{mn},+)$.
\item
$(\Zmod n,+,\times)$, $[x]\mapsto[mx]$ taraf\i ndan $(\Zmod{mn},+,\times)$ halkas\i na g\"om\"ulmez.
\item
Tan\i ma g\"ore $(a,b)\mathrel E(x,y)\iff ay=bx$ ise
\begin{equation*}
\Qp=(\Z\times\Z)/E
\end{equation*}
olsun.  Bu k\"umenin her $[(a,b)]$ eleman\i, $a/b$ veya
\begin{equation*}
\frac ab
\end{equation*}
olarak yaz\i labilir.
O zaman okuldaki gibe toplama ve \c carpma $\Qp$ k\"umesinde tan\i mlanabilir, ve
\begin{equation*}
x\mapsto\frac x1\colon(\N,+,\times)\embedded(\Qp,+,\times).
\end{equation*}
\item
$(x,y)\mapsto\begin{bmatrix}x&y\\-y&x\end{bmatrix}\colon
(\R\times\R,+)\embedded(\Mat^{2\times2}(\R),+)$.
\item
$x+\mi y\mapsto\begin{bmatrix}x&y\\-y&x\end{bmatrix}\colon
(\C,+,\times)\embedded(\Mat^{2\times2}(\R),+,\times)$.
\end{asparaenum}
\end{example}

E\u ger $h\colon\str A\embedded\str B$ ve $h$,
$x\mapsto x$ \"ozde\c slik g\"ondermesiyse $\str A$,
$\str B$'nin \textbf{altyap\i s\i d\i r,}
ve bu durumda
\begin{equation*}
\str A\included\str B
\end{equation*}
ifadesini yazar\i z.

\begin{theorem}\label{thm:emb}
$h\colon\str A\embedded\str B$ ve
$\phi$, $\sig$'in niceleyicisiz bir form\"ul ise
\begin{equation}\label{eqn:emb-iff}
h[\phi^{\str A}]=\phi^{\str B}\cap h[A^n].
\end{equation}
\end{theorem}

\begin{proof}
\Teoreme{thm:qf} g\"ore t\"umevar\i m kullanaca\u g\i z.
  \begin{asparaenum}
    \item
\c Simdiki durumda \Teoremin{thm:hom} ad\i mlar\i\ tersilenebilir,
dolay\i s\i yla $\phi$ b\"ol\"unemez ise
\eqref{eqn:emb-iff} iddias\i\ do\u grudur.
\item
\.Iddia $\phi$'nin $\psi$ oldu\u gu durumda do\u gru ise
\begin{align*}
\vec a\in\lnot\phi^{\str A}
&\iff\vec a\in A^n\setminus\phi^{\str A}&&\text{[\eqref{eqn:not-phi}]}\\
&\iff h(\vec a)\in h[A^n]\setminus h[\phi^{\str A}]&&\text{[$h$ birebir]}\\
&\iff h(\vec a)\in h[A^n]\setminus(\phi^{\str B}\cap h[A^n])&&\text{[hipotez]}\\
&\iff h(\vec a)\in h[A^n]\setminus\phi^{\str B}&&\\
&\iff h(\vec a)\in B^n\setminus\phi^{\str B}&&\\
&\iff h(\vec a)\in\lnot\phi^{\str B}.&&\text{[\eqref{eqn:not-phi}]}
\end{align*}
\item
\.Iddia $\phi$'nin $\psi$ veya $\theta$ oldu\u gu durumda do\u gru ise
\begin{align*}
\vec a\in(\psi\land\theta)^{\str A}
&\iff\vec a\in\psi^{\str A}\cap\theta^{\str A}&&\text{[]}\\
&\iff h(\vec a)\in h[\psi^{\str A}]\cap h[\theta^{\str A}]&&\text{[]}\\
&\iff h(\vec a)\in \psi^{\str B}\cap\theta^{\str B}&&\text{[]}\\
&\iff h(\vec a)\in(\psi\land\theta)^{\str B}.
\end{align*}
  \end{asparaenum}
  Sonu\c c olarak iddia her niceleyicisiz $\phi$ i\c cin do\u grudur.
\end{proof}

\subsection{\.Izomorfizimler}

E\u ger $h\colon\str A\embedded\str B$ ve $h[A]=B$ ise, o zaman $h\inv$ de bir g\"ommedir,
ve $h$ bir
\textbf{e\c syap\i\ d\"on\"u\c s\"um\"u} veya \textbf{izomorf\/izimdir.}
Bu durumda
\begin{equation*}
h:\str A\isom\str B
\end{equation*}
ifadesini yazal\i m.

\begin{example}\mbox{}
\begin{asparaenum}
\item
$\gcd(m,n)=1$ ise
\begin{equation*}
(x,y)\mapsto mx+ny\colon(\Zmod m\oplus\Zmod n,+)\isom(\Zmod{mn},+).
\end{equation*}
\item
\textbf{\c Cin Kalan Teoremi.}
$\gcd(m,n)=1$, $an\equiv 1\pmod m$ ve $bm\equiv 1\pmod n$ ise
\begin{equation*}
(x,y)\mapsto anx+bmy\colon(\Zmod m\oplus\Zmod n,+,\times)\isom(\Zmod{mn},+,\times).
\end{equation*}
\end{asparaenum}
\end{example}


\begin{theorem}
  $h\colon\str A\isom\str B$ ise
	$\sig$'in her $\phi$ form\"ul\"u i\c cin
\begin{equation*}
h[\phi^{\str A}]=\phi^{\str B}.
\end{equation*}
\end{theorem}

\begin{proof}
$\psi\in\Fm^{n+1}(\sig)$ ve iddia $\phi$'nin $\psi$ oldu\u gu durumda do\u gru olsun,
ve $\vec a\in A^n$ olsun.
O zaman $h$ e\c sleme oldu\u gundan
$A$'n\i n bir $b$ eleman\i\ i\c cin $(\vec a,b)\in\psi^{\str A}$
ancak ve ancak $B$'nin bir $c$ eleman\i\ i\c cin
$(h(\vec a),c)\in\psi^{\str A}$.
K\i saca
\begin{equation*}
h[\Exists{x_n}\phi^{\str A}]=\Exists{x_n}\phi^{\str B}.
\end{equation*}
Kan\i t\i n kalan\i,
\Teoremin{thm:emb} kan\i t\i\ gibidir.
\end{proof}

\begingroup\small

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\endgroup
\newpage
\appendix

\section{Yunan Harf\/leri}
%\begin{table}%\larger
%  \captionabove{Yunan Harf\/leri}
%\addcontentsline{toc}{section}{Yunan Harf\/leri}%
\begingroup
\renewcommand{\arraystretch}{1.1}
%  \centering  % doesn't affect the table
\mbox{}\hfill\begin{tabular}{|c c c c|}\hline
b\"uy\"uk&min\"usk\"ul&\c ceviri&ad\\\hline
\Gk{A}& \Gk{a} & a & alpha \\ 
\Gk{B}& \Gk{b} & b & b\^eta   \\ 
\Gk{G}& \Gk{g} & g & gamma \\ 
\Gk{D}& \Gk{d} & d & delta \\ 
\Gk{E}& \Gk{e} & e & epsilon (basit e)\\ 
%&\Gk{\stigma}& stigma\\
\Gk{Z}& \Gk{z} & z & z\^eta   \\ 
\Gk{H}& \Gk{h} & \^e & \^eta \\ 
 \Gk{J}& \Gk{j} & th & th\^eta \\
%\hline 
 \Gk{I}& \Gk{i} & i & i\^ota\\ 
\Gk{K}& \Gk{k} & k & kappa\\ 
 \Gk{L}& \Gk{l} & l & lambda\\ 
 \Gk{M}& \Gk{m} & m & m\"u \\
 \Gk{N}& \Gk{n} & n & n\"u \\ 
 \Gk{X}& \Gk{x} & x, ks & xi \\ 
 \Gk{O}& \Gk{o} & o & omikron (k\"u\c c\"uk o)\\ 
 \Gk{P}& \Gk{p} & p & pi\\ 
%&\Gk{\qoppa}& koppa\\
%\hline
 \Gk{R}& \Gk{r} & r & rh\^o\\ 
 \Gk{S}& \Gk{sv, c} & s & sigma \\
 \Gk{T}& \Gk{t} & t & ta\"u \\ 
 \Gk{U}& \Gk{u} & y, \"u & \"upsilon (basit \"u) \\ 
 \Gk{F}& \Gk{f} & ph, f & phi\\ 
 \Gk{Q}& \Gk{q} & kh, ch & khi\\ 
 \Gk{Y}& \Gk{y} & ps & psi\\ 
 \Gk{W}& \Gk{w} & \^o & \^omega (b\"uy\"uk \^o)\\
%&\Gk{\sampi}& sampi\\
\hline
      \end{tabular}\hfill\mbox{}
			
\noindent Epsilon ve \"upsilon ``basittir''
\c c\"unk\"u Orta \c Ca\u g'da \Gk{ai} ve \Gk{oi} birle\c simlerinin telaffuzlar\i\
ayn\i ym\i\c s \cite{Smyth}.
%\end{table}
\endgroup

\section{Alman Harf\/leri}\label{German}
%\begin{table}
%  \caption[Alman Harf\/leri]{Alman Harf\/leri}
%\addcontentsline{toc}{section}{Alman Harf\/leri}%
\begingroup
\centering
{\relscale{1.6}
$\begin{array}{ccccccc}
\mathfrak{A\,a}&\mathfrak{B\,b}&\mathfrak{C\,c}&\mathfrak{D\,d}&\mathfrak{E\,e}&\mathfrak{F\,f}&\mathfrak{G\,g}\\
\mathfrak{H\,h}&\mathfrak{I\,i}&\mathfrak{J\,j}&\mathfrak{K\,k}&\mathfrak{L\,l}&\mathfrak{M\,m}&\mathfrak{N\,n}\\
\mathfrak{O\,o}&\mathfrak{P\,p}&\mathfrak{Q\,q}&\mathfrak{R\,r}&\mathfrak{S\,s}&\mathfrak{T\,t}&\mathfrak{U\,u}\\
           &\mathfrak{V\,v}&\mathfrak{W\,w}&\mathfrak{X\,x}&\mathfrak{Y\,y}&\mathfrak{Z\,z}&
 \end{array}$}

\mbox{}

A\c sa\u g\i daki yaz\i l\i\ bi\c cimleri
Heffner'in \cite{German} kitab\i ndan al\i n\i r:


\mbox{}

\includegraphics%[width=4in]%
{../../german-script-cropped.eps}

\mbox{}

%\end{table}
\endgroup


\end{document}