\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*{elimination of imaginaries}

[[!redirects imaginary element]]

\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{the__construction}{The $(-)^{\operatorname{eq}}$ construction}\dotfill \pageref*{the__construction} \linebreak
\noindent\hyperlink{relation_to_indrepresentable_power_objects}{Relation to ind-representable power objects}\dotfill \pageref*{relation_to_indrepresentable_power_objects} \linebreak
\noindent\hyperlink{imaginaries_and_automorphisms}{Imaginaries and automorphisms}\dotfill \pageref*{imaginaries_and_automorphisms} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the [[category of sets]], we have a [[bijection|bijective correspondence]] (up to [[isomorphism]]) between [[surjective functions]] on $X$ and [[equivalence relations]] on $X$, where a surjective function $\pi \colon X \to Y$ is taken to the equivalence relation $E_{\pi}$ gotten by pulling back the [[diagonal]] of $Y$, and an equivalence relation $E$ is taken to the function $f_E$ which [[projection|projects]] $X$ onto the set of $E$-classes. If we replace [[Set]] with the category of [[definable sets]] $\mathbf{Def}(T)$ of a [[first-order theory]] $T$, this correspondence generally fails, because when $E$ is a definable equivalence relation, there may not be a corresponding definable $f_E$. That is to say, [[internal congruences|internal congruences]] in $\mathbf{Def}(T)$ are not generally effective.

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

We say that:

\begin{enumerate}%
\item $T$ has \textbf{uniform elimination of imaginaries} if the above correspondence does hold, that is, if internal congruences in $\mathbf{Def}(T)$ are effective.


\item $T$ has \textbf{elimination of imaginaries} if for every definable set $X = \{m \in \mathbb{M} \operatorname{ | } \varphi(m,b)\}$, there exists a formula $\psi(x,y)$ and a tuple $c$ with the same sort as $y$ such that $c$ uniquely satisfies $X = \{m \in M \operatorname{ | } \psi(m,c)\}.$



\end{enumerate}
Inside a saturated model $\mathbb{M} \models T$, elimination of imaginaries is equivalent to the following statement: for every definable set $X$, there exists a tuple $c$ such that for all $\sigma \in \operatorname{Aut}(\mathbb{M})$, $\sigma(X) = X$ setwise if and only if $\sigma(c) = c$ pointwise. This is sometimes called the \textbf{coding of definable sets}.

Poizat noticed that $T$ having elimination of imaginaries allows one to develop a classical Galois theory classifying definably-closed extensions of small parameter sets in terms of the closed subgroups of a profinite automorphism group in any sufficiently saturated model of $T$.

The \emph{imaginary elements} of a theory $T$ are precisely $E$-classes, where $E$ is a $0$-definable equivalence relation on $T$.

\hypertarget{the__construction}{}\subsection*{{The $(-)^{\operatorname{eq}}$ construction}}\label{the__construction}

In the process of inventing [[stability theory]], i.e. classifying theories according to the number of isomorphism classes of their models in each cardinality, Shelah found he needed to eliminate imaginaries in a universal way, so he found a way to conservatively interpret each theory $T$ in a theory $T^{\operatorname{eq}}$ which eliminated all the imaginaries of $T$. The process just involves throwing in for each congruence $E$ on $X$ a new sort ([[type]] in type theory) for $X/E$ and a projection map $X \twoheadrightarrow X/E$.

$T^{\operatorname{eq}}$ satisfies the following universal property: any theory which interprets $T$ and eliminates imaginaries must uniquely interpret $T^{\operatorname{eq}}$ as theories under $T$ (in the $2$-category of theories and interpretations.)

Makkai realized not long after that the $(-)^{\operatorname{eq}}$ construction corresponded to the [[pretopos completion]] of the [[syntactic category]] of $T$.

(Indeed, pretoposes are just toposes that might be missing some power objects. According to the next section, inside a Boolean pretopos these power objects will still be there, just generally only formally.)

\hypertarget{relation_to_indrepresentable_power_objects}{}\subsection*{{Relation to ind-representable power objects}}\label{relation_to_indrepresentable_power_objects}

If a first-order theory eliminates imaginaries, it interprets infinitely many constants. This is because it interprets at least two constants: a code for the diagonal relation and its complement. Taking binary sequences of these two constants in higher and higher Cartesian products of the model with itself produces infinitely many constants.

In particular, when $T$ eliminates imaginaries, its [[syntactic category]] has finite [[coproducts]]. In particular, $\mathbf{2}$ exists, and all definable subsets have definable classifying maps.

The following characterization is due to Moshe Kamensky, and can be found \href{https://arxiv.org/abs/1012.3185}{here}.

\begin{prop}
\label{}\hypertarget{}{}
Let (T) be a first-order theory which interprets two constants. Then $T$ eliminates imaginaries if and only if for all $X$ and $Y$ in $\mathbf{Def}(T)$, the presheaf $Z \mapsto \operatorname{Hom}_{\mathbf{Def}(T)}(Z \times X, Y)$ is ind-representable.

\end{prop}
\begin{proof}
Suppose first that the presheaves $Y^X \overset{\operatorname{df}}{=}Z \mapsto \operatorname{Hom}_{\mathbf{Def}(T)}(Z \times X,Y)$ are ind-representable. Let $E \overset{s}{\underset{t}{\rightrightarrows}} X$ be a definable equivalence relation on $X$. Let $\phi_E : X \times X \to 2$ classify $E \overset{(s,t)}{\hookrightarrow} X \times X$. Let $\mathbf{y} : \mathbf{Def}(T) \hookrightarrow \widehat{\mathbf{Def}(T)}$ be the Yoneda embedding. Then $\mathbf{y} \phi_E$ has a product-exponential transpose $\overline{\mathbf{y} \phi_E} : \mathbf{y}X \to 2^X$. The Yoneda lemma says that $\operatorname{Hom}_{\widehat{\mathbf{Def}(T)}}(\mathbf{y}X, 2^X) \simeq 2^X(X)$, and by assumption $2^X(X) \simeq \underset{\longrightarrow}{\lim}\operatorname{Hom}(X,J(i))$ for some $J : I \to \mathbf{Def}(T)$ a filtered diagram of definable sets. Since this is filtered, the colimit as a set can be computed as

\begin{displaymath}
\bigsqcup_{i \in I} J(i) / \sim,
\end{displaymath}
where the equivalence relation $\sim$ identifies elements that are eventually sent to the same element by transition maps in the diagram. Hence, we can identify $\overline{\mathbf{y}\phi_E}$ with its equivalence class of maps $\{\phi_i : X \to J(i)\}_{i \in I}$. Now, $\mathbf{y}E$ is the kernel pair of $\overline{\mathbf{y} \phi_E}$ in $\widehat{\mathbf{Def}(T)}$, so $E = \bigvee_{i \in I} \ker(\phi_i)$. By compactness, we can replace $I$ with a finite subset $I_0 \subseteq I$. Since $I$ is filtered, there exists a weak coproduct $j$ to $I_0$. Since the $\{\phi_i\}_{i \in I}$ turn $X$ into a cone to $J$, for each $i \in I_0$ there is a transition map $h_{ij} : J(i) \to J(j)$, so $\ker(\phi_j) = \ker(h_{ij} \circ \phi_i) \supseteq \ker(\phi_i)$. Replacing each $\ker(\phi_i)$ with $\ker(\phi_j)$, we see $E = \ker(\phi_j)$, and $\phi_j$ is definable.

On the other hand, suppose we have elimination of imaginaries. Fix $X$ and $Y$. For each $Z$ and $f : Z \times X \to Y$, we may form the equivalence relation $E_f$ on $Z$ which says $z_1 \sim_{E_f} z_2$ if and only if they define the same fiber of $\Gamma(f)$. The projection map induces a map

\begin{displaymath}
(-/E_f) \times X : Z \times X \to Z/E_f \times X
\end{displaymath}
which is also a map $f \to f/E_f$ of objects over $Y$.

Now, the [[category of elements]] $\operatorname{Pt}(1, Y^X)$ of the presheaf $\operatorname{Hom}(- \times X, Y)$ has objects maps $f : Z \times X \to Y$ for $Z$ ranging over $\mathbf{Def}(T)$, and morphisms from $f_1 : Z_1 \times X \to Y$ to $f_2 : Z_2 \times X \to Y$ the maps $g \times X : Z_1 \times X \to Z_2 \times X$ for $g : Z_1 \to Z_2$ such that $f_1 = f_2 \circ g$. For each object $f \in \operatorname{Pt}(1, Y^X)$, $-/ E_f \times X$ as above coequalizes all pairs of maps in $\operatorname{Pt}(1, Y^X)$ into $f$. Furthermore, the natural projection $p$ from $\operatorname{Pt}(1, Y^X) \to \mathbf{Def}(T)$ by sending $f : Z \times X \to Y \mapsto Z$ creates colimits, in particular finite coproducts, which by assumption exist in $\mathbf{Def}(T)$. So $\operatorname{Pt}(1, Y^X)$ is filtered, and since for any presheaf $P$, we have that

\begin{displaymath}
P \simeq \underset{\longrightarrow}{\lim}\left(\operatorname{Pt}(1,P) \overset{p}{\to} \mathbf{Def}(T) \overset{\mathbf{y}}{\hookrightarrow} \widehat{\mathbf{Def}(T)}\right),
\end{displaymath}
so $Y^X$ is ind-representable.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
In the above proof, complementation is required to characterize compactness as ``every covering of a definable set by an infinite family of definable sets is finitely supported.'' The same proof works if we replace inner homs with power objects.

\end{remark}
\hypertarget{imaginaries_and_automorphisms}{}\subsection*{{Imaginaries and automorphisms}}\label{imaginaries_and_automorphisms}

Let $M \models T$ be a model. If a formula $\varphi(x_1, x_2)$ is a $0$-definable equivalence relation $E$ (i.e. an [[internal congruence]] in $\mathbf{Def}(T)$) on a definable set $X$, any automorphism $\sigma$ of $M$ must preserve and reflect $E$-equivalence, so that $\sigma$ extends along $X \twoheadrightarrow X/E$ to a permutation on $X/E$. So automorphisms of models ``already see'' imaginaries. This can be stated rigorously as: the expansion of a monster model $\mathbb{M}$ of $T$ to a monster model $\mathbb{M}^{\operatorname{eq}}$ of $T^{\operatorname{eq}}$ will have the same automorphism group as $\mathbb{M}$.

To make things concrete, let's consider the structure $(\mathbb{Z}, +)$. Since there is an automorphism which switches signs, the only constant is $0$. So the theory of this structure cannot eliminate imaginaries, since such theories must interpret infinitely many distinct constants.

However, we can tell that this structure has many imaginaries by its rigidity: the above automorphism is the only nontrivial one. This is because any such automorphism of $(\mathbb{Z},+)$ must also act on the internal congruences $E_n$ which say

\begin{displaymath}
x \sim_{E_n} y \iff x \equiv y (\operatorname{mod} n) \iff (\exists \ell)\left[x - y = \ell + \dots \text{ (n times) } \dots + \ell\right],
\end{displaymath}
that is, must extend along the abelian group epimorphisms $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/n\mathbb{Z}$ for each $n$, so that $k \mapsto -k$ is the only one that works.

(It's essentially for this reason that the automorphism group of the [[forgetful functor]] $\mathbf{Grp} \to \mathbf{Set}$ is also $\mathbb{Z}/2 \mathbb{Z}$.)

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[pretopos completion]]


\item [[Barr-exact category]]


\item [[hyperimaginary element]]


\item [[Galois theory]]



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

\begin{itemize}%
\item Bruno Poizat, \emph{Une th\'e{}orie de Galois imaginaire}, J. Symbolic Logic \textbf{48} (1984), no.4, 1151-1170, \href{http://www.ams.org/mathscinet-getitem?mr=727805}{MR85e:03083}, \href{http://dx.doi.org/10.2307/2273680}{doi}
\item wikipedia \href{http://en.wikipedia.org/wiki/Imaginary_element}{imaginary element}
\item Anand Pillay, \emph{Some remarks on definable equivalence relations in O-minimal structures}, J. Symbolic Logic \textbf{51} (1986), 709-714, \href{http://www.ams.org/mathscinet-getitem?mr=853850}{MR87h:03046}, \href{http://dx.doi.org/10.2307/2274024}{doi}
\item Jan Holly, \emph{Definable operations on sets and elimination of imaginaries}, Proc. Amer. Math. Soc. \textbf{117} (1993), no. 4, 1149--1157, \href{http://www.ams.org/mathscinet-getitem?mr=1116261}{MR93e:03052}, \href{http://dx.doi.org/10.2307/2159546}{doi}, \href{http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116261-6/S0002-9939-1993-1116261-6.pdf}{pdf}
\item [[Ehud Hrushovski]], \emph{Groupoids, imaginaries and internal covers}, \href{http://arxiv.org/abs/math/0603413}{arxiv/math.LO/0603413}; \emph{On finite imaginaries}, \href{http://arxiv.org/abs/0902.0842}{arxiv/0902.0842}
\item D. Haskell, E. Hrushovski, H.D.Macpherson, \emph{Definable sets in algebraically closed valued fields: elimination of imaginaries}, J. reine und angewandte Mathematik \textbf{597} (2006)
\item [[Saharon Shelah]], \emph{Classification theory and the number of non-isomorphic models}, Studies in Logic and the Foundations of Mathematics \textbf{92}, North Holland, Amsterdam 1978 [[!redirects elimination of imaginaries]]
\item Moshe Kamensky, \href{http://arxiv.org/abs/1012.3185}{\emph{A categorical approach to internality}}

\end{itemize}
[[!redirects imaginary element]] [[!redirects imaginaries]]



\end{document}