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\begin{document}

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\section*{hyperimaginary 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{sec:orgcc9169c}{Definition}\dotfill \pageref*{sec:orgcc9169c} \linebreak
\noindent\hyperlink{sec:orgdc24af7}{Examples}\dotfill \pageref*{sec:orgdc24af7} \linebreak
\noindent\hyperlink{sec:org59c5525}{Elimination of hyperimaginaries}\dotfill \pageref*{sec:org59c5525} \linebreak
\noindent\hyperlink{sec:org623f6fa}{Properties}\dotfill \pageref*{sec:org623f6fa} \linebreak
\noindent\hyperlink{sec:org1bb2b22}{Related concepts}\dotfill \pageref*{sec:org1bb2b22} \linebreak
\noindent\hyperlink{sec:org00f5b3c}{References}\dotfill \pageref*{sec:org00f5b3c} \linebreak


\hypertarget{sec:orgcc9169c}{}\subsection*{{Definition}}\label{sec:orgcc9169c}

An [[imaginary element]] of a (finitary) [[first-order theory]] is an equivalence class $X/E$ of a (finitary first-order) [[definable set]] $X$ quotiented by a (finitary first-order) [[definable groupoid|definable equivalence relation]] $E$. A \textbf{hyperimaginary} is an equivalence class $X/E$ of a [[definable set\#type-definable-set | type-definable]] $X$ quotiented by a [[definable set\#type-definable-set | type-definable]] equivalence relation $E$.

\hypertarget{sec:orgdc24af7}{}\subsection*{{Examples}}\label{sec:orgdc24af7}

The prototypical example of a hyperimaginary is an equivalence class of the quotient of the [[hyperreal numbers]] by the equivalence relation ``$x$ is infinitesimally close to $y$'', which is definable by the countable conjunction $\left( \left| x - y \right| &lt; \frac{1}{n} \right)_{n \in \mathbb{N}}$.

\hypertarget{sec:org59c5525}{}\subsection*{{Elimination of hyperimaginaries}}\label{sec:org59c5525}

Let $X/E$ be a hyperimaginary. $T$ is said to \textbf{eliminate} the hyperimaginary $X/E$ (c.f. [[elimination of imaginaries]]) if $X/E$ is interdefinable with a sequence of imaginaries, i.e. if every type-definable equivalence relation is, in fact, a conjunction of definable equivalence relations.

[[stable theory|Stable theories]] eliminate hyperimaginaries. This is related to a result of Hrushovski's that in a stable theory, (pro-definable) groupoids are pro-(definable groupoids).

\hypertarget{sec:org623f6fa}{}\subsection*{{Properties}}\label{sec:org623f6fa}

\begin{itemize}%
\item In the same way that the [[automorphism group]] $\operatorname{Aut}(M)$ of a model acts on [[imaginary element|imaginaries]], $\operatorname{Aut}(M)$ acts on hyperimaginaries.


\item Any hyperimaginary $X/E$ in the sense of the previous paragraph can be extended to a quotient of the entire model $M^{\alpha}$ (in the (possibly infinitary) [[type|sort]] $\alpha$ of $E$) by a type-definable equivalence relation $E'$ obtained by preserving $E$ on $X$ and extending it to be the discrete equivalence relation on $M^{\alpha} \backslash X$. In symbols, $E' \overset{\operatorname{df}}{=} (E(x,y) \wedge (x,y) \in X \times X) \vee (x = y)$. (This is a special case of the fact that finite disjunctions of type-definable sets are again type-definable. It is not true that arbitrary disjunctions of type-definable sets are again type-definable, since $\operatorname{Aut}(\mathbb{M})$-invariant subsets of $\mathbb{M}$ the [[monster model]] have this form.)


\item The process of taking points of hyperimaginaries in models ($M \mapsto M(X/E)$) does not commute with [[ultraproducts]]. For example, if $E$ is the equivalence relation in the reals of being infinitesimally close together, it's easy to see that $\mathbb{R}^{\mathcal{U}}/E \not \simeq \left(\mathbb{R}/E \right)^{\mathcal{U}}$, since $\mathbb{R}/E \simeq \mathbb{R}$ (after all, the real numbers are Hausdorff) so that the right hand side is the nonstandard reals $^* \mathbb{R}$ again (so every number has a [[halo|cloud]] of infinitesimals around it) while the left hand side has no points which are infinitesimally close together.



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
This is actually an excellent example of why we shouldn't expect in general the process of taking points of hyperimaginaries in a model to commute with ultraproducts: on one hand, if $E$ is a type-definable equivalence relation on $X$, and $\mathcal{U}$ an [[ultrafilter]], then computing $\left(\prod_{i \in I} M_i/\mathcal{U} \right)(X/E)$ yields $E$-equivalence classes of [[germs]] $[m_i]_{\mathcal{U}}$. That is, for every condition $\varphi(x_1, x_2)$ in $E$ that needs to be checked to conclude that two tuples are $E$-equivalent, every pair of $E$-equivalent germs $[m_i]_{\mathcal{U}}$ and $[n_i]_{\mathcal{U}}$ agree on $\varphi$ on some subset $P \subseteq I$ for $P$ in $\mathcal{U}$.But these $P$s depend on $\varphi$. They aren't uniform, and there's possibly no set $P'$ in the ultrafilter (the intersection of all the $P$`s might be empty) on which one could decide if $[m_i]$ and $[n_i]$ are $E$-equivalent for every condition $\varphi \in E$.

On the other hand, one could try computing $\left( \prod_{i \in I} M_i(X/E) \right) /\mathcal{U}$. One ends up instead with germs of hyperimaginaries, as in $[[m_i]_E]_{\mathcal{U}}$, but this is a tighter equivalence relation: $([x_i]_E)_i$ and $([y_i]_E)_i)$ are $\mathcal{U}$-equivalent if and only if there is a single $P \in \mathcal{U}$, uniform for every $i \in P$ and for every $\varphi \in E$ so that $M_i \models \varphi(x_i, y_i)$. This means that the commutation of these two procedures should fail precisely when the $P$`s on the left hand side intersect to something outside of the ultrafilter.

Indeed, in the example we first mentioned, consider the sequences $(1, 2, 3, \dots)$ and $(1 + 1/2, 2 + 1/3, 3 + 1/4, \dots)$. If we take them mod-$\mathcal{U}$, then mod-$E$, then they are equivalent as hyperimaginaries in $^*\mathbb{R}$, since for each condition ``x is less than $1/n$ away from y'', these two sequences agree cofinitely many times (and every ultrafilter contains the cofinite filter). But these cofinite sets of indices are getting smaller and smaller and intersect to $\emptyset$. And if we take them mod-$E$, then mod-$\mathcal{U}$, they remain distinct.

\end{remark}
\hypertarget{sec:org1bb2b22}{}\subsection*{{Related concepts}}\label{sec:org1bb2b22}

\begin{itemize}%
\item [[imaginary element]]


\item [[infinitesimal]]


\item [[simple theory]]



\end{itemize}
\hypertarget{sec:org00f5b3c}{}\subsection*{{References}}\label{sec:org00f5b3c}

\begin{itemize}%
\item Frank Wagner, \emph{Simple Theories} (2000)


\item Byunghan Kim, \emph{Simplicity Theory} (2014)


\item Enrique Casanovas, \emph{Simple theories and hyperimaginaries} (2011)



\end{itemize}
[[!redirects hyperimaginary]] [[!redirects hyperimaginaries]] [[!redirects hyperimaginary elements]] [[!redirects elimination of hyperimaginaries]]



\end{document}