\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{substructure complete theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{equivalence_with_quantifier_elimination}{Equivalence with quantifier elimination}\dotfill \pageref*{equivalence_with_quantifier_elimination} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

One wants an equivalent and purely [[semantics|semantic]] characterization of [[quantifier elimination]].

A [[first-order theory|theory]] $T$ is substructure complete if after fixing a \emph{substructure}, the theory of the [[elementary class]] of all $T$-models into which that substructure embeds is \emph{complete}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[first-order theory]] $T$ in a language $\mathcal{L}$ is \textbf{substructure complete} if for any $M \in \mathbf{Mod}(T)$ and any $\mathcal{L}$-[[first-order structure|substructure]] $A \hookrightarrow M,$ the theory $T_{\mathsf{Diag}(A)}$ (expansion by $A$`s [[quantifier-free diagram]])---whose models are precisely those models $N \in \mathbf{Mod}(T)$ such that there exists an embedding of $\mathcal{L}$-structures $A \hookrightarrow N$---is complete.

\end{defn}
\hypertarget{equivalence_with_quantifier_elimination}{}\subsection*{{Equivalence with quantifier elimination}}\label{equivalence_with_quantifier_elimination}

\begin{theorem}
\label{}\hypertarget{}{}
$T$ [[quantifier elimination|eliminates quantifiers]] if and only if $T$ is substructure-complete.

\end{theorem}
\begin{proof}
$(\Rightarrow)$ Suppose $T$ eliminates quantifiers. Fix a substructure $A$. Since quantifier-free sentences are automatically transferred by embeddings (whereas without the additional property of quantifier-freeness, we would require the additional property that an embedding be [[elementary embedding|elementary]]), then for any two models $M, N \in \mathbf{Mod}(T)$

\begin{displaymath}
M \leftarrow A \rightarrow N
\end{displaymath}
containing $A$, we have that for every sentence $\varphi$ satisfied by $N$, $T \models \varphi \leftrightarrow \psi$ where $\psi$ is quantifier-free. Therefore, $\psi$ transfers to $A$ and $M$, so $M \models \varphi$ as well, and vice-versa.

$(\Leftarrow)$ Now suppose $T$ is substructure complete. Fix $\varphi(x)$ an $\mathcal{L}$-formula. Consider the [[type (in model theory)|type]] $\Phi$ of its quantifier-free consequences,i .e.

\begin{displaymath}
\Phi(x) \overset{\operatorname{df}}{=} \{\varphi^*(x) \text{ quantifier-free } \operatorname{|} T \models \varphi \implies \varphi^*\}.
\end{displaymath}
(This thing is like the quantifier-free principal ultrafilter on $\varphi$ in the Boolean algebra of definable sets.)

Let $d_1, \dots, d_n$ be distinct new constant symbols. Write $d = \vec{d} = (d_1, \dots, d_n)$.

Claim: $T \cup \Phi(d) \models \varphi(d).$

(This claim says, roughly, that knowing the principal ultrafilter of quantifier-free definable sets containing $\varphi(x)$ suffices to pin down $\varphi(x)$. After establishing the claim, it will be easy to improve it to full quantifier elimination.)

Proof of claim: suppose towards a contradiction that there exists an $A \models T \cup \Phi(d) \cup \{\not \varphi(d)\}.$ (This is a structure in the language $\mathcal{L}$ of $T$, expanded by $d$.) Let $d^A$ be the interpretation of the new constants $d$ in $A$. Consider the finitely-generated $\mathcal{L}$-substructure of $A$ around $d^A$:

\begin{displaymath}
C \overset{\operatorname{df}}{=} \langle d^A \rangle_{\mathcal{L}}.
\end{displaymath}
Subclaim: $T \cup \{\varphi(d^A)\} \cup \mathsf{Diag}(C)$ \emph{is} satisfiable.

(This subclaim amounts to saying that while $A$ models $\neg \varphi(d^A)$, we can find another model $B \models T$, embedding $C$, such that $B$ \emph{does} models $\varphi(d^A)$.)

Proof of subclaim: this will be a standard [[compactness theorem|compactness argument]]. Suppose not; by compactness there exist finitely many sentences $\theta_1, \dots, \theta_m \in \mathsf{Diag}(C)$ such that

\begin{displaymath}
T \cup \{\varphi(d^A)\} \cup \{\theta_i\}_{1 \leq i \leq m}
\end{displaymath}
is not satisfiable. By the definition of [[quantifier-free diagram]], we can write $\theta_i = \theta'_i(d^A)$, where $\theta'_i$ is a quantifier-free $\mathcal{L}$-formula.

Therefore,

\begin{displaymath}
T \cup \{\varphi(d^A), \theta'_0(d^A), \theta'_1(d^A), \dots, \theta_m'(d^A)\}
\end{displaymath}
is not satisfiable, where (remembering that $d^A$ is now just a tuple of constant symbols) $\theta'_0(x)$ specifies that all the entries of $x$ are distinct.

Therefore,

\begin{displaymath}
T \models \varphi(d^A) \rightarrow \bigvee_{i = 0}^m \neg \theta'_i(d^A).
\end{displaymath}
Since there are no constraints on the constant symbols $d^A$ and $\theta'_0$ specifies that they are all distinct, we may generalize them, so that

\begin{displaymath}
T \models \forall x \left( \varphi(x) \to \bigvee_{i = 0}^m \neg \theta'_i(x) \right)
\end{displaymath}
hence

\begin{displaymath}
\bigvee_{i = 0}^m \neg \theta'_i(x) \in \Phi(x).
\end{displaymath}
Now, by assumption, $A \models \Phi(d)$. This means in particular that $A \models \bigvee_{i = 0}^m \neg \theta'_i(d^A).$

Since that disjunction is quantifier-free, it transfers down to $C$. Therefore, there is some $0 \leq j \leq m$ such that $C \models \neg \theta'_j(d^A).$ Since the $d^A$ were distinct when we obtained $A$ (from which we obtained $C$), we can rule out $j \neq 0$. But for each $j = 1, \dots, m$, $\theta_j = \theta'_j(d^A) \in \mathsf{Diag}(C)$. This is a contradiction.

This proves the subclaim.

Now we proceed with proving the claim.

Let $\mathbf{B} \models T \cup \{\varphi(d^A)\} \cup \mathsf{Diag}(C).$

Then:

\begin{displaymath}
A \models \neg \varphi(d^A), \text{ and } B \models \varphi(d^A),
\end{displaymath}
which contradicts substructure completeness over $C$.

This proves the claim.

Now we proceed with proving the theorem.

With the claim in hand, the [[compactness theorem]] tells us that the entailment $T \cup \Phi(x) \models \varphi(d)$ is finitely supported, so that there are finitely many $\varphi^*_1, \dots, \varphi^*_m \in \Phi(x)$ such that

\begin{displaymath}
T \cup \{\varphi_i^*(d)\}_{i \leq m} \models \varphi(d).
\end{displaymath}
Write $\varphi^* \overset{\operatorname{df}}{=} \bigvee_{i \leq m} \varphi_i^*.$ Again, we generalize the constants $d$, obtaining

\begin{displaymath}
T \models \forall x \left( \varphi \leftrightarrows \varphi^* \right).
\end{displaymath}
Since the $\varphi^*_i$ are all quantifier-free, and $\varphi$ was arbitrary, we have proved that $T$ eliminates quantifiers.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item By the above theorem, any theory which [[quantifier elimination|eliminates quantifiers]] is substructure complete.


\item So e.g. the [[countable random graph]], the theory [[ACF]] of algebraically closed fields, the theory [[DLO]] of dense linear orders\ldots{}



\end{itemize}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item The analogous (and weaker) property where one asks that the theory of the elementary class of $T$-models embedding a fixed $T$-model is complete is called [[model complete theory | model completeness]].


\item Of course, substructure-completeness implies model-completeness.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[quantifier elimination]]


\item [[model complete theory]]


\item [[existentially closed model]]


\item [[diagram of a first-order structure]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Dave Marker, \emph{Model theory: an introduction}, theorem 3.14

\end{itemize}
[[!redirects substructure-complete]]

[[!redirects substructure completeness]]

[[!redirects substructure-completeness]]

[[!redirects substructure complete theories]]

[[!redirects substructure-complete theories]]

[[!redirects substructure complete]]



\end{document}